Physics 561 Condensed Matter Physics II

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Eduardo Fradkin

Physics 561 Condensed Matter Physics II

University of Illinois at Urbana-Champaign 2009

Chapter 1

Second Quantization 1.1

Creation and Annihilation Operators in Quantum Mechanics

We will begin with a quick review of creation and annihilation operators in the non-relativistic linear harmonic oscillator. Let a and a† be two operators acting on an abstract Hilbert space of states, and satisfying the commutation relation 

 a, a† = 1

(1.1)

where by “1” we mean the identity operator of this Hilbert space. The operators a and a† are not self-adjoint but are the adjoint of each other. Let |αi be a state which we will take to be an eigenvector of the Hermitian operators a† a with eigenvalue α which is a real number, a† a |αi = α|αi

(1.2)

α = hα|a† a|αi = ka|αik2 ≥ 0

(1.3)

Hence, where we used the fundamental axiom of Quantum Mechanics that the norm of all states in the physical Hilbert space is positive. As a result, the eigenvalues α of the eigenstates of a† a must be non-negative real numbers. Furthermore, since for all operators A, B and C [AB, C] = A [B, C] + [A, C] B

(1.4)

we get 

 a† a, a = −a  †  a a, a† = a† 1

(1.5) (1.6)

2

CHAPTER 1. SECOND QUANTIZATION

i.e., a and a† are “eigen-operators” of a† a. Hence,

a† a a = a a† a − 1

a† a a† = a† a† a + 1

(1.7) (1.8)

Consequently we find



a† a a|αi = a a† a − 1 |αi = (α − 1) a|αi

(1.9)

Hence the state a|αi is an eigenstate of a† a with eigenvalue α − 1, provided a|αi = 6 0. Similarly, a† |αi is an eigenstate of a† a with eigenvalue α + 1, provided a† |αi = 6 0. This also implies that |α − 1i = |α + 1i =

1 √ a|αi α 1 √ a† |αi α+1

(1.10) (1.11)

Let us assume that an |αi = 6 0,

∀n ∈ Z+

(1.12)

Hence, an |αi is an eigenstate of a† a with eigenvalue α − n. However, α − n < 0 if α < n, which contradicts our earlier result that all these eigenvalues must be non-negative real numbers. Hence, for a given α there must exist an integer n such that an |αi = 6 0 but an+1 |αi = 0, where n ∈ Z+ . Let |α − ni =

1 an |αi ⇒ a† a |α − ni = (α − n) |α − ni kan |αik

where kan |α − nik = But a|α − ni = −

√ α−n

1 an+1 |αi = 0 ⇒ kan |αik

(1.13)

(1.14) α=n

(1.15)

In other words the allowed eigenvalues of a† a are the non-negative integers. Let us now define the ground state |0i, as the state annihilated by a, a|0i = 0

(1.16)

1 |ni = √ (a† )n |0i n!

(1.17)

hn|mi = δn,m n!

(1.18)

Then, an arbitrary state |ni is

which has the inner product

1.1. CREATION AND ANNIHILATION OPERATORS IN QUANTUM MECHANICS3 In summary, we found that creation and annihilation operators obey √ n + 1 |n + 1i (1.19) a† |ni = √ a |ni = n |n − 1i (1.20) † a a |ni = n |ni (1.21) (1.22)

and thus their matrix elements are √ hm|a† |ni = n + 1 δm,n+1

1.1.1

hm|a|ni =

√ n δm,n−1

(1.23)

The Linear harmonic Oscillator

The Hamiltonian of the Linear Harmonic Oscillator is H=

P2 1 + mω 2 X 2 2m 2

(1.24)

where X and P , the coordinate and momentum Hermitian operators satisfy canonical commutation relations, [X, P ] = i~ We now define the creation and annihilation operators a† and a as r  P 1 mω X + i√ a = √ ~ 2 mω~ r  P 1 mω † X − i√ a = √ ~ 2 mω~ which satisfy



Since X P

 a, a† = 1

r


~ a + a† 2mω r
mω~ a − a† = 2 i

=

the Hamiltonian takes the simple form   1 † H = ~ω a a + 2

(1.25)

(1.26) (1.27)

(1.28)

(1.29) (1.30)

(1.31)

The eigenstates of the Hamiltonian are constructed easily using our results since all eigenstates of a† a are eigenstates of H. Thus, the eigenstates of H are the eigenstates of a† a,   1 H|ni = ~ω n + |ni (1.32) 2

4

CHAPTER 1. SECOND QUANTIZATION

with eigenvalues

  1 En = ~ω n + 2

(1.33)

where n = 0, 1, . . .. The ground state |0i is the state annihilated by a, 1 a|0i ≡ 2

r

 P mω X + i√ |0i = 0 ~ mω~

Since hx|P |φi = −i~

d hx|φi dx

we find that ψ0 (x) = hx|0i satisfies   ~ d ψ0 (x) = 0 x+ mω dx

(1.34)

(1.35)

(1.36)

whose (normalized) solution is  mω 1/4 − mω x2 ψ0 (x) = e 2~ π~

(1.37)

The wave functions ψn (x) of the excited states |ni are ψn (x)

1 = hx|ni = √ hx|(a† )n |0i n!  n ~ d 1  mω n/2 x− ψ0 (x) = √ 2~ mω dx n!

(1.38)

Creation and annihilation operators are very useful. Let us consider for instance the anharmonic oscillator whose Hamiltonian is H=

1 P2 + mω 2 X 2 + λX 4 2m 2

(1.39)

Let us compute the eigenvalues En to lowest order in perturbation theory in powers of λ. The first order shift ∆En is = λ hn|X 4 |ni + O(λ2 )  2 ~ = λ hn|(a + a† )4 |ni + . . . 2mω 2   ~ hn|a† a† aa|ni + other terms with two a′ s and two a†′ s + . . . =λ 2mω 2 
~ = λ 6n2 + 6n + 3 + . . . (1.40) 2mω

∆En

1.1. CREATION AND ANNIHILATION OPERATORS IN QUANTUM MECHANICS5

1.1.2

Many Harmonic Oscillators

It is trivial to extend these ideas to the case of many harmonic oscillators, which is a crude model of an elastic solid. Consider a system of N identical linear harmonic oscillators of mass M and frequency ω, with coordinates {Qi } and momenta {Pi }, where i = 1, . . . , N . These operators satisfy the commutation relations [Qj , Qk ] = [Pj , Pk ] = 0, [Qj , Pk ] = i~δjk (1.41) where j, k = 1, . . . , N . The Hamiltonian is H=

N N X 1 X Pi2 Vij Qi Qj + 2Mi 2 i,j=1 i=1

(1.42)

where Vij is a symmetric positive definite matrix, Vij = Vji . We will find the spectrum (and eigenstates) of this system by changing variables to normal modes and using creation and annihilation operators for the normal modes. To this end we will first rescale coordinates and momenta so as to absorb the particle mass M : p xi = M i Qi (1.43) Pi (1.44) pi = √ Mi 1 Uij = p Vij (1.45) Mi Mj

which also satisfy

[xj , pj ] = i~δjk and H=

(1.46)

N X p2

N 1 X Uij xi xj + 2 2 i,j=1 i

i=1

(1.47)

We now got to normal mode variables by means of an orthogonal transformation Cjk , i.e. [C −1 ]jk = Ckj , x ek =

N X

N X

Ckj xj ,

j=1

Cki Cji = δkj ,

i=1

pek =

N X

N X

Ckj pj

j=1

Cik Cij = δkj

(1.48)

i=1

Sine the matrix Uij is real symmetric and positive definite, its eigenvalues, which we will denote by ωk2 (with k = 1, . . . , N ), are all non-negative, ωk2 ≥ 0. The eigenvalue equation is N X

i,j=1

Cki Cℓj Uij = ωk2 δkℓ ,

(no sum over k)

(1.49)

6

CHAPTER 1. SECOND QUANTIZATION

Since the transformation is orthogonal, it preserves the commutation relations [e xj , x ek ] = [e pj , pek ] = 0,

and the Hamiltonian is now diagonal

[e xj , pek ] = i~δjk

(1.50)

N

H=


1X 2 pej + ωj2 x e2j 2 i=1

(1.51)

We now define creation and annihilation operators for the normal modes   1 i √ ωj x aj = √ ej + √ pej ωj 2~   1 i √ a†j = √ ωj x ej − √ pej ωj 2~ s  ~  x ej = aj + a†j 2ωj r  ~ωj  (1.52) pej = −i aj − a†j 2 where, once again,

i h [aj , ak ] a†j , a†k = 0,

h i aj , a†k = δjk

(1.53)

and the normal mode Hamiltonian takes the standard form H=

  1 ~ωj a†j aj + 2 j=1

N X

(1.54)

The eigenstates of the Hamiltonian are labelled by the eigenvalues of a†j aj for each normal mode j, |n1 , . . . , nN i ≡ |{nj }i. Hence, H|n1 , . . . , nN i = where

  1 ~ωj nj + |n1 , . . . , nN i 2 j=1

N X



(1.55)

 N Y (a†j )nj  |0, . . . , 0i p |n1 , . . . , nN i =  n ! j j=1

(1.56)

|0i ≡ |0, . . . , 0i

(1.57)

The ground state of the system, which we will denote by |0i, is the state in which all normal modes are in their ground state,

1.1. CREATION AND ANNIHILATION OPERATORS IN QUANTUM MECHANICS7 Thus, the ground state |0i is annihilated by the annihilation operators of all normal modes, aj |0i = 0,

∀j

(1.58)

and the ground state energy of the system Egnd is Egnd =

N X 1 j=1

2

(1.59)

~ωj

The energy of the excited states is E(n1 , . . . , nN ) =

N X

~ωj nj + Egnd

(1.60)

j=1

We can now regard the state |0i as the vacuum state and the excited states |n1 , . . . , nN i as a state with nj excitations (or particles) of type j. In this context, the excitations are called phonons. A single-phonon state of type j will be denoted by |ji, is |ji = a†j |0i = |0, . . . , 0, 1j , 0, . . . , 0i

(1.61)

This state is an eigenstate of H, H|ji = Ha†j |0i = (~ωj + Egnd ) a†j |0i = (~ωj + Egnd ) |ji

(1.62)

with excitation energy ~ωj . Hence, an arbitrary state |n1 , . . . , nN i can also be regarded as a collection of non-interacting particles (or excitations), each carrying an energy equal to the excitation energy (relative to the ground state energy). The total number of phonons in a given state is measured by the number operator ˆ = N

N X

a†j aj

(1.63)

j=1

Notice that although the number of oscillators is fixed (and equal to N ) the number of excitations may differ greatly from one state to another. We now note that the state |n1 , . . . , nN i can also be represented as n1

n2

z }| { z }| { 1 | 1 . . . 1, 2 . . . 2 . . .i |n1 , . . . , nk , . . .i ≡ √ n1 !n2 ! . . .

(1.64)

We will see below that this form appears naturally in the quantization of systems of identical particles. Eq.(1.64) is symmetric under the exchange of labels of the phonons. Thus, phonons are bosons.

8

1.2

CHAPTER 1. SECOND QUANTIZATION

The Quantized Elastic Solid

We will now consider the problem of an elastic solid in the approximation of continuum elasticity, i.e. we will be interested in vibrations on wavelengths long compared to the inter-atomic spacing. At the classical level, the physical state of this system is determined by specifying the local three-component vector displacement field ~u(~r, t), which describes the local displacement of the atoms away from their equilibrium positions, and by the velocities of the atoms, ∂~ u r , t). The classical Lagrangian L for an isotropic solid is ∂t (~ L=

Z

ρ d r 2 3



∂u ∂t

2

−

1 2

Z

d3 r [K∇i uj ∇i uj + Γ∇i ui ∇j uj ]

where ρ is the mass density, K and Γ are two elastic moduli. The classical equations of motion are   δL ∂ δL = δui ∂t δ u˙ i

(1.65)

(1.66)

which have the explicit form of a wave equation ρ

∂ 2 ui ~ · ~u = 0 − K∇2 ui − Γ∇i ∇ ∂t2

(1.67)

We now define the canonical momentum Πi (~r, t) Πi (~r, t) =

δL = ρu˙ i δ u˙ i

(1.68)

which in this case coincides with the linear momentum density of the particles. The classical Hamiltonian H is # " Z Z 2 ~2 ∂~u K Γ ~ Π 2 3 3 H = d rΠi (~r, t) (~r, t) − L = d r ∇ · ~u + (∇i ~u) + ∂t 2ρ 2 2 (1.69) In the quantum theory, the displacement field ~u(~r) and the canonical momen~ r ) become operators acting on a Hilbert space, obeying the equal-time tum Π(~ commutation relations h i h i h i ~ = Πi (~r), Πj (R) ~ = 0, ~ = i~δ 3 (~r − R)δ ~ ij ui (~r), uj (R) ui (~r), Πj (R)

(1.70) Due to the translational invariance of the continuum solid, the canonical transformation to normal modes is a Fourier transform. Thus we write Z u ei (~ p) = d3 r e−i~p·~r ui (~r) (1.71) Z e i (~ Π p) = d3 r e−i~p·~r Πi (~r) (1.72)

9

1.2. THE QUANTIZED ELASTIC SOLID Using the representations of the Dirac δ-function Z d3 p i~p·(~r−R) ~ ~ = e δ 3 (~r − R) (2π)3 Z (2π)3 δ 3 (~ p − ~q) = d3 r e−i(~p−~q)·~r

(1.73) (1.74)

we can write 1 ui (~r) = √ ρ Πi (~r) =

√

ρ

Z

Z

d3 p i~p·~r e u ei (~ p) (2π)3

d3 p i~p·~r e e Πi (~ p) (2π)3

(1.75) (1.76)

where we have scaled out the density ρ or future convenience. On the other hand, since ui (~r) and Πi (~r) are real (and Hermitian) ui (~r) = u†i (~r),

Πi (~r) = Π†i (~r)

e i (~ their Fourier transformed fields u ei (~ p) and Π p) obey u e†i (~ p) = u ei (−~ p),

e † (~ e Π p) i p) = Πi (−~

(1.77)

(1.78)

with equal-time commutation relations h i h i e j (~ e k (~ e k (~q) = i~(2π)3 δ 3 (~ [e uj (~ p), u ek (~ q )] = Π p), Π q ) = 0, u ej (~ p), Π p + ~q)δjk In terms of the Fourier transformed fields  Z d3 p 1e e i (~ H= Πi (−~ p)Π p) + (2π)3 2

where

2 ωij (~ p) =

(1.79) the Hamiltonian has the form  1 2 ω (~ p)e ui (−~ p)e uj (~ p) (1.80) 2 ij

K 2 Γ p~ δij + pi pj ρ ρ

(1.81)

2 The 3 × 3 matrix ωij has two eigenvalues:

1. 2 ωL (~ p)

=



K +Γ ρ



~2 p

(1.82)

with eigenvector parallel to the unit vector p~/|~ p| 2. ωT2 (~ p) =



K ρ



p~ 2

(1.83)

with a two-dimensional degenerate space spanned by the mutually orthogonal unit vectors ~e1 (~ p) and ~e2 (~ p), both orthogonal to p~: ~eα (~ p) · p~ = 0, (α = 1, 2),

~e1 (~ p) · ~e2 (~ p) = 0

(1.84)

10

CHAPTER 1. SECOND QUANTIZATION

We can now expand the displacement field u e(~ p) into a longitudinal component pi (1.85) u eL p) = u eL (~ p) i (~ |~ p| and two transverse components

u eTi (~ p) =

X

α=1,2

u eTα (~ p)eα p) i (~

(1.86)

e i (~ The canonical momenta Π p) can also be expanded into one longitudinal come e Tα (~ ponent ΠL (~ p) and two transverse components Π p). As a result the Hamiltonian can be decomposed into a sum of two terms, H = HL + HT

(1.87)

• HL involves only the longitudinal component of the field and momenta Z o 1 d3 p n e 2 e L (~ HL = Π (−~ p ) Π p ) + ω (~ p ) u e (−~ p )e u (~ p ) (1.88) L L L L 2 (2π)3

• HT involves only the transverse components of the field and momenta Z o 1 d3 p X n e T 2 T T e T (~ HT = Π (−~ p ) Π p ) + ω (~ p ) u e (−~ p )e u (~ p ) (1.89) α α T α α 2 (2π)3 α=1,2

We can now define creation and annihilation operators for both longitudinal and transverse components ! p i 1 e L (~ Π p) ωL (~ p) u eL (~ p) + p aL (~ p) = √ 2~ ωL (~ p) ! p i 1 † e L (−~ ωL (~ p) u eL (−~ p) − p aL (~ p) = √ Π p) 2~ ωL (~ p) ! p i 1 eα ωT (~ p) u eα p) + p aα p) = √ Π p) T (~ T (~ T (~ 2~ ωT (~ p) ! p i 1 α α α † e ΠT (−~ p) ωT (~ p) u eT (−~ p) − p aT (~ p) = √ 2~ ωT (~ p) (1.90)

which obey standard commutation relations:   [aL (~ p), aL (~q)] = aL (~ p)† , aL (~q)† ) = 0   aL (~ p), aL (~q)† = (2π)3 δ 3 (~ p − ~q) h i h i β α α † β aT (~ p), aT (~q) = aT (~ p) , aT (~q)† ) = 0 i h = (2π)3 δ 3 (~ p − ~q)δαβ aα p), aβT (~q)† T (~   [aL (~ p), aα q )] = aL (~ p), aα q )† ) = 0 T (~ T (~

(1.91)

11

1.2. THE QUANTIZED ELASTIC SOLID where α, β = 1, 2. Conversely we also have u eL (~ p) =

e L (~ Π p) = u eα p) = T (~

e α (~ Π T p) =

s


~ aL (~ p) + aL (−~ p)† 2ωL (~ p) r
~ωL (~ p) aL (~ p) − aL (−~ p)† −i 2 s
~ aα p ) + aα p)† T (~ T (−~ 2ωT (~ p) r
~ωT (~ p) p ) − aα p)† aα −i T (~ T (−~ 2

(1.92)

The Hamiltonian now reads Z
d3 p 1 ~ωL (~ p) aL (~ p)† aL (~ p) + aL (~ p)aL (~ p)† H = 3 2 (2π) Z X
1 d3 p + aα p)† aα p ) + aα p)aα p)† ~ωT (~ p) T (~ T (~ T (~ T (~ 3 2 (2π) α=1,2 = Egnd +

Z

d3 p (2π)3

†

~ωL (~ p) aL (~ p) aL (~ p) +

X

~ωT (~ p)

α=1,2

The ground state |0i has energy Egnd , Z d3 p 1 Egnd = V (~ωL (~ p) + 2~ωT (~ p)) (2π)3 2 where we have used that lim δ 3 (~ p) =

p ~→0

!

aα p)† aα p) T (~ T (~

V (2π)3

(1.93)

(1.94)

(1.95)

As before the ground state is annihilated by all annihilation operators aL (~ p)|0i = 0,

aα p)|0i = 0 T (~

(1.96)

and it will be regarded as the state without phonons. There are three one-phonon states with wave vector p~: • One longitudinal phonon state |L, ~pi = aL (~ p)† |0i

(1.97)

EL (~ p) = ~ωL (~ p) = vL ~|~ p|

(1.98)

with energy EL (~ p)

12

CHAPTER 1. SECOND QUANTIZATION where vL =

s

K +Γ ρ

(1.99)

is the speed of the longitudinal phonon, • Two transverse phonon states |T, α, ~pi = aα p)† |0i T (~

(1.100)

one for each polarization, with energy ET (~ p) ET (~ p) = ~ωT (~ p) = vT ~|~ p| where vT =

s

K ρ

(1.101)

(1.102)

is the speed of the transverse phonons. The energies of the longitudinal and transverse phonon we found vanish as p~ → 0. These are acoustic phonons and the speeds vL and vT are speeds of sound. Notice that if the elastic modulus Γ = 0 all three states become degenerate. If in addition we were to have considered the effects of lattice anisotropies, the two transverse branches may no longer be degenerate as in this case. Similarly we can define multi-phonon states, with either polarization. For instance a state with two longitudinal phonons with momenta p~ and ~q is |L, ~p, ~qi = aL (~ p)† aL (~q)† |0i

(1.103)

This state has energy ~ωL (~ p) + ~ωL (~q) above the ground state. Finally, let us define the linear momentum operator P~ , for phonons of either longitudinal or transverse polarization, ! Z 3 X d p † α † α P~ = ~~ p aL (~ p) aL (~ p) + aT (~ p) aT (~ p) (1.104) (2π)3 α=1,2 This operator obeys the commutation relations i h P~ , aL (~k)† = ~~kaL (~k)† i h ~k)† = ~~kaα (~k)† P~ , aα ( T T h i P~ , aL (~k) = −~~kaL (~k) i h ~ ~ α ~ P~ , aα T (k) = −~kaT (k)

(1.105)

1.2. THE QUANTIZED ELASTIC SOLID and commutes with the Hamiltonian h i P~ , H = 0

13

(1.106)

Hence P~ is a conserved quantity. Moreover, using the commutation relations and the expressions for the displacement fields it is easy to show that h i ~ i (~x) P~ , ui (~x) = i~∇u

(1.107)

which implies that P~ is the generator of infinitesimal displacements. Hence, it is the linear momentum operator. It is easy to see that P~ annihilates the ground state P~ |0i = 0

(1.108)

which means that the ground state has zero momentum. In other terms, the ground state is translationally invariant (as it should). Using the commutation relations it is easy to show that P~ |L, ~ki = ~~k |L, ~ki,

P~ |T, α, ~ki = ~~k |T, α, ~ki

(1.109)

which allows us to identify the momentum carried by a phonon with ~~k where ~k is the label of the Fourier transform. Finally, we notice that we can easily write down an expression for the displacement field ui (~r) and the canonical momentum Πi (~r) in terms of creation and annihilation operators for longitudinal and transverse phonons: ui (~r) =

s
~ pi d3 p aL (~ p)ei~p·~r − aL (~ p)† e−i~p·~r 3 (2π) 2ωL (p) |~ p| s Z X
~ 1 d3 p +√ eα p ) aα p)ei~p·~r − aα p)† e−i~p·~r i (~ T (~ T (~ 3 ρ (2π) 2ωT (p) α=1,2 1 √ ρ

Z

(1.110)

which is known as a mode expansion. There is a similar expression for the canonical momentum Πi (~r). In summary, after quantizing the elastic solid we found that the quantum states of this system can be classified in terms if a set of excitations, the longitudinal and transverse phonons. These states carry energy and momentum (as well as polarization) and hence behave as particles. For these reason we will regard these excitations as the quasi-particles of this system. We will see that quasi-particles arise generically in interacting many-body system. One problem we will be interested in is in understanding the relation between the properties of the ground state and the quantum numbers of the quasiparticles.

14

CHAPTER 1. SECOND QUANTIZATION

1.3

Indistinguishable Particles

Let us consider now the problem of a system of N identical non-relativistic particles. For the sake of simplicity I will assume that the physical state of each particle j is described by its position ~xj relative to some reference frame. This case is easy to generalize. The wave function for this system is Ψ(x1 , . . . , xN ). If the particles are identical then the probability density, |Ψ(x1 , . . . , xN )|2 , must be invariant (i.e., unchanged) under arbitrary exchanges of the labels that we use to identify (or designate) the particles. In quantum mechanics, particles do not have well defined trajectories. Only the states of a physical system are well defined. Thus, even though at some initial time t0 the N particles may be localized at a set of well defined positions x1 , . . . , xN , they will become delocalized as the system evolves. Furthermore the Hamiltonian itself is invariant under a permutation of the particle labels. Hence, permutations constitute a symmetry of a manyparticle quantum mechanical system. In other terms, identical particles are indistinguishable in quantum mechanics. In particular, the probability density of any eigenstate must remain invariant if the labels of any pair of particles are exchanged. If we denote by Pjk the operator that exchanges the labels of particles j and k, the wave functions must change under the action of this operator at most by a phase factor. Hence, we must require that Pjk Ψ(x1 , . . . , xj , . . . , xk , . . . , xN ) = eiφ Ψ(x1 , . . . , xj , . . . , xk , . . . , xN )

(1.111)

Under a further exchange operation, the particles return to their initial labels and we recover the original state. This sample argument then requires that φ = 0, π since 2φ must not be an observable phase. We then conclude that there are two possibilities: either Ψ is even under permutation and P Ψ = Ψ, or Ψ is odd under permutation and P Ψ = −Ψ. Systems of identical particles which have wave functions which are even under a pairwise permutation of the particle labels are called bosons. In the other case, Ψ odd under pairwise permutation, they are Fermions. It must be stressed that these arguments only show that the requirement that the state Ψ be either even or odd is only a sufficient condition. It turns out that under special circumstances (e.g in one and twodimensional systems) other options become available and the phase factor φ may take values different from 0 or π. These particles are called anyons. For the moment the only cases in which they may exist appears to be in situations in which the particles are restricted to move on a line or on a plane. In the case of relativistic quantum field theories, the requirement that the states have well defined statistics (or symmetry) is demanded by a very deep and fundamental theorem which links the statistics of the states of the spin of the field. This is known as the spin-statistics theorem and it is actually an axiom of relativistic quantum mechanics (and field theory).

1.4.

1.4

15

FOCK SPACE

Fock Space

We will now discuss a procedure, known as Second Quantization, which will enable us to keep track of the symmetry of the states of systems of many identical particles in a simple way. Let us consider a system of N identical non-relativistic particles. The wave functions in the coordinate representation are Ψ(x1 , . . . , xN ) where the labels x1 , . . . , xN denote both the coordinates and the spin states of the particles in the state |Ψi. For the sake of definiteness we will discuss physical ˆ of the form systems describable by Hamiltonians H N N 2 X X X ˆ =−~ V (xj ) + U (xj − xk ) + . . . ▽2j + H 2m j=1 j=1

(1.112)

j,k

Let {φn (x)} be the wave functions for a complete set of one-particle states. Then an arbitrary N -particle state can be expanded in a basis which is the tensor product of the one-particle states, namely Ψ(x1 , . . . , xN ) =

X

C(n1 , . . . nN )φn1 (x1 ) . . . φnN (xN )

(1.113)

{nj}

Thus, if Ψ is symmetric (antisymmetric) under an arbitrary exchange xj ↔ xk , the coefficients C(n1 , . . . , nN ) must be symmetric (antisymmetric) under the exchange nj ↔ nk . A set of N -particle basis states with well defined permutation symmetry is the properly symmetrized or antisymmetrized tensor product 1 X P |Ψ1 , . . . ΨN i ≡ |Ψ1 i × |Ψ2 i × · · · × |ΨN i = √ ξ |ΨP (1) i × · · · × |ΨP (N ) i N! P (1.114) where the sum runs over the set of all possible permutations P . The weight factor ξ is +1 for bosons and −1 for fermions. Notice that, for fermions, the N -particle state vanishes if two particles are in the same one-particle state. This is the Pauli Exclusion Principle. The inner product of two N -particle states is hχ1 , . . . χN |ψ1 , . . . , ψN i = =

1 X P +Q ξ hχQ(1) |ψP (1) i · · · hχQ(N ) |ψP (1) i = N! P,Q X ′ ξ P hχ1 |ψP (1) i · · · hχN |ψP (N ) i P′

(1.115)

where P ′ = P + Q denotes the permutation resulting from the composition of the permutations P and Q. Since P and Q are arbitrary permutations, P ′ spans the set of all possible permutations as well.

16

CHAPTER 1. SECOND QUANTIZATION

It is easy to see that Eq.(1.115) is nothing but the permanent (determinant) of the matrix hχj |ψk i for symmetric (antisymmetric) states, i.e., hχ1 |ψ1 i . . . hχ1 |ψN i .. .. hχ1 , . . . χN |ψ1 , . . . ψN i = (1.116) . . hχN |ψ1 i . . . hχN |ψN i ξ

In the case of antisymmetric states, the inner product is the familiar Slater determinant. Let us denote by {|αi} a complete set of orthonormal one-particle states. They satisfy X hα|βi = δαβ |αihα| = 1 (1.117) α

The N -particle states are {|α1 , . . . αN i}. Because of the symmetry requirements, the labels αj can be arranged in the form of a monotonic sequence α1 ≤ α2 ≤ · · · ≤ αN for bosons, or in the form of a strict monotonic sequence α1 < α2 < · · · < αN for fermions. Let nj be an integer which counts how many particles are in the j-th one-particle state. The boson states |α1 , . . . αN i must be normalized by a factor of the form 1 √ |α1 , . . . , αN i n 1 ! . . . nN !

(α1 ≤ α2 ≤ · · · ≤ αN )

(1.118)

and nj are non-negative integers. For fermions the states are |α1 , . . . , αN i

(α1 < α2 < · · · < αN )

(1.119)

and nj = 0 > 1. These N -particle states are complete and orthonormal X 1 |α1 , . . . , αN ihα1 , . . . , αN | = Iˆ N ! α ,...,α 1

(1.120)

N

where the sum over the α’s is unrestricted and the operator Iˆ is the identity operator in the space of N -particle states. We will now consider the more general problem in which the number of particles N is not fixed a-priori. Rather, we will consider an enlarged space of states in which the number of particles is allowed to fluctuate. In the language of Statistical Physics what we are doing is to go from the Canonical Ensemble to the Grand Canonical Ensemble. Thus, let us denote by H0 the Hilbert space with no particles, H1 the Hilbert space with only one particle and, in general, HN the Hilbert space for N -particles. The direct sum of these spaces H H = H0 ⊕ H1 ⊕ · · · ⊕ HN ⊕ · · ·

(1.121)

is usually called Fock space. An arbitrary state |ψi in Fock space is the sum over the subspaces HN , |ψi = |ψ (0) i + |ψ (1) i + · · · + |ψ (N ) i + · · ·

(1.122)

17

1.5. CREATION AND ANNIHILATION OPERATORS

The subspace with no particles is a one-dimensional space spanned by the vector |0i which we will call the vacuum. The subspaces with well defined number of particles are defined to be orthogonal to each other in the sense that the inner product in Fock space ∞ X hχ(j) |ψ (j) i (1.123) hχ|ψi ≡ j=0

vanishes if |χi and |ψi belong to different subspaces.

1.5

Creation and Annihilation Operators

Let |φi be an arbitrary one-particle state, i.e. |φi ∈ H1 . Let us define the creation operator a ˆ† (φ) by its action on an arbitrary state in Fock space a ˆ† (φ)|ψ1 , . . . , ψN i ≡ |φ, ψ1 , . . . , ψN i

(1.124)

Clearly, a ˆ† (φ) maps the N -particle state with proper symmetry |ψ1 , . . . , ψN i to the N + 1-particle state |φ, ψ, . . . , ψN i, also with proper symmetry . The destruction or annihilation operator a ˆ(φ) is then defined as the adjoint of a ˆ† (φ), ∗

hχ1 , . . . , χN −1 |ˆ a(φ)|ψ1 , . . . , ψN i ≡ hψ1 , . . . , ψN |ˆ a† (φ)|χ1 , . . . , χN −1 i

(1.125)

Hence hχ1 , . . . , χN −1 |ˆ a(φ)|ψ1 , . . . , ψN i = hψ1 , . . . , ψN |φ, χ1 , . . . , χN −1 i∗ = hψ1 |φi hψ1 |χ1 i · · · hψ1 |χN −1 i .. .. .. = . . . hψN |φi hψN |χ1 i · · · hψN |χN −1 i

∗ ξ

(1.126)

We can now expand the permanent (or determinant) along the first column to get hχ1 , . . . , χN −1 |ˆ a(φ)|ψ1 , . . . , ψN i =

hψ1 |χ1 i ... N .. X . = ξ k−1 hψk |φi . .. (no ψk ) k=1 hψN |χ1 i ... =

N X

k=1

∗ ... hψN |χN −1 i ξ hψ1 |χN −1 i .. .

ξ k−1 hψk |φi hχ1 , . . . , χN −1 |ψ1 , . . . ψˆk . . . , ψN i

where ψˆk indicates that ψk is absent.

(1.127)

18

CHAPTER 1. SECOND QUANTIZATION Thus, the destruction operator is given by a ˆ(φ)|ψ1 , . . . , ψN i =

N X

k=1

ξ k−1 hφ|ψk i|ψ1 , . . . , ψˆk , . . . , ψN i

(1.128)

With these definitions, we can easily see that the operators a ˆ† (φ) and a ˆ(φ) obey the commutation relations a ˆ† (φ1 )ˆ a† (φ2 ) = ξ a ˆ† (φ2 )ˆ a† (φ1 ) Let us introduce the notation h i ˆ B ˆ A,

−ξ

(1.129)

ˆ −ξ B ˆ Aˆ ≡ AˆB

(1.130)

ˆ are two arbitrary operators. For ξ = +1 (bosons) we have the where Aˆ and B commutator  †   †  ˆ (φ1 ), a ˆ† (φ2 ) = 0 (1.131) a ˆ (φ1 ), a ˆ† (φ2 ) +1 ≡ a while for ξ = −1 it is the anticommutator  †  †  ˆ (φ1 ), a ˆ† (φ2 ) = 0 a ˆ (φ1 ), a ˆ† (φ2 ) −1 ≡ a

(1.132)

Similarly for any pair of arbitrary one-particle states |φ1 i and |φ2 i we get [ˆ a(φ1 ), a ˆ(φ2 )]−ξ = 0

(1.133)

It is also easy to check that the following identity generally holds   a ˆ(φ1 ), a ˆ† (φ2 ) −ξ = hφ1 |φ2 i

(1.134)

So far we have not picked any particular representation. Let us consider the occupation number representation in which the states are labelled by the number of particles nk in the single-particle state k. In this case, we have n1

n2

z }| { z }| { 1 | 1 . . . 1, 2 . . . 2 . . .i |n1 , . . . , nk , . . .i ≡ √ n1 !n2 ! . . .

(1.135)

In the case of bosons, the nj ’s can be any non-negative integer, while for fermions they can only be equal to zero or one. In general we have that if |αi is the αth single-particle state, then √ a ˆ†α |n1 , . . . , nα , . . .i = nα + 1 |n1 , . . . , nα + 1, . . .i √ a ˆα |n1 , . . . , nα , . . .i = nα |n1 , . . . , nα − 1, . . .i

(1.136)

Thus for both fermions and bosons, a ˆα annihilates all states with nα = 0, while for fermions a ˆ†α annihilates all states with nα = 1.

1.5. CREATION AND ANNIHILATION OPERATORS

19

We conclude that for bosons the following commutation relations hold h i h i [ˆ aα , a ˆβ ] = a ˆ†α , a ˆ†β = 0 a ˆα , a ˆ†β = δαβ (1.137)

whereas for fermions we obtain instead anticommutation relations n o n o {ˆ aα a ˆβ } = a ˆ†β , a ˆ†β = 0 a ˆα a ˆ†β = δαβ

(1.138)

If a unitary transformation is performed in the space of one-particle state vectors, then a unitary transformation is induced in the space of the operators themselves, i.e., |χi = α|ψi + β|φi

a ˆ(χ) = α∗ a ˆ(ψ) + β ∗ a ˆ(φ) † a ˆ (χ) = αˆ a† (ψ) + βˆ a† (φ)

⇒

(1.139)

and we say that a ˆ† (χ) transforms like the ket |χi while a ˆ(χ) transforms like the bra hχ|. For example, we can pick as the complete set of one-particle states the momentum states {|~ pi}. This is “momentum space”. With this choice the commutation relations are  †  a ˆ (~ p), a ˆ† (~ q ) −ξ = [ˆ a(~ p), a ˆ(~q)]−ξ = 0   d d † p − ~q) a ˆ(~ p), a ˆ (~ q ) −ξ = (2π) δ (~

(1.140)

where d is the dimensionality of space. In this representation, an N -particle state is |~ p1 , . . . , ~ pN i = a ˆ† (~ p1 ) . . . a ˆ† (~ pN )|0i (1.141)

On the other hand, we can also pick the one-particle states to be eigenstates of the position operators, i.e., |~x1 , . . . ~xN i = a ˆ† (~x1 ) . . . a ˆ† (~xN )|0i In position space, the operators satisfy  †  a ˆ (~x1 ), a ˆ† (~x2 ) −ξ =   a ˆ(~x1 ), a ˆ† (~x2 ) −ξ =

(1.142)

[ˆ a(~x1 ), a ˆ(~x2 )]−ξ = 0 δ d (~x1 − ~x2 )

(1.143)

This is the position space or coordinate representation. A transformation from position space to momentum space is the Fourier transform Z Z d |~ pi = d x |~xih~x|~ pi = dd x |~xiei~p·~x (1.144) and, conversely |~xi =

Z

dd p |~ pie−i~p·~x (2π)d

(1.145)

20

CHAPTER 1. SECOND QUANTIZATION

Then, the operators themselves obey Z † a ˆ (~ p) = dd x a ˆ† (~x)ei~p·~x Z dd p † † a ˆ (~ p)e−i~p·~x a ˆ (~x) = (2π)d

1.6

(1.146)

General Operators in Fock Space

Let A(1) be an operator acting on one-particle states. We can always define an extension Aˆ of A(1) acting on any arbitrary state |ψi of the N -particle Hilbert space HN as follows: N X

ˆ A|ψi ≡

j=1

|ψ1 i × . . . × A(1) |ψj i × . . . × |ψN i

(1.147)

For instance, if the one-particle basis states {|ψj i} are eigenstates of Aˆ with eigenvalues {aj } we get   N X ˆ (1.148) A|ψi = aj  |ψi j=1

We wish to find an expression for an arbitrary operator Aˆ in terms of creation (1) and annihilation operators. Let us first consider the operator Aαβ = |αihβ| (1)

which acts on one-particle states. The operators Aαβ form a basis of the space of operators acting on one-particle states. Then, the N -particle extension of (1) Aαβ is Aˆαβ |ψi =

N X j=1

|ψ1 i × · · · × |αi × · · · × |ψN ihβ|ψj i

(1.149)

Thus Aˆαβ |ψi =

N X j=1

j

z}|{ |ψ1 , . . . , α , . . . , ψN i hβ|ψj i

(1.150)

In other words, we can replace the one-particle state |ψj i from the basis with the state |αi at the price of a weight factor, the overlap hβ|ψj i. This operator has a very simple expression in terms of creation and annihilation operators. Indeed, a ˆ† (α)ˆ a(β)|ψi =

N X

k=1

ξ k−1 hβ|ψk i |α, ψ1 , . . . , ψk−1 , ψk+1 , . . . , ψN i

(1.151)

1.6.

21

GENERAL OPERATORS IN FOCK SPACE

We can now use the symmetry of the state to write k

z}|{ ξ k−1 |α, ψ1 , . . . , ψk−1 , ψk+1 , . . . , ψN i = |ψ1 , . . . , α , . . . , ψN i

(1.152)

Thus the operator Aˆαβ , the extension of |αihβ| to the N -particle space, coincides with a ˆ† (α)ˆ a(β) Aˆαβ ≡ a ˆ† (α)ˆ a(β) (1.153) We can use this result to find the extension for an arbitrary operator A(1) of the form X A(1) = |αihα|A(1) |βi hβ| (1.154) α,β

we find Aˆ =

X α,β

a ˆ† (α)ˆ a(β)hα|A(1) |βi

(1.155)

Hence the coefficients of the expansion are the matrix elements of A(1) between arbitrary one-particle states. We now discuss a few operators of interest. 1. The Identity Operator: The Identity Operator ˆ 1 of the one-particle Hilbert space X ˆ1 = |αihα|

(1.156)

α

ˆ in Fock space becomes the number operator N X ˆ = N a ˆ† (α)ˆ a(α)

(1.157)

α

In momentum and in position space it takes the form ˆ = N

Z

dd p † a ˆ (~ p)ˆ a(~ p) = (2π)d

Z

d

†

d xa ˆ (~x)ˆ a(~x) =

Z

dd x ρˆ(~x)

(1.158)

where ρˆ(x) = a ˆ† (~x)ˆ a(~x) is the particle density operator. 2. The Linear Momentum Operator: In the space H1 , the linear momentum operator is (1)

pˆj

=

Z

dd p pj |~ pih~ p| = (2π)d

Z

dd x |~xi

~ ∂j h~x| i

(1.159)

Thus, we get that the total linear momentum operator Pˆj is Pˆj =

Z

dd p pj a ˆ† (~ p)ˆ a(~ p) = (2π)d

Z

~ dd x a ˆ† (~x) ∂j a ˆ(~x) i

(1.160)

22

CHAPTER 1. SECOND QUANTIZATION

a ˆ† (~ p + ~q)

p~ + ~q

~q

Ve (~q)

p~

a ˆ(~ p)

Figure 1.1: One-body scattering. 3. Hamiltonian: The one-particle Hamiltonian H (1) p~ 2 + V (~x) 2m

(1.161)

~2 2 d ▽ δ (~x − ~y ) + V (~x)δ d (~x − ~y) 2m

(1.162)

H (1) = has the matrix elements h~x|H (1) |~y i = −

Thus, in Fock space we get Z   ~2 2 ˆ ▽ +V (~x) a ˆ(~x) H = dd x a ˆ† (~x) − 2m in position space. In momentum space we can define Z e V (~q) = dd x V (~x)e−i~q·~x

(1.163)

(1.164)

the Fourier transform of the potential V (x), and get Z Z Z dd p p~ 2 † dd p dd q e ˆ = H V (~q)ˆ a† (~ p + ~q)ˆ a(~ p) a ˆ (~ p )ˆ a (~ p ) + d d (2π) 2m (2π) (2π)d (1.165) The last term has a very simple physical interpretation. When acting on a one-particle state with well-defined momentum, say |~ pi, the potential term yields another one-particle state with momentum p~ + ~q, where ~q is the momentum transfer, with amplitude Ve (~q). This process is usually depicted by the diagram of Fig.1.1. Two-Body Interactions: A two-particle interaction is an operator Vˆ (2) which acts on the space of

1.6.

23

GENERAL OPERATORS IN FOCK SPACE two-particle states H2 , which has the form 1X |α, βiV (2) (α, β)hα, β| V (2) = 2

(1.166)

α,β

The methods developed above yield an extension of V (2) to Fock space of the form 1X † a ˆ (α)ˆ a† (β)ˆ a(β)ˆ a(α) V (2) (α, β) (1.167) Vˆ = 2 α,β

In position space, ignoring spin, we get Z Z 1 d ˆ V = d x dd y a ˆ† (~x) a ˆ† (~y ) a ˆ(~y ) a ˆ(~x) V (2) (~x, ~y) 2 Z Z Z 1 1 ≡ dd x dd y ρˆ(~x)V (2) (~x, ~y )ˆ ρ(~y ) + dd x V (2) (~x, ~x) ρˆ(~x) 2 2 (1.168)

where we have used the commutation relations. In momentum space we find instead Z Z Z 1 dd p dd q dd k e ~ † ˆ V = V (k) a ˆ (~ p + ~k)ˆ a† (~q − ~k)ˆ a(~q)ˆ a(~ p) 2 (2π)d (2π)d (2π)d (1.169) where V˜ (~k) is only a function of the momentum transfer ~k. This is a consequence of translation invariance. In particular for a Coulomb interaction, e2 (1.170) V (2) (~x, ~y ) = |~x − ~y| for which 4πe2 V˜ (~k) = (1.171) ~k 2

a ˆ (~ p + ~k)

p~ + ~k

~q − ~k

†

a ˆ† (~q − ~k)

~k Ve (~k) a ˆ(~ p) 4.

p~

~q

a ˆ(~q)

Figure 1.2: Two-body interaction.

24

1.7

CHAPTER 1. SECOND QUANTIZATION

Non-Relativistic Field Theory and Second Quantization

We can now reformulate the problem of an N -particle system as a non-relativistic field theory. The procedure described in the previous section is commonly known as Second Quantization. If the (identical) particles are bosons, the operators a ˆ(φ) obey canonical commutation relations. If the (identical) particles are Fermions, the operators a ˆ(φ) obey canonical anticommutation relations. In ˆ x) which position space, it is customary to represent a ˆ† (φ) by the operator ψ(~ obeys the equal-time algebra h i ˆ x), ψˆ† (~y ) ψ(~ = δ d (~x − ~y ) −ξ i h i h ˆ x), ψ(~ ˆ y) =0 = ψˆ† (~x), ψˆ† (~y ) ψ(~ −ξ

−ξ

(1.172)

In this framework, the one-particle Schr¨odinger equation becomes the classical field equation   ~2 2 ∂ ▽ −V (~x) ψ = 0 (1.173) i~ + ∂t 2m Can we find a Lagrangian density L from which the one-particle Schr¨odinger equation follows as its classical equation of motion? The answer is yes and L is given by ↔ ~2 ~ † ▽ψ · ▽ψ − V (~x)ψ † ψ (1.174) L = i~ψ † ∂t ψ − 2m Its Euler-Lagrange equations are ∂t

δL ~ · δL + δL = −▽ ~ † δ∂t ψ † δψ † δ ▽ψ

(1.175)

which are equivalent to the field Equation Eq. 1.173. The canonical momenta Π(x) and Π† (y) are δL Πψ = = −i~ψ (1.176) δ∂t ψ † and Π†ψ =

δL = i~ψ † δ∂t ψ

Thus, the (equal-time) canonical commutation relations are h i ˆ x), Π(~ ˆ y) ψ(~ = i~δ(~x − ~y) −ξ

which require that

h

i ˆ x), ψˆ† (~y ) ψ(~

−ξ

= δ d (~x − ~y)

(1.177)

(1.178)

(1.179)

1.8.

1.8

NON-RELATIVISTIC FERMIONS AT ZERO TEMPERATURE

25

Non-Relativistic Fermions at Zero Temperature

The results of the previous sections tell us that the action for non-relativistic fermions (with two-body interactions) is (in D = d + 1 space-time dimensions) S

= −

 ~2 ~ ˆ† ~ ˆ † † ˆ ˆ ˆ ˆ ▽ψ · ▽ψ − V (~x)ψ (x)ψ(x) d x ψ i~∂t ψ − 2m Z Z 1 ˆ dD x dD x′ ψˆ† (x)ψˆ† (x′ )U (x − x′ )ψˆ† (x)ψ(x)] 2

Z

D



(1.180)

where U (x − x′ ) represents instantaneous (two-body) interactions, U (x − x′ ) ≡ U (~x − ~x′ )δ(x0 − x′0 )

(1.181)

ˆ for this system is The Hamiltonian H ˆ H

= +

Z

~2 ~ ˆ† ~ ˆ ▽ψ · ▽ψ + V (~x)ψˆ† (~x)ψ(~x)] dd x [ 2m Z Z 1 d ˆ ′ )ψ(x) ˆ d x dd x′ ψˆ† (x)ψˆ† (x′ )U (x − x′ )ψ(x 2

(1.182)

ˆ For Fermions the fields ψ(x) and ψˆ† (x) satisfy equal-time canonical anticommutation relations ˆ x), ψˆ† (~x)} = δ D (~x − ~x′ ) {ψ(~ (1.183) while for Bosons they satisfy ˆ x), ψˆ† (~x′ )] = δ D (~x − ~x′ ) [ψ(~

(1.184)

b In both R cases, the Hamiltonian H commutes with the total number operator ˆ ˆ = dd xψˆ† (x)ψ(x) ˆ conserves the total number of particles. N since H The Fock space picture of the many-body problem is equivalent to the Grand Canonical Ensemble of Statistical Mechanics. Thus, instead of fixing the number of particles we will introduce a Lagrange multiplier µ, the chemical potential, to weigh contributions from different parts of the Fock space. Thus, we define e the operator H, e ≡H b − µN b H (1.185)

b this amounts to a shift of the energy by µN . In a Hilbert space with fixed N We will now allow the system to choose the sector of the Fock space but with b i is fixed to be some the requirement that the average number of particles hN ¯ . In the thermodynamic limit (N → ∞), µ represents the difference number N

26

CHAPTER 1. SECOND QUANTIZATION

of the ground state energies between two sectors with N + 1 and N particles e is (for spinless fermions) respectively. The modified Hamiltonian H e H

1.9

=

+

Z

~2 2 ˆ x) ▽ +V (~x) − µ]ψ(~ dd x ψˆ† (~x)[− 2m Z Z 1 ˆ y )ψ(~ ˆ x) dd x dd y ψˆ† (~x)ψˆ† (~y)U (~x − ~y)ψ(~ 2

(1.186)

The Ground State of a System of Free Fermions

E

single particle energy

Fermi Energy

EF

single particle states occupied states Fermi Sea

empty states

Figure 1.3: The Fermi Sea. Let us discuss now the very simple problem of finding the ground state for a system of N spinless free fermions. In this case, the interaction potential vanishes and, if the system is isolated, so does the one-body potential V (~x). In general there will be a complete set of one-particle states {|αi} and, in this ˆ is basis, H X ˆ = H Eα a ˆ†α aα (1.187) α

where the index α labels the one-particle states by increasing order of their single-particle energies E1 ≤ E2 ≤ · · · ≤ En ≤ · · ·

(1.188)

27

1.10. EXCITED STATES

Since were are dealing with fermions, we cannot put more than one particle in each state. Thus the state the lowest energy is obtained by filling up all the first N single particle states. Let |gndi denote this ground state |gndi =

N Y

α=1

N

a ˆ†α |0i

≡

a ˆ†1

···a ˆ†N |0i

z }| { = | 1 . . . 1, 00 . . .i

(1.189)

The energy of this state is Egnd with Egnd = E1 + · · · + EN

(1.190)

The energy of the top-most occupied single particle state, EN , is called the Fermi energy of the system and the set of occupied states is called the filled Fermi sea.

1.10

Excited States

A state like |ψi

N −1

z }| { |ψi = | 1 . . . 1 010 . . .i

(1.191)

is an excited state. It is obtained by removing one particle from the single particle state N (thus leaving a hole behind) and putting the particle in the unoccupied single particle state N + 1. This is a state with one particle-hole

|Groundi

N N +1

single particle states

a ˆ†N +1 a ˆN |Ψi

single particle states

Figure 1.4: An excited (particle-hole) state. pair, and it has the form |1 . . . 1010 . . .i = a ˆ†N +1 a ˆN |gndi

(1.192)

The energy of this state is Eψ = E1 + · · · + EN −1 + EN +1

(1.193)

28

CHAPTER 1. SECOND QUANTIZATION

Hence Eψ = Egnd + EN +1 − EN

(1.194)

and, since EN +1 ≥ EN , Eψ ≥ Egnd . The excitation energy ǫψ = Eψ − Egnd is ǫψ = EN +1 − EN ≥ 0

1.11

(1.195)

Construction of the Physical Hilbert Space

It is apparent that, instead of using the empty state |0i for reference state, it is physically more reasonable to use instead the filled Fermi sea |gndi as the physical reference state or vacuum state. Thus this state is a vacuum in the sense of absence of excitations. These arguments motivate the introduction of the particle-hole transformation. Let us introduce the fermion operators bα such that ˆbα = a ˆ†α

for α ≤ N

(1.196)

Since a ˆ†α |gndi = 0 (for α ≤ N ) the operators ˆbα annihilate the ground state |gndi, i.e., ˆbα |gndi = 0 (1.197) The following anticommutation relations hold o n o n o n ˆbβ , ˆb′ = a ˆα , ˆb†β = 0 {ˆ aα , a ˆ′α } = a ˆα , ˆbβ = β n o o n ˆbβ , ˆb† ′ = δββ ′ a ˆα , a ˆ†α′ = δαα′ β

(1.198)

where α, α′ > N and β, β ′ ≤ N . Thus, relative to the state |gndi, a ˆ†α and ˆb†β behave like creation operators. An arbitrary excited state has the form |α1 . . . αm , β1 . . . βn ; gndi ≡ a ˆ†α1 . . . a ˆ†αm ˆb†β1 . . . ˆb†βn |gndi

(1.199)

This state has m particles (in the single-particle states α1 , . . . , αm ) and n holes (in the single-particle states β1 , . . . , βn ). The ground state is annihilated by the operators a ˆα and ˆbβ a ˆα |gndi = ˆbβ |gndi = 0

(α > N β ≤ N )

(1.200)

ˆ is normal ordered relative to the empty state |0i, i.e. The Hamiltonian H ˆ H|0i = 0,

(1.201)

but it is not normal ordered relative to the physical ground state |gndi. The ˆ relative to |gndi. particle-hole transformation enables us to normal order H X X X X ˆ = H Eα a ˆ†α a ˆα = Eα + Eα a ˆ†α a ˆα − (1.202) Eβ ˆb†β ˆbβ α

α≤N

α>N

β≤N

1.11. CONSTRUCTION OF THE PHYSICAL HILBERT SPACE Thus

ˆ = Egnd + : H ˆ: H

where Egnd =

N X

29

(1.203)

Eα

(1.204)

α=1

is the ground state energy, and the normal ordered Hamiltonian is X X † ˆb ˆbβ Eβ ˆ := :H Eα a ˆ†a a ˆα − β α>N

(1.205)

β≤N

ˆ is not normal-ordered relative to |gndi either. Thus, The number operator N we write X X X † ˆb ˆbβ ˆ= N a ˆ†α a ˆα = N + a ˆ†α aα − (1.206) β α

α>N

β≤N

We see that particles raise the energy while holes reduce it. However, if we deal ˆ (i.e., [N, ˆ H] ˆ = 0) for with Hamiltonians that conserve the particle number N every particle that is removed a hole must be created. Hence particles and holes can only be created in pairs. A particle-hole state |α, β; gndi is |α, β; gndi ≡ a ˆ†αˆb†β |gndi

It is an eigenstate with an energy ˆ H|α, β; gndi = =



(1.207)

 ˆ: a Egnd + : H ˆ†αˆb†β |gndi

(Egnd + Eα − Eβ ) |α, β; gndi

This state has exactly N particles since ˆ |α, β; gndi = (N + 1 − 1)|α, β; gndi = N |α, β; gndi N

Let us finally notice that the field operator ψˆ† (x) in position space is X X X ψˆ† (~x) = h~x|αiˆ a†α = φα (~x)ˆ a†α + φβ (~x)ˆbβ α

α>N

(1.208) (1.209)

(1.210)

β≤N

where {φα (~x)} are the single particle wave functions. The procedure of normal ordering allows us to define the physical Hilbert space. The physical meaning of this approach becomes more transparent in the thermodynamic limit N → ∞ and V → ∞ at constant density ρ. In this limit, the space of Hilbert space is the set of states which is obtained by acting with creation and annihilation operators finitely on the ground state. The spectrum of states that results from this approach consists on the set of states with finite excitation energy. Hilbert spaces which are built on reference states with macroscopically different number of particles are effectively disconnected from each other. Thus, the normal ordering of a Hamiltonian of a system with an infinite number of degrees of freedom amounts to a choice of the Hilbert space. This restriction becomes of fundamental importance when interactions are taken into account.

30

1.12

CHAPTER 1. SECOND QUANTIZATION

The Free Fermi Gas

Let us consider the case of free spin one-half electrons moving in free space. The Hamiltonian for this system is Z   X ~2 2 ˜ = dd x (1.211) ▽ −µ ψˆσ (~x) H ψˆσ† (~x) − 2m σ=↑,↓

where the label σ =↑, ↓ indicates the z-projection of the spin of the electron. The value of the chemical potential µ will be determined once we satisfy that the electron density is equal to some fixed value ρ¯. In momentum space, we get Z p ~·~ x dd p ˆ ˆ ψσ (~x) = (1.212) ψσ (~ p) e−i ~ d (2π~) where the operators ψˆσ (~ p) and ψˆσ† (~ p) satisfy o n ψˆσ (~ p), ψˆσ† ′ (~ p) = (2π~)d δσσ′ δ d (~ p − p~′ ) n o ˆ p), ψˆσ′ (~ ψˆσ† (~ p), ψˆσ† ′ (~ p) = {ψ(~ p′ )} = 0 The Hamiltonian has the very simple form Z  dd p X  ˜ = H ǫ(~ p ) − µ ψˆσ† (~ p)ψˆσ (~ p) (2π~)d

(1.213)

(1.214)

σ=ˆi,↓

where ε(~ p) is given by p~2 (1.215) 2m For this simple case, ε(~ p) is independent of the spin state. It is convenient to define the energy of the one-particle states measured relative to the chemical potential (or Fermi energy) µ = EF , E(~ p) as ε(~ p) =

E(~ p) = ε(~ p) − µ

(1.216)

Hence, E(~ p) is the excitation energy measured from the Fermi energy EF = µ. The energy E(~ p) does not have a definite sign since there are states with ε(~ p) > µ as well as states with ε(~ p) < µ. Let us define by pF the value of |~ p| for which E(pF ) = ε(pF ) − µ = 0

(1.217)

This is the Fermi momentum. Thus, for |~ p| < pF , E(~ p) is negative while for |~ p| > pF , E(~ p) is positive. We can construct the ground state of the system by finding the state with lowest energy at fixed µ. Since E(~ p) is negative for |~ p| ≤ pF , we see that by

1.12.

31

THE FREE FERMI GAS

filling up all of those states we get the lowest possible energy. It is then natural to normal order the system relative to a state in which all one-particle states with |~ p| ≤ pF are occupied. Hence we make the particle- hole transformation ˆbσ (~ p) = ψˆσ† (~ p) for |~ p| ≤ pF ˆ a ˆσ (~ p) = ψσ (~ p) for |~ p| > pF

(1.218)

In terms of the operators a ˆσ and ˆbσ , the Hamiltonian is i X Z dd p h † ˆbσ (~ ˆb† (~ ˜ = E(~ p )θ(|~ p | − p )ˆ a (~ p )ˆ a (~ p ) + θ(p − |~ p |)E(~ p ) p ) p ) H F σ F σ σ (2π~)d σ=↑,↓

(1.219)

where θ(x) is the step function θ(x) =



1 x>0 0 x≤0

(1.220)

Using the anticommutation relations in the last term we get h i X Z dd p † ˆb† (~ ˆbσ (~ ˜ ˜gnd H= E(~ p ) θ(|~ p | − p )ˆ a (~ p )ˆ a (~ p ) − θ(p − |~ p |) p ) p ) +E F σ F σ σ (2π~)d σ=↑,↓

(1.221) ˜gnd , the ground state energy measured from the chemical potential µ, where E is given by X Z dd p θ(pF − |~ p|)E(~ p)(2π)d δ d (0) = Egnd − µN (1.222) E˜gnd = (2π~)d σ=↑,↓

Recall that (2π~)d δ(0) is equal to d d

d (d)

(2π~) δ (0) = lim (2π~) δ p ~→0

(~ p) = lim

p ~→0

Z

dd x ei~p·~x/~ = V

(1.223)

where V is the volume of the system. Thus, E˜gnd is extensive E˜gnd = V ε˜gnd and the ground state energy density ε˜gnd is Z dd p ε˜gnd = 2 E(~ p) = εgnd − µ¯ ρ d |~ p|≤pF (2π~)

(1.224)

(1.225)

where the factor of 2 comes from the two spin orientations. Putting everything together we get  2  2   Z pF Z Sd p ~ p dd p d−1 dp p −µ =2 −µ ε˜gnd = 2 d 2m (2π~)d 2m 0 |~ p|≤pF (2π~) (1.226)

32

CHAPTER 1. SECOND QUANTIZATION

where Sd is the area of the d-dimensional hypersphere. Our definitions tell us p2F ≡ EF where EF , is the Fermi energy. that the chemical potential is µ = 2m Thus the ground state energy density εgnd ( measured from the empty state) is equal to Z pF pd+2 1 Sd Sd pdF Sd d+1 F Egnd = = 2E dp p = F m (2π~)d 0 m(d + 2) (2π~)d (d + 2)(2π~)d (1.227) How many particles does this state have? To find that out we need to look at the number operator. The number operator can also be normal-ordered with respect to this state Z dd p X ˆ† ˆ N = ψσ (~ p)ψˆσ (~ p) = (2π~)d σ=↑,↓ Z o dd p X n † † ˆ ˆ θ(|~ p | − p )ˆ a (~ p )ˆ a (~ p ) + θ(p − |~ p |) b (~ p ) b (~ p ) = F σ F σ σ σ (2π~)d σ=↑,↓

(1.228)

ˆ can also be written in the form Hence, N ˆ =: N ˆ : +N N

(1.229)

ˆ : is where the normal-ordered number operator : N Z i dd p X h † ˆb† (~ ˆbσ (~ ˆ θ(|~ p | − p )ˆ a (~ p )ˆ a (~ p ) − θ(p − |~ p |) p ) p ) (1.230) : N := F σ F σ σ (2π~)d σ=↑,↓

and N , the number of particles in the reference state |gndi, is Z dd p X Sd 2 N= V θ(pF − |~ p|)(2π~)d δ d (0) = pdF (2π~)d d (2π~)d

(1.231)

σ=↑,↓

Therefore, the particle density ρ¯ = ρ¯ =

N V

is

2 Sd pd d (2π~)d F

(1.232)

This equation determines the Fermi momentum pF in terms of the density ρ¯. Egnd d Similarly we find that the ground state energy per particle is N = d+2 EF . The excited states can be constructed in a similar fashion. The state |+, σ, p~i |+, σ, p~i = a ˆ†α (~ p)|gndi

(1.233)

is a state which represents an electron with spin σ and momentum p~ while the state |−, σ, p~i |−, σ, p~i = ˆb†σ (~ p)|gndi (1.234)

1.13. FREE BOSE AND FERMI GASES AT T > 0

33

represents a hole with spin σ and momentum ~p. From our previous discussion we see that electrons have momentum ~p with |~ p| > pF while holes have momentum ~p with |~ p| < pF . The excitation energy of a one-electron state is E(~ p) ≥ 0(for |~ p| > pF ), while the excitation energy of a one-hole state is −E(~ p) ≥ 0 (for |~ p| < pF ). Similarly, an electron-hole pair is a state of the form |σ~ p, σ ′ p~′ i = a ˆ†σ (~ p)ˆb†σ′ (~ p′ )|gndi

(1.235)

with |~ p| > pF and |~ p′ | < pF . This state has excitation energy E(~ p)−E(~ p′ ), which is positive. Hence, states which are obtained from the ground state without changing the density, can only increase the energy. This proves that |gndi is indeed the ground state. However, if the density is allowed to change, we can always construct states with energy less than Egnd by creating a number of holes without creating an equal number of particles.

1.13

Free Bose and Fermi Gases at T > 0

We will now quickly review the properties of free Fermi and Bose gases and Bose condensation, and their thermodynamic properties. We’ll work in the Grand Canonical Ensemble in which particle number is not exactly conserved, i.e. we couple the system to a bath at temperature T and fixed chemical potential µ. The Grand Partition Function at fixed T and µ is ˆ

ˆ

ZG = e−βΩ = tr eβ(H−µN )

(1.236)

ˆ is the Hamiltonian, N ˆ is the total particle number operator, β = where H (kT )−1 , where T is the temperature and k is the Boltzmann constant, and Ω is the thermodynamic potential Ω = Ω(T, V, µ). For a free particle system the total Hamiltonian H is a sum of one-particle Hamiltonians, X H= εℓ a ˆ†ℓ a ˆℓ (1.237) ℓ

where ℓ = 0, 1, . . . labels the states of the complete set of single-particle states {|ℓi}, and {εℓ } are the eigenvalues of the single-particle Hamiltonian. The Grand Partition Function ZG can be computed easily in the occupation number representation of the eigenstates of H: P X ZG = hn1 . . . nk . . . |eβ ℓ (µ − εℓ )nℓ |n1 . . . nk . . .i (1.238) n1 ...nk ...

where nℓ is the occupation number of the ℓ-th single particle state. The allowed values of the occupation numbers nℓ for fermions and bosons is nℓ = 0, 1 nℓ = 0, 1, . . .

fermions bosons

(1.239)

34

CHAPTER 1. SECOND QUANTIZATION Thus, the Grand Partition Function is ZG =

∞ X Y nℓ

ℓ=0

hnℓ |eβ(µ−εℓ )nℓ |nℓ i

(1.240)

For fermions we get F

F ZG = e−βΩG =

∞ Y X

eβ(µ−εℓ ) =

ℓ=0 nℓ =0,1

∞ h Y

1 + eβ(µ−εℓ )

ℓ=0

i

(1.241)

or, what is the same, the thermodynamic potential for a system of free fermions ΩF G at fixed T and µ is given by an expression of the form ΩF G = −kT

h i ln 1 + eβ(µ−εℓ )

∞ X ℓ=0

Fermions

(1.242)

For bosons we find instead, B

B ZG = e−βΩG =

∞ Y

∞ X

eβ(µ−εℓ ) =

nℓ =0

ℓ=0

∞ Y

ℓ=0

1 1−

eβ(µ−εℓ )

(1.243)

Hence, the thermodynamic potential ΩB G for a system of free bosons at fixed T and µ is ∞ i h X B ΩG = kT ln 1 − eβ(µ−εℓ ) Bosons (1.244) ℓ=0

The average number of particles hN i is ˆ

hN i =

ˆ

ˆ e−β(H−µN ) trN ˆ ˆ) N tre−β(H−µ

=

hN i = ρV =

1 ∂ 1 1 ∂ZG = ln ZG β ZG ∂µ β ∂µ

1 ∂ ∂ΩG [−βΩG ] = − β ∂µ ∂µ

(1.245)

(1.246)

where ρ is the particle density and V is the volume. For bosons hN i is hN i =

∞ 1 1 X (−1) eβ(µ−εℓ ) β β(µ−ε ℓ) β 1 − e ℓ=0

(1.247)

Hence, hN i =

∞ X ℓ=0

1 eβ(εℓ −µ) − 1

Bosons

(1.248)

1.13. FREE BOSE AND FERMI GASES AT T > 0

35

Whereas for fermions we find, ∞ X

hN i =

1 eβ(εℓ −µ)

ℓ=0

Fermions

+1

(1.249)

In general the average number of particles hN i is hN i =

∞ X ℓ=0

hnℓ i =

∞ X ℓ=0

1 eβ(εℓ −µ) ± 1

(1.250)

where + holds for fermions and − holds for bosons. The internal energy U of the system is U = hHi = hH − µN i + µhN i

(1.251)

∂ ln ZG ∂ 1 ∂ZG = − = (βΩG ) ZG ∂β ∂β ∂β

(1.252)

where hH − µN i = − Hence, U=

∂ (βΩG ) + µhN i ∂β

(1.253)

As usual, we will need to find the value of the chemical potential µ in terms of the average number of particles hN i (or in terms of the average density ρ) at fixed temperature T and volume V . Once this is done we can compute the thermodynamic variables of the systems, such as the pressure P and the entropy S by using standard thermodynamic relations, e.g. G Pressure: P = − ∂Ω ∂V µ,T G Entropy: S = − ∂Ω ∂T µ,V The following integrals will be helpful below Z +∞ a 2 1 b2 dx I(a, b) = e− 2 x −bx √ = √ e 2a a 2π −∞

and

1.13.1

 

n! a−(n+1)/2 In (a) = (n/2)! 2n/2  0

for n even

(1.254)

(1.255)

for n odd

Bose Case

Let ε0 = 0, εℓ ≥ 0, and µ ≤ 0. The thermodynamic potential for bosons ΩB G is ΩB G = kT

∞ X ℓ=0

i h ln 1 − eβ(µ−εℓ )

(1.256)

36

CHAPTER 1. SECOND QUANTIZATION

The number of modes in the box of volume V with momenta in the infinitesimal d3 k region d3 p is sV (2π~) 3 , and for bosons of spin S we have set s = 2S + 1. Thus for spin-1 bosons there are 3 states. (However, in the case of photons S = 1 but have only 2 polarization states and hence s = 2.) The single-particle energies ε(~ p) are ε(~ p) =

p2 2m

(1.257)

where we have approximated the discrete levels by a continuum. In these terms, ΩB G becomes   Z p2 d3 p ) β(µ− 2m (1.258) ln 1 − e ΩB = kT sV G (2π~)3 and ρ = hN i/V is given by ρ=

hN i 1 ∂Ω = − = s V V ∂µ

Z

βp2

e− 2m eβµ 1−e

2 − βp 2m

+βµ

d3 p (2π~)3

(1.259)

It is convenient to introduce the fugacity z and the variable x z = eβµ

x2 =

βp2 2m

in terms of which the volume element of momentum space becomes 3/2  d3 p 2m 4πp2 x2 dx → dp = (2π~)3 (2π~)3 2π 2 ~3 β

(1.260)

(1.261)

where we have carried out the angular integration. The density ρ is " 3/2 #  Z ∞ 2 x2 ze−x 2m ρ = s dx 2π 2 ~3 β 1 − ze−x2 o 3/2 Z +∞  ∞ X 2 s 2m = dx z n+1 e−(n+1)x x2 2 3 4π ~ β −∞ n=0 3/2 X  ∞ Z +∞ 2 2m s dx z n x2 e−nx = = 4π 2 ~3 β n=1 −∞ 3/2 X  Z +∞ ∞ 2 s zn 2m = dy y 2 e−y 2 3 3/2 4π ~ β n −∞ n=1 3/2 X  √ ∞ π zn 2m s (1.262) = 2 3 3/2 4π ~ β 2 n n=1 We now introduce the (generalized) Riemann ζ-function, ζr (z) ∞ X zn ζr (z) = nr n=1

(1.263)

1.13. FREE BOSE AND FERMI GASES AT T > 0

37

(which is well defined for r > 0 and Re z > 0) in terms of which the expression for the density takes the more compact form 3/2  mkT ζ3/2 (z), where z = eβµ (1.264) ρ = s 2π~2 This equation must inverted to determine µ(ρ). Likewise, the internal energy U can also be written in terms of a Riemann ζ-function: 3/2  ∂ 3 mkT U = V ζ5/2 (z) (1.265) (βΩG ) + µV ρ = s kT ∂β 2 2π~2 ρ low) ⇒ ζ3/2 (z) is small and can be approximated If ρ is low and T high (or T 3/2 by a few terms of its power series expansion

ζ3/2 (z) =

z2 z + 3/2 + . . . ⇒ z small (i.e. µ large) 1 2

(1.266)

Hence, in the low density (or high temperature) limit we can approximate ζ3/2 (z) ≈ z and ζ5/2 (z) ≈ z

(1.267)

and, in this limit, the internal energy density is 3 U = ε ≃ kT ρ, V 2

(1.268)

which is the classical result. ρ Conversely if T 3/2 is large, i.e. large ρ and low T , the fugacity z is no longer small. Furthermore the function ζr (z) has a singularity at |z| = 1. For r = 32 , 25 the ζr (1)-function takes the finite values ζ3/2 (1) = 2.612 . . . ζ5/2 (1) = 1.341 . . .

(1.269)

′ ′′ although the derivatives ζ3/2 (1) and ζ5/2 (1) diverge. Let us define the critical temperature Tc as the temperature at which the fugacity takes the value 1 z(Tc ) = 1 (1.270)

In other terms, at T = Tc the expressions are at the radius of convergence of the ζ-function. Since  3/2 mkTc ρ=s ζ3/2 (1) (1.271) 2π~2

we find the Tc is given by Tc =

2π~2 mk



ρ/s ζ3/2 (1)

2/3

Critical Temperature

(1.272)

38

CHAPTER 1. SECOND QUANTIZATION

Our results work only for T ≥ Tc . Why do we have a problem for T < Tc ? Let’s reexamine the sum ρ=−

∞ X 1 ∂Ω = ∂µ e−β(µ−εℓ ) − 1

(1.273)

ℓ=0

µ

T

Tc

Figure 1.5: The chemical potential as a function of temperature in a free Bose gas. Here Tc is the Bose-Einstein condensation temperature. For small z, e−βµ = e+β|µ| is large, and each term has a small contribution. But for |z| → 1 the first few terms may have a large contribution. In particular hn1 i =

1 eβ(ε1 −µ)

−1

1

≪

eβ(ε0 −µ)

If we now set ε0 = 0, we find that hn0 i =

1 e−βµ

−1

 ⇒ µ = −kT ln 1 +

−1

1 hn0 i



= hn0 i

(1.274)

−kT hn0 i

(1.275)

≈

for hn0 i ≫ 1. For low T , µ approaches zero as N → ∞ (at fixed ρ), and e which are not in the εℓ − µ ≈ εℓ , for ℓ > 0. Hence, the number of particles N lowest energy state ε0 = 0 is e N

= ≃ =

hN − n0 i = Z

3

∞ X ℓ=1

1 eβεi − 1

1 d p (2π~)3 eβp2 /2m − 1  3/2 mkT sV ζ3/2 (1) 2π~2

sV

Hence, e = hN i N



T Tc

3/2

(1.276)

(1.277)

1.13. FREE BOSE AND FERMI GASES AT T > 0 and

"

˜ = hN i hn0 i = hN i − N

1−



39

T Tc

3/2 #

(1.278)

Then, the average number of particles in the “single-particle ground state” |0i is "  3/2 # T hn0 (T )i = hN i 1 − (1.279) Tc and their density (the “condensate fraction”) is    3/2  ρ 1 − TTc ρ0 (T ) =  0

T < Tc

(1.280)

T > Tc

z

T −3/2

Tc

T

Figure 1.6: The fugacity as a function of temperature in a free Bose gas. Tc is the Bose-Einstein condensation temperature. This is a phase transition known as Bose-Einstein Condensation (BEC). Thus, for T < Tc , the lowest energy state is macroscopically occupied, which is why this phenomenon is called Bose-condensation. It simply means that for a Bose system the ground state has finite fraction bosons in the same singleparticle ground state. Let us examine the behavior of the thermodynamic quantities near and below the phase transition.The total energy is U=

3 kT s 2



mkT 2π~2

3/2

V ζ5/2 (1) =

Hence, U=

ζ5/2 (1) 3 kT hN i 2 ρ3/2 (1)



T Tc

3 e ζ5/2 (1) kT N 2 ζ3/2 (1) 3/2

∝ T 5/2

(1.281)

(1.282)

40

CHAPTER 1. SECOND QUANTIZATION

hn0 i V

ρ

Tc

T

Figure 1.7: The condensate fraction hn0 i/V a function of temperature in a free Bose gas. The specific heat at constant volume (and particle number) Cv is    3/2 ∂U T Cv = ∝ , for T < Tc ∂T V,N Tc On the other hand, since for T > Tc 3/2  mkT ζ3/2 (z)sV hN i = 2π~2

(1.283)

(1.284)

we find that the internal energy above Tc is U =

ζ5/2 (z) 3 kT hN i 2 ζ3/2 (z)

(1.285)

To use this expression we must determine first z = z(T ) at fixed density ρ. How do the thermodynamic quantities of a Bose system behave near Tc ? From the above discussion we see that at Tc there must be a singularity in most thermodynamic quantities. However it turns out that while the free Bose gas does have a phase transition at Tc , the behavior near Tc is dominated by critical fluctuations which are governed by inter-particle interactions which are not included in a system of free bosons. Thus, while the specific heat Cv of a real system of bosons has a divergence as T → Tc (both from above and from below), a system of free bosons has a mild singularity in the form of a jump in the slope of Cv (T ) at Tc . The study of the behavior of singularities at critical points is the subject of the theory of Critical Phenomena. The free Bose gas is also pathological in other ways. Below Tc a free Bose gas exhibits Bose condensation but it is not a superfluid. We will discuss this issue later on this semester. For now we note that there are no true free Bose gases in nature since all atoms interact with each other however weakly. These interactions govern the superfluid properties of the Bose fluid.

1.13. FREE BOSE AND FERMI GASES AT T > 0

1.13.2

41

Fermi Case

In the case of fermions, due to the Pauli Principle there is no condensation in a single-particle state. In the presence of interactions the ground state of a system of fermions may be in a highly non-trivial phase including superconductivity, charge density waves and other more exotic possibilities. Contrary to the case of a free Bose system, a system of free fermions does not have a phase transition at any temperature. The thermodynamic potential for free fermions ΩF G (T, V, µ) is ΩF G (T, V, µ)

∞ X

= −kT

ℓ=0

  ln 1 + e−β(εℓ −µ)

(1.286)

where, once again, the integers ℓ = 0, 1, . . . label a complete set of one-particle states. No restriction on the sign of µ is now necessary since the argument of the logarithm is now positive as all terms are manifestly positive. Thus there are no vanishing denominators in the expressions for the thermodynamic quantities as in the Bose case. Thus, in the case of (free) fermions we can take the thermodynamic limit without difficulty, and use the integral expressions right away. For spin S fermions, s = 2S + 1, the thermodynamic potential is  Z ” “ 2 p d3 p −µ −β 2m (1.287) ln 1 + e ΩF = −kT s V G (2π~)3 and the density ρ is given by hN i 1 ∂Ω ρ= = − =s V V ∂µ

Z

βp2

d3 p e− 2m eβµ 2 3 (2π~) 1 + e− βp 2m eβµ

(1.288)

The average occupation number of a general one-particle state |ℓi is hnℓ i =

e−β(εℓ −µ) ∂Ω = ∂εℓ 1 + e−β(εℓ −µ)

=

1 eβ(εℓ −µ)

+1

(1.289)

which is the Fermi-Dirac distribution function. Most of the expressions of interest for fermions involve the the Fermi function f (x) 1 f (x) = x (1.290) e +1 In particular, if we introduce the one-particle density of states N0 (ε), N0 (ε) = s

2π(2m)3/2 √ ε (2π~)3

(1.291)

the expression for the particle density ρ can be written in the more compact form Z ∞ ρ=s dε N0 (ε) f (β(ε − µ)) (1.292) 0

42

CHAPTER 1. SECOND QUANTIZATION

hnℓ i

hnℓ i

1

1

ε

µ0

(a) Occupancy nℓ at T = 0

µ0

ε

(b) Occupancy nℓ for T > 0

Figure 1.8: The occupation number of eigenstate |ℓi in a free Fermi gas, (a) at T = 0 and (b) for T > 0. Here, µ0 = EF is the Fermi energy. where we have assumed that the one-particle spectrum begins at ε0 = 0. Similarly, the internal energy density for a system of free fermions of spin S is   Z p2 d3 p p2 e−β( 2m −µ) 1 ∂βΩF U G = + µN = s u= p2 −µ) V V ∂β (2π~)3 2m 1 + e−β( 2m Z ∞ ≡ s dε ε N0 (ε) f (β(ε − µ)) (1.293) 0

In particular, at T = 0 we get u(0) =

3 EF ρ 5

(1.294)

The pressure P at temperature T is obtained from the thermodynamic relation  F ∂ΩG P =− (1.295) ∂V T,µ For a free Fermi system we obtain Z ∞   P = kT s dε N0 (ε) ln 1 + e−β(ε − µ)

(1.296)

0

This result implies that there is a non-zero pressure P0 in a Fermi gas at T = 0 even in the absence of interactions: Z µ0 2 dε N0 (ε)(µ0 − ε) = µ0 ρ − u(0) = EF ρ > 0 P0 = lim P (T, µ, V ) = T →0 5 0 (1.297) which is known as the Fermi pressure. Thus, the pressure of a system of free fermions is non-zero (and positive) even at T = 0 due to the effects of the Pauli Principle which keeps fermions from occupying the same single-particle state.

1.13. FREE BOSE AND FERMI GASES AT T > 0

43

For a free Fermi gas at T = 0, µ0 = EF . Hence, at T = 0 all states with εℓ < µ0 are occupied and all other states are empty. At low temperatures kT ≪ EF , most of the states below EF will remain occupied while most of the states above EF will remain empty, and only a small fraction of states with single particle energies close to the Fermi energy will be affected by thermal fluctuations. This observation motivates the Sommerfeld expansion which is useful to determine the low temperature behavior of a free Fermi system. Consider an expression of the form Z ∞ I= dε f (β(ε − µ)) g(ε) (1.298) 0

where g(ε) is a smooth function of the energy. Eq.(1.298) can be written in the equivalent form Z µ Z µ Z ∞ g(ε) g(ε) − + dε dε g(ε) (1.299) I= dε 0 0 µ eβ(ε − µ) + 1 e−β(ε − µ) + 1 If we now make the change of variables x = β(ε − µ) in the first integral of eq.(1.299) and x = −β(ε − µ) in the second integral of Eq.(1.299), we get Z µ Z ∞ Z βµ g(µ + x/β) g(µ − x/β) dε g(ε) + kT dx I= − kT dx (1.300) x+1 e ex + 1 0 0 0 Since the function g(x) is a smooth differentiable function of its argument we can approximate x x (1.301) g(µ ± ) = g(µ) ± g ′ (µ) + . . . β β At low temperatures βµ ≫ 1, or kT ≪ µ, with exponential precision we can extend the upper end of the integral in the last term of Eq.(1.300) to infinity, and obtain the asymptotic result Z µ Z ∞ Z µ 2 π2 x I= dε g(ε)+ 2 g ′ (µ) dx+. . . = (kT )2 g ′ (µ)+. . . dε g(ε)+ x β e + 1 6 0 0 0 (1.302) where we have neglected terms O(e−βµ ) and O((kT )4 ), and used the integral: Z ∞ x π2 dx x = (1.303) e +1 12 0 Using these results we can now determine the low temperature behavior of all thermodynamic quantities of interest. Thus we obtain !  2 π 2 kT µ(T ) = µ0 1 − + ... (1.304) 12 µ0 u(T ) = u(0) + γ T 2 + . . . where γ=

2π(2m)3/2 π 2 √ s µ0 k 2 (2π~)3 6

(1.305)

(1.306)

44

CHAPTER 1. SECOND QUANTIZATION

from where we find that the low-temperature specific heat Cv for free fermions is Cv = 2γT + . . . (1.307) A similar line of argument shows that the thermodynamic potential ΩF G at low temperatures, kT ≪ µ, is ΩF G V

π2 N0 (EF )(kT )2 + . . . 6 ΩF π2 ρ G (0) + O((kT )4 ) − (kT )2 V 6 EF

= u(0) − µρ − s =

(1.308)

There is a simple and intuitive way to understand the T 2 dependence (or scaling) of the thermodynamic potential. First we note that the thermal fluctuations only affect a small number of single particle states all contained within a range of the order of kT around the Fermi energy, EF , multiplied by the density of single particle states, N0 (EF ). This number is thus kT N0 (EF ). On the other hand, the temperature dependent part of the thermodynamic potential has a factor of kT in front. Thus, we obtain the scaling N0 (EF )(kT )2 . Similar considerations apply to all other quantities.

Chapter 2 Green Functions and Observables 2.1

Green Functions in Quantum Mechanics

We will be interested in studying the properties of the ground state of a quantum mechanical many particle system. We will also be interested in understanding the physical properties of its low energy excitations. In order to accomplish these goals we will consider the following problem. Let |Gi be the ground state of the system. We will imagine that at some time t in the past we have acted on the system with a local perturbation, which we will b r , t)† , and created a particular initial describe in terms of a local operator O(~ b r, t)† |Gi. We may want to ask what is the quantum mechanical state O(~ b r ′ )† |Gi at time t + T . In other amplitude to find the system in state O(~ words, we will prepare the system in some initial state which differs from the ground state by a few local excitations, a disturbance, and we will watch how this disturbance evolves in space and time. Any local measurement done on a quantum many-particle system can be described by a process essentially of this type. Since we are going to be talking about time dependence it is useful to use the Heisenberg representation of the quantum evolution in which the operators evolve according to the law b b bH (t) = eiHt/~ b e−iHt/~ O O

1

(2.1)

2

CHAPTER 2. GREEN FUNCTIONS AND OBSERVABLES

bH (t) obeys the equation of motion Hence, the Heisenberg operator O b ∂O i h bi H, O = ∂t ~

(2.2)

To simplify the notation we will drop the H label of the Heisenberg operators from now on. Consider now a measurement in which at some time t we prepare the sysb †(~r, t)|Gi and that we wish to ask for the amplitude tem in the initial state O b † (~r, t′ )|Gi. The that at some later time t′ the system is in the final state O quantum mechanical amplitude of interest is: b r ′ , t′ ) O b † (~r, t)|Gi Amplitude = hG|O(~

(2.3)

We will be interested in considering two combinations of amplitudes of this type: • A time-ordered amplitude • A causal amplitude As we shall see, the results of physical measurements is given in terms of a causal amplitude but we will be able to compute more directly the timeordered amplitude instead. In general these two amplitudes are related by a well defined analytic continuation procedure. b 1) We will define the time-ordered product of two Heisenberg operators A(t b and B(t2 ) to be ( b 1 )B(t b 2 ), for t1 > t2 A(t b 1 )B(t b 2 )) = T (A(t (2.4) b 2 )A(t b 1 ), for t2 > t1 ±B(t

where the plus (+) sign applies for bosons and the minus (−) sign applies for fermions. Note that operators with an even number of fermion operators are bosonic. Let us define the Heaviside (or step) function θ(t), ( 1, for t ≥ 0 (2.5) θ(t) = 0, for t < 0 in terms of which the time-ordered product of two operators is   b b b 1 ) A(t b 2 ) ± θ(t2 − t1 ) B(t b 2 ) A(t b 1) T A(t1 )B(t2 ) = θ(t1 − t2 ) A(t

(2.6)

2.2. THE FEYNMAN PROPAGATOR AT FINITE DENSITY

3

In this language, the time-ordered amplitude, or Feynman propagator for the b is operator O, b r , t)O b † (~r ′ , t′ ))|Gi GF (~r, t; ~r ′ , t′ ) = −ihG|T (O(~

(2.7)

while the causal propagator is

h i b r , t), O b †(~r ′ , t′ ) |Gi Gc (~r, t; ~r ′ , t′ ) = −iθ(t − t′ )hG| O(~

(2.8)

±

where

[A, B]± = AB ∓ BA

(2.9)

where, − applies for bosons and + applies for fermions respectively. In contrast with the time-ordered amplitude, the causal amplitude vanishes unless t′ > t.

2.2

The Feynman propagator at finite density

Let us consider first the simplest example: the amplitude for a particle of spin σ at (~r, t) to propagate freely to (r ′ , t′ ). We will assume that the system has a finite density ρ. Let us denote by ψσ† (~r, t) and ψσ (~r, t) the creation and annihilation operators for a particle at ~r at time t. For the moment we will consider both the cases of fermions and bosons on the same footing. The one-particle Feynman (time-ordered) propagator is GF (~r, t, σ; ~r ′ , t′ , σ ′ ) = −i hG|T ψσ (~r, t) ψσ† ′ (~r ′ , t′ ))|Gi

(2.10)

and the causal function is   Gc (~r, t, σ; ~r ′ , t′ , σ ′ ) = −i θ(t − t′ )hG| ψσ (~r, t), ψσ+ , (r ′ , t′ ) −ζ |Gi

(2.11)

where θ(t) is the Heaviside function, and ζ = ±, for bosons and fermions respectively, i.e. a commutator for bosons and an anticommutator for fermions. Let us calculate the Feynman propagator GF for a system of non-interacting particles at finite density ρ. From the definition of a time-ordered product we get GF (~r, t, σ; ~r ′ , t′ , σ ′ ) = − iθ(t − t′ ) hG|ψσ (~r, t)ψσ† ′ (~r ′ , t′ )|Gi

− i θ(t′ − t) ζ hG|ψσ† ′ (~r ′ , t′ )ψσ (~r, t)|Gi

(2.12)

4

CHAPTER 2. GREEN FUNCTIONS AND OBSERVABLES

We now will use the evolution equation for a Heisenberg operator, ψσ (~r, t) = eiHt/~ψσ (~r)e−iHt/~

(2.13)

and plug it back into the expression for the two amplitudes in the Feynman propagator. For the first amplitude in Eq.(2.12), which holds for t > t′ , we get: ′

′

hG|ψσ (~r, t)ψσ† ′ (~r ′ , t′ )|Gi = hG|eiHt/~ψσ (~r)e−iHt/~ eiHt /~ψσ† (~r ′ )e−iHt /~|Gi (2.14) Here |Gi is the normalized ground state of the system and EG is the ground state energy H|Gi = EG |Gi, hG|Gi = 1 (2.15) in terms of which the first amplitude in Eq.(2.12) becomes ′

′

hG|ψσ (~r, t)ψσ† ′ (~r ′ , t′ )|Gi = eiEG (t−t )/~hG|ψσ (~r)e−iH(t−t )/~ψσ† ′ (~r ′ )|Gi (2.16) The operator ψσ (~r) has the mode expansion ψσ (~r) =

X

cα,σ ϕn (~r)

(2.17)

α

where {ϕn (~r)} are the wave functions of a complete set of one-particle states {|αi}, labeled by an index α, and cα,σ† and cα,σ are the associated creation and annihilation operators in Fock space, which obey i i h h [cα,σ , cα′ ,σ′ ]−ζ = c†α,σ , c†α′ ,σ′ = 0, cα,σ , c†α′ ,σ′ = δαα′ δσσ′ (2.18) −ζ

−ζ

To make further progress we will need to specify the ground state. We will restrict ourselves for the moment to the case of fermions. We will discuss the case of bosons at finite density later on. The ground state |Gi of a system of fermions at finite density is the filled Fermi sea Y Y |Gi = c†ασ |0i (2.19) α≤G σ=↑,↓

We will now perform a particle-hole transformation for α ≤ G, and define new creation and annihilation operators such that the annihilation operators

2.2. THE FEYNMAN PROPAGATOR AT FINITE DENSITY

5

destroy the filled Fermi sea |Gi, i.e. we will define new operators which are normal ordered with respect to the filled Fermi sea: bασ = a†ασ ,

b†ασ = aασ ,

for α ≤ G

(2.20)

which satisfy aασ |Gi = 0, for α > G bασ |Gi = 0, for α ≤ G

(2.21)

Thus, we write mode expansion now as X X X ϕα (~r) cασ = ϕα (~r) aασ + ϕα (~r) b†ασ ψσ (~r) = α

ψσ† (~r)

=

X

α>G

ϕ∗α (~r)

c†ασ

=

α

X

α≤G

ϕ∗α (~r)

a†ασ

+

α>G

X

ϕ∗α (~r) bασ

α≤G

(2.22)

Let us consider the first term in the Feynman propagator, Eq.(2.12). Upon inserting an complete set {|ni} of eigenstates of the Hamiltonian H (not just the one-particle states!), H|ni = En |ni

(2.23)

we find that the first amplitude in Eq.(2.12) can be expanded as X ′ ′ hG|ψσ (~r)e−iH(t−t )/~ψσ† ′ (~r ′ )|Gi = hG|ψσ (~r)|nie−i(En −EG )(t−t )/~hn|ψσ† ′ (~r ′ )|Gi n

(2.24)

The matrix element hG|ψσ (~r)|ni has the explicit form X X hG|ψσ (~r)|ni = hG|aασ |niϕα (~r) + hG|b†ασ |niϕα (~r) α>G

=

X

α>G

α≤G

hG|aασ |niϕα (~r)

(2.25)

since bασ |Gi = 0. Similarly we find X X hn|bα′ σ′ |Giϕ∗α′ (~r ′ ) hn|a†α′ σ′ |Gi ϕ∗α′ (~r′ ) + hn|ψσ† ′ (~r ′ )|Gi = α′ ≤G

α′ >G

=

X

α′ >G

hn|a†α′ σ′ |Gi ϕ∗α′ (~r′ )

(2.26)

CHAPTER 2. GREEN FUNCTIONS AND OBSERVABLES

6

Thus, we see that only the states with α > G, which we shall call particle states, contribute to Eq.(2.25) and Eq.(2.26). It will be convenient to simplify the notation by introducing the symbol  1 α>G (2.27) θ(α − G) = 0 α≤G which is analogous to the Heaviside function θ(t) introduced above. With this notation the mode expansion becomes X  ψσ (~r) = θ(α − G)aασ ϕα (~r) + θ(G − α)b†ασ ϕα (~r) (2.28) α

and the first amplitude in the Feynman propagator, Eq.(2.12), becomes X ′ ′ hG|ψσ (~r)e−iH(t−t ) ψσ† ′ (~r ′ )|Gi = hG|ψσ (~r)|ni hn|ψσ† ′ (~r ′ )|Gie−i(En −EG )(t−t )/~ =

XX αα′

n

′ e−i(En −EG )(t−t )/~ϕ∗α (~r)ϕα′ (~r ′ )

n

n

× θ(α − G)θ(α′ − G)hG|aασ |nihn|a†α′ σ′ |Gi + θ(G − α)θ(G − α′ )hG|b†ασ |ni hn|bα′ σ′ |Gi + θ(α − G)θ(G − α′ )hG|aασ |nihn|bα′ σ′ |Gi o + θ(G − α)θ(α′ − G)hG|b†ασ |nihn|a†α′ σ′ |Gi (2.29)

The matrix elements in Eq.(2.29) vanish unless the intermediate states |ni satisfy |ni = a†ασ |Gi = a†α′ σ′ |Gi |ni = bασ |Gi = bα′ σ′ |Gi = 0

(2.30)

both of which require that α = α′ and σ = σ ′ , and α > G. In other terms, the only intermediate states |ni which contribute to the expansion are states which differ from the ground state |Gi by a single particle state |α, σi ≡ |α, σ; Gi, of either spin component. We conclude that the first amplitude in the Feynman propagator, Eq.(2.12), is X ′ ′ hG|ψσ (~r)e−iH(t−t ) ψσ† ′ (~r ′ )|Gi = δσσ′ e−i(Eα −EG )(t−t )/~ ϕ∗α (r) ϕα (r ′ )θ(α−G) α

(2.31)

2.2. THE FEYNMAN PROPAGATOR AT FINITE DENSITY

7

where Eα − EG is the excitation energy of the particle-like state |α, σi = a†ασ |0i, with α > G. Likewise, the second amplitude in the Feynman propagator Eq.(2.12), which applies for t < t′ , involves the same operators but acting now in reverse order. The only change is that now the contributing intermediate states are |ni = b†ασ |Gi = b†α′ σ′ |Gi |ni = aασ |Gi = aα′ σ′ |Gi = 0

(2.32)

which once again require that α = α′ and σ = σ ′ , but now α ≤ G. Thus, in this case only the single hole states (of either spin component) contribute to the amplitude. Therefore, we can write the second amplitude in the Feynman propagator, Eq.(2.12), as ′

hG|ψσ† (~r ′ )e−iH(t−t )/~ ψσ′ (~r)|Gi = δσσ′

X

′

e−i(Eα −EG )(t−t )/~ϕ∗α (~r)ϕα (~r ′ )θ(G−α)

α

(2.33) b†α,σ |0i,

where Eα − EG is the excitation energy of the hole-like state |α, σi = with α ≤ G. In conclusion, we find that the Feynman propagator, Eq.(2.12), has the explicit form X GF (~r, t, σ; ~r ′ , t′ , σ ′ ) = −iδσσ′ ϕ∗α (~r)ϕα (r ′ )

α h i ′ ′ × θ(α − G)e−i(Eα −EG )(t−t )/~θ(t − t′ ) − θ(G − α)e−i(Eα −EG )(t−t )/~θ(t′ − t)

(2.34)

We will now perform a Fourier transform in time and define ′ ′

GF (~r σ, ~r σ ; Ω) =

Z

+∞

′

dt GF (~r, t, σ; ~r ′ , t′ , σ ′ )eiΩ(t−t )

(2.35)

−∞

The Heaviside function θ(t) has the following representation as a contour integral in the complex frequency plane I dω eiωt θ(t) = lim+ (2.36) δ→0 Γ 2πi ω − iδ

CHAPTER 2. GREEN FUNCTIONS AND OBSERVABLES

8

where Γ = Γ+ is the counter-clockwise oriented contour obtained by closing the contour on the upper half plane as shown in Fig.2.2, while Γ = Γ− is the negatively oriented contour obtained by closing the contour on the lower half plane, also shown in Fig.2.2. The integral on the real axis can be extended to the large arc Γ+ of radius R → ∞ provided that t > 0, and to the large arc Γ− for t < 0, as in these cases the integrals over the arcs vanishes. That this is a representation of the function θ(t) can be seen by noting that the integrand of Eq.(2.36) has a pole only in the upper half plane, located at ω = iδ (where δ > 0), which implies that if we choose the contour Γ+ we pick up the pole but if we choose the contour Γ− we do not. Using the Theorem of the Residues we get

θ(t) = lim+ δ→0

 i h iωt e   , iδ = e−ωδ −→ 1 (closing upward) Res  ω − iδ   0

(2.37)

(closing downward)

Im ω Γ+

for t > 0

R

Re ω

Γ−

for t < 0

Figure 2.1: Contours on the complex frequency plane used in Eq.(2.36)

2.2. THE FEYNMAN PROPAGATOR AT FINITE DENSITY

Z

9

We will now use this integral representation to write I Z ∞ ∞ dω −i(Eα −EG )t/~ eiωt iΩt −i(Eα −EG )t/~ iΩt dt e θ(t) e = dt e e ω − iδ Γ 2πi −∞ −∞ Z I ∞ 1 dω dt ei(ω+Ω−(Eα −EG )/~)t = 2πi ω − iδ −∞ Z ∞ dω 2πδ (ω + Ω − (Eα − EG ) /~) ≡ ω − iδ −∞ 2πi i
= Eα −EG + iδ Ω− ~ (2.38)

On the other hand, a similar line of argument shows that Z ∞ i
dt eiΩt θ(−t) e−i(Eα −EG )t/~ = G − iδ Ω − Eα −E −∞ ~

(2.39)

Hence,

GF (~rσ, ~r ′ σ ′ ; ω) =

= δσσ′

X α





    θ(G − α) θ(α − G)   ∗ ′

+ ϕα (~r)ϕα (~r )   E −E E −E α α G G ω − + iδ ω − − iδ  ~ ~ | {z } | {z } particle

hole

contribution

contribution

(2.40)

Thus, particles contribute with poles in the lower half plane with positive excitation energy, and holes contribute with poles in the upper half plane also with positive excitation energy. The representation of the Feynman propagator of Eq.(2.40), known as the Lehmann representation, is correct for any free fermion system. (We will see later on that it also holds for interacting systems provided the physical eigenstates have finite overlap with the states created by free fermion operators.) For the case of free fermions in free space the one-particle eigenstates are just momentum eigenstates, |αi ≡ |~pi, the excitation energies are ε(~p) = E(~p) − EG ≡

p~ 2 2m

(2.41)

10

CHAPTER 2. GREEN FUNCTIONS AND OBSERVABLES

and the wave functions ϕp~ (~r) are just plane waves, ϕp~ (~r) =

1 ei~p·~r/~ (2π~)3/2

(2.42)

In this case the label G corresponds to the Fermi momentum pF " # Z 3 θ(| ~ p | −p ) θ(p − | p ~ |) d p ′ F F + GF (~rσ, ~r ′ σ ′ ; ω) = ~δσσ′ ei~p·(~r−~r )/~ p ~2 p ~2 (2π~)3 − iδ ~ω − 2m + iδ ~ω − 2m (2.43) where the Fermi momentum pF and the Fermi energy EF are determined in terms of the density ρ = N/V ,
2/3 ~2 6π 2 ρ (2.44) EF = 2m We can use these results to write an expression for the Fourier transform eF (~p, ω) G eF (~p, ω) = ~Θ(|~p| − pF ) + ~Θ(pF − |~p|) (2.45) G ~ω − E(~p) + iǫ ~ω − E(~p) − iǫ where E(~p) = ǫ(~p) − µ. An equivalent (and more compact) expression is
1/3 pF = ~ 6π ρ , 2

eF (~p, ω) = G

~ ~ω − E(~p) + iǫ sign(|~p| − pF )

(2.46)

Notice that eF (~p, ω) = −~πΘ(|~p| − pF )δ(~ω − ǫ(~p) + µ) + ~πΘ(pF − |~p|)δ(~ω − ǫ(~p) + µ) ImG = −~πδ(~ω − ǫ(~p) + µ) [Θ (|~p| − pF ) − Θ(pF − |~p|)] (2.47) The last identity shows that e F (˜ sign ImG p, ω) = −sign ω

eF (p, ω) as Hence, we may also write G eF (~p, ω) = G

~ ~ω − E(~p) + iǫ sign ω

(2.48)

(2.49)

eF (~p, ω) has poles at ω = E(~p) and that With this expression, we see that G all the poles with ω > 0 are infinitesimally shifted downwards to the lower half-plane, while the others are raised upwards to the upper half-plane by the same amount. Since E(p) = ǫ(p) − µ, all poles with ǫ(p) > µ are shifted downwards, while all poles with ǫ(p) < µ are shifted upwards.

2.2. THE FEYNMAN PROPAGATOR AT FINITE DENSITY

11

Im ω

Re ω xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

µ

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Figure 2.2: Analytic structure of the Green function at finite chemical potential µ in the complex frequency plane.

2.2.1

Propagators and observables

In a non-interacting system, the propagator is known explicitly and it can be used to compute expectation values of physical observables. However the following relations hold for a general system regardless of the interactions. Thus, the ground state expectation value of the density operator hρ(~r)i is X GF (~r, t, σ; ~r ′ t′ σ ′ ) (2.50) hρ(~r)i = −i lim lim ′ ′ − ~r →~ r t →t

σ

while the spin density ( or local magnetization) is X ~σ′ σ GF (~r, t, σ; ~r ′ t′ σ ′ ) ~ r)i = −i lim lim S hS(~ ′ ′ − ~ r →~ r t →t

(2.51)

σ,σ′

where ~ = ~ ~τ S 2

(2.52)

CHAPTER 2. GREEN FUNCTIONS AND OBSERVABLES

12

and τ a (with a = 1, 2, 3) are the three Pauli matrices. The second quantized local current density operator Ji (~r) is  ~ X † ψσ (~r)∇i ψσ (~r) − ∇i ψσ† (~r)ψσ (~r) 2mi σ ~ X † ← → ≡ ψσ (~r) ∇i ψσ (~r) 2mi σ  ~ X † ′ i i † ′ ψ (~ r ) ∇ ψ (~ r ) − ∇ r )ψ (~ r ) ≡ lim ′ ψσ (~ σ σ σ ~ r ~ r ~ r ′ →~ r 2mi σ

Ji (~r) =

(2.53) (2.54) (2.55)

and its ground state expectation value is  ~ X hG|ψσ† (~r ′ ) ∇~ri ψσ (~r)|Gi − hG|∇~ri ′ ψσ† (~r ′ )ψσ (~r)|Gi ~ r →~ r 2mi σ (2.56)  i  ~ ′ ′ ′ i ′ ′ ′ ∇ G (~ r , t, σ; ~ r , t , σ ) − ∇ r , t, σ; ~ r , t , σ ) lim lim =− ′ GF (~ F ~ r ~ r 2m ~r ′ →~r t′ →t− (2.57)

hG|Ji(~r)|Gi = lim ′

2.3

Interacting Systems

In the Schr¨odinger representation, or Schr¨odinger Picture, the states |ΨS (t)i are time-dependent functions whose evolution is governed by the Schr¨odinger’s equation i~ ∂t |ΨS (t)i = H|ΨS (t)i (2.58) which can be solved (formally) by |ΨS (t)i = e−iH(t−t0 )/~ |ΨS (t0 )i

(2.59)

where |ΨS (t0 )i is the initial state. The problem with this statement is that the (unitary) evolution operator U(t, t0 ) U(t, t0 ) = e−iH(t−t0 )/~

(2.60)

is very complicated and we don’t really know how to construct it in almost all cases.

2.3. INTERACTING SYSTEMS

13

Alternatively we can use the Heisenberg Picture in which the states are fixed and the operators obey equations of motion with ∂t AH (t) =

i [H, AH (t)] , ~

AH (t) = eiH(t−t0 )/~AS (t0 )e−iH(t−t0 ) (2.61)

where AS (t0 ) is a fixed (initial) operator. Again we need to solve these generally non-linear equations which are also very complex. However, let us suppose that H = H0 + H1 and that we know how to solve for H0 . Still we want to include the effects of H1 . This is the standard problem of perturbation theory. This situation motivates the introduction of the Interaction Picture, in which the states are |ΨI (t)i |ΨI (t)i = eiH0 t/~ |ΨS (t)i

(2.62)

and satisfy the evolution equation

where H1 (t), is

i~|ΨI (t)i = H1 (t)|ψI (t)i

(2.63)

H1 (t) = eiH0 t/~H1 e−H0 t/~

(2.64)

and thus obeys the equation of motion ∂t H1 (t) =

i [H0 , H1 (t)] ~

(2.65)

Thus, in the interaction representation the time evolution of the states is governed by the perturbation H1 (t) while all possible operators A(t), the perturbation H1 among them, obey the Heisenberg equations of motion associated with H0 AI (t) = eiH0 t/~Ae−iH0 t/~,

i~∂t AI (t) = [H0 , AI (t)]

(2.66)

Hence the interaction representation is the Heisenberg representation of H0 . In particular, the time evolution operator U(t, t0 ), in the Interaction Representation satisfies |ΨI (t)i = UI (t, t0 )|ΨI (t0 )i (2.67) and obeys the evolution equation i~∂t UI (t, t0 ) = HI (t)UI (t, t0 )

(2.68)

14

CHAPTER 2. GREEN FUNCTIONS AND OBSERVABLES

The evolution operators have the properties U(t1 , t2 )U(t2 , t3 ) = U(t1 , t3 ) (form a group) U(t1 , t2 )−1 = U(t1 , t2 )† (unitarity) U(t1 , t2 )−1 = U(t2 , t1 ) (inverse)

(2.69)

Furthermore, the Heisenberg and Interaction Representations are related as follows |ΨH i = eiHt/~e−iH0 t/~|ΨI (t)i = U(t, t0 )† |ΨI (t)i AH (t) = U(t, t0 )† AI (t)U(t, t0 )

(2.70) (2.71)

The equation of motion of the evolution operator U(I (t, t0 ), Eq.(2.68) can also be written as an integral equation (we will drop the label “I” from now on): Z t i dt′ H1 (t′ )U(t′ , t0 ) (2.72) U(t, t0 ) = U(t0 , t0 ) − ~ t0 with U(t0 , t0 ) = 1 i U(t, t0 ) = 1 − ~

Z

t

dt′ H1 (t′ )U(t′ , t0 )

(2.73)

t0

We will solve this equation by iteration. For this approach to converge we need to require that H1 (t) be switched on and off very slowly (adiabatically). This amounts to making the replacement H1 → e−ǫ|t| H1 (t)

(2.74)

and taking the limit ǫ → 0+ afterwords. Hence, we can write a formal iterative solution of the form  Z t  2 Z t Z t′ −i −i ′ ′ ′ ⇒ U(t, t0 ) = 1+ dt′′ H1 (t′ )H1 (t′′ )+. . . dt H1 (t ) + dt ~ ~ t0 t0 t0 (2.75) However it is easy to verify that Z Z Z t′ Z t 1 t ′ t ′′ ′ ′′ ′ ′′ dt T (H1 (t′ )H1 (t′′ )) (2.76) dt dt dt H1 (t )H1 (t ) = 2 t0 t0 t0 t0 where T (H1 (t′ )H1 (t′′ )) is the time-ordered-product of the two operators.

2.3. INTERACTING SYSTEMS

15

Moreover, all terms involving nested time integrations can be written in a more compact form in terms of time-ordered products: Z t Z t1 Z tn−1 dt1 dt2 . . . dtn H1 (t1 )H1 (t2 ) . . . H1 (tn ) = t0 t0 t0 Z Z t Z t 1 t dt1 dt2 . . . dtn T (H1 (t1 )H1 (t2 ) . . . H1 (tn )) = n! t0 t0 t0 (2.77) Thus, the evolution operator can be written as an expansion in terms of time-ordered products: n Z t Z t ∞  X 1 −i U(t1 t0 ) = dt1 . . . dtn T (H1 (t1 ) . . . H1 (tn )) (2.78) ~ n! t t 0 0 n=0 Hence, the evolution operator U(t, t0 ) can be represented formally as a timeordered exponential: Z i t ′ − dt H1 (t′ ) U(t, t0 ) ≡ T e ~ t0 (2.79) We should be careful to note that Eq.(2.79) is simply a short-hand notation used to write the formal expression for the evolution operator given in Eq.(2.78) in a more compact form. (In other words we have not shown that the series exponentiates!) The evolution operator U(t, t0 ) will play a central role in much of what we will do. In particular, it can be used to determine the exact (Heisenberg) eigenstates which are given by |ψH i = |ψI (0)i = Uˆ (0, t0 )|ΨI (t0 )i

(2.80)

Once again, in order to use expressions of this type we must we imagine switching the interaction adiabatically slowly, i.e. H1 → e−ǫ|t| H1 (t)) Let us consider now two states, |Φ0 i which is an exact non-degenerate eigenstate of H0 , and |Ψ0 i which is an exact (also non-degenerate) eigenstate of H = H0 + H1 . We will assume once again that the interaction H1 was absent in the remote past and that it was turned on adiabatically slowly, and that during this process there is no level crossing involved. Hence, the

16

CHAPTER 2. GREEN FUNCTIONS AND OBSERVABLES

state remains non-degenerate during the process. Then the following identity, known as the Gell-Mann Low Theorem, holds: Uǫ (0, −∞)|Φ0 i |Ψ0 i = lim+ hΦ0 |Ψ0 i ǫ→0 hΦ0 |Uǫ (0, −∞|Φ0 i

(2.81)

where we have Uǫ (t, t′ ) is the evolution operator for an adiabatically switched on perturbation H1 (t). The Gell-Mann Low Theorem, Eq.(2.81), simply states that the eigenstate |Φ0 i of H0 evolves smoothly into the exact eigenstate |Ψ0 i of H. In particular it means that the exact expectation value of a Heisenberg operator AH (t) in the state |Ψ0 i, an exact eigenstate of H, in terms of expectation values in the interaction representation is given by hΨ0 |AH (t)|Ψ0 i hΦ0 |T AI (t)S|Φ0 i = hΨ0 |Ψ0 i hΦ0 |S|Φ0 i

(2.82)

where the left hand side is an expectation value in the Heisenberg representation while the right hand side involves only expectation values in the interaction representation. In Eq.(2.82) where we introduced the S-matrix, S = Uǫ (∞, −∞), whose expectation value in the state |Φ0 i is Z +∞ i − dt′ H1 (t′ ) ~ −∞ hΦ0 |S|Φ0 i = hΦ0 |T e |Φ0 i (2.83) In particular, these results tell us how to write an expression for the Feynman propagator, which is an expectation value in the Heisenberg Representation, in terms of expectation values in the Interaction Representation. Thus, (ignoring spin indices) we find that the Feynman propagator has the representation Z +∞ i − dt′′ H1 (t′′ ) † ~ ′ ′ −∞ hΦ0 |T ψI (~xt)ψI (~x t ) e |Φ0 i GF (~x, t; ~x′ , t′ ) = −i (2.84) Z +∞ i ′′ ′′ − dt H1 (t ) ~ −∞ hΦ |T e |Φ i 0

0

where |Φ0 i is the ground state of H0 . However, since the evolution operator is a time-ordered exponential, which is defined by a series in powers of the perturbation H1 , Eq.(2.82) and Eq.(2.83) are nothing but a perturbative evaluation of the propagator.

2.4. PERTURBATION THEORY AND FEYNMAN DIAGRAMS

2.4

17

Perturbation Theory and Feynman Diagrams

Thus we have reduced the calculation of GF , the Feynman propagator (or Green function) for the electron, to a power series expansion in which each term is an expectation value calculated in the Interaction Representation. This is just perturbation theory. The factor in the denominator, 0 hG|S|Gi0 , is simply the consequence of the renormalization of norm of the ground state |Gi0 due to the effects of the perturbation H1 . We have seen that fermion field operator ψ(~x, t) has a mode expansion of the form X   ψ(x) = θ(α − G)ϕα (x)aα + θ(G − α)ϕα (x)b†α (2.85) α

where aα |Gi0 = 0 and bα |Gi0 = 0, and {ϕα } are a complete set of singleparticle wave functions. Then, we can split the operator ψ(~x, t) into two pieces, ψ (±) (~x, t), ψ(x) = ψ (+) (x) + ψ (−)

(2.86)

where both ψ (+) and ψ (−)† annihilate the ground state |Gi0 : ψ (+) |Gi0 = 0,

and

ψ (−)† |Gi0 = 0

(2.87)

We have seen that inside a time-ordered product, fermionic operators anti-commute, T (AB) = −T (BA) (2.88) which will play a central role in what follows. We will now define the useful concept of a normal ordered product. The normal ordered product of any number of operators is an ordered product in which all annihilation operators appear to the right of all creation operators. Furthermore, inside a normal ordered product operators can be reordered up to a sign determined by the parity of the permutation P used to reorder the operators. We define the normal ordered product of a set of n Heisenberg operators {Aj }, with j = 1, . . . , n, which we will denote by : A1 A2 . . . An :, to satisfy : A1 A2 . . . An : |Gi0 = 0 : A1 . . . An : = (−1)P : A3 A1 A2 . . . An :

(2.89) (2.90)

CHAPTER 2. GREEN FUNCTIONS AND OBSERVABLES

18

where the factor of (−1)P holds only for fermionic operators. Furthermore, : ψ (+) (x)ψ (−) (y) : = −ψ (−) (y)ψ (+) (x) †

†

: ψ (+) (x)ψ (+) (y) : = −ψ (+) (y)ψ (+) (x)

(2.91)

We also note that, by definition, the expectation value of any normal ordered product vanishes: (2.92) 0 hG| : A1 . . . An : |Gi0 = 0 We will now define the contraction of two operators as the difference of their time-ordered and normal-ordered products: AB ≡ T (AB)− : AB :

(2.93)

Since T (ψ

(+)

(x)ψ

(−)

′

(x )) =

(

ψ (+) (x)ψ (−) (x′ ) for tx > tx′ −ψ (−) (x)ψ (+) (x′ ) for tx′ > tx

(2.94)

we see that T (ψ (+) (x)ψ (−) (x′ )) = −ψ (−) (x′ )ψ (+) (x)

(2.95)

Similarly, we also get : ψ (+) (x)ψ (−) (x′ ) := −ψ (−) (x′ )ψ (+) (x)

(2.96)

Thus the contraction of a ψ (+) operator and a ψ (−) operator vanishes: ψ (+) (x)ψ (−) (x′ ) = 0 The only non-vanishing contractions are ( i G0F (x, x′ ) † ψ (+) (x)ψ (+) (x′ ) = 0 ( 0 † ψ (−) (x)ψ (−) (x′ ) = i G0F (x, x′ )

(2.97)

for tx > tx′ for tx < tx′ for tx > tx′ for tx < tx′ (2.98)

2.4. PERTURBATION THEORY AND FEYNMAN DIAGRAMS

19

Hence, the ground state expectation value of the time-ordered product of two operators can be related to the expectation value of the contraction of the operators. Indeed, 0 hG|T (AB)|Gi0

=

0 hG|AB|Gi0

+ 0 hG| : AB : |Gi0

≡

0 hG|AB|Gi0

≡ AB

(2.99)

Hence, 0 hG|T (AB)|Gi0

= AB

(2.100)

This result allows us to identify the contraction of the field operators with the Feynman propagator: ψ(x)ψ † (x′ ) = i G0F (x, x′ )

(2.101)

We will also need to know how to compute normal ordered products with partially contracted operators. For instance it easy to show that : ABCD . . . := − : ACBD . . . := −ACN(BD . . .)

(2.102)

These results lead to the following identity, known as Wick’s Theorem, which relates time-ordered products to normal ordered products and contractions. Let A1 , . . . , AN be a set of N operators. Then, T (A1 . . . AN ) =: A1 . . . AN : + : A1 A2 A3 A4 . . . AN : +other normal ordered terms with one contraction + : A1 A2 A3 A4 . . . AN : +other normal ordered terms with two contractions +... +A1 A2 A3 A4 . . . AN −1 AN + all other products of pairs of contractions (2.103) It is also easy to verify that : A1 . . . AN : B = : A1 . . . AN B : + : A1 . . . AN −1 AN B : + . . . + : A1 . . . AN B : + + : A1 . . . AN B : (2.104)

20

CHAPTER 2. GREEN FUNCTIONS AND OBSERVABLES

In particular, we find that the ground state expectation value of a timeordered product of operators is equal to the sum of all possible products of contractions of pairs of operators: hG|T (A1 . . . AN )|Gi = A1 A2 A3 A4 . . . AN −1 AN + . . .

(2.105)

where the ellipsis . . . represents the sum of all other products of pairs of contractions. Notice that if we are dealing with fermionic operators, each term in this sum will have a sign determined by the number of permutations done in the process of contacting the operators.

Chapter 3 Linear Response Theory 3.1

Measurements and Correlation Functions

We will now discuss how to represent physical measurements. All local physical measurements of a macroscopic quantum (many-body) system amount in practice to a process in which a localized disturbance is created by an applied external force, in the neighborhood of some point ~r at some time t, and the response of the system is then measured at some other point ~r ′ at tome later time t′ > t. On the other hand, in many cases we also will be interested in the global response of a system to an uniform perturbation, as in the case of most thermodynamic measurements. There are many examples of these type of measurements. For example, in a typical optical experiment one has an external electromagnetic wave impinging on a system (for instance, a metal). In the case of an electromagnetic field with sufficiently low energy we can ignore its quantum mechanical nature and treat it as a classical wave. It will interact with the degrees of freedom in a variety of ways. Thus, the scalar potential A0 couplesR to the local charge density through a term in the Hamiltonian of the form d3 xA0 (~x, t)ρ(~x, t), where ρ(~x, t) is the second quantized charge density operator (in the Heisen~ x, t) couples berg representation). On the other hand, the vector potential A(~ ~ to the particle current operator j (~x, t) (the full form of the coupling is actually more complex, as we shall see below). Thus, at this level, we see that an external classical electromagnetic field can induce density waves and currents in a system. However, at higher energies the quantum nature of the electromagnetic 1

2

CHAPTER 3. LINEAR RESPONSE THEORY

field cannot be ignored, and in reality what one is considering are the scattering processes of external photons with the internal degrees of freedom of the system of interest. Photons couple to the atoms and through this coupling we can gain information on the phonon degrees of freedom. Thus, Raman scattering of photons provide a direct window to the dynamics of phonons (among many other things). On the other hand, X-ray photons are more energetic and sufficiently so that they can in fact eject an electron from the system, leaving behind a hole. Thus, by measuring the properties of the ejected photo-electron it is possible to infer the spectral properties of the hole. This is an example of a photoemission experiment. If the momentum and energy of the photoelectron are measured, this is an angle-resolved photo-emission (ARPES) experiment, which essentially measures the retarded Green function of the hole. Other scattering processes couple instead to the spin of the degrees of freedom. For example in neutron scattering experiments, the initial state has neutrons with well defined energy, momentum and spin polarization which scatter off the spin degrees of freedom into a final state also with well defined energy momentum and spin polarization. The leading coupling is due to the effectively short-range interaction between the spin of the neutron and ~ x, t) of the degrees of freedoms of the system. The the local spin density S(~ interaction between the neutron magnetic moment and the local spin den~ x, t). Neutron sity leads to a term on the Hamiltonian which is linear in S(~ scattering experiments measure the spin-spin correlation function. We will now discuss a general theory of these type of measurements. Let H be the full Hamiltonian describing the system in isolation. One way to test its properties is to couple the system to a weak external perturbation (assuming that the ground state is stable) and to determine how the ground state and the excited states are affected by the perturbation. Let O(~x, t) be a local observable, such as the local density, the charge current, or the local magnetization. The (total) Hamiltonian HT for the system weakly coupled to an external perturbation, which we will represent by a Hamiltonian Hext , is HT = H + Hext (3.1) We will assume that the perturbation is not only weak, in which case its effects should be describable in perturbation theory, but it is also adiabatically switched on and off. Then as we discussed above, the Heisenberg

3.1. MEASUREMENTS AND CORRELATION FUNCTIONS

3

representation of the isolated system, with Hamiltonian H, is the interaction representation of the system coupled to the source. Thus, the physical observables of the coupled system will evolve according to the Hamiltonian of the isolated system but the states follow the external perturbation, which we will refer to as the source. Hence, the expectation value of the observable O(~x, t) in the exact ground state |Gi of H, under the action of the weak perturbation Hex is modified as hG|O(~x, t)|Gi → hG|U −1 (t)O(~x, t)U(t)|Gi

(3.2)

where U(t) is the evolution operator in the interaction representation of HT . As we have seen, U(t) is given by the time-ordered exponential i − U(t) = T e ~

Z

t

dt′ Hext (t′ )

−∞

(3.3)

By expanding U(t) to linear order in the perturbation Hext , we see that to lowest (linear) order the change in the expectation value δhG|O(~x, t)|Gi is i δhG|O(~x, t)|Gi = ~

Z

t

dt′ hG| [Hext (t′ ), O(~x, t)] |Gi

(3.4)

−∞

This change represents the linear response of the system to the external perturbation. It is given in terms of the ground state expectation value of the commutator of the perturbation and the observable. For this reason, this approach is called Linear Response Theory. Notice that in Eq.(3.4), there is an ordering of the times t and t′ : the time t at which the change is observed is always later than the time(s) t′ during which the external perturbation acted, t > t′ . Hence, Eq.(3.4) explicitly obeys causality. In general, if O(~x, t) is a local observable, Hext (t) represents an external source which couples linearly to the observable, Z Hext (t) = d3 x O(~x, t)f (~x, t) (3.5) To simplify matters, we will assume that the observable O(~x, t) is normal ordered with respect to the exact ground state |Gi, i.e. we will require that hG|O(x, t)|Gi = 0; in other terms, we are considering operators which measure the fluctuations of the observable away from the expectation value.

4

CHAPTER 3. LINEAR RESPONSE THEORY

Hence, the linear change (from zero) in the expectation value of the observable induced by the source f (~x, t) is i hG|O(x, t)|Gif = ~

Z

t ′

dt −∞

Z

d3 x′ hG| [O(~x′ , t′ ), O(~x, t)] |Gi f (~x ′ , t′ ) (3.6)

Thus, the response is linear in the force. The “coefficient” of proportionality between the change in the expectation value hG|O(~x, t)|Gi and the force f (~x ′ t′ ) defines a generalized susceptibility χ(~xt, ~x ′ t′ ) through the definition hG|O(~x, t)|Gi ≡ “χ · f ” =

Z

3 ′

dx

Z

t

dt′ χ(~xt, ~x ′ t′ )f (x′ , t′ )

(3.7)

−∞

By inspection we see that we can identify the generalized susceptibility χ(~xt; ~x ′ t′ ) with the retarded propagator, or correlation function, of the observable O(~x, t): i χ(~xt; ~x ′ t′ ) ≡ − θ(t − t′ )hG| [O(~x, t), O(~x ′ , t′ )] |Gi ~

(3.8)

That is, the susceptibility is the retarded Green function of the observable. Let f (~x, ω) be the time Fourier transform of the force f (~x, t), Z ∞ dω iωt e f (~x, t) (3.9) f (~x, t) = −∞ 2π The time Fourier transform of Eq.(3.7) is Z hG|O(~x, ω)|Gi = d3~x′ χ(~x, ~x ′ ; ω) f (~x ′ , ω)

(3.10)

However, since hG|O(~x, t)|Gi =   Z ∞ Z t Z dω iωt −i iω(t′ −t) 3 ′ ′ = e hG| [O(~x, t), O(~x ′ , t′ )] |Gif (~x ′ , ω) d x dt e 2π ~ −∞ −∞   Z ∞ Z 0 Z dω iωt −i iωt′ 3 ′ ′ = e hG| [O(~x, t), O(~x ′ , t + t′ )] |Gif (~x ′ , ω) d x e dt ~ −∞ 2π −∞ (3.11)

3.1. MEASUREMENTS AND CORRELATION FUNCTIONS

5

we find hG|O(~x, ω)|Gi =

Z

3 ′

dx



−i ~

0

Z

dt′ hG| [O(~x, t), O(~x ′ , t + t′ )] |Gif (~x ′ , ω)

−∞

(3.12)

Hence, the time-Fourier transform of the susceptibility becomes Z 0 i dτ eiωτ hG|[O(~x, t), O(~x ′ ; t + τ )]|Gi χ(ω; ~x, ~x ) = − ~ −∞ Z i 0 ≡ − dτ eiωτ hG|[O(~x, 0), O(~x ′ , τ )]|Gi ~ −∞ ′

(3.13)

Therefore, we find that the susceptibility χ(~x, ~x ′ ; ω) is given by i χ(~x, ~x ; ω) = − ~ ′

Z

0

dτ eiωτ hG|[O(~x, 0), O(~x ′ , τ )]|Gi

(3.14)

−∞

This result is known as the Kubo Formula. We just showed that the retarded Green function of the observable O(~x, t), ret DO (x, x′ ) = −iθ(t − t′ )hG|[O(x), O(x′)]|Gi

is related to the generalized susceptibility, Z ∞ 1 ret ′ dτ eiωτ DO (x, x′ ) χ(ω; ~x, ~x ) = ~ −∞

(3.15)

(3.16)

Hence, 1 hG|O(x)|Gi = ~

Z

∞ ′

dt

−∞

1 ≡ ~

Z

Z

ret d3 x′ DO (x, x′ )f (~x ′ , t′ )

ret d4 x′ DO (x, x′ )f (x′ )

(3.17)

where x ≡ (~x, t). After a Fourier transform we get hG|O(~p, ω)|Gi =

1 ret D (~p, ω)f (~p, ω) ~ O

(3.18)

6

CHAPTER 3. LINEAR RESPONSE THEORY

In Fourier space, the generalized susceptibility χ(~p, ω) is hG|O(~p, ω)|Gi 1 ret = DO (~p, ω) f (~p, ω) ~

“response” “force” (3.19) In conclusion, the response functions of physical interest, and the associated measurable susceptibilities, are given in terms of the retarded Green functions of the physical observables. However, as we saw in the last section, what we can calculate more directly are the time-ordered Green functions, i.e. the propagators of the observables. we will show next that the analytic properties of these functions in the complex frequency (or energy) plane are such that there is an analytic continuation procedure relating the timeordered and the retarded Green functions. In addition, in practice we really are interested in knowing the physical susceptibilities and propagators not only in the ground state but also at non-zero temperature. Fortunately the analytic continuation will enable us to determine all these functions also at non-zero temperature. χ(~p, ω) =

3.2

⇐⇒

χ=

Finite Temperature

All physical systems in thermal equilibrium are actually at finite temperature T . We will now show that there is a simple and straightforward way to adapt the T = 0 methods to account for the effects of thermal fluctuations. We will assume throughout that the system is in thermal equilibrium with a heat bath at temperature T and at a fixed chemical potential µ, i.e. we will treat the system in the Grand Canonical Ensemble. The Gibbs density matrix ρG and the Grand partition Function ZG are ρG ≡ e−β (H − µN) ,

ZG = Tr ρG = e−βΩG

(3.20)

where β = 1/(kT ), H is the Hamiltonian, N is the particle number operator, which commutes with the Hamiltonian, [N, H] = 0, ZG is the grand partition function and ΩG is the grand potential (or thermodynamic potential). 1 The thermal expectation value, i.e. the expectation value in the the grand 1

Recall that a chemical potential can only be defined for a conserved quantity, in this case the particle number operator N .

3.2. FINITE TEMPERATURE

7

canonical ensemble, of a physical local observable A(~x, t) is given by D

E Tr (A(~x, t) ρ ) G A(~x, t) = Tr ρG

(3.21)

Let {|λi} be a complete set of eigenstates of the full Hamiltonian H and of the particle number operator N, with eigenvalues {Eλ } and {Nλ }. (Notice that since we are working in the grand canonical ensemble, necessarily the states {|λi} are states in Fock space.) Then, we can write the thermal average as D

E

A(~x, t) =

P

λ

hλ|A(~x, t)|λi e−β(Eλ − µNλ ) P −β(E − µN ) λ λ λe

(3.22)

Thus, in principle in order to compute a thermal average we will have to know first the quantum mechanical expectation value in each state and then compute the thermal average. Although this sounds rather laborious there are very direct ways to compute the thermal average directly. In what follows, as we did before, we will include the chemical potential term in the Hamiltonian, H → H − µN, which now depends explicitly on the chemical potential µ. To calculate thermal averages it useful to define temperature correlation functions. Let A be the Schr¨odinger operator for an observable. We define a temperature variable τ , with 0 ≤ τ ≤ β~ and the τ -dependent operator A(τ ) A(τ ) = eHτ /~ A e−Hτ /~

(3.23)

which is formally obtained by the analytic continuation to imaginary time τ = it of the Heisenberg operator A(t) A(t) = eiHt/~ A e−iHt/~

(3.24)

(Notice that in general A† (τ ) 6= A(τ )† .) Likewise, the evolution operator U(t) of a quantum system in the Schr¨odinger picture, now becomes U(t) = e−iHt/~ → Ue (τ ) = e−Hτ /~

(3.25)

In particular the Gibbs density matrix ρG (β) is ρG (β) = e−βH = Ue (β~)

(3.26)

8

CHAPTER 3. LINEAR RESPONSE THEORY

Thus a quantum system in equilibrium at temperature T can be regarded (formally) as a the imaginary time evolution of a quantum system. In addition since the ensemble average is a trace in Fock space, the initial and finite states are the same state. In other words, the states obey periodic boundary conditions in the imaginary time coordinate τ . Thus we will restrict the imaginary time variable to the range 0 ≤ τ ≤ β~. Much of what we did at T = 0 can be done also at T 6= 0. In particular we can similarly define imaginary time ordered products of operators, Tτ (A(τ1 )B(τ2 )), by analogy with time ordered products. The temperature Green function for the electron operator is defined as E D ′ † ′ ′ ′ ′ Gσσ (~ x , τ )) (~ x , τ ; ~ x , τ ) = − T (ψ (~ x , τ )ψ ′ τ σ T σ   Tr Tτ (ψσ (~x, τ )ψσ† ′ (~x ′ , τ ′ )) ρG = − (3.27) Tr ρG Due to the fact that we are computing a trace it is straightforward to show that, as a consequence of the fermionic statistics (and of the anticommutation relations of fermionic operators), the imaginary time propagator, the temperature Green function, for fermionic operators obeys antiperiodic boundary conditions in imaginary time (independently on both τ and τ ′ ). Instead, the temperature Green function for bosonic operators obey periodic boundary conditions in imaginary time. Thus GT (~x, τ + β~; ~x ′ , τ ′ ) = ∓GT (~x, τ ; ~x ′ , τ ′ )

(3.28)

where (−) holds for fermions and (+) holds for bosons. These periodic and anti-periodic boundary conditions have important consequences. In particular, since the imaginary time interval is finite, the propagators can be expanded in Fourier series as X ′ GT (~x − ~x ′ , τ − τ ′ ) = GT (~x − ~x ′ , ωn ) eiωn (τ − τ ) (3.29) n∈Z

Periodic and anti-periodic boundary conditions in τ and in τ ′ are satisfied if the frequencies {ωn }, where n ∈ Z, are respectively given by 2π n, β~   1 2π n+ , ωn = β~ 2

ωn =

for bosons for fermions

(3.30)

3.3. GREEN FUNCTIONS AT T 6= 0

9

Im ω

Im ω

2π (n + 1/2) i β~

n i 2π β~

2π β~ π i β~

2π β~

Re ω

(a)

Re ω

(b)

Figure 3.1: Poles on the imaginary frequency axis for (a) fermions and (b) bosons. These considerations also apply to correlation functions which are always bosonic, even in a theory of fermions as they involve operators which are typically bilinear in fermions. Since the temperature Green function is formally the analytic continuation of the time-ordered Green function to imaginary time, restricted to the imaginary time interval 0 ≤ τ ≤ β~, its imaginary time Fourier transform (or series), GF (~x −~x ′ , ωn ) can also be regarded as the analytic continuation from real frequency ω to the imaginary frequency axis, restricted to the values iωn discussed above for fermions and bosons (See Fig.3.1).

3.3

Green Functions at T 6= 0

We will see now that there is a close relationship between the time-ordered, the retarded and the temperature propagators. The same considerations apply for the response and correlation functions. The key connection is the concept of the spectral function which describes the fluctuation spectrum. In particular, for the case of the response functions we will find a connection between the fluctuations and the dissipation in the system. We will begin with the connection between the fermion propagator, the fermion retarded Green function and the fermion temperature propagator.

10

CHAPTER 3. LINEAR RESPONSE THEORY

To simplify matters I will drop the spin indices (which are easily restored) and focus on the time dependence. Except for straightforward sign changes the same ideas apply to the boson propagator. Here we will follow closely the discussion in Doniach and Sondheimer.

3.3.1

Spectral Function

Let us define the fermion correlator G> (~x − ~x ′ , t) D E Tr e−βH eiHt/~ ψ(~x)e−iHt/~ ψ † (~x ′ )
G> (~x − ~x ′ , t) = ψ(~x, t) ψ † (~x ′ , 0) = Tr e−βH (3.31) Notice that this is a thermal average of a product (not necessarily ordered!) of Heisenberg operators. Let {|λi} be a complete set of eigenstates of the Hamiltonian in Fock space. Completeness means that the identity operator in Fock space can be expanded as X |λihλ| (3.32) 1= λ

Then, we can write the correlator as D E 1 X † ′ ψ(~x, t) ψ (~x , 0) = hλ|ψ † (~x ′ )e−βH |λ′ ihλ′ |eiHt/~ ψ(~x)e−iHt/~ |λi ZG ′ λλ 1 X −βEλ′ e hλ|ψ † (~x ′ )|λ′ihλ′ |ψ(~x)|λiei(Eλ′ −Eλ )t/~ = ZG λλ′

(3.33)

where ZG =

X

e−βEλ

(3.34)

λ

is the (grand) partition function. D E ′ † ′ Let J1 (~x − ~x ; ω) be the time Fourier transform of ψ(~x, t) ψ (~x , 0) , Z ∞D E ′ J1 (~x − ~x ; ω) = ψ(~x, t) ψ † (~x ′ , 0) eiωt dt −∞

1 X −βEλ′ Eλ′ − Eλ = e hλ|ψ † (~x ′ )|λ′ ihλ′ |ψ(~x)|λi 2π δ( + ω) ZG ′ ~ λλ

(3.35)

3.3. GREEN FUNCTIONS AT T 6= 0

11

† ′ J1 (~x −~x ′ ; ω) is the spectral function of the time correlator hψ(~ D x, t) ψ (~x , 0)i.E Similarly we can also write the correlator G< (~x−~x ′ , t) = ψ † (~x ′ , 0)ψ(~x, t) and its time Fourier transform J2 (~x, ~x ′ ; ω): Z ∞D E ′ J2 (~x − ~x ; ω) = ψ † (~x ′ , 0)ψ(~x, t) eiωt dt −∞

Eλ′ − Eλ 1 X −βEλ e hλ|ψ † (~x ′ )|λ′ ihλ′ |ψ(~x)|λi 2π δ( + ω) = ZG ′ ~ λλ

(3.36)

Using the properties of the Dirac δ-function we get J2 (~x − ~x ′ ; ω) =

Eλ′ − Eλ 1 X −β(Eλ′ + ~ω) e hλ|ψ † (~x ′ )|λ′ ihλ′ |ψ(~x)|λi 2π δ( + ω) ZG λλ′ ~

= e−β~ω J1 (~x − ~x ′ ; ω)

(3.37)

Notice that at T = 0, only the state |λ′ i = |Gi (the ground state) survives in sum over λ′ . Thus, at T = 0, J1 (~x − ~x ′ ; ω) vanishes for ω < 0, while J2 (~x − ~x ′ ; ω) vanishes (at T = 0) for ω > 0.

3.3.2

The Retarded Green Function

Let us consider now the retarded fermion Green function GR (~x − ~x ′ , t) (at finite temperature), D E GR (~x − ~x ′ , t) = −iθ(t) ψ(~x, t), ψ † (~x ′ , 0)
= −iθ(t) hψ(~x, t) ψ † (~x ′ , 0)i + hψ † (~x ′ , 0)ψ(~x, t) i   ′ ′ = −iθ(t) G> (~x − ~x , t) + G< (~x − ~x , t)

(3.38)

where we have used anti-commutators because we are dealing with fermion operators. For a bosonic operator we must replace the anticommutator by a commutator. For a translationally invariant system we can clearly express the retarded Green function in terms of the spectral functions J1 (~k, ω) and J2 (~k, ω), the space Fourier transforms of the functions J1 (~x, ~x ′ ; ω) and J2 (~x, ~x ′ ; ω) (where

12

CHAPTER 3. LINEAR RESPONSE THEORY

~k is the wave vector) defined above. To do that we will note that for the case of translationally invariant systems the Hamiltonian also commutes with the total linear momentum operator P~ , [P~ , H] = 0, and hence the states {|λi} can also be chosen to be eigenstates of the linear momentum P~ , with eigenvalues {P~λ }. If we now recall that since P~ is the generator of infinitesimal translations in space, we can also write ~

~

ψ(~x) = eiP ·~x/~ ψ(0)e−iP ·~x/~

(3.39)

Hence, in Fourier space, the spectral function J1 (~k, ω) becomes P~λ′ − P~λ ~ Eλ′ − Eλ 1 X −βEλ′ ′ e |hλ |ψ(0)|λi|2 2π δ( +ω) (2π)3 δ (3) ( +k) J1 (~k, ω) = ZG ′ ~ ~ λλ (3.40) ~ ~ where Pλ and Pλ′ are the total linear momentum of the states |λi and |λ′ i. Notice that J1 (~k, ω) is a real function. By the same argument used above we also get J2 (~k, ω) = e−β~ω J1 (~k, ω)

(3.41)

Then, the retarded fermion Green function at T > 0 is Z Z ∞  ~ d3 k ′ ′ dω ′  ~ ′ R ′ ′ ~ G (~x, t; ~x , 0) = −iθ(t) J1 (k, ω ) + J2 (k, ω ) ei(k·(~x−~x )−ω t) 3 (2π) −∞ 2π Z ∞ Z  dω ′  ′ ′ d3 k ~ −β~ω ′ J1 (ω ′) ei(k·(~x−~x )−ω t) = −iθ(t) 1 + e 3 (2π) −∞ 2π (3.42) Its Fourier transform GR (~k, ω) is G (~k, ω) = R

Z

∞ −∞

 J (~k, ω ′) dω ′  ′ 1 1 + e−β~ω 2π ω − ω ′ + iη

(3.43)

(where η → 0+ ). This function has poles in the lower half plane Imω < 0. We will now use the fact that J1 (~k, ω) is a real function and the property lim+

η→0

1 1 = P − iπδ(x) x + iη x

(3.44)

3.3. GREEN FUNCTIONS AT T 6= 0 where x is real and P

13

x 1 = lim+ 2 x η→0 x + η 2

is the principal value of 1/x, to show that Z ∞  dω ′  ′ R ~ Im G (k, ω) = −π 1 + e−β~ω J1 (~k, ω ′) δ(ω − ω ′ ) −∞ 2π
1 = − 1 + e−β~ω J1 (~k, ω) 2

(3.45)

(3.46)

Hence we obtain J1 (~k, ω) = −



2 1 + e−β~ω



Im GR (~k, ω)

(3.47)

For a bosonic operator, and in particular this will apply to the generalized susceptibilities, the corresponding relation between the spectral function J1 (~k, ω) and the imaginary part of the Green function is   2 ~ Im GR (~k, ω) (3.48) J1 (k, ω) = − 1 − e−β~ω

3.3.3

The Time-Ordered (Feynman) Fermion Propagator

Let us consider now the Feynman fermion propagator (also at T > 0) D E GF (~x − ~x ′ , t) = −i T (ψ(~x, t)ψ † (~x ′ , 0) (3.49) where we use a fermion time-ordered product. It can also be expressed in terms of the spectral function J1 (~x − ~x ′ ; ω) and of its Fourier transform (in space) J1 (~k, ω): Z ∞ dω ′ ′ ′ GF (~x − ~x , t) = −iθ(t) J1 (~x − ~x ′ , ω ′)e−iω t −∞ 2π Z ∞ dω ′ ′ +iθ(−t) J2 (~x − ~x ′ , ω ′)e−iω t −∞ 2π (3.50)

14

CHAPTER 3. LINEAR RESPONSE THEORY

and its time (and space) Fourier transform GF (~k, ω) =

Z

=

Z

∞

−∞ ∞

−∞

) J2 (~k, ω ′ ) J1 (~k, ω ′) + ω − ω ′ + iη ω − ω ′ − iη   −β~ω ′ dω ′ 1 e ′ J1 (~k, ω ) + 2π ω − ω ′ + iη ω − ω ′ − iη

dω ′ 2π

(

(3.51)

which has poles both in the upper and in the lower half of the complex ω plane. Thus, contrary to the retarded Green function, the Feynman propagator is not analytic on either half plane. Since J1 (~k, ω) is real, we get Re G (~k, ω) = Re GF (~k, ω) = P R

Z

∞ −∞

  J (~k, ω ′) dω ′ ′ 1 1 + e−β~ω ω − ω ′ 2π

(3.52)

and
1 1 + e−β~ω J1 (~k, ω) 2
1 Im GF (~k, ω) = − 1 − e−β~ω J1 (~k, ω) 2 Im GR (~k, ω) = −

(3.53)

which imply that the real and imaginary parts of the retarded and time ordered functions are related by Re GR (~k, ω) = Re GF (~k, ω) β~ω Im GR (~k, ω) = coth( ) Im GF (~k, ω) 2

(3.54)

At T = 0, Im GR (~k, ω) = sign(ω) Im GF (~k, ω)

(3.55)

Finally the real and imaginary parts of the retarded (and time-ordered) Green function are related by the Kramers-Kronig relation Re G (~k, ω) = −P R

Z

∞

−∞

dω ′ Im GR (~k, ω ′ ) π ω − ω′

(3.56)

3.3. GREEN FUNCTIONS AT T 6= 0

3.3.4

15

The temperature Green Function

The temperature Green function GT (~x −~x ′ , τ ) is the time-ordered propagator in imaginary time τ , D E ′ † ′ GT (~x − ~x , τ ) = − Tτ (ψ(~x, τ )ψ (~x , 0)) (3.57) where Tτ is the imaginary time ordering operator, and 0 ≤ τ ≤ β~. Hence

GT (~x − ~x ′ , τ ) = −θ(τ )hψ(~x, τ )ψ † (~x ′ , 0)i + θ(−τ )hψ † (~x ′ , 0)ψ(~x, τ )i (3.58) Once again we expand the thermal expectation values in terms of complete sets of states {|λi}, to obtain D E 1 X −βEλ′ † ′ ψ(~x, τ )ψ (~x , 0) = e hλ|ψ † (~x ′ )|λ′ ihλ′ |ψ(~x)|λieτ (Eλ′ −Eλ )/~ ZG ′ λλ D E X 1 e−βEλ hλ|ψ † (~x ′ )|λ′ ihλ′ |ψ(~x)|λieτ (Eλ′ −Eλ )/~ ψ † (~x ′ , 0)ψ(~x, τ ) = ZG ′ λλ

(3.59)

Since the temperature (or imaginary time) fermion Green function is antiperiodic in time, GT (~x − ~x ′ , τ ) = −GT (~x − ~x ′ , τ + β~), it can be expanded in Fourier series with coefficients Z β~ 1 ′ dτ e−iωn τ GT (~x − ~x ′ , τ ) GT (~x − ~x , ωn ) = β~ 0 Z β~ 1 = dτ e−iωn τ hψ(~x, τ )ψ †~x ′ )i β~ 0  −βEλ  e + e−βEλ′ 1 X † ′ ′ ′ hλ|ψ (~x )|λ ihλ |ψ(~x)|λi = − β~ZG λλ′ Eλ′ − Eλ − i~ωn

(3.60)

where ωn = 2π (n + 1/2). β~ In Fourier space we find 1 GT (~k, ωn ) = − β~

Z

∞

1 β~

Z

∞

= −

−∞

−∞

! J1 (~k, ω ′ ) + J2 (~k, ω ′) −iωn − ω ′  J (~k, ω ′ ) dω ′  ′ 1 1 + e−β~ω 2π −iωn − ω ′ dω ′ 2π

(3.61)

16

CHAPTER 3. LINEAR RESPONSE THEORY Im ω 2π (n + 1/2) i β~

ω + iη

π i β~

Re ω

Figure 3.2: Analytic continuation to real frequency at finite temperature. Thus, the temperature Green function GT (~k, ωn ) can also be determined from the spectral function J1 (~k, ω). Furthermore, if we now compare this result with the analogous expression for the retarded Green function GR (~k, ω) we see that GR (~k, −iωn ) = −β~GT (~k, ωn )

(3.62)

Hence, GR (~k, ω) is the analytic continuation of −β~GT (~k, ωn ) to a patch of the complex plane, that includes the imaginary axis: −iωn → ω + iη ⇒ −β~GT (~k, ωn ) → GR (~k, ω)

(3.63)

Since GT (~k, ωn ) is known only for a discrete set of frequencies on the imaginary axis, {iωn }, this procedure amounts to an analytic continuation from the imaginary frequency axis to the whole complex frequency plane, and thus to the real axis. (See Fig.3.2) Thus we have now a direct way to compute these functions 1. We first compute the temperature Green function GT (~k, ωn ) 2. Next, by analytic continuation we find the retarded Green function GR (~k, ω) 3. The spectral function J1 (~k, ω) is obtained from the imaginary part of the retarded Green function.

3.4. DISSIPATION AND RESPONSE

17

4. The time-ordered and the retarded Green functions are both determined in terms on the spectral function

3.4

Dissipation and Response

We will now extend these ideas to response functions. We will find along the way a very important result known as the fluctuation-dissipation theorem. We will begin by considering first a very simple problem. Imagine that we have a classical linear harmonic oscillator, represented by a coordinate x(t). The oscillator has mass m and natural frequency ω0 . The oscillator is in thermal equilibrium with a bath represented by a set of random timedependent “internal forces” Fint (t), which represent the collisions between the degrees of freedom of the bath with the oscillator. Thus in equilibrium the classical equation of motion of the oscillator is m

d2 x(t) + mω02 x(t) = Fint (t) 2 dt

(3.64)

We will now imagine that we act with an external force Fext (t) on the oscillator. 2 The average non-equilibrium displacement (averaged with respect to the random internal forces) which we will denote by hx(t)in.e. , is linearly related to the external force by an expression of the form Z ∞ hx(t)in.e. = χ(t, ¯ t′ )Fext (t′ )dt′ (3.65) −∞

The equation of motion in the presence of the external force is m

d2 x(t) + mω02 x(t) = Fint (t) + Fext (t) 2 dt

(3.66)

so that on average we have D d2 x(t) E + mω02 hx(t)in.e. = hFint (t)in.e. + Fext (t) m dt2 n.e.

(3.67)

where we have used the fact that, in the presence of the external force, the average effect of the internal forces, which we define as friction, does not 2

Notice that in the presence of the external force, the system (the oscillator) is no longer in equilibrium.

18

CHAPTER 3. LINEAR RESPONSE THEORY

vanish, hFint (t)in.e. 6= 0. In simple phenomenological models we normally use the “constitutive relation” which asserts that the friction force is a linear function of the velocity of the oscillator hFint (t)in.e. = −mγ

D dx(t) E dt

(3.68)

n.e.

where γ is the friction constant. The equation of motion now is D dx(t) E D d2 x(t) E + mω02 hx(t)in.e. = Fext (t) + mγ m dt2 n.e. dt n.e.

(3.69)

d2 χ(t, ¯ t′ ) dχ(t, ¯ t′ ) m + mγ + mω02 χ(t, ¯ t′ ) = δ(t − t′ ) 2 dt dt

(3.70)

Hence, χ(t, ¯ t′ ) satisfies

which is to say that χ(t, ¯ t′ ) = χ(t ¯ − t′ ) is the retarded Green function for this non-equilibrium system. It is straightforward to show that χ(t ¯ − t′ ) has the integral representation 1 χ(t ¯ −t)=− m ′

Z

∞

−∞

′

e−iω(t−t ) dω 2π ω 2 − ω02 + iγω

(3.71)

where the integration runs along the real axis on any path in the upper half plane. Thus, the real and imaginary parts of its Fourier transform, χ′ (ω) and χ′′ (ω), in this simple problem are, respectively, given by ω02 − ω 2 i h 2 m (ω 2 − ω02) + (γω)2 ωγ i h χ′′ (ω) = 2 m (ω 2 − ω02) + (γω)2 χ′ (ω) =

(3.72) (3.73)

As usual, χ′ (ω) and χ′′ (ω) obey the Kramers-Kronig relation. Using Cauchy’s Theorem we can define a function χ(z) where z is a complex number, Z ∞ Z ∞ dω χ(ω) dω χ′′ (ω) χ(z) = = (3.74) −∞ 2π ω − z −∞ π ω − z

19

3.5. THE FLUCTUATION-DISSIPATION THEOREM

where z is in the upper half plane, Imz > 0. By explicit calculation we find χ(z) = −

1 1 m z 2 − ω02 − iγz

(3.75)

which does not have a pole on either half plane but instead a branch cut along the real axis, with discontinuity 2χ′′ (ω). Notice that the branch cut arises only in the presence of a finite damping coefficient γ. In other words, it is due to the existence of friction forces. We will now show that χ′′ (ω) represents the dissipation. Let us compute the work done by the external force, per unit time, Z ∞ D dx(t) E dW dχ − = Fext (t) = Fext (t − t′ ) Fext (t′ ) dt′ dt dt n.e. dt −∞ Z ∞ Z ∞ dω ′ −iωt ′ ′ ′ ′ = e Fext (t) (−iω )χ(ω ) dt′ eiω t Fext (t′ ) −∞ −∞ 2π (3.76) For a “monochromatic” force Fext (t) = Re (Fext eiωt ), the power dissipated, averaged over one cycle, is −

dW ω 2 = χ′′ (ω) Fext (t) dt 2

(3.77)

where we used that χ′′ (ω) = −χ′′ (−ω), which holds for this simple model but it is more generally true. Hence, we see that the dissipated power is proportional to χ′′ (ω), i.e. to the discontinuity across the branch cut. In particular this simple model suggests that there is a direct connection between dissipation and susceptibilities. This result is actually quite general and we will identify the imaginary part of the response functions with the dissipative response.

3.5

The Fluctuation-Dissipation Theorem

Let us now return to the discussion of the response functions. As we saw above, even in Fermi systems the response functions are related to correlators of bosonic operators (e.g. bilinears of Fermi operators). In particular, for a general bosonic operator O(~x, t), the generalized susceptibility χ(~x, t; ~x ′ , t′ ) is

20

CHAPTER 3. LINEAR RESPONSE THEORY

simply related to the retarded Green function D ret (~x, t; ~x ′ , t′ ) for the operator O(~x, t): 1 (3.78) χ(~x, t; ~x ′ , t′ ) = D ret (~x, t; ~x ′ , t′ ) ~ This relation is also correct at finite temperature T . Thus, just as in the case of the fermion correlators, we will need the connection between the retarded, time-ordered and temperature correlation functions. These relations are very similar expect for minor changes due to Bose statistics. We will not repeat all the arguments and we will instead just write down the important results. Just as in the fermionic case, the key concept is the spectral function. Thus, for a general bosonic operator O(~x, t) we define the correlators D> (~x − ~x ′ , t), D< (~x − ~x ′ , t), D ret (~x − ~x ′ , t), DF (~x − ~x ′ , t) and DT (~x − ~x ′ , τ ), by the thermal averages D E D> (~x − ~x ′ , t) = O(~x, t)O(~x ′ , 0) D E ′ ′ D< (~x − ~x , t) = O(~x , 0) O(~x, t) D E D ret (~x − ~x ′ , t) = −iθ(t) [O(~x ′ , t), O(~x, 0)] D E DF (~x − ~x ′ , t) = −i T (O(~x ′ , t), O(~x, 0)) D E ′ ′ DT (~x − ~x , τ ) = Tτ (O(~x , τ ), O(~x, 0))

(3.79)

The spectral functions J1 (~k, ω) and J2 (~k, ω) are once again the space and time Fourier transforms of the correlators D> (~x − ~x ′ , t) and D< (~x − ~x ′ , t) respectively:

which obey

J1 (~k, ω) = D> (~k, ω) J2 (~k, ω) = D< (~k, ω)

(3.80)

J2 (~k, ω) = e−β~ω J1 (~k, ω)

(3.81)

However, since the operators O(~x, t) are bosonic (that is, they obey commutation relations), the temperature Green function obeys periodic boundary conditions in imaginary time, DT (~x − ~x ′ , τ + β~) = DT (~x − ~x ′ , τ ). This condition, which implies that the Matusbara frequencies now take the val2π n, where n ∈ Z, has important consequences. In particular, ues ωn = β~

21

3.5. THE FLUCTUATION-DISSIPATION THEOREM

paraphrasing what we did for the Fermi case, we now find that the spectral function J1 (~k, ω) and the imaginary part of the Fourier transform of the retarded Green function Im D ret (~k, ω) are related by J1 (~k, ω) = −

2 Im D ret (~k, ω) −β~ω 1−e

(3.82)

In particular, this relation implies that the dissipative component of the dynamical susceptibility χ′′ (~k, ω) is proportional to the spectral function J1 (~k, ω),
~ (3.83) χ′′ (~k, ω) = − 1 − e−β~ω J1 (~k, ω) 2

This relation is known as the Fluctuation-Dissipation Theorem. On the other hand the retarded and time ordered Green functions are now related by ReD ret (~k, ω) = ReDF (~k, ω) β~ω ImD ret (~k, ω) = tanh( )ImDF (~k, ω) 2

(3.84) (3.85)

Also, just as in the fermionic case, the real and imaginary parts of the retarded correlation function are related by the usual Kramers-Kronig relation Re D (~k, ω) = −P ret

Z

∞ −∞

dω ′ Im D ret (~k, ω ′) π ω − ω′

Finally, the temperature Green function DT (~k, ωn ), where ωn = related to the spectral function by the integral transform 1 DT (~k, ωn ) = − β~

Z

∞

−∞

 J (~k, ω ′ ) dω ′  1 −β~ω ′ 1−e 2π −iωn − ω ′

(3.86) 2π n, β~

is

(3.87)

Thus, here too, all the correlation functions of interest can be found by the same analytic continuation procedure we already discussed for fermions. The bosonic nature of the correlation functions follows form (a) the physically correct spectral functions and (b) the temperature-dependent prefactors in the integrand of this equation.

22

3.6

CHAPTER 3. LINEAR RESPONSE THEORY

Perturbation Theory for T > 0

We will now discuss briefly perturbation theory at finite temperature T . The approach is very similar to what we did at T = 0. Let is consider a system with Hamiltonian H = H0 + H1 . We will be interested in the fermion temperature Green function GT (~x − ~x ′ , τ ) D E ′ ′ † ′ ′ GT (~x − ~x , τ − τ ) = Tτ (ψ(~x, τ )ψ (~x , τ ) (3.88) for a system described by the Hamiltonian H. We now introduce a representation analogous to the interaction representation but for T > 0. Let U(τ, τ ′ ) be the operator U(τ, τ ′ ) = eτ H0 /~ e−(τ −τ

′ )(H

0 +H1 )/~

e−τ

′H

0 /~

(3.89)

Notice that although this operator is not unitary, it satisfies the group property U(τ, τ ′ )U(τ ′ , τ ′′ ) = U(τ, τ ′′ ) (3.90) and it obeys the (imaginary time) equation of motion −~

∂U (τ, τ ′ ) = H1 (τ )U(τ, τ ′ ) ∂τ

(3.91)

where H1 (τ ) = eτ H0 /~ H1 e−τ H0 /~

(3.92)

In particular the full density matrix is given by e−βH = e−βH0 U(β~, 0)

(3.93)

It is straightforward to show that the temperature Green function of the full system is given by the following expression in terms of thermal averages of the unperturbed system: D E Tτ (U(β~, 0)ψ(~x, τ )ψ † (~x ′ , τ ′ )) 0 D E GT (~x − ~x ′ , τ − τ ′ ) = (3.94) U(β~, 0) 0

Here we used the notation

hAi0 =

Tr (e−βH0 A) Tr e−βH0

(3.95)

3.7. THE ELECTRICAL CONDUCTIVITY OF A METAL

23

where A is an arbitrary operator. Notice that the operator U(β~, 0)) plays here a role analogous to the S-matrix. Hence, using the same line of reasoning that we used for the S-matrix, we can write Z Z β~ ∞ X 1 (−1)n β~ dτ1 . . . dτn Tτ (H1 (τ1 ) . . . H1 (τn )) U(β~, 0) = n! ~n 0 0 n=0 Z 1 β~ − dτ H1 (τ ) ≡ Tτ e ~ 0 (3.96) Thus we can carry out perturbation theory much in the same way as we did at T = 0. The only missing ingredient is a generalization of Wick’s Theorem for T > 0. It turns out that this works exactly in the same way provided that the contraction of an arbitrary pair of operators A(τ1 ) and B(τ2 ) is identified with imaginary time ordered thermal expectation values of the operators with respect to H0 : A(τ1 )B(τ2 ) = hTτ (A(τ1 )B(τ2 ))i0 (3.97) which is a thermal contraction. Notice that we are using the same notation for the thermal contraction that we used at T = 0 although it now has a different meaning. Wick’s Theorem simply states that the thermal average of the product of any number of operators is equal to the sum of the products of all possible pair-wise thermal contractions. Consequently, the structure of the perturbation series is the same as the one we found at T = 0. In particular we will also find that disconnected diagrams, which at T = 0 are called the “vacuum diagrams”, also cancel out exactly at T > 0. They only survive in the computation of the grand partition function. Thus, all of the Feynman rules used at T = 0 carry over to T > 0 with the only change that the factor of −i/~ now simply becomes −1/~. All other rules remain intact, except that the propagators are now temperature Green functions, which obey specific periodic and anti-periodic boundary conditions in imaginary time.

3.7

The Electrical Conductivity of a Metal

We will now consider the response of an electron gas to weak external electromagnetic fields Aµ (x). The formalism can be generalized easily to other

24

CHAPTER 3. LINEAR RESPONSE THEORY

systems and responses. In particular, we will consider the electrical conductivity of a metal. There are three effects (and couplings) that we need to take into consideration: a) electrostatic, b) diamagnetic (or orbital) and c) paramagnetic. The electrostatic coupling is simply the coupling to an external potential with Hext given by Z XZ 3 † d x eφ(x, t) ψσ (x)ψσ (x) ≡ d3 x J0 (~x)A0 (~x) (3.98) Hext = σ=↑,↓

where φ ≡ A0 is the scalar potential ( or time component of the vector potential Aµ ). The diamagnetic coupling (or orbital) follows from the minimal ~ The kinetic energy term H coupling to the external vector potential A. kin is modified following the minimal coupling prescription to become     Z 2 ie ~ ie ~ 3 ~ † ~ ~ Hkin (A) = d x ▽ + A(x) ψ (x) · ▽ − A(~x) ψ(x) (3.99) 2m ~c ~c which can be written as a sum of two terms Hkin (A) = Hkin (0) + Hext (A)

(3.100)

where Hkin (0) is the kinetic energy term of the Hamiltonian in the absence of the field, and Hext (A) is the total perturbation, i.e., Z i h e ~2 3 ~ ~ A (~x)J0 (x) Hext (A) = d x J0 (~x)A0 (~x) − J (~x) · A(~x) − 2mc2 Z h i e ~2 = d3 x Jµ (x)Aµ (x) − A (x)J (x) (3.101) 0 2mc2 P ~ x) is the Here J0 (x) = eρ(x) = e σ ψσ† ψσ is the local charge density, and J(~ gauge-invariant charge current defined as 2 X ie~ X † ~ σ (~x) − ▽ψ ~ † (~x)ψ(~x)] − e A(~ ~ x) [ψσ (~x)▽ψ ψσ† (~x)ψσ (~x) 2mc σ mc2 σ  † ie~ X † ~ σ (~x) − Dψ ~ σ (~x) ψσ (~x)] [ψ (~x)Dψ ≡ 2mc σ σ

~ x) = J(~

(3.102)

3.7. THE ELECTRICAL CONDUCTIVITY OF A METAL

25

~ is the (spatial) covariant derivative. We are using here ~ =∇ ~ −i e A where D ~c ~ and Jµ Aµ = J0 A0 − J~ · A. ~ a 4-vector notation Jµ = (J0 , J) ~ x) is the sum of the two terms, one which represents the mass Clearly J(~ e2 ~ † current and the diamagnetic term, mc 2 Aψ ψ. We can write the total perturbation, including the scalar potential A0 , if we write Z e ~2 Hext = d3 x[Jµ (x)Aµ (x) − A J0 (x)] (3.103) 2mc2

Finally, we can also consider a paramagnetic coupling to the spin degrees of freedom which has the Zeeman form Z Zeeman ~ x) · ψ † (~x)S ~σσ′ ψσ′ (~x) Hext = d3 x g B(~ (3.104) σ

~ = ~ for spin where g is typically of the order of the Bohr magneton µB and S 2 one-half systems. We will discuss the Zeeman term later in the context of the magnetically ordered states. For now will focus on the purely electromagnetic response of a general interacting Fermi system. A straightforward application of the Linear Response formulas derived above yields an expression for the current hJµ i′ in the presence of the perturbation. Z Z i t ′ ′ d3 x′ hG|[Jν (x′ ), Jµ (x)]|GiAν (x′ ) + . . . dt hJµ (x)i = hJµ (x)iG − ~ −∞ (3.105) This formula suggests that we should define the retarded current correlation ret function Dµν (x, x′ ) ret Dµν (x, x′ ) = −iΘ(x0 − x′0 )hG|[Jµ(x), Jν (x′ )]|Gi

(3.106)

The induced current hJk iind hJk iind = hJk i′ − hjk iG

(3.107)

(where jk is the “paramagnetic” component of the current) has a very simple ret form in terms of Dµν (x, x′ ), namely Z D E 1 R 1 R d4 x′ Dkℓ (x, x′ )Aℓ (x′ ) − Dk0 (x, x′ )A0 (x′ ) Jk (x) = ~ ~ ind e (3.108) − 2 J0 (x)Ak (x) + . . . mc

26

3.7.1

CHAPTER 3. LINEAR RESPONSE THEORY

The dielectric tensor and the conductivity tensor

The induced current has two important properties: a) it is conserved, namely the the induced current and the induced density satisfy a continuity equation ∂ρind ~ ~ + ∇ · Jind = 0 ∂t

(3.109)

and b) it is gauge invariant. This means that we should be able express the induced current directly in terms of gauge invariant quantities such as the electric and magnetic fields. Since hJµ (x)iind is gauge invariant, we can compute its form in any gauge. In the gauge A0 = 0 the spacial components of hJµ (x)iind are Z 1 e2 ρ ret d4 x′ Dkℓ (x − x′ )Aℓ (x′ ) + . . . (3.110) hJk (x)iind = − 2 Ak (x) + mc ~ ~ ext and magnetic field H ~ are In this gauge, the external electric field E ~ ext = −∂0 A ~ E

~ = ∇× ~ A ~ H

(3.111)

Now, in Fourier space, we can write 1 ret e2 ρ (~p, ω)Aℓ (~p, ω) hJk (~p, ω)iind = − 2 Ak (~p, ω) + Dkℓ ~  mc  1 ret e2 ρ Eℓext ≡ Dkℓ (~p, ω) − δ (p, ω) kℓ ~ mc2 iω

(3.112)

This expression is almost the conductivity. It is not quite that it since the conductivity is a relation between the total current J~ = J~ind + J~ext and the ~ instead of one between the induced current and the total electric field E, external electric field. In order to take these electromagnetic effects into account, we must use Maxwell’s equations in a medium which involve the ~ D, ~ B ~ and H ~ fields E, ~ ·D ~ ×E ~ =ρ ▽ ~ = − ∂ B~ ▽ ∂t ~ ·B ~ ×H ~ =0 ▽ ~ = ∂ D~ + J~ ▽ ∂t where

~ =H ~ +M ~ B

~ =E ~ ext + E ~ ind E

(3.113)

(3.114)

3.7. THE ELECTRICAL CONDUCTIVITY OF A METAL

27

~ and E ~ ind are the magnetic and electric polarization vectors. In Here M particular ~ ind J~ind = ∂t E (3.115) and

~ = ∂t E ~ + J~ind ∂t D

(3.116)

~ must be proportional to E ~ Linear Response theory is the statement that D Dj = εjk Ek

(3.117)

~ and E ~ ext satisfy similar equations where εjk is the dielectric tensor. Since E ~ ×▽ ~ ×E ~ ~ + ∂t J~ −▽ = ∂t2 E ~ ×▽ ~ ×E ~ ext = ∂ 2 E ~ ext + ∂t J~ext −▽ t (3.118) ~ ×▽ ~ ×E ~ ▽ ~ · E) ~ = ▽( ~ − ▽2 E, ~ we can write, for the Fourier transforms, and ▽ the equations pi pj Ej (~p, ω) − p~2 Ei (~p, ω) = −ω 2 Ej (~p, ω) − iωJi (~p, ω) pi pj Ejext (~p, ω) − p~2 Eiext (~p, ω) = −ω 2 Eiext (~p, ω) − iωJiext (~p, ω) (3.119) Thus, we get pi pj Ej (~p, ω) − p~2 Ei (~p, ω) + ω 2 Ei (~p, ω) = −iωJiind (~p, ω) + pi pj Ejext (~p, ω) − p~2 Eiext (~p, ω) + ω 2 Eiext (~p, ω) (3.120) Since we showed above that   1 ret e2 ρ ind −iωJi (~p, ω) = δij 2 − Dij (~p, ω) Ejext (~p, ω) mc ~

(3.121)

we conclude that (pi pj − ~p2 δij + ω 2δij ) Ej (~p, ω) = e2 ρ 1 (δij 2 − DijR (~p, ω) + pi pj − p~2 δij + ω 2δij ) Ejext (~p, ω) mc ~ (3.122)

28

CHAPTER 3. LINEAR RESPONSE THEORY

In matrix form, these equations have the simpler form   2 1 R e ρ 2 2 2 2 ~ ext (3.123) ~ I − D + p ⊗ p − p~ I + ω I E (p ⊗ p − ~p I + ω I)E = mc2 ~ ~ ~ This equation allows us to write E ext in terms of E. Hence we find that the induced current is related to the total field by   2 e2 ρ e ρ 1 ret ~ iω Jind = D − 1 [ 2 I − D ret + p ⊗ p − p~2 I + ω 2 1]−1 2 mc mc ~
~ p ⊗ p − ~p2 I + ω 2 I E

(3.124)

It follows that the conductivity tensor σ is     e2 ρ e2 ρ 1 ret 1 ret D (~p, ω) − I + D (~p, ω) − I iωσ(~p, ω) = ~ mc2 ~ mc2 1 e2 ρ [ 2 I − D ret (~p, ω) + p ⊗ p − p~2 I + ω 2 I]−1 ~  mc e2 ρ 1 ret D (~p, ω) − I ~ mc2 (3.125) ~ = εE, ~ the dielectric tensor ε is Also, since D ε=I+

i σ ω

(3.126)

We conclude that both the conductivity tensor and the dielectric tensor can be determined from the retarded current and density correlation functions.

3.7.2

Correlation Functions and Conservation Laws

In the problem discussed in the previous section, we saw that we had to consider a correlation function of currents. Since the currents are conserved, i.e., ∂µ J µ = 0, we expect that the correlation function Dµν (x, x′ ) should obey a similar equation. Let us compute the divergence of the retarded correlation µν function, ∂µx Dret (x, x′ ), µν ∂µx Dret (x, x′ ) = ∂µx [−iΘ(x0 − x′0 )hG|[J µ (x), J ν (x′ )]|Gi

(3.127)

3.7. THE ELECTRICAL CONDUCTIVITY OF A METAL

29

Except for the contribution coming from the step function, we see that we can operate with the derivative inside the expectation value to get
µν ∂µx Dret (x, x′ ) = −i ∂µx Θ(x0 − x′0 ) hG|[J µ (x), J ν (x′ )]|Gi −iΘ(x0 − x′0 )hG|[∂µx J µ (x), J ν (x′ )]|Gi (3.128) The second term vanishes since J µ (x) is a conserved current and the first term is non zero only if µ = 0. Hence we find µν ∂µx Dret (x, x′ ) = −iδ(x0 − x′0 )hG|[J 0 (x), J ν (x′ )]|Gi

(3.129)

which is the v.e.v. of an equal-time commutator. These commutators are given by hG|[J 0 (~x, x0 ), J 0 (~x′ , x0 )]|Gi = 0 ie2 x ∂ [δ(~x − ~x′ )hρ(~x)i] hG|[J 0 (~x, x0 ), J i (~x′ , x0 )]Gi = mc2 k (3.130) ret Hence, the divergence of Dµν is ret ∂xµ Dµk (x, x′ )

e2 x 4 ∂k [δ (x − x′ )hρ(x)i] ; = 2 mc

ret ∂xµ′ D0µ (x, x′ ) = 0

(3.131)

and e2 x 4 ∂ [δ (x − x′ )hρ(x′ )i] mc2 k Notice that the time-ordered functions also identities can be used to prove that hJ~ind i conserved. Furthermore, in momentum and become ret ∂xν′ Dkν (x, x′ ) = −

;

ret ∂xµ′ D0µ (x, x′ ) = 0 (3.132)

satisfy these identities. These is indeed gauge-invariant and frequency space, the identities

ret ret −iωD00 (~p, ω) − ipk Dk0 (~p, ω) = 0 ret ret −iωD0k (~p, ω) − ipℓ Dℓk (~p, ω) = −

e2 ρ ipk mc2

ret ret −iωD00 (p, ω) − ipk D0k (~p, ω) = 0 ret ret −iωDk0 (~p, ω) − ipℓ Dkℓ (~p, ω) = −

e2 ρ ipk mc2 (3.133)

30

CHAPTER 3. LINEAR RESPONSE THEORY

We can combine these identities to get ret ret ω 2D00 (~p, ω) − pℓ pk Dℓk (~p, ω) = −

e2 ρ 2 p~ mc2

(3.134)

Hence, the density-density and the current-current correlation functions are not independent. A number of interesting identities follow from this equation. In particular if we take the static limit ω → 0 at fixed momentum p~, we get ret lim pℓ pk Dℓk (~p, ω) =

ω→0

e2 ρ 2 p~ mc2

(3.135)

ret provided that limω→0 D00 (~p, ω) is not singular for ~p 6= 0. Also from the equal-time commutator

ie2 x ∂ (δ(~x − ~x′ )hρ(x)i) mc2 k

(3.136)

e2 ▽2 (δ(x − x′ )hρ(x)i) mc2 x

(3.137)

hG|[Jk (~x, x0 ), J0 (~x′ , x0 )]|Gi = we get ret lim ∂kx Dk0 (x, x′ ) = ′

x0 →x0

If the system is uniform, hρ(x)i = ρ, we can Fourier transform this identity to get Z +∞ e2 ρ dω ret ipk Dk0 (~p, ω) = − 2 ~p 2 (3.138) mc −∞ 2π The conservation laws yield the alternative expression Z +∞ e2 ρ 2 dω ret iωD00 (~p, ω) = p~ mc2 −∞ 2π

(3.139)

This identity is known as the f -sum rule. If the system is isotropic, these relations can be used to yield a simpler ret form for the conductivity tensor. Indeed, if the system is isotropic, Dkℓ (~p, ω) ret ret is a sum of a longitudinal part Dk and a transverse part D⊥   pℓ pk pℓ pk ret ret ret − δℓk (3.140) Dℓk (~p, ω) = Dk (~p, ω) 2 + D⊥ (~p, ω) p~ ~p 2 ret Thus, we get a relation between D00 and the longitudinal part Dkret ret ω 2 D00 (p, ω) − ~p 2 Dkret (~p, ω) = −

e2 ρ 2 p~ mc2

(3.141)

3.7. THE ELECTRICAL CONDUCTIVITY OF A METAL Hence ret D00 (~p, ω)

p~ 2 = 2 ω



Dkret (~p, ω)

and lim Dkret (~p, ω) =

ω→0

e2 ρ − mc2



31

(3.142)

e2 ρ mc2

(3.143)

for all ~p. The conductivity tensor can similarly be split into longitudinal σk and transverse σ⊥ components   pi pj pi pj − δij (3.144) σij = σk 2 + σ⊥ p~ p~ 2 We find

" # e2 ρ Dkret − mc 2 1 σk = iω −Dkret + e2 ρ2 + ω 2 mc

and 1 σ⊥ = iω

 " 2 e ρ ret D⊥ − 1+ mc2

e2 ρ mc2 ret D⊥ + ω2

(3.145)

ret D⊥ − e2 ρ mc2

−

− ~p 2

#

(3.146)

These relations tell us that the real part of σk is determined by the imaginary part of Dkret . Thus, the resistive part of σk (which is responsible for dissipation in the system) is determined by the imaginary part of a response function. This is generally the case as it follows from the fluctuation-dissipation theorem.

3.7.3

Consequences of the f -sum rule

We will now assume that there is a stable collective mode, i.e., a plasmon R must branch. Therefore the retarded density-density correlation function D00 have a pole at the plasmon dispersion, ~ωpl (~p): R D00 (p, ω) =

 R A(p, ω) + D00 reg 2 (p) ω 2 − ωpl

(3.147)

 R where A(~p, ω) is the residue, and D00 is regular (analytic) near the plasma reg R branch. In order to enforce the condition that D00 is the retarded correlation

32

CHAPTER 3. LINEAR RESPONSE THEORY

function we will perform the analytic continuation ω → ω + iε. This analytic structure of the correlation function holds for ω ≈ |ωpℓ (p)|. We will nowsee that the sum rule determines the residue A(p, ω). HowR ever, since D00 is non-singular, we have reg Z

+∞

−∞

 R dω ω D00 (ω, p) = 0 reg

(3.148)

since we can close the contour on the upper half plane without enclosing any singularities. Thus, in order to satisfy the f -sum rule we must demand that Z +∞ e2 ρ 2 A(p, ω) = p (3.149) dω iω 2 (p) (ω + iε)2 − ωpℓ mc2 −∞ We will now close the contour on the pole and compute the residue. The pole is at ω = ±|ωpℓ (p)| + iǫ, and the residue is residue : = −πA(p, ±|ωpℓ (p)|)

(3.150)

which leads to the a condition for the amplitude A(p, ω): −π(A(p, +|ωpℓ (p)|) + A(p, −|ωpℓ (p)|)) = Hence, A=−

e2 ρ 2 p mc2

1 e2 ρ 2 p 2π mc2

(3.151)

(3.152)

(typically A(ω) = A∗ (−ω)). Thus we conclude that for frequencies close to the plasma branch the retarded density-density correlation function must have the form R (p, ω) D00

 R 1 e2 ρ p2 + D00 reg (p, ω) =− 2 (p) 2π mc2 ω 2 − ωpℓ

(3.153)

Notice that the residue is determined by the exact density and by the bare mass. This is a consequence of Galilean Invariance.

Chapter 4 The Weakly Interacting Electron Gas 4.1

The Electron Gas in Perturbation Theory

Consider an electron gas with a spin-independent instantaneous electronelectron interaction U(x − y) described by the Hamiltonian H1 : Z Z 1 3 d x d3 y n(x)n(y) U(x − y) (4.1) H1 = 2 where n(x) =

X

ψσ† (x)ψσ (x)

(4.2)

Z

(4.3)

σ=↑,↓

Then,

where

Z

+∞ −∞

1 dtH1 (t) = 2

Z

4

dx

d4 y n(x)n(y) V (x − y)

V (x − y) = U(x − y)δ(t − t′ )

(4.4)

† ′ ′ Gνν F (x, x ) = −ihG|T (ψν (x)ψν ′ (x )|Gi

(4.5)

In the perturbative evaluation of the Feynman electron propagator ′

where |Gi is the exact ground state of the interacting system, we will need to evaluate the expectation value the matrix element in the unperturbed ground 1

2

CHAPTER 4. THE WEAKLY INTERACTING ELECTRON GAS

state:

i − † ′ ~ 0 hG|T (ψν (x)ψν ′ (x ) e

Z

∞

dt′′ H1 (t′′ ) −∞

)|Gi0

(4.6)

where |Gi0 is the ground state of the unperturbed system. To lowest order in perturbation theory in H1 , we will need to compute the expectation value Z Z X 1 4 V (z − z ′ ) 0 hG|T (ψν (x)ψν† ′ (x′ )ψσ† (z)ψσ (z)ψσ† ′ (z ′ )ψσ′ (z))|Gi0 d z d4 z ′ 2 σσ′ =↑,↓

(4.7)

We will now use Wick’s theorem to express this expectation value in terms of products of pair-wise contractions. We will denote a contraction by ′ ψν (x)ψν† ′ (x′ ) = 0 hG|T (ψν (x)ψν† ′ (x′ ))|Gi0 = iGνν 0 (x, x ) ′

(4.8)

which is the Feynman propagator of the non-interacting system. There is a total of six non vanishing terms that contribute to the expectation value. Each non-vanishing contribution is given by a particular way of contracting the six Fermi operators in the expectation values. The six contributions are: 1.

1 = 2

1 2 Z

Z

4

dz 4

d z

Z

Z

d4 z ′

X σσ′

4 ′

dz

X σσ′

V (z − z ′ )ψν (x)ψν† ′ (x′ )ψσ† (z)ψσ (z)ψσ† ′ (z ′ )ψσ′ (z ′ ) ′

V (z − z )



′ ′ iGνν 0 (x, x )



(−iGσσ 0 (z, z))



′ ′ −iG0σ σ (z ′ , z ′ )



(4.9)

2.

=

1 2

1 2 Z

Z

4

dz

d4 z

Z

Z

d4 z ′

d4 z ′

X σσ′

X σσ′

V (z − z ′ )ψν (x)ψν† ′ (x′ )ψσ† (z)ψσ (z)ψσ† ′ (z ′ )ψσ′ (z ′ )

      ′ ′ σ′ σ ′ σσ′ ′ V (z − z ′ ) iGνν (x, x ) −iG (z , z) iG (z, z ) 0 0 0

(4.10)

3

4.1. THE ELECTRON GAS IN PERTURBATION THEORY 3.

=

1 2

1 2 Z

Z

4

dz

d4 z

Z

Z

d4 z ′

X σσ′

d4 z ′

X σσ′

V (z − z ′ ) ψν (x)ψν† ′ (x′ )ψσ† (z)ψσ (z)ψσ† ′ (z ′ )ψσ′ (z ′ )

V (z − z ′ ) (−iGνσ 0 (x, z))



′

′ −iGσν 0 (z, x )

 

′ ′

−iG0σ σ (z ′ , z ′ )



(4.11)

4.

1 = 2

1 2 Z

Z

4

dz 4

dz

Z

Z

d4 z ′

X σσ′

4 ′

dz

X σσ′

V (z − z ′ )ψν (x)ψν† ′ (x′ )ψσ† (z)ψσ (z)ψσ† ′ (z ′ )ψσ′ (z ′ )

V (z − z

′

) (−iGνσ 0 (x, z))



′ ′ −iG0σ ν (z ′ , x′ )

 

′ ′ iGσσ 0 (z, z )



(4.12)

5.

=

1 2

1 2 Z

Z

4

dz

d4 z

Z

Z

d4 z ′

X σσ′

d4 z ′

X σσ′

V (z − z ′ )ψν (x)ψν† ′ (x′ )ψσ† (z)ψσ (z)ψσ† ′ (z ′ )ψσ′ (z ′ )

      ′ ′ σν ′ ′ σ′ σ ′ V (z − z ′ ) −iGνσ (x, z ) iG (z, x ) −iG (z , z) 0 0 0

(4.13)

6.

=

1 2

1 2 Z

Z

4

dz

d4 z

Z

Z

d4 z ′

d4 z ′

X σσ′

X σσ′

V (z − z ′ )ψν (x)ψν† ′ (x′ )ψσ† (z)ψσ (z)ψσ† ′ (z ′ )ψσ′ (z ′ )

    ′ ′ σσ σ′ ν ′ ′ ′ V (z − z ′ ) −iGνσ (x, z ) (−iG (z, z)) −iG (z , x ) 0 0 0

(4.14)

4

CHAPTER 4. THE WEAKLY INTERACTING ELECTRON GAS

Hence, to lowest order in perturbation theory, i.e. linear in the pair interaction potential V , we get

+

1 (−i) 1! ~

+

1 (−i) 1! ~

+

1 (−i) 1! ~

+

1 (−i) 1! ~

+

1 (−i) 1! ~

+

1 (−i) 1! ~

′

′

′ νν ′ iGνν F (x, x ) = iG0 (x, x )+ Z Z X 1 ′ ′ σσ σ′ σ′ ′ ′ dz dz ′ V (z − z ′ )i3 Gνν (z , z ) 0 (x, x )G0 (z, z)G0 2 ′ σσ Z Z X 1 ′ ′ σσ′ ′ σ′ σ ′ dz dz ′ V (z − z ′ )i3 (−1) Gνν 0 (x, x ) G0 (z, z )G0 (z , z) 2 σσ′ Z Z X 1 σν ′ ′ σ′ σ′ ′ ′ V (z − z ′ )i3 (−1)Gνσ dz dz ′ 0 (x, z)G0 (z, x )G0 (z , z ) 2 σσ′ Z Z X 1 σ′ ν ′ ′ ′ σσ′ ′ dz dz ′ V (z − z ′ )i3 Gνσ 0 (x, z)G0 (z , x )G0 (z, z ) 2 σσ′ Z Z X 1 ′ ′ σ′ σ ′ σν ′ ′ dz dz ′ V (z − z ′ )i3 Gνσ 0 (x, z )G0 (z , z)G0 (z, x ) 2 σσ′ Z Z X 1 ′ ′ σ′ ν ′ ′ 0 V (z − z ′ )i3 (−1)Gνσ dz dz ′ 0 (x, z )G0 (z , y)Gσσ (z, z) 2 σσ′

0 hG|S|Gi0

+ ...

(4.15)

We will use Feynman diagrams to do the bookkeeping for us. Thus to ′ ′ each factor of iGνν 0 (x, x ) we associate an oriented line running from x to x′ , as shown in Fig.4.2(a). The line is oriented since the state created (or destroyed) by the field operator is either an electron or a hole, which both charged. Thus the orientation of the propagator lines follows from charge conservation. Similarly, each interaction is given in terms of the potential V (x−x′ ), which we will call a vertex and it is shown in Fig.4.2(b). In this case the vertex is spin-independent but in other cases , e.g. exchange interactions, the vertex has a non-trivial spin structure. We will now assign a Feynman diagram to each contribution in Eq.(4.15). Let us consider the first contribution to Eq.(4.15), which we have denoted by (a). In this term • There are two external points, x and x′ (and their spin labels ν and ν ′ ), and (in this case) one pair of internal points, z and z ′ (and the internal spin labels σ and σ ′ ). • there is an interaction factor of 12 V (z − z ′ ), which in this case is spinindependent

(a) (b) (c) (d) (e) (f )

5

4.1. THE ELECTRON GAS IN PERTURBATION THEORY x′

x′

ν′ z′ σ′

σz x

ν

x′

ν′

ν′

(+)

z ′σ ′

zσ

(−)

ν

x

(b)

(a)

x′ ν ′ z′

z

(−)

σ

z′ σ′

zσ

(+)

xν

x ν (c)

x′ ν ′

(d)

x′ ν ′

(+)

z′ σ′

z σ

zσ

z′ σ′

(−)

xν

xν (e)

(f)

Figure 4.1: Feynman diagrams for the first order contributions to the electron propagator, Eq.(4.15); the contributions of the diagrams (d) and (e) are equal, and so are the diagrams (c) and (f ). The (±) signs denote the fermionic sign of the diagram.

′ ′ iGνν 0 (x, x )

= xν

(a) Propagator

x′ ν ′

V (z − z ′ ) =

z

(b) Vertex

Figure 4.2: Feynman rules

z′

6

CHAPTER 4. THE WEAKLY INTERACTING ELECTRON GAS

(a)

(b)

Figure 4.3: Vacuum Diagrams • There is a sum over the internal spin labels and integrals over the internal space-time coordinates   1 −i • there is an overall factor of . 1! ~ • There are three operator contractions, and each contraction is a factor of i multiplying a Feynman propagator. • The contractions link the external points x and x′ (and their spin labels ν and ν ′ ) to each other and to the internal points z and z ′ (and to the internal spin labels σ and σ ′ ). • There is an overall sign that results from the anticommutation rules We will draw a Feynman diagram for this contribution by drawing an oriented line for each contraction. These three lines connect the external points x and x′ to each other and to the internal coordinates z and z ′ . We will draw a broken line connecting the coordinates z and z ′ . The spin labels must also be contracted to each other as in this theory the propagators are diagonal in spin. The result is shown in Fig.4.1(a). Notice that in this contribution the external points are contracted to each other and that the internal points are connected to each other but not to the external points. This is an example of a disconnected diagram: this diagram can be split in two separate pieces (each corresponding to a separate factor in the actual expression) by drawing a line that does not cross (or cut) any propagator line. In fact, the wto factor are just a free propagator factor ′ ′ iGνν 0 (x, x ) and a diagram obtained by contracting only the internal vertices (the two loops in Fig.4.1(a)).

7

4.1. THE ELECTRON GAS IN PERTURBATION THEORY

This factor also shows up in the perturbative expansion of the denominator 0 hG|S|Gi0 : 

 Z −i 1 hG|S|Gi = 1 + V (z − z ′ ) G0σσ (z, z)G0σ′ σ′ (z ′ , z ′ ) ~ 2 zz ′  Z 1 ′ 0 ′ 0 ′ V (z − z )(−1)Gσσ′ (z, z )Gσ′ σ (z , z) + . . . + 2 zz ′ ≡1 +

−

+ ...

(4.16)

which have a diagrammatic for shown in Fig.4.3. We can now assign a diagram to each of the contributions to Eq.(4.15) and Eq.(4.16). The corresponding diagrams are shown in Fig.4.1(b)-(f), and in Fig.4.3(a) and (b). We now notice that we can write (in compact form)

0 hG|S|Gi0

= =

        

iGF = iGF

−2

(

1 +

+2 −2

−

+ +2

)

+ ...

−    (   + ... 1 +    

+ ... −

)

+ ...

Hence, the disconnected diagrams are canceled exactly by the denominator 0 hG|S|Gi0 (the “vacuum graphs”). This result is true to all orders in perturbation theory. Thus when calculating a Green’s function we only need to calculate connected diagrams. To first order in perturbation theory, we

(4.17)

8

CHAPTER 4. THE WEAKLY INTERACTING ELECTRON GAS

find that the electron Green function, the Feynman propagator, is

iGF (x, x′ ) =

−2

+2

+ ... (4.18) The two leading perturbative contributions are the Hartree term (the second term above) and the Fock term (the third term.) Notice that the term with one fermion loop has a negative sign.1 We will now summarize the Feynman Rules (in position space) for a fermion N-point function, which has N external legs (attached to the N external points). The Feynman diagrams for an n-th order contribution in perturbation theory are obtained as follows: 1. Pair up all vertices (internal and external) one with an outgoing arrow to one with an incoming arrow 2. Assign a contraction to each pair and a factor of iG0 to each. 3. Weight of the diagram: 1 n!



−i ~

n Y n

j,k=1

V (zj − zk ) × products of factors of iG0

(4.19)

4. Assign a factor of (−1)F where F is the number of fermionic loops in the diagram 5. Integrate over space-time coordinates of internal vertices 6. Sum over all internal (contracted) spin labels 7. Consider only connected diagrams. 1

The factors of 2 come about because that interaction is non-local. These factors cancel against the factors of 1/2 of the interaction mediated by the pair potential. With this in mind we will drop both factors and count each diagram only once.

4.2. EFFECTIVE INTERACTION AND SCREENING

4.2

9

Effective interaction and Screening

We can calculate the effective interaction by calculating the density-density correlation function (or density propagator) i Π(x, x′ ) = − hG|T (n(x)n(x′ ))|Gi ~ iX hG|T ψσ† (x)ψσ (x)ψσ† ′ (x′ ) ψσ′ (x′ )|Gi =− ~ ′

(4.20)

σσ

To zeroth-order in perturbation theory, i.e. in the absence of interactions, we can compute this expectation value by using Wick’s Theorem: ( ) i X † † ′ ′ ′ ′ † † Π0 (x, x ) = − ψσ (x)ψσ (x)ψσ′ (x )ψσ′ (x ) + ψσ (x)ψσ (x)ψσ′ (x )ψσ′ (x ) ~ σσ′ (4.21) Hence we get ′






′ ′ 0 ′ Π0 (x, x ) = 0 hG|n(x)|Gi0 0 hG|n(x )|Gi0 + tr G0 (x, x )G (x , x)   −i  2 ≡ ρ + tr G0 (x, x′ )G0 (x′ , x) ~ −i 2 ≡ ρ + Π0c (x, x′ ) (4.22) ~ ′

−i ~

where we have introduced the connected density-density correlation function Π0c (x, x′ ) = −

i tr [ G0 (x, x′ )G0 (x′ , x) ] ~

(4.23)

We will consider now the connected two particle Green function (or 4point function) † † G(2) c (x1 σ1 ; x2 σ2 ; x3 σ3 , x4 σ4 ) = hG|T ψ(1)ψ(2)ψ (3)ψ (4)|Gic

(4.24)

where we have denoted by k = 1, 2, 3, 4, both the space-time coordinate xk and the spin label σk . To zeroth order in perturbation theory there are no connected pieces since both non-vanishing contributions to this expectation

10

CHAPTER 4. THE WEAKLY INTERACTING ELECTRON GAS

value factorize: ψ(1)ψ(2)ψ † (3)ψ † (4) + ψ(1)ψ(2)ψ † (3)ψ † (4) = iG0 (1, 4) iG0 (2, 3) − iG0 (1, 3) iG0 (2, 4)

=

− (4.25)

The first disconnected contribution is known as the direct term, while the second disconnected contribution is known as the exchange term. The first connected contribution to the two-particle Green function (or 4-point function) appears in first order in perturbation theory, whose contribution to the two-particle Green function is   Z Z X −i 1 V (z1 − z2 ) dz1 dz2 ~ 2 α α 1 2

×0 hG|T ψσ1 (x1 )ψσ2 (x2 )ψσ† 3 (x3 )ψσ† 4 (x4 )ψα† 1 (z1 )ψα1 (z1 )ψα† 2 (z2 )ψα2 (z2 )|Gi0 (4.26)

Again, considering only connected pieces and ignoring vacuum diagrams (which cancel out exactly), we find that to lowest order in perturbation theory the connected two-particle Green function is given by two fully connected (and identical) contributions2

G(2) c (1, 2; 3, 4) =

=



−i ~



2 2

Z

+

dz1

Z

+ ...

dz2 V (z1 − z2 ) iG0 (1, z1 )iG0 (2, z2 )iG0 (z2 , 3)iG0 (z, 4)

+...

(4.27) (2)

Thus, is we ignore the contributions from the external legs Gc , we see that the connected two-particle Green function Gc (1, 2; 3, 4)c is just the interaction 2

Notice again the cancelation of the factors of 1/2.

11

4.2. EFFECTIVE INTERACTION AND SCREENING t3

time

t2 t1

Figure 4.4: Intermediate states with particle-hole pairs in two-particle scattering processes potential. This is then a way to measure the effective interactions in the system including screening the electrons! What happens at higher orders in perturbation theory? The lowest order contributions schematically look as follows:

+ (−1)

+ (−1)2

+ ...

(4.28)

Let’s examine this a little closer. Let us consider that we act on the ground state by removing two electrons (or creating two holes) and that in the final state we also have two holes. Then, to second order in perturbation theory, the interaction can mix this state with an intermediate state in which there is an extra particle-hole pair in the intermediate state. This is shown in Fig.4.4. In higher orders we will have processes, among others, which in the intermediate states have, in addition to the two incoming particles (or holes), a number of particle-hole pairs. There are also processes, shown in Fig.4.5, in which two particles (or two holes) scatter each into intermediate states without additional particle-hole pairs. However we notice that there are higher order terms which consist basically in stringing together diagrams that include one fermion loop, one after each other, as in Fig. 4.6 (a) and (b). These sum over all of these diagrams can be viewed as an effective interaction.

12

CHAPTER 4. THE WEAKLY INTERACTING ELECTRON GAS

(a)

(b)

Figure 4.5: Processes not included in RPA

(a)

(b)

Figure 4.6: Two-loop bubble diagrams This motivates us to consider the sum over all such processes, leading to an effective interaction Veff (z − z ′ ), which is no longer instantaneous. When the intermediate process is represented by an elementary bubble diagram, Fig.4.7, one obtains an approximate form of the effective interaction known as the Random Phase Approximation, which is asymptotically exact in the high density limit (see below).

Figure 4.7: One-loop bubble diagram We will now sum over all these bubble diagrams. The n-th order term comes in n! copies which are just permutations an thus leads to an overall

13

4.3. FEYNMAN RULES IN MOMENTUM SPACE

factor of n!. We will do the sum by amputating the external legs (propagators) from each diagram. We find Z 1 ′ ′ d1d2V (z − 1) Π0c (1, 2)V (2 − z ′ ) Veff (z − z ) = V (z − z ) + 2! Z Z 1 + 2! d1d2 d1′ d2′ V (z − 1)Π0c (1, 1′)V (1′ − 2′ )Π0c (2′ , 2)V (2 − z ′ ) 2! +... (4.29) where we have used Π0c (x, x′ ), the connected density-density correlation function of the non-interacting system, defined in Eq.(4.23). Hence the effective interaction is the solution of the integral equation Veff (x1 − x2 ) = V (x1 − x2 ) +

Z

dz1

Z

dz2 V (x1 − z1 ) Π0c (z1 , z2 )Veff (z2 − x2 ) (4.30)

We will see that in momentum space it is quite easy to solve this equation.

4.3

Feynman Rules in Momentum Space

We will now discuss briefly the Feynman rules in momentum space. Let us begin by recalling the form of the Hamiltonian in momentum space. The non-interacting Hamiltonian H0 ≡ H0 − µN is simply Z d3 p X H0 = E(~p) ψσ† (~p)ψσ (~p) (4.31) (2π~)3 σ where E(~p) is the excitation energy measured from the Fermi energy EF , E(~p) ≡

~p 2 − EF 2m

(4.32)

The interaction term H1 is Z Z Z X 1 d3 p d3 q d3 H1 = Ve (~k) ψσ† (~p+~k)ψν† (~q−~k)ψν (~q)ψσ (~p) 2 (2π~)3 (2π~)3 (2π~)3 σ,ν=↑,↓

(4.33)

14

CHAPTER 4. THE WEAKLY INTERACTING ELECTRON GAS

where Ve (~k) is the Fourier transform of the two-particle interaction potential Z ~ Ve (~k) = d3 x V (~x) e−ik · ~x/~ (4.34) Recall that for the Coulomb interaction in three dimensions Ve (~k) is 4πe2 ~2 Ve (~k) = ε0 | ~p |2

(4.35)

where ε0 is the dielectric constant of the background medium in which the electrons move. Similarly, the Fourier transform (in space and time) of the Feynman propagator Z Z dω d3 p σσ′ ′ ′ i~p · (~x − ~x ′ )/~ e−iω(t − t′ )/~ Gσσ′ (~p, ω) G0 (~x−~x , t−t ) = e 0 2π~ (2π~)3 (4.36) where # " θ(pF − | p~ |) θ(| p~ | −pF ) σσ′ + G0 (~p, ω) = δσσ′ p) ω − E(p) + iδ ω − E(~ − iδ ~ ~ δσσ′ (4.37) ≡ + i sign(| p | −pF ) δ ω − E(p) ~ 2

p ~ − EF > 0 for | ~p |> pF and viceversa, On the other hand, since E(~p) = 2m we can also write the propagators as

δσσ′

′

Gσσ p, ω) = 0 (~

ω−

E(p) ~

+ i sign(ω) δ

≡ δσσ′ G0 (p)

(4.38)

Thus, for ω < 0 (i.e. below the Fermi energy) the unperturbed Feynman propagator G0 (ω, ~p) has poles on the upper half of the complex frequency plane (corresponding to holes) while for ω > 0 (i.e. above the Fermi energy) it has poles on the lower half of the complex frequency plane, as shown in Fig.4.8. There is a pole on the real axis for each momentum ~p (and spin projection) at E(~p)/~. The distance between poles vanishes in the thermodynamic limit: the poles coalesce into a branch cut. This is characteristic of a Fermi system without an energy gap.

15

4.3. FEYNMAN RULES IN MOMENTUM SPACE Im ω

iδ Re ω −iδ

Figure 4.8: Singularities of the Electron Feynman propagator on the complex frequency plane. The “bullets” are the poles for the single-particle state of momentum p~. The Feynman rules in momentum space are the essentially same as in position space except that now we have to ensure that energy and momentum is conserved at every interaction vertex. Thus given any diagram in the position space representation well can always use a Fourier transform to get the momentum space representation. (In what follows we will use the notation p = (ω, ~p)). Thus the rules now are • every fermion line carries energy momentum p • every interaction line carries a momentum transfer q • every fermion contraction is a factor of iG0 (p) • every interaction with momentum transfer q is a factor Ve (q) energy and momentum must be conserved at every vertex. p

q

(a)

(b)

′

Figure 4.9: (a) Gσσ 0 (p); (b) V (q).

16

CHAPTER 4. THE WEAKLY INTERACTING ELECTRON GAS • there is a factor of (−1)F for a diagram with F fermion loops • internal momenta and spin labels must be integrated and summed over • there is the same factor of 1/n! and (−i/~)n to order n in perturbation theory p σ qν

p σ

For example, the Feynman diagram shown in Eq.(4.1)(c), which in momentum space is shown in Fig.4.3, in momentum space represents the following contribution to the propagator   Z −i σσ′ e 2δσσ′ G0 (p)G0 (p)V (0) G0 (q) (4.39) δG(1) (p) = ~ q where

Ve (0) = lim Ve (q)

(4.40)

lim Ve (q) = ∞

(4.41)

q→0

and the overall factor of 2 is due to the spin trace in the internal fermion loop. Note that, although for the case of Coulomb interactions q→0

if we include the effects of the neutralizing positive background charge of the ions, which amounts to to subtract the average charge density from the local electronic density operator ρ(x) (or, equivalently, to normal order the density operator), this term divergent term exactly cancels out. As another example we will consider the one loop correction to the twoparticle scattering process shown in Fig.4.10, which yields the expression:  2 h i2 Z 1 −i ′ ′ δσσ′ δνν ′ 2×2(−1) G0 (p)G0 (p+k)G0 (p )G0 (p −k) Ve (k) G0 (q)G0 (k+q) 2! ~ q (4.42)

4.4. THE DYSON EQUATION AND THE SELF ENERGY

17

Here, one factor of 2 in front is the spin contribution from the internal fermion loop; the second factor of 2 comes from the topologically in-equivalent diagram. σ′

p+k

ν′

p′ − k

ν

p′

q k

k λ

σ

q+k p

Figure 4.10: One loop correction in momentum space

4.4

The Dyson Equation and The Self Energy

The structure of the equation for the effective interaction Veff , eq.(4.30), is quite generic. The quantity we called Π0c is just the leading correction to the polarization function. In momentum space, Eq.(4.30) becomes Veff (k) = V (k) + V (k)Π(k)Veff (k)

(4.43)

In Eq.(4.30), Π(k) is just Π0c (k) to lowest order in perturbation theory. Eq.(4.30) (and Eq.(4.74)) is an example of a Dyson Equation and the quantity Π(k) is a self-energy. In terms of Π(k), the explicit solution of Eq.(4.74) is V (k) Veff (k) = (4.44) 1 − V (k)Π(k) We will shortly analyze the behavior of Veff (k). The (connected) density-density correlation function D00 (p, ω) ≡ Πc (p, ω)(can also be computed within this (RPA) approximation. Indeed we saw that, to zeroth-order in perturbation theory, it is just Πc0 (p, ω). Within RPA it is replaced by the bubble sum: Πc (p, ω) = Π0c (p, ω) + Π0c (p, ω)V (k)Πc (p, ω)

(4.45)

18

CHAPTER 4. THE WEAKLY INTERACTING ELECTRON GAS

or,what is the same Πc (p, ω) =

Π0c (p, ω) 1 − V (k)Πc (p, ω)

(4.46)

This result already tells us, at this lowest order of approximation, what is the induced charge density ρind (p, ω) due to an external perturbation due to a static external charge which interacts with the scalar potential V (p, ω) (the Fourier transform of the Coulomb potential). We find V (p, ω)Π0c (p, ω) 1 − V (p, ω)Π0c (p, ω) (4.47) For a static probe, the potential is independent of time and and hence the frequency has to be set to zero, ω → 0. We will see in the next section what this implies in an explicit computation. The same analysis can be applied to the Fermion Green Function. In momentum space, GF (k) has the perturbative expansion hρind (p, ω)i = Πc (p, ω)V (p, ω) = Π0c (p, ω)Veff (p, ω) =

GF (p) =

=

+

+

+ ... (4.48)

In higher orders in perturbation theory we will find diagrams such as those of Fig.4.11(a) and (b). In Fig.4.11 we see that there are two types of diagrams: (a) diagrams that can be split in two by cutting a single propagator line inside the diagram (carrying the external momentum p), and (b) diagrams that cannot be split in two by cutting a single internal propagator line. The first type of diagrams are said to be one-particle reducible; the second type of diagrams are one-particle irreducible. It is easy to see that summing all the one-particle reducible diagrams of all types lead us to rewriting the perturbation series as =

+

111111 000000 000000 111111 000000 111111 000000 111111 000000 111111

(4.49)

4.4. THE DYSON EQUATION AND THE SELF ENERGY k

q

p

p−q

p

p−q

p

p−k

(a)

q

p

19

k

p

p−q−k p−k (b)

Figure 4.11: One-particle reducible (a) and irreducible (b) Feynman diagrams. which is known as the Dyson equation for the full Feynman propagator. Here the filled “blob”, which we will cal the self energy, is the sum of all one-particle irreducible diagrams (upon amputating their external legs). The Dyson equation which has the explicit form ′

′

′

′ ′

σσ σν νν Gσσ (k) GFν σ (k) F (k) = G0 (k) + G0 (k) Σ

(4.50)

νν ′

where Σ (k) is the self energy operator. The formal solution of the Dyson equation is
νν ′
σσ′ ′ (4.51) = G0 (k)σσ − Σ−1 (k) G−1 (k) In a paramagnetic metal, the ground state is spin unpolarized. Thus G0 and GF must be independent of the spin polarization, and G0 , GF and the self energy Σ have the form ′

Gσσ 0 (k) = δσσ′ G0 (k),

′

Gσσ F (k) = δσσ′ GF (k),

′

Σσσ (k) = δσσ′ Σ(k) (4.52)

where −1 G−1 F (k) = G0 (k) − Σ(k)

(4.53)

Σ(k) = Σ′ (k) + i Σ′′ (k)

(4.54)

The self-energy Σ(k) can be split into its real and imaginary parts

Since GF (k) is the (full) Feynman propagator, the imaginary part of the self energy obeys   ′′ ~ sign Σ (k) = −sign | k | −pF (4.55)

20

CHAPTER 4. THE WEAKLY INTERACTING ELECTRON GAS

Since the free propagator obeys   ~k | −pF G−1 (k) = ω − E (k) + iδ sign | 0 0

(4.56)

~k 2 − EF is the free particle spectrum, the full propagator 2m GF (k) must obey

where E0 (k) =

Re G−1 (k) = ω − E0 (k) − Σ′ (k)



Im G (k) = (δ− | Σ (k) |) sign | ~k | −pF −1

′′



(4.57)

Now, if lim Σ′′ (k) 9 0, then the poles of the full propagator move away from the real axis. The poles of GF (k) are the quasiparticles of the fully interacting system. Hence, if the poles move away from the real axis, this means that the quasiparticle states are not necessarily sharp in energy and have a finite width, or what is equivalent, Σ′′ (k) is the quasiparticle decay rate and Σ′′ (k)−1 is the quasiparticle lifetime. Moreover, the equation for zeros of the real part of GF (k)−1 ω(~k) − E0 (k) − Σ′ (ω(~k), ~k) = 0

(4.58)

define the effective quasiparticle dispersion. In the non-interacting case, for | ~k | close enough to the Fermi momentum pF , and for energies close enough to the Fermi surface, ω → 0, we can use a linearized spectrum E0 (~k) ∼ = vF (|~k| − pF )

(4.59)

where vF is the Fermi velocity ∂E0 ~vF = ∂~k p

(4.60) F

Only a momentum change in the direction normal to the Fermi surface costs energy. The imaginary part of Σ′′ (k) is the width of the quasiparticle state at momentum ~k. However, the quasiparticles are physically meaningful only if they are stable, i..e if they are long lived. Thus, there must be restrictions on the allowed behavior of Σ′′ (k) for the quasiparticles to remain stable even after

4.4. THE DYSON EQUATION AND THE SELF ENERGY

21

~vF ~kF

Figure 4.12: The Fermi surface interactions are taken into account. Physically we expect that a very energetic quasiparticle can decay into a collection of quasiparticles and quasiholes and thus that there should be a finite width at finite frequency ω. However, for the low-energy excitations this cannot be the case since otherwise would would have to conclude that the perturbed electron gas would be completely different than the free Fermi gas. We will see that, in a certain limit that we will discuss later, the interacting Fermi gas can behave much in the same way as the non-interacting Fermi gas (up to important corrections). This is the basis of the Landau Theory of the Fermi Liquid. However we will see that in some cases, such as in one space dimension, this picture breaks down even for infinitesimally weak interactions. This is the case of the 1D Luttinger liquid. Hence, the quasiparticles are stable provided lim Σ′′ (~k, ω) = 0

ω→0

(4.61)

However, since the real part also vanishes, linearly, as the Fermi surface is approached, the quasiparticles can remain sharply defined states only is Σ′′ vanishes faster than linear as ω → 0. If we assume that the frequency dependence of Σ′′ is analytic, then a behavior Σ′′ ∼ ω 2 as ω → 0 and |~k| → pF is consistent with the stability of the quasiparticles. This is the behavior found in the weakly interacting electron gas, and it is the key assumption of the Landau Theory of the Fermi Liquid, which we will discuss shortly. In particular, this arguments also imply that in the asymptotic low energy regime the behavior full Green function is be dominated by the pole. Hence at low energies the full propagator GF must have a singular part Gsing (~k, ω), dominated by the pole, and a regular part GF (~k, ω) = Gsing (~k, ω) + Greg (~k, ω)

(4.62)

22

CHAPTER 4. THE WEAKLY INTERACTING ELECTRON GAS

where Gsing (~k, ω) =

Z(~k, ω)   ~ ω − k · ~vF − pF + i signω δ

(4.63)

which holds asymptotically for ω → 0 and | ~k |→ pF . The quantity lim lim Z(~k, ω) ≡ Z

(4.64)

ω→0 |~k|→pF

is called the residue, which turns out to have the bounds 0 pF and |~k| > pF ; this is the particle-particle channel. In this case the poles are located at Ω=

E(~k + ~q) E(~k) − iδ, −ω + − iδ ~ ~

(4.81)

and both are located below the real axis. Closing the contour on the upper half plane, we find that the integral vanishes (Fig.4.14). 2. |~k + ~q| < pF , |~k| < pF . Again both singularities are on the same side of the real axis, this time above the real axis. Thus closing on the lower half plane, Fig.4.15, the integral is found to be zero. This is the hole-hole channel.

26

CHAPTER 4. THE WEAKLY INTERACTING ELECTRON GAS Im k0

E(~k + ~q)/~ − ω + iδ iδ

E(k)/~ + iδ Re k0

Figure 4.15: Contour used in the hole-hole channel 3. |~k + ~q| > pF , |~k| < pF (case (a)), and |~k + ~q| < pF , |~k| > pF (case (b)); this is the particle-hole channel. Now, in both cases, the two singularities lay on opposite sides of the real axis. The result of the contour integration is the same no matter where we close, but we get two contributions, one from case (a) and the other from case (b). Thus for case (a) we get the result (see Fig.4.16(a)) ( ) ~k) 2πi E( θ(|k + q| − pF )θ(pF − |~k|)Res G0 (k + q)G0 (k), + iδ 2π ~ =

iθ(|~k + ~q| − pF )θ(pF − |~k|) ω+

E(~k)−E(~k+~ q) ~

+ iδ

(4.82) Instead in case (b) (see Fig.4.16(b)) we get ( ) ~k) 2πi E( − θ(|k + q| − pF )θ(pF − |~k|)Res G0 (k + q)G0 (k), − iδ 2π ~ =

−iθ(pF − |~k + ~q|)θ(|~k| − pF ) ω+

E(~k)−E(~k+~ q) ~

− iδ

(4.83)

27

4.5. THE DIELECTRIC FUNCTION

Therefore, we find that Π0c (~q, ω) is given by the integral   Z  3 dk θ(|~k + ~q| − pF )θ(pF − |~k|) θ(pF − |~k + ~q|)θ(|~k| − pF )  2     ~ − Π0c (~q, ω) = ~ ~ (2π~)3  ω + E(~k)−E(~k+~q) + iδ ω + E(k)−E(k+~q) − iδ  ~

~

(4.84)

Upon a change in the integration variables, we get Z 2 d3 k 0 Πc (~q, ω) = θ(|~k + ~q| − pF )θ(pF − |k|) ~ (2π~)3     1 1     − ×  ω + E(k)−E(k+q) + iδ ω − E(k)−E(k+q) − iδ  ~ ~

(4.85)

4.5.1

The Static Limit, ω = 0

In the static limit, ω = 0, the polarization operator Π0c (~q, ω) becomes Π0c (~q, 0)

8m =− 2 ~

Z

For Coulomb interactions

d3 k θ(|~k + ~q| − pF )θ(pF − |~k|) (2π~)3 ~q 2 + 2~k · ~q 4π~2 e2 Ve (|~q|) = ε0 |~q|2

we obtain the (static) dielectric function ε(~q, 0)  1/3 Z 4me2 4 d3 k u(x) ε(~q, 0) = 1 + 2 =1+ rs 2 2 π ε0 |~q| R ~q 2 + 2~k · ~q 9π x

(4.86)

(4.87)

(4.88)

where the integration is restricted to the region R = {|~k| < pF , |~k + ~q| > pF }, and x is the dimensionless variable x = 2p|~qF| (which measures the the momentum transfer in units of the diameter of the Fermi surface). In Eq.(4.88) we introduced rs , the radius of a sphere of volume v = V /N = 1/ρ, measured in units of the (effective) Bohr radius, which in this case is a0 = ~2 ε0 /me2 . It is easy to see that rs satisfies  1/3 a0 9π pF rs = (4.89) ~ 4

28

CHAPTER 4. THE WEAKLY INTERACTING ELECTRON GAS

Im k0

iδ

E(k)/~ + iδ Re k0

−iδ ~ E(k + ~q)/~ − ω − iδ

(a) Im k0

E(~k + ~q)/~ − ω − iδ iδ

Re k0

−iδ E(k)/~ + iδ

(b)

Figure 4.16: Contours used in he particle-hole channel.

29

4.5. THE DIELECTRIC FUNCTION or, alternatively,



e2 ε0 a0 EF





4 =2 9π

2/3

rs2

(4.90)

Hence, rs is a measure of the typical value of the potential energy in units of the Fermi energy. Therefore, the limit of weak interactions is rs ≪ 1, which is also achieved at (very!) high densities. In these regimes the kinetic energy dominates over the potential energy. The dimensionless function u(x) use in Eq.(4.88) is given by   1 + x 1 1 2 u(x) = 1 + (1 − x ) log (4.91) 2 2x 1 − x For q → 0, i.e. x → 0, u → 1 and the static dielectric function becomes ε(~q, 0) ≃ 1 + λ2T F

λ2 |q|2 

(4.92) 1/3

 p 2

(4.93)

| ~r | − e2 d3 q 4πe2 ~2 /ε0 i~q·~r/~ e = e ξT F (2π~)3 q 2 + λ2 ε0 | ~r |

(4.95)

4 4 me2 pF ~2 = ≡ 2 = 2 ξT F π ε0 ~ π

4 9π

rs

F

~

where ξT F is the Thomas-Fermi screening length. This is the (semi-classical) Thomas-Fermi approximation. Within this approximation the effective static interaction potential becomes   4πe2 ~2 4π~2 e2 /ε0 ε0 q 2 VeTF (~q, 0) ≃ (4.94) = λ2 q 2 + λ2 1+ 2 q which in position space is a Yukawa potential:

VTF (~r) =

Z

Thus, at least in this approximation, the interactions have become shortranged. Actually this calculation is incorrect since u(x) has a weak singularity as x → 1 (| ~q |→ 2pF ). Thus, if the Fermi surface is sharp we expect extra

30

CHAPTER 4. THE WEAKLY INTERACTING ELECTRON GAS

contributions to the dielectric function. For large |~r|, the correct asymptotic behavior has the power law form (Friedel, Langer and Vosko) Veff (|~r|) ∝

cos(2pF |~r|) |~r|3

(4.96)

instead of the (much faster) exponential decay law predicted by the ThomasFermi approximation. This result implies that if the Fermi surface is sharp the induced charge density caused by an external static (Coulomb) probe has the oscillatory behavior (with the power law prefactor) given above. This behavior is known as Friedel Oscillations and it is observed in simple metals (such as Mo) in scanning tunneling microscopy (STM) experiments. However, we will see below that that at finite temperatures we will recover exponential screening, albeit with a temperature-dependent screening length. This is correct since at finite temperature the Fermi surface is smeared. At high temperatures the screening length approaches the classical Debye screening length.

4.5.2

Dynamic Behavior

I will not do a full analysis of Π0c (~q, ω), which can be found in standard textbooks (e.g. Fetter and Walecka). We will discuss here its salient physical properties. The two particle Green function in the RPA approximation has the form Ve (q) × external legs (4.97) G(2) c (q) = e 0 (q) 1 − Ve (q)Π c

As usual the poles of the Green function determine the spectrum of elementary excitations. Within this approximation, the poles are determined my the zeros of the denominator of Eq.(4.97). Hence, the spectrum of collective excitations (particle-hole bound states) is determined by the condition: e 0 (q) 1 = Ve (q)Π c

(4.98)

The explicit form of this condition is      X  1 1 − 1 = Ve (q) E(~p) − E(~p + ~q) E(~p) − E(~p + ~q)   |~ p|pF (4.99)

31

4.5. THE DIELECTRIC FUNCTION

e 0 (q) Ve (q)Π c

plasma branch

1

ωpl ωmax

Figure 4.17: Roots of Eq.(4.99)

The roots of this equation determine the particle-hole excitations. e 0c has poles at ω = ±(E(~p) − E(~p + ~q))/~, which is the energy of a Π free particle-hole pair, and the distance between two nearby poles goes to zero as the thermodynamic limit V → ∞. There’s a maximum frequency for such scattering states given by the largest value of (E(~p) − E(~p + ~q))/~ compatible with the restrictions | ~p |< pF and | ~p + ~q |> pF . This happens q~ 2 when |~p| = pF and ~p is parallel to ~q. This implies that ~ωmax = pFm|~q| + 2m . Thus the continuum of particle-hole states has the range 0 ≤ ω ≤ ωmax and it shrinks to zero as ~q → 0. The extra root at ω = ωpl , known as the plasma frequency, represents a collective excitation which survives as ~q → 0. This root lies outside the particle-hole continuum and hence it represents a stable particle-hole bound e 0 (~q, ω) is state. Thus in this frequency regime the polarization operatorΠ c purely real. By keeping the small ~q behavior only we can approximate e 0 (q) = Π c

X

|~ p| PF → • hole (|p| < PF ) • charge ±e, spin

Interacting • quasiparticle |~p| > pF • quasihole |~p| < pF • charge ±e, spin

1 2

1 2

• stable only at low energies (ω → 0) The state of the interacting systems can be parameterized by the actual distribution function n(~p). Let δn(~p) ≡ n(~p) − n0 (~p). For the system to be stable δn(p) must be non-zero only for |~p| ≈ pF and the ground state energy is determined by δn(p). If n0 (p) → n(p) = n0 (p) + δn(p) with δn/n ≪ 1(for all p~ close to the fermi surface), the total energy isE = E0 + δE and we can expand the excitation energy δE in powers of the change of the distribution function δn(~p) as X εp δn(p) + · · · (5.1) δE = • always stable

p

The excitation energy δE should also tell us how much energy does it cost to add an excitation of momentum p~ close to the Fermi surface. Thus,

3

5.1. ADIABATIC CONTINUITY empty

empty

00000000000000000000000000000000 11111111111111111111111111111111 11111111111111111111111111111111 00000000000000000000000000000000 00000000000000000000000000000000 11111111111111111111111111111111 0000000 1111111 00000000000000000000000000000000 11111111111111111111111111111111 0000000 1111111 00000000000000000000000000000000 11111111111111111111111111111111 0000000 1111111 00000000000000000000000000000000 11111111111111111111111111111111 0000000 1111111 00000000000000000000000000000000 11111111111111111111111111111111 0000000 1111111 00000000000000000000000000000000 11111111111111111111111111111111 0000000 1111111 00000000000000000000000000000000 11111111111111111111111111111111 0000000 1111111 00000000000000000000000000000000 11111111111111111111111111111111 0000000 1111111 00000000000000000000000000000000 11111111111111111111111111111111 0000000 1111111 00000000000000000000000000000000 11111111111111111111111111111111 0000000 1111111 00000000000000000000000000000000 11111111111111111111111111111111 0000000 1111111 00000000000000000000000000000000 11111111111111111111111111111111 0000000 1111111 00000000000000000000000000000000 11111111111111111111111111111111 0000000 1111111 00000000000000000000000000000000 11111111111111111111111111111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111

occupied

occupied

pF

pF

Λ

(b) Interacting system

(a) Non-interacting system

Figure 5.2: Fermi surface in a) a non-interacting system and b) in an interacting system. Only the states in the narrow region Λ ≪ pF contribute in the presence of interactions. the quasiparticle energy ε(p) should be given by ε(p) =

δE δn(p)

(5.2)

The difference between the energy of the ground state with N + 1 particles and N particles is (by definition) the chemical potential µ of the system, E(N + 1) − E(N) = µ

(5.3)

Hence, we see that µ = ε(~p), or, what is the same,

with, |~p| = pF

(5.4)

∂E = ε(pF ) (5.5) ∂N The higher order corrections to δE in powers of δn(~p) know about the interactions among quasiparticles. This is the Landau expansion. Is ε(p) independent of the existence of other quasiparticles? If so ε(p) would be independent of δn(p). In that case ε(p) = ε0 (p)! This is obviously not true. Thus, to low orders in δn(~p) we must have terms of the form X 1X f (~p, p~′ )δn(~p)δn(~p′ ) + O((δn(~p))3 ) (5.6) E(δn) = ε(p)δn(~p) + 2 ′ µ=

p ~

p ~,~ p

where we have introduced the Landau parameters, the symmetric function f (~p, ~p′ ) = f (~p′ , ~p).

4

CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID

Near |~p| = pF the Fermi velocity is given by ~vF (p~~) = ∂p~ ε(~p). Hence, we can define an effective mass m∗ =

pF |~vF (pF )|

(5.7)

which is isotropic only if the Fermi surface is isotropic. Hence we conclude that ε(p) , which is defined by ε(p) = has the form ε(p) = ε0 (p) +

X p ~′

δE δn(p)

f (~p, ~p′ )δn(~p′ ) + · · ·

(5.8)

(5.9)

The correction term gives a measure of the change of the quasiparticle energy due to the presence of other quasiparticles. The function f (~p, p~′ ) measures the strength of quasiparticle-quasiparticle interactions. Hence f (~p, p~′ ) is an effective interaction for excitations arbitrarily close to the Fermi surface. What about spin effects? If the system is isotropic and there are no magnetic fields present, the quasiparticle with up spin (↑) has the same energy as the quasiparticle with down spin (↓). Hence ε↑ (~p) = ε↓ (~p) (Note: This relation is changed in the presence of an external magnetic field by the Zeeman effect). Likewise, the interactions between quasiparticles depends only on the relative orientation of the spins σ and σ ′ . The Landau interaction term is modified by spin effects as X p ~,~ p′

f (~p, ~p′ )δn(~p) δn(~p′ ) →

XX p ~,~ p′

fσσ′ (~p, p~′ )δnσ (~p)δnσ (~p′ )

(5.10)

σ,σ′

By symmetry considerations we expect that f↑↑ =f↓↓ ≡ f (S) + f (A)

f↑↓ =f↓↑ ≡ f (S) − f (A)

(5.11)

Thus we can also write the quasiparticle interaction term as the sum of a

5

5.1. ADIABATIC CONTINUITY symmetric and an antisymmetric (or exchange) term XX fσσ′ (p, p′)δnσ (p) δnσ′ (p′ ) = p ~,~ p′ σ,σ′

symmetric →

X p ~,~ p′

antisymmetric →

f (S) (p, p′ )(δn↑ (p) + δn↓ (p))(δn↑ (p′ ) + δn↓ (p′ )) + X p ~,~ p′

f (A) (p, p′ )(δn↑ (p) − δn↓ (p))(δn↑ (p′ ) − δn↓ (p′ )) (5.12)

The density of quasiparticle states at Fermi surface, N(0), is N(0) =

1 X ∂np0~σ 1 X 0 δ(εp~σ − µ) = − V V ∂εp~σ p ~,σ

(5.13)

p ~σ

where we have used the fact that np0~,σ is the fermi function at T = 0 (a step function). Hence, we find m∗ pF (5.14) N(0) = 2 3 π ~ where m∗ is the effective mass. To avoid confusion, note that εp0~σ represents energy of quasiparticles at the Fermi-surface, while ε0 (p) represents the energy of the non-interacting system, as indicated previously. In a general scenario, where all of the quasiparticle spins are not quantized along the same axis, the spin polarization of the Fermi liquid is (in the following equations τ represents Pauli matrices, and α, α represents the matrix indices) XX σi = (τi )αα [n(~p)]αα (5.15) p

αα

Equivalently,

1X (np~)αα 2 α

(5.16)

1X (~τ )αα · [n(~p)]αα 2 αα

(5.17)

1 X X fp~αα,~p ′ α′ α′ (δn(p))αα (δn(p′ ))αα V 2 pp′ ′ ′

(5.18)

n(~p) = and ~σ (~p) = Consequently δ2E =

ααα α

6

CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID

where fp~αα,~p′ α′ α′ = fp~Sp~′ δαα δα′ α′ + fp~Ap~′ ~ταα~τα′ α′

(5.19)

which can also be written more compactly as (S)

(A)

fp~p~′ ≡ fpp′ + fpp′ ~τ · ~τ

(no trace)

(5.20)

For a rotationally invariant system, the interaction functions fp~S,A ,~ p ′ must depend only on the angle θ defined by ~p · ~p ′ cos θ(~p, ~p ) = 2 pF ′

(5.21)

Hence we should be able to use an expansion of the form fp~S,A ,~ p′

=

∞ X

fℓS,A Pℓ (cos θ)

(5.22)

ℓ=0

where

Z 2ℓ + 1 1 1 = dxPℓ (x) (fp~↑,~p ′ ↑ ± fp~↑,~p ′ ↓ ) (5.23) 2 2 0 which leads to the definition of the Landau parameters in terms of angular momentum channels FℓS,A ≡ N(0)fℓS,A (5.24) flS,A

5.2 5.2.1

Equilibrium Properties Specific Heat:

The low temperature specific heat of a Fermi liquid, just as in the case of non-interacting fermions, is linear in T with a coefficient determined by the effective mass m∗ of the quasiparticles at pF . Let’s compute the low temperature entropy, or rather the variation of the quasiparticle entropy (per unit volume) as T → T + δT kB X [nσ (~p) ln nσ (~p) + (1 − nσ (~p)) ln(1 − nσ (~p))] (5.25) S=− V p ~σ

where nσ (~p) is the Fermi-Dirac distribution nσ (~p) =

1 e(εσ (~p)−µ)/kB T

+1

(5.26)

5.2. EQUILIBRIUM PROPERTIES

7

and εσ (~p) is the quasiparticle excitation energy, i.e., δF = εσ (~p) δnσ (~p)

(5.27)

where F (δn(~p) is the free energy. Thus, the variation of the entropy is δS =

1 X (εσ (~p) − µ)δnσ (~p) TV

(5.28)

p ~σ

where

  (εσ (~p) − µ) ∂nσ (~p) − δnσ (~p) = δT + δεσ (~p) − δµ {z } | ∂εσ (~p) T

(5.29)

where the term in braces is due to the quasiparticle interactions. Here, contribution from the first term is δS1 = − Since

∂nσ (~ p) ∂εσ (~ p)

δT 1 X ∂nσ (~p) (εσ (~p) − µ) 2 V ∂εσ (~p) T

(5.30)

p ~σ

is non-zero only within kB T of the Fermi energy, we find

  2 1 ε−µ ∂ 4π δS = − p dε δT 3 (ε−µ)/kB T + 1 dε (2π~) ∂ε e T σ   Z +∞ ∂ 1 2 ≡ −kB N(0) dx x2 δT x+1 ∂x e −∞ XZ

2 dp

(5.31)

Hence we find that the low temperature contribution of the quasiparticles is (to leading order) given by π2 2 S1 = N(0)kB T 3 and that the specific heat is   π2 m∗ pF 2 ∂S 2 = S1 = N(0)kB T = k T Cv = T ∂T V 3 3~3 B

(5.32)

(5.33)

8

CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID

We now introduce the Fermi temperature TF =

εF p2F ≡ ∗ 2m kB kB

(5.34)

as the Fermi energy in temperature units, in terms of which the specific heat becomes T π2 T 0 π2 = C (5.35) CV = nkB 2 TF 3 TF V where CV0 = 23 nkB is the specific heat of a classical ideal gas, and n is the particle density. Using these results we find that the low temperature correction to the Free Energy, F = E − T S, is δF ≈ −SδT (to lowest order), i.e., F ≈ E0 −

π2 T2 nkB 4 TF

(5.36)

where E0 is the ground state energy. The chemical potential µ is (note that TF is a function of m∗ and hence a function of n) µ(n, T ) = −



∂F ∂n



T

π2 = µ(n, 0) − kB 4



1 n ∂m∗ + ∗ 3 m ∂n



T2 TF

(5.37)

Let us now compute the compressibility: κ=−

1 ∂n 1 ∂V = 2 V ∂P n ∂µ

(5.38)

where P is the pressure. At T = 0, δnσ (~p) =

∂nσ (~p) (δεσ (~p) − δµ) ∂εσ (~p)

(5.39)

The quasiparticle energy εσ (~p) depends on µ only through its dependence on and δnσ′ (~p′ ) (i.e., quasiparticle interactions, see Eq. 5.9). As T → 0 both ∂n ∂ε δnσ (p) vanish unless all momenta are at the Fermi-surface. δεσ (p) = f0S

1 X δnσ′ (~p′ ) ≡ f0S δn V ′ ′ σ ,~ p

(5.40)

9

5.3. SPIN SUSCEPTIBILITY where f0S is Landau parameter with with l = 0. Hence, we have δnσ (~p) = and δn =

∂nσ (~p) S (f δn − δµ) ∂εσ (~p) 0

1 X ∂nσ (~p) S 1 X δnσ (~p) = (f δn − δµ) V V ∂εσ (~p) 0

(5.42)

∂nσ (~p) −→ −δ(|p| − pF ) for T → 0 ∂εσ (~p)

(5.43)

δn = −N(0)(f0S δn − δµ)

(5.44)

δn[1 + N(0)f0S ] = N(0)δµ

(5.45)

σ,~ p

Similarly,

(5.41)

σ,~ p

Thus, and

using the expression for the (s-wave) symmetric (singlet) Landau parameter, F0S = N(0)f0S

(5.46)

we can write

N(0) ∂n = ∂µ 1 + F0S which leads to an expression for the compressibility κ: κ=

1 N(0) n2 1 + F0S

(5.47)

(5.48)

which includes the Fermi liquid correction expressed in terms of the Landau parameter F0S .

5.3

Spin Susceptibility

We will now determine the (spin) magnetic susceptibility of a Fermi liquid. Thus we need to consider its response to an external magnetic field. Here we will be interested in the effect of the Zeeman coupling, which causes the quasiparticle energy to change by an amount that depends on the spin polarization: 1 (5.49) − ~γσz H 2

10

CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID

where γ is the gyromagnetic ratio, σz is the diagonal Pauli matrix, and H is the external (uniform) magnetic field. By taking into account also the change caused to the distribution functions we find 1 1 X δεp~,σ = − ~γσz H + fσ,σ′ (~p, ~p′ )δnp~′ ,σ′ (5.50) 2 V ′ ′ p ~ ,σ

where, as before,

∂np~,σ (δεp~,σ − δµ) (5.51) ∂εp~,σ The chemical potential is a scalar (and time reversal invariant ) quantity and as such it cannot have a linear variation with the magnetic field. Hence the only possible dependence of µ with H must be an even power and (at least) of order H 2 . Hence it does not contribute to the magnetic susceptibility (within linear response). We will neglect this contribution. Hence, δnp~,σ ∝ δεp~,σ , are independent of the direction of the momentum p~, and have opposite sign for ↑ and ↓ quasiparticles. Since δnp~,σ 6= 0 only is ~p is on the Fermi surface (which we will assume to be isotropic), we find 1 X fσ,σ′ (~p, p~′ )δnp~′ ,σ′ = 2f0A δnσ = σz f0A (δn↑ − δn↓ ) (5.52) V ′ ′ δnp~,σ =

p ~ ,σ

where δσ is the change in the total number of particles (per unit volume) with spin σ. Hence,   1 1 A ~γσz H − 2f0 δnσ (5.53) δnσ = N(0) 2 2 The net spin polarization is

~ N(0)H γ 2 1 + F0A

(5.54)


2 ~ γ ~2 N(0) H M =γ 2 1 + F0A

(5.55)

δn↑ − δn↓ = and the total magnetization M is

We can thus identify the spin susceptibility χ with χ=

~2 γ 2 N(0) 4 1 + F0S

(5.56)

which is the (Pauli) spin susceptibility of a free Fermi gas with mass m∗ , with the Fermi liquid correction.

11

5.4. EFFECTIVE MASS AND GALILEAN INVARIANCE

5.4

Effective mass and Galilean Invariance

In Galilean invariant systems, there is a simple relation between m∗ , the bare mass m and the Landau Fermi liquid parameter F1S , given by m∗ 1 (S) = 1 + F1 m 3

(5.57)

To see how this comes about we will consider a Galilean transformation to a frame at speed ~v . The Hamiltonian of the system transforms as follows 1 H → H ′ = H − P~ · ~v + M~v 2 2

(5.58)

where P~ is the total momentum operator in the laboratory frame and M = Nm is the total mass of the system. Hence the transformed total energy and total momentum are 1 E ′ = E − P~ · v + M~v 2 2 ′ ~ ~ P = P − M~v

(5.59)

Consider the change in energy due to adding a quasiparticle of momentum p~ in the lab frame. The total mass changes as M → M + m where m is the bare mass. The addition of one quasiparticle involves the addition of one bare particle. In lab frame the momentum increases by ~p and the energy by εσ (~p). In the moving frame the momentum increases by ~p − m~v

(5.60)

1 εp~ − ~p · ~v + m~v 2 2

(5.61)

and the energy increases by

Therefore the quasiparticle energy in the moving frame is given by 1 εp′~−m~v = εp~ − p~ · ~v + m~v 2 2 1 ′ ⇒ εp~ = εp~+m~v − p~ · ~v − m~v 2 2

(5.62)

12

CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID

which is a consequence of Galilean invariance. Expanding to order ~v, and using the definition of the effective mass m∗ , we have:   m − m∗ ′ p~ · ~v (5.63) εp~−m~v ≈ εp~ + m∗ From the moving frame the ground state looks like a Fermi surface centered at p~ = −m~v , hence the occupation numbers change as follows np′~ = np0~+m~v = np0~ + m~v · ∂p~ np0~ + · · ·

(5.64)

where np0~+m~v refers to the lab frame. The quasiparticle energy in the moving frame is εp′~ = εp~ {np′~′ } = εp~{np0~+m~v }

(5.65)

Note that this is valid for one-component systems only. We have ∂p~′ np~′ = ∂p~′ εp~′ ⇒ ε′p = εp + = εp −

∂n0p′ ∂εp′ 1 X

V

p′ σ ′

S fpp v· ′ m~

p~′ ∂n0p′ m∗ ∂εp′

F1S m p~ · ~v 3 m∗

(5.66)

For |~p| = pF , the Fermi momentum, we have m − m∗ m F1S = − m∗ m∗ 3 ∗ m 1 (S) ⇒ = 1 + F1 m 3

(5.67) (S)

This implies that the relative deviation of m∗ from m is determined by F1 .

5.5

Thermodynamic Stability

The ground state should be a minimum of the (Gibbs) free energy which implies that there should be restrictions of the Landau parameters. Consider

13

5.5. THERMODYNAMIC STABILITY

a distortion of the Fermi Surface characterized by a direction dependent Fermi momentum pF (θ).2 nσ (~p) = θ(pF (θ) − |~p|) For a stable system, the thermodynamic potential a minimum. Therefore the change in the (Gibbs) distortion is 1 X 0 (E − µn) − (E − µn)0 = (ε − µ)δnσ (p) + V p,σ p

(5.68) G = E − µn must be free energy due to the 1 X fpσ,p′ σ′ δnσ (p)δnσ′ (p′ ) 2V 2 ′ pp σσ′

(5.69) where δnσ (p) = nσ (p) − n0σ (p)

1 ∂ = δpF δ(pF − |p|) − (δpF )2 δ(pF − p) 2 ∂p

(5.70)

where the change of the Fermi momentum is δpF = pF (θ) − p0F . The first term in Eq. 5.67 is XZ 1 1 1 1 0 2 εpσ (εp − µ)δnσ (p) = N(0)vF d(cos θ) (δpF (θ, σ))2 (5.71) V 4 2 −1 σ where

vF δpF (θ) ≡

∞ X

vℓ,σ Pℓ (cos θ)

(5.72)

ℓ=0

The second term in Eq. 5.67 is X Z 1 d cos θ Z 1 d cos θ′ 1 2 (N(0)vF ) fpσ,p′ σ′ δpF (θ, σ)δpF (θ′ , σ ′ ) 8 2 2 −1 σσ′ −1

(5.73)

which implies that

" ! (S) F N(0) (vℓ↑ + vℓ↓ )2 1 + ℓ δE − µδn = 8(2ℓ + 1) 2ℓ + 1 ℓ=0 !# (A) Fℓ 2 +(vℓ↑ − vℓ↓ ) 1 + 2ℓ + 1 ∞ X

2

(5.74)

For simplicity we assume that the distorted Fermi surface has azimuthal symmetry.

14

CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID

The Fermi liquid is stable under the deformation if δE − µδn > 0

(5.75)

which requires that FℓS,A 1+ ≥0 2ℓ + 1 FℓS,A ≥ −(2ℓ + 1) Notice that for ℓ = 1

(Pormeranchuk)

  1 (S) ≥0 N(0) 1 + Fℓ 3

(5.76)

(5.77)

is always satisfied for a Galilean invariant one-component system. What happens if a Pomeranchuk inequality is violated? Clearly if this happens the system will gain energy by distorting the Fermi surface. Thus, the ground state of a system that violates the Pomeranchuk bound has a broken rotational invariance. The simplest example of such a state in the nematic Fermi fluid in which the symmetry is broken in the quadrupolar (ℓ = 2 or d-wave) channel.3

5.6

Non-Equilibrium Properties

In practice we are also interested in dynamical effects, involving the propagation of excitations. Thus we need to consider systems slightly away from thermal equilibrium and slightly inhomogeneous. We wish to generalize the previous discussion to this case and to define position and time dependent distributions nσ~p (~r, t). Clearly there is a problem with the uncertainty principle since we cannot define both p~ and ~r with arbitrary precision. At temperature T the momentum fluctuates with a characteristic value ∆p ∼ kvBFT . If we wish to define localized quasiparticles a typical length λ, we must have λ∆p ≫ ~ for the “classical” picture to work, which implies that λ≫ 3

~ ~vF ∼ ∆p kB T

(5.78)

To stabilize this state one needs to consider contributions to the Gibbs free energy at orders higher than (δpF (θ))2 .

15

5.6. NON-EQUILIBRIUM PROPERTIES ~q

σ′

σ

p~

Figure 5.3: A particle-pair with relative momentum ~q and spin polarizations σ and σ ′ , on the FS at p~. As T → 0 only macroscopic excitations can be described by a classical picture (λ → ∞ as T1 ). In general we will have to use a Wigner distribution function W (~r1 σ1 , ~r2 σ2 ; t), i.e., the amplitude for removing a particle at ~r1 with spin σ1 at time t and at the same time to add a particle at ~r2 with spin σ2 . The Wigner function is defined as Z Z i d 3 p2 d 3 p1 e ~ (~p1 ·~r1 −~p2 ·~r2 ) ha†p2 σ2 (t)ap1 σ1 (t)i W (~r1 σ1 , ~r2 σ2 ; t) = 3 3 (2π~) (2π~) (5.79) Define   Z ~r′ ′ ~r′ 3 ′ − ~i p ~·~ r′ [np~ (~r, t)]σσ′ = d ~r e W ~r + , σ; ~r − , σ ; t 2 2 (5.80) Z 3 dq − ~i q~·~ r † e ≡ hap~+ ~q ,σ′ (t)ap~− ~q ,σ (t)i 2 (2π~)3 2 where |F Si is the Filled Fermi Sea or the ground state against which the expectation values are evaluated, and ap~+ ~q ,σ′ (t)ap~− ~q ,σ (t)|F Si is a particle2 2 hole pair with relative momentum ~q localized at ~p on the Fermi Surface. Clearly a smooth distortion of the Fermi Sea requires a large number of such pairs leading to coherent states of particle-hole pairs. (We’ll come back to this later). The quasi-particle density is X Z d3 p X [n (~ r , t)] = W (~rσ, ~rσ) (5.81) p ~ σσ (2π~)3 σ σ

16

CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID

and the number of quasi-particles with momentum p~ is Z XZ XZ ~ r1 −~ r2 3 3 d r[np (~r, t)]σσ = d r1 d3 r2 e−i~p· ~ W (~r1 σ, ~r2 σ). σ

(5.82)

σ

For λ ≫ k~vBFT , note that [np (r, t)]σσ′ becomes a classical distribution. If the system is inhomogeneous, the total energy E(t) may vary with time. We can still define εσ~p (~r, t) as the quasiparticle energy at position ~r Z Z XZ d3 p 3 3 Ep~σ (~r, t)δnpσ (r, t) (5.83) δE(t) = d rδE(~r, t) = dr (2π~)3 σ where δE(~r, t) is the local energy density. We have Z XZ d 3 p′ 3 ′ dr δEp~σ (~r, t) = f~σ,~p′ σ′ (~r, ~r′ , t)δnp~′ σ′ (~r′ , t) 3 p (2π~) σ′

(5.84)

If the system is neutral the interactions are typically (assumed to be!) short-ranged and vary only over microscopic scales of the order of p~F . Therefore, we can replace fp~σ,~p′ σ′ (~r, ~r′ , t) by a local form. If there are Coulomb forces e2 δ(t − t′ ) + fp~σ,~p′ σ′ δ(r − r ′ ) (5.85) fp~σ,~p′ σ′ (~r, ~r′ , t) ≈ |r − r ′ |

However many of these assumptions (concerning the existence and stability of quasiparticles) fails for transverse interactions (mediated by gauge fields) and in one-dimension. We will come back to this problem later.

5.7

Kinetic Equation

We now turn to the problem of the evolution (i.e. dynamics) of the quasiparticle disturbances. We will use Landau’s “quantum” kinetic theory. We begin by looking at the regime in which δnp~σ (~r, t) can be regarded as a classical distribution where it should obey a kinetic equation, i.e. the Boltzmann equation. As usual this equation is simply the continuity equation for δ~np~,σ (~r, t) and embodies the condition of local charge conservation in the fluid. In the absence of collisions between quasiparticles their number must be constant. Hence, d δnσ~p (~r, t) = 0 (5.86) dt

17

5.7. KINETIC EQUATION This implies that ∂ ∂ ∂ δnp~ (~r, t) + · (~vp~ δnp~ (~r, t)) + · (f~p~ (~r, t)δnp~ (r, t)) = 0 ∂t ∂~r ∂~p

(5.87)

The quasiparticle group velocity in space is ~vp~ (~nt) =

∂ εp~(~r, t) ∂~p

(5.88)

The rate of change of quasiparticle momentum (the force) is d~p ∂ εp~(~r, t) ≡ f~p~ (~r, t) = − ∂~r dt

(5.89)

Hence we obtain Landau’s kinetic equation ∂ δnp~ − {εp~, δnp~ }PB = I[δnp′ ] ∂t

(5.90)

where I[δnp′ ] is the collision integral, and P B denotes the Poisson bracket {εp , δnp }P B =

∂ ∂εp ∂δnp~ ∂ εp · δnp − · ∂~r ∂~p ∂~p ∂~r

(5.91)

Landau’s kinetic equation differs from the Boltzmann equation in that 1. εp can be a function of ~r, t 2.

∂ ε~ ∂~ r p

includes effective field contributions.

For example, if the system interacts with an external probe of the form of a R potential U(~r, t), then the total energy is increased by d~r · U(~r, t)δn(~r, t). Therefore Z ∂ ∂ ∂ d 3 p′ εp (~r, t) = U(~r, t) + fp~p~′ δnp~′ (r, t). (5.92) 3 ∂~r ∂~r (2π~) ∂~r The first term on the right hand side in the above equation is present in dilute gases, wheras the second term arises from self-consistency condition as effects of other quasiparticles. In the quantum case one needs to use Wigner functions, giving rise to the quantum mechanical version of the kinetic equation. In Landau’s approach

18

CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID

1 multiplied by commutators, proone replaces the Poisson Brackets by i~ ducing the quantum mechanical equations of motion (see Baym and Pethick, pages 19-20). Near equilibrium we can linearize the transport equation, which then becomes the Landau-Silin Equation   ∂n0p ∂δnp ∂ δnp − + vp~ · δεp = I[np′ ] (5.93) ∂t ∂~r ∂ε

where ~vp and tion is

∂n0p ∂εp

are equilibrium functions. The Fourier transformed equa-

(ω − ~q · ~vp )δnp~(~q, ω) − ~q · ~vp

5.8

∂n0p δεp (~q, ω) = iI[np′ ] ∂εp

(5.94)

Conservation Laws

Let us first define n(r, t) ≡

XZ σ

d3 p np~σ (~r, t) (2π~)3

(5.95)

By integrating the Landau equation over p~ (and σ) we get ! ! Z Z Z X X ∂ XZ ∂ ∂ X npσ + · ~vp npσ + (f~p np ) = Ip [np ] ∂t ∂~ r ∂~ p p p p p σ σ σ σ

(5.96)

The number of quasiparticles is conserved upon collisions, therefore XZ I[np ] = 0 σ

p

(5.97) ∂ · [fp np ] = 0 p ∂p P R Let us define the current ~j(r, t) ≡ ~v n~σ . The continuity equation, σ p ~ p p which is just charge conservation, is Z

where ~j =

XZ σ

∂n(~r, t) ~ ~ + ∇ · j(~r, t) = 0 ∂t

(5.98)

XZ

(5.99)

~vp~ np~σ = p

σ

p

∇p~ εpσ (r, t)np~σ (r, t)

19

5.8. CONSERVATION LAWS

On linearizing around equilibrium we get an expression for the current density (superscript zero denotes equilibrium quantities) XZ ~j = (5.100) (∇p ε0pσ δnσp + ∇p δεp n0p ) p

σ

Since

δnpσ = δnpσ −

∂n0pσ δεpσ ∂εpσ

which allows us to write the current as i XZ h 0 0 ~ ~ ~j = ∇p εpσ δnpσ − ∇p np δεp σ

∂n0pσ ~ p ε0pσ ∇ ∂εpσ

(5.103)

~ p ε0pσ )δnpσ (∇

(5.104)

fpσ,p′ σ′ δnp′ σ′ (r, t)

(5.105)

~ p n0pσ = ∇ the current now becomes ~j =

XZ σ

δεpσ =

XZ σ

δnpσ

(5.102)

p

Using that

Similarly we can write

(5.101)

p

p′

Z ∂n0pσ X fpσ,p′ σ′ δnp′ σ′ = δnpσ − ∂εpσ ′ p′

(5.106)

σ

I’ll keep only the Fermi surface contribution, and find that the current takes the form   XZ 1 (S) 0 ~ p ε )δnpσ ~j(r, t) = (∇ r , t) 1 + F1 ~ (~ p 3 p σ (5.107)   Z 1 + 31 F1S X = p~ δnp~σ (r, t) m∗ p σ Galilean invariance implies 1 +

1 3

(S)

F1

X ~j = 1 m σ

Z

=

p

m∗ . m

Hence we have

p~δnp~σ =

~ P m

(5.108)

20

CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID

where the last ratio is momentum density over the bare mass. Otherwise, the general result is   1 + 31 F1S ~ ~j = P (5.109) m∗

5.8.1

Momentum Conservation

The local momentum density ~g (~r, t) is given by XZ ~g (r, t) = ~pnp~σ (r, t)

(5.110)

p

σ

It obeys the local conservation law X ∂~gi + ∇j Tij + ∂t σ where Tij =

XZ σ

Z

pi p

p

∂εpσ npσ = 0 ∂ri

∂εpσ npσ ∂pj

(5.111)

(5.112)

is (almost!) the stress tensor of the Fermi fluid. Using that ∂ ∂npσ ∂εpσ npσ = (εpσ npσ ) − εpσ ∂ri ∂ri ∂ri and

XZ σ

p

εpσ

∂ ∂npσ = E − n(~r, t)∇i U(~r, t) ∂ri ∂ri

(5.113)

(5.114)

we can define the stress tensor Πij Πij = Tij + δij

XZ σ

p

εpσ npσ − E

!

(5.115)

which implies that ∂gi (~r, t) + ∇j Πij (~r, t) + n(~r, t)∇i U(~r, t) = 0 ∂t

(5.116)

21

5.9. COLLECTIVE MODES: ZERO SOUND

5.8.2

Energy Conservation

Multiplying by εpσ and XZ σ

P R σ

p

we obtain

∂npσ ~ X +∇· εpσ ∂t p σ

Z

(∇p εpσ ) εpσ npσ = 0

We can now define the energy current density XZ ~jE = (∇p εpσ )(εpσ − U)npσ σ

(5.117)

p

(5.118)

p

which obeys the energy conservation equation ∂ ~ · ~jE = −~j · ∇U ~ (E − Un) + ∇ ∂t

5.9

(5.119)

Collective modes: Zero sound

We will now look at the solutions of the Landau-Silin kinetic equation, Eq.(5.91), in the limit T → 0 in which the collision integral can be neglected. By linearizing this equation we obtain 

 ∂n0p ∂ ~ p (r, t) = 0 ~ ~vp · ∇δε + ~vp · ∇ δnp (r, t) − ∂t ∂εp

with δεp (r, t) = U(r, t) +

Z

fpp′ δnp′ (r, t)

(5.120)

(5.121)

p′

We now go to its Fourier transform and assume that the external potential is monocromatic U(r, t) ≡ Uei(~q·~r−ωt) (5.122) and δnp (r, t) = δnp (q, ω)ei(~q·~r−ωt) we obtain

∂n0p (ω − ~q · ~vp )δnp + ~q · ~vp (U + ∂εp

Z

p′

fpp′ δnp′ ) = 0

(5.123)

(5.124)

22

CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID θ ~q p~

Figure 5.4: A particle-hole fluctuation with momentum propagating with ~q near a point ~p on the Fermi surface. Let us write δnp in terms of νp defined by δnp ≡ −

∂n0p νp ∂εp

(5.125)

where we have assumed that only the Fermi surface matters. Within this notation we find that the linearized kinetic equation takes the form Z ∂np0~′ ~q · ~vp~ ~q · ~vp~ νp~ + f (~p, ~p′ ) νp~′ = U (5.126) ω − ~q · ~vp~ p~′ ∂εp~′ ω − ~q · ~vp with ~p on the Fermi surface. We will now make use of the azimuthal symmetry to expand the fluctuation in partial waves ∞ X νp~ = νℓ Pℓ (cos θ) (5.127) ℓ=0

and write

Z

p′

f (~p, p~′ )

∞ X ∂np0~′ 1 (S) νp~′ = − Fℓ Pℓ (cos θ)νℓ ∂εp~′ 2ℓ + 1 ℓ=0

Let us now define the dimensionless parameter s ω s= qvF

(5.128)

(5.129)

23

5.9. COLLECTIVE MODES: ZERO SOUND and

1 Ωℓℓ′ (s) = Ωℓ′ ℓ (s) = 2

Z

1

dxPℓ (x) −1

The Landau-Silin Equation now takes the form

x Pℓ′ (x) x−s

(5.130)

∞

X νℓ′ νℓ + Ωℓℓ′ (s)Fℓs′ ′ = −Ωℓ0 (s)U 2ℓ + 1 2ℓ + 1 ′

(5.131)

ℓ =0

This equation has solutions of the form νℓ (s). Let us consider first the s-wave channel, ℓ = 0. In this channel we find s s − 1 s s − 1 π Ω00 (s) = 1 + ln = 1 + ln (5.132) + i s θ(1 − |s|) 2 s+1 2 s+1 2

and similar expressions in the other channels. For example, if we assume that the only non-vanishing Landau parameter (s) is in the s-wave channel, F0s 6= 0 and Fℓ = 0(ℓ ≥ 1), we find the simple equation (s) ν0 (s) + Ω00 (s)F0 ν0 (s) = −Ω00 (s)U (5.133) whose solution is ν0 (s) = −

Ω00 (s)U

(5.134)

(s)

1 + F0 Ω00

The equation for the other angular momentum modes, with ℓ ≥ 1, is νℓ (s) + Ωℓ0 (s)F0 ν0 = −Ωℓ0 (s)U 2ℓ + 1

(5.135)

(s)

νℓ Ωℓ0 F0 (s)Ω00 U − = −Ωℓ0 (s)U 2ℓ + 1 1 + F0s Ω00 (s) (s)

(s)

 

(s)

(5.136)  

  νℓ Ω U[F  Ω00 (s) − 1 −  F0 Ω00 (s)] = ℓ0 0 (s) 2ℓ + 1 1 + F0 Ω00 (s)

(5.137)

νℓ Ωℓ0 (s) = ν0 (s) 2ℓ + 1 Ω00 (s)

(5.138)

Hence,

The equation ν0 (s) = −

Ω00 (s)U 1 + F0s Ω00 (s)

(5.139)

24

CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID zero sound

ω

qvF

particle-hole continuum

~q Figure 5.5: Spectrum of collective modes has poles at the zeros, s0 , of 1 + F0s Ω00 (s0 ) = 0

(5.140)

We have the following regimes • For 0 ≤ s < 1, Ω00 (s) is complex. This solution corresponds to the particle-hole continuum. • For 1 ≤ s < ∞, Ω00 (s) is a real and monotonically increasing function of s. (S)

In particular, in the latter case we find that if F0 > 0, then there is a simple pole with s0 > 1  1 + e− F10 , for F0 ≪ 1 ω s0 = = q (s) (5.141)  F0 , |~q|vF for F0 ≫ 1 3

This solution corresponds to an (undamped) sound mode with dispersion ω = |~q|vF s0 and a sound velocity c0 = vF s0 . This collective mode is known as zero sound. Notice that the edge of the particle-hole continuum is at ω = qvF . (See Fig.5.5).

5.10

The Quasiparticle Lifetime

The Landau interactions can in principle give rise to a finite lifetime τ for the quasiparticles. However, if the lifetime remains finite as ω → µ = EF ⇒

25

5.10. THE QUASIPARTICLE LIFETIME hole

particle

f

particle Figure 5.6: A process that leads to a finite quasiparticle lifetime; f denotes a Landau interaction. the whole theory breaks down. Thus, stability of the Fermi liquid requires that the lifetime to diverge , τ → ∞ (at T = 0) as ω → µ. Similarly, if T > 0 ⇒ the lifetime must also diverge as T → 0(ω = µ). In principle the lifetime τ is a function of ω − µ and T . The rate of decay is τ1 = Γ(ω − µ, T ). We can compute this function for |ω − µ| ≪ µ and T ≪ µ(µ = εF = kB TF ). In terms of the Green function the decay rate shows up as an imaginary part of the self energy, namely 1 1 = ω − (ε0 (~p) − µ) − Σ(~p, ω) − Σ(~p, ω) 1 = (5.142) ω − (ε0 (~p) − µ + ReΣ(~p, ω)) − iImΣ(~p, ω)

G(~p, ω) =

G−1 p, ω) 0 (~

where sgn(ℑΣ(p, ω)) = sgn(ω − µ)

Hence the rate is given by

Γ(~p, ω) = ImΣ(~p, ω) =

at |~p| = pF

(5.143)

1 τ

(5.144)

and in general it is a function of both p~ and ω. In terms of diagrams, the lifetime arises because there is a finite amplitude for a process with a quasiparticle at (~p, ω) in the initial state and a quasiparticle (with some momentum and frequency) and some particle-hole pairs in the finite state (see Fig.5.6). The amplitude is determined by the Landau interaction parameters f that we defined above. How does this process enter in the computation of the self-energy? There is a term in perturbation theory of the form shown in Fig. 5.7. Alternately,

26

CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID

Figure 5.7: Feynman diagram with a contribution to ImΣ. the imaginary part comes from effects of the collision integral (see Baym and Pethick p. 87) and is given in terms of the t-matrix: 1 = τp~

Z

Z d3 q d 3 p′ (2π)3 (2π)3 2π|hp − q, p′ + q|t|p, p′ i|2 δ(ε(p) + ε(p′ ) − ε(p − q) − ε(p′ − q)) ×[n0 (p′ )(1 − n0 (p′ + q))(1 − n0 (p − q)) + (1 − n0 (p′ ))n0 (p′ + q)n0 (p′ − q)] (5.145)

which follows from using Fermi’s Golden Rule. The first term in the expression in brackets in Eq.(5.143) represents the rate at which quasiparticles are scattered into new unoccupied states while the second term represents the blocking of such processes due to occupied states. The t-matrix amplitude hp − q, p′ + q|t|p, p′ i is represented by summing up particle-particle (or particle-hole) ladder diagrams, scattering processes of the type shown in Fig.5.8, given by the solution of the Bethe-salpeter Equation (see Baym and Pethick, p. 77): ~ p′′ n0′′ ~q · ∇ p fp′′ ,p′ tp~p~′ (q, ω + iη) = fp,p′ − tp′′ p′ (q, ω + iη) ω + iη − ~q · ~vp′′ p′′ 6=p′ X

(5.146)

The matrix elements of the t-matrix can also be split into a singlet tS and triplet tA channels, and further be expanded in angular momentum components, tS,A ℓ : X tpp′ (q, 0) = tℓ (q, 0)Pℓ (cos θ) (5.147) ℓ

27

5.10. THE QUASIPARTICLE LIFETIME whose coefficients are given by tSℓ = tA = ℓ

fℓS 1+

|q| ≪ pF

(5.148)

|q| ≪ pF

(5.149)

1

Fℓs 2ℓ+1 fℓA FℓA + 2ℓ+1

After some algebra on finds that the decay rate at finite temperature T and frequency ε at zero momentum transfer (i.e. on the Fermi surface) is given by 1 π(N(0)t(0))2 ≈ [(ε − µ)2 + (πkB T )2 ] + · · · (5.150) τp 8πvF (p2F /qc ) (qc ∼ Λ is a momentum cutoff) which satisfies Landau’s assumptions. The Landau picture we discussed works very well in neutral Fermi fluids (such as the normal phase of 3 He) and in most (simple) metals. However we will see that it fails in a number of important situations. In particular it fails in one dimension (for any value of the interaction coupling constants) and also near a quantum phase transition. It also fails in systems with strong correlation.

Figure 5.8: Feynman diagram that contributes to tp~p~′ (q, ω + iη) in the BetheSalpeter Equation (5.144).

Chapter 6 Theory of the Luttinger Liquid 6.1

One-dimensional Fermi systems

We will now consider the case of one-dimensional (1D) Fermi systems where the Landau theory fails. The way it fails is quite instructive as it reveals that in 1D these systems are generally at a (quantum) critical point, and it will also teach as valuable lessons on quantum criticality. It will also turn out that the problem of 1D Fermi systems is closely related to the problem of magnetism in 1D, quantum spin chains. One-dimensional (and quasi-one-dimensional) systems of fermions occur in several experimentally accessible systems. The simplest one to visualize is a quantum wire. A quantum wire is a system of electrons in a semiconductor, typically a GaAs-AlAs heterostructure built by molecular beam epitaxy (MBE), in which the electronic motion is laterally confined along two directions, but not along the third. An example of such a channel of length L and width d (here shown as a 2D system) is seen in Fig.6.1a. Systems of this type can be made with high degree of purity with very long (elastic) mean free paths, often tens of microns or even longer. The resulting electronic system is a one-dimensional electron gas (1DEG). In addition to quantum wires, 1DEGs also arise naturally in carbon nanotubes, where they are typically multicomponent (with the number of components being determined by the diameter of the nanotube.) Other one-dimensional Fermi systems include the edge states of twodimensional electron gases (2DEG) in large magnetic fields in the regime in which the quantum Hall effects are seen. (We will discuss this problem 1

2

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

later on.) This case is rather special as these edge states can only propagate in one direction, determined by the sign of the perpendicular magnetic field. Other (quasi) one-dimensional systems occur in organic compounds, such as TTFTCNQ, some Betchgaard salts (commonly called as the “BEDT’s” and the “ET’s”) and TMSF’s. There are also quasi-1D chalcogenide materials, e.g. NbSe3 (and others) as well as complex oxides. The some of the oxides, e.g. Sr14−x Cax Cu24 O41 which can be regarded as a set of weakly coupled ladders (as opposed to chains). Quasi-1D Fermi systems are often used to describe complex ordered states in 2D strongly correlated systems. A typical example are the stripe phases of the copper oxide high Tc superconductors, such as La2−x Srx CuO4 , La2−x Bax CuO4 , and others. ε ε2

L ε1 ~2 2md2

d

ε0 (a)

(b)

Figure 6.1: a) A long quantum wire of length L and transverse width d (L ≫ d): a channel for the electron fluid. Only 2D case for simplicity. b) Square well spectrum of the transverse quantum single particle states confined by the finite width d of the wire. We will consider first the conceptually simpler example of the quantum wire. We will assume that the electron density is such that the Fermi energy lies below the energy of the first excited state. The result is that the singleparticle states with momenta in the range −pF < p < pF are occupied and the states outside this range are empty. Thus the Fermi “surface” of this system reduces to two Fermi points at ±pF . We will assume that the wire is long enough, L ≫ d, so that the single particle states fill up densely the momentum axis, and that the density is high enough so that ∆p = 2π~/L ≪ pF . On the other hand, we will assume that the wire is narrow enough so that the next band of (excited) states can effectively be neglected,

3

6.1. ONE-DIMENSIONAL FERMI SYSTEMS

εF ≪ ~2 /(2md2 ).∗ In practice this means that the following inequality holds d L ≫1≫ d λF

(6.1)

where λF = ~/pF is the Fermi wavelength. ε ε1 (p) ε0 (p)

∆E =

~2 2md2

2π~ v L F

εF p −pF

pF ∆p =

2π~ L

Figure 6.2: Energy-momentum relation of the two lowest bands of propagating non-relativistic free fermions along the length of the quantum wire; εF is the Fermi energy, ±pF are the two Fermi points, vF = pF /m is the Fermi velocity. The filled Fermi sea (occupied states) are shown in blue; ∆E and ∆p are the level spacings in a finite wire of length L. We have shifted the minimum of the energy of the lowest band to be at zero. The Hamiltonian for the 1DEG is H = H0 + Hint where   XZ L ~2 ∂ 2 † dx ψσ (x) − H0 = − µ ψσ (x) 2m ∂x2 σ=↑,↓ 0 if the free fermion Hamiltonian, and X Z L Z L Hint = dx dx′ ψσ† (x)ψσ (x) U(x − x′ ) ψσ† ′ (x′ )ψσ′ (x′ ) σ,σ′ =↑,↓

∗

0

(6.2)

(6.3)

0

If the Fermi energy εF & ~2 /(2md2 ) the two lowest bands will be partially occupied and there will be four Fermi points.

4

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

where U(x − x′ ) is the interaction potential, which can be Coulomb or short ranged, depending of the physical situation. In what follows for simplicity we will use periodic boundary conditions, that require ψσ (x + L) = ψσ (x)

(6.4)

which amount to wrap the system (‘compactification’) on a circle. Sometimes we may want to use the more physical open boundary conditions. In many cases we will be interested in lattice systems. So, consider a one dimensional chain of N sites (atoms) and lattice spacing a, and total length L = Na. The lattice Hamiltonian is H=

N X X

j=1 σ=↑,↓

+

N  X j=1

  t ψσ† (j)ψσ (j + 1) + h.c.

 Un↑ (j)n↓ (j) + V n(j)n(j + 1)

(6.5)

where nσ (j) = ψσ† (j)ψ P σ (j) is the fermion occupation number with spin σ at site j, and n(j) = σ nσ (j) is the total occupation number (i.e. the charge) at site j. This Hamiltonian is known as the extended Hubbard model. Here U is the on-site interaction and V is the nearest neighbor repulsion. This model describes a system of electrons with hopping only between nearest neighboring sites; t, the hopping amplitude, is the local kinetic energy. This system has only one band of single particle states with dispersion relation ε(p) = 2t cos(pa)

(6.6)

In the thermodynamic limit, N → ∞, the momenta p lie on the first Brillouin Zone, − πa < p ≤ πa . In general we will be interested in a system either at fixed chemical potential µ or at fixed density n = Ne /N. The effective model for interacting systems that we will discuss will describe equally (with minor changes) the low energy physics of both continuum and lattice systems. What is special about one dimension? • In the Landau theory of the Fermi liquid we considered the low energy states and we saw that they can be described in terms of particlehole pairs. In dimensions D > 1 the momentum δ~q of the pair is not necessarily parallel to the Fermi wave vector pF of the location of the

5

6.1. ONE-DIMENSIONAL FERMI SYSTEMS pF −pF

0

δq pF

Figure 6.3: One-dimensional kinematics: the momentum of a particle-hole pair of momentum q is always parallel (or anti-parallel) to the Fermi wave vector pF . Fermi surface where the pair is excited. However in 1D δq must be either parallel or anti-parallel to the Fermi momentum pF as the FS has collapsed to just two (or more) points. • This kinematic restriction implies that particle-hole pairs form effectively long lived bound states, the collective modes, since the particle and the hole move with the same speed (the Fermi velocity). We will see that this implies that the low energy effective theory is a theory of bosons. This is the main reason why the non-perturbative theory of 1D fermions, bosonization, works. • Another insight can be gleaned by looking at the density correlators, whose singularities are the collective modes. In D > 1 the retarded density-density correlation function D R (q, ω) of a free fermion is Z np − np+q dD p R (6.7) D (q, ω) = D (2π) ω − ε(p + q) + ε(p) + iη

For low momenta |q| ≪ pF and at low energies ω ≪ EF , D R (q, ω) can be written as an integral on the Fermi surface Z bF dD p q·p R D (q, ω) ≃ δ (|p| − pF ) D b F ) vF + iη (2π) ω − (q · p I bF pD−1 q·p F = db pF (6.8) D bF ) vF + iη (2π) F S ω − (q · p

For D > 1 the angular integration is a function of q and ω which has branch cuts. For instance, in 3D we saw that the result is ω − qv ω F R (6.9) ln D (q, ω) ∼ 1 + + ... 2qvF ω + qvF We saw before that the branch cuts mean that the collective modes (zero sound) eventually becomes Landau damped.

6

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID • However, in 1D there is no such angular integration (the FS is just two points!) and the result is  q  1 1 R D (q, ω) ∼ (6.10) − 2π ω − qvF + iη ω + qvF + iη

This expression contains two singularities, two poles, representing bosonic states that move to the right (the first term) or to the left (the second term). It is easy to check that this result is consistent with f -sum rule.

Furthermore, these results suggests that a theory of free fermions in 1D must be, in some sense, equivalent to a theory of a Bose field whose excitations obey the dispersion relation ω = pvF . In other terms, the bosons are density fluctuations, which in this case are just sound waves.

6.2

Dirac fermions and the Luttinger Model

We will now proceed to construct an effective low energy theory by following a procedure similar to what led to the Landau theory of the Fermi liquid. The result, however, will be quite different in 1D. To this end we will first look at the free fermion system and focus on the low energy excitations. In 1D instead of a Fermi surface we have (at least) two Fermi points at ±pF . The low energy fermionic states have thus momenta p ∼ ±pF and single particle energy close to εF : ε(p) ≃ εF + (|p| − pF )vF + . . .

(6.11)

We are interested in the electronic states near the Fermi energy. Thus, consider the fermion operator ψσ (x), whose Fourier expansion is (we will set ~ = 1 from now on) in the thermodynamic limit (L → ∞) Z dp ψσ (x) = ψσ (p) eipx (6.12) 2π Only its Fourier components near ±pF describe low energy states. This suggests that we restrict ourselves to the modes of the momentum expansion in a neighborhood of ±pF of width 2Λ, and that we write Z Λ Z Λ dp i(p+pF )x dp i(p−pF )x ψσ (x) ≃ e ψσ (p + pF ) + e ψσ (p − pF ) (6.13) −Λ 2π −Λ 2π

6.2. DIRAC FERMIONS AND THE LUTTINGER MODEL

7

and that we define right and left moving fields ψσ,R (x) and ψσ,L (x) such that ψσ (x) ≃ eipF x ψσ,R (x) + e−ipF x ψσ,L (x)

(6.14)

Thus we have split off the rapidly oscillating piece of the field and we focus on the slowly varying parts, ψσ,R (x) and ψσ,L (x), whose Fourier transforms are ψσ,R (p) = ψσ (p + pF ),

and

ψσ,L (p) = ψσ (p − pF )

respectively. The free fermion Hamiltonian X Z dp ε(p) ψσ† (p)ψσ (p) H0 = 2π σ

(6.15)

(6.16)

becomes H0 =

XZ σ

Λ

−Λ

  dp † † pvF ψσ,R (p)ψσ,R (p) − ψσ,L (p)ψσ,L (p) 2π

(6.17)

where we have linearized the dispersion ε(p) near the Fermi momenta ±pF . Let us define the two-component spinor   ψσ,R (x) ψσ (x) = (6.18) ψσ,L (x) in terms of which the free fermion Hamiltonian is XZ X Z dp † ψσ (p)σ3 pvF ψσ (p) = dx ψσ† (x)σ3 ivF ∂x ψσ (x) H0 = 2π σ σ where

  1 0 σ3 = 0 −1

(6.19)

(6.20)

In this form the effective low energy Hamiltonian reduces to the (massless) Dirac Hamiltonian in 1D. In Fig.6.2 we show the dispersion in the spinor notation. Most of the interaction terms we discussed above can be expressed in terms of the local densities of right and left moving fermions † jR,σ (x) = ψR,σ (x)ψR,σ (x),

† jL,σ (x) = ψL,σ (x)ψL,σ (x)

(6.21)

8

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID ε D

p

-Λ

Λ

−D

Figure 6.4: The Dirac dispersion; the slope is the Fermi velocity vF . The momentum cutoff is Λ and the energy cutoff is D = vF Λ.

from which we can write the slowly varying part of the charge density operator j0 (x) (i.e. with Fourier component with wave vectors close to p = 0) and the charge current density j1 (x) as j0 (x) = jR (x) + jL (x),

j1 (x) = jR (x) − jL (x)

(6.22)

which is a 2-vector of the form jµ (x) = (j0 , j1 )

(6.23)

with µ = 0, 1 (not to be confused with the chemical potential!). Thus, the coupling to a slowly varying external electromagnetic field Aµ (x) = (A0 , A1 ) is represented by a term of the form Z   e (6.24) Hem = dx − eA0 (x)j0 (x) + A1 (x)j1 (x) c The actual particle density operator of the microscopic system, X ρ(x) = ψσ† (x)ψσ (x) σ

(6.25)

6.2. DIRAC FERMIONS AND THE LUTTINGER MODEL

9

can be written in the form (using the decomposition of the Fermi field in right and left movers)  X † † ρ(x) = ρ0 + j0 (x) + e2ipF x ψR,σ (x)ψL,σ (x) + e−2ipF x ψL,σ (x)ψR,σ (x) + . . . σ

(6.26) where ρ0 = = is the average total density of electrons (including spin), and where . . . represent terms that oscillate more rapidly for large pF . The significance of the oscillatory terms can be seen by adding a coupling to a periodic potential V (x) (with wave vector 2pF ) of the form∗ Ne L

2pF π

V (x) = V0 cos(2pF x) to the Hamiltonian, leading to a new term† Z Hpot = dxV (x)ρ(x) Z Z dq dp V (q)ψ † (p + q)ψ(p) = −e 2π 2π Z   dp = (−eV0 ) ψR† (p)ψL (p) + ψL† (p)ψR (p) 2π Z   = dx (−eV0 ) ψR† (x)ψL (x) + ψL† (x)ψR (x)

(6.27)

(6.28)

In other terms, a periodic potential of wave vector K = 2pF causes backscattering: it scatters a right moving fermion into a left moving fermion (and viceversa). Similarly, a periodic potential of wave vector K ≪ 2pF scatters right movers into right movers (and left movers into left movers). Thus, in the case of a free fermion coupled to a periodic potential V (x) with wave vector K = 2pF , the potential induces backscattering that mixed the two Fermi points at ±pF . This leads to the existence of an energy gap at the Fermi energy. In terms of the Dirac Hamiltonian, the periodic potential V (x) leads to the Hamiltonian Z  H = dx ψ † (x) ivF σ3 ∂x + eV0 σ1 )ψ(x) (6.29) ∗

For simplicity here we consider only potentials with wave vectors commensurate with the fermion density, K = 2pF . More general cases can also be considered and lead to interesting physical effects. † We will drop the spin indices from now on.

10

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

where

  0 1 σ1 = 1 0

(6.30)

where eV0 is the energy gap ∆, and is usually denoted by ∆ = mvF2 (instead of using the speed of light, that is). In the Dirac theory it is useful to define the matrices α = σ3 (in 1D) and β = σ1 , such that the Dirac Hamiltonian reads Z   † H = dx ψ (x) αivF ∂x + β∆ ψ(x) (6.31) The single-particle spectrum is consists of particles and holes with energy p ε(p) = vF2 p2 + ∆2 . In the Dirac theory it is customary to define Dirac’s γ-matrices. In this 1D case there are just two of them, γ0 = β = σ1 and γ1 = βα = iσ2 . They satisfy the algebra {γ0 , γ1 } = 0,

γ02 = 1,

If we define ψ¯ = ψ † γ0 , the fermion mass term is

γ12 = −1

¯ ψR† ψL + ψL† ψR = ψ † γ0 ψ = ψψ

6.3

(6.32)

(6.33)

Order parameters of the 1DEG

Similarly we can also define the matrix γ 5 = γ0 γ1 = σ3 , and the bilinear ¯ 5ψ ψγ ¯ 5 ψ = ψ † ψL − ψ † ψR ψγ (6.34) R L

It is straightforward to see that the density ρ(x) can be written as

5 ¯ ¯ ρ(x) = ρ0 + j0 (x) + cos(2pF x) ψ(x)ψ(x) + i sin(2pF x) ψ(x)γ ψ(x) (6.35)

¯ ¯ From here we see that if hψ(x)ψ(x)i = 6 0 (or hψ(x)γ 6 0), then the 5 ψ(x)i = expectation value of the charge density hρ(x)i has a modulated component (over the background ρ0 ). If this were to occur spontaneously (i.e. in the absence of an external periodic potential) then the ground state of the system ¯ and hiψγ ¯ 5 ψi play the would be a charge density wave (CDW). Hence, hψψi role of the order parameters of the CDW state.∗ ∗

We can also see that the expectation value of the density will be even (invariant) under 5 ¯ inversion, x → −x (i.e. parity), if hψ(x)γ ψ(x)i = 0; conversely, if this expectation value is not zero, the density will not be even under parity, which amounts to a phase shift.

6.3. ORDER PARAMETERS OF THE 1DEG

11

In the absence of the periodic potential the original system is translationally invariant. The periodic potential breaks translation invariance. To see how that works we define the transformation   iθ e ψR (x) iθγ 5 (6.36) ψ(x) → e ψ(x) = −iθ e ψL (x) which is known as a chiral transformation. Under this transformation the two component vector   ¯ ψψ (6.37) ¯ 5ψ iψγ transforms as a rotation      ¯ ¯ ψψ cos(2θ) − sin(2θ) ψψ ¯ 5 ψ → sin(2θ) cos(2θ) ¯ 5ψ iψγ iψγ

(6.38)

under which the density operator becomes ρ(x) → ρ0 + j0 (x) + ei2(pF x−θ) ψR† (x)ψL (x) + e−i2(pF x−θ) ψL† (x)ψR (x)   θ (6.39) = ρ x− pF Therefore, a chiral transformation by an angle θ is equivalent to a translation of the charge density by a displacement d = pθF . Notice that transformations by θ = nπ have no physical effect as they amount to translations by a distance n pπF = 2n Kπ = nℓ, i.e. an integer number of periods ℓ = 2π of the CDW. K Thus only chiral transformations modulo π are observable. In a similar fashion we can define an operator corresponding to a spin density wave (SDW). Indeed, the local magnetization (or spin polarization) density a ma (x) = ψσ† (x)τσ,σ (6.40) ′ ψσ ′ (x) (where τ a are the three Pauli matrices, acting only on the spin indices σ, σ ′ ), which can be expressed as † a ma (x) = j0a (x) + e2ipF x ψR,σ (x)τσ,σ ′ ψL,σ ′ (x) + h.c.

(6.41)

where j0a (x) is the slowly varying spin density † † a a j0a (x) = ψR,σ (x)τσ,σ ′ ψR,σ ′ (x) + ψL,σ (x)τσ,σ ′ ψL,σ ′ (x)

(6.42)

12

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

and, similarly, the spin current is † † a a j1a (x) = ψR,σ (x)τσ,σ ′ ψR,σ ′ (x) − ψL,σ (x)τσ,σ ′ ψL,σ ′ (x)

(6.43)

The SDW order parameters are a N a (x) = ψ¯s,σ (x)τσ,σ ′ ψs,σ ′ (x),

a 5 Nca (x) = iψ¯s,σ (x)τσ,σ (6.44) ′ γs,s′ ψs′ ,σ ′ (x)

(where s, s′ = R, L) and describe modulations of the local spin polarization with wave vector K = 2pF . Finally let us discuss pairing operators. We will see later in this class that they are associated with superconductivity. Here we will be interested in pairing operators associated with uniform ground states (although modulated states are also possible). Pairing operators that create a pair of quasiparticles with total momentum (close to) zero have the form † † OSP (x) = hψR,σ (x)ψL,−σ (x)i,

† † OT P (x) = hψR,σ (x)ψL,σ (x)i

(6.45)

where OSP (x) corresponds to (spin) single pairing, and OT P (x) to (spin) triplet pairing. Differently from all the operators we discussed so far, the pairing operators do not conserve particle number. We will see below that all of these order parameters break some (generally continuous) symmetry of the system: translation invariance for th CDW, spin rotations and translation invariance for the SDW, and (global) gauge invariance (associated with particle number conservation) for the superconducting case. There is a theorem, known and the Mermin-Wagner Theorem, that states that in a 1D quantum system continuous symmetries cannot be spontaneously broken.† More precisely, this theorem states that correlation functions of order parameters that transform under a continuous global symmetry cannot decay more slowly than a power law function of distance (or time). We will now see that in the case of the Luttinger model the behavior is a exactly power law. We will interpret this as saying that the system is at a (quantum) critical point.

6.4

The Luttinger Model: Bosonization

We will now consider the Luttinger (Tomonaga) model.∗ We will consider first the case of spinless fermions. The Hamiltonian density H of the Lut† ∗

In high energy physics this theorem is often attributed to S. Coleman. Also known as the massless Thirring model in high energy physics.

13

6.4. THE LUTTINGER MODEL: BOSONIZATION

tinger model is     † 2 2 H = ψ (x) αivF ∂x +β∆ ψ(x)+2g2 ρR (x)ρL (x)+g4 ρR (x) +ρL (x) (6.46) where ρR (x) ≡ ψR† (x)ψR (x) and ρL (x) ≡ ψL† (x)ψL (x) denote the densities of right and left movers, respectively. Here g2 = Ve (0) − Ve (2pF ) and g4 = Ve (0)/2, where Ve (q) is the Fourier transform of the interaction potential. Hence, g2 measures the strength of the backscattering interactions and g4 the forward scattering interactions.† For a model of spinless fermions on a lattice near half-filling with nearest neighbor interactions with coupling constant V , the coupling constants become g2 = 2V and g4 = V . Notice that the Hamiltonian of the Luttinger model has the same form as the Landau theory of the Fermi liquid in which the quasiparticles have only forward scattering interactions, here represented by g4 . Here we have also included backscattering process labeled by g2 , with a wave vector 2pF (i.e. across the “Fermi surface”). Due to the kinematical restrictions of a curved Fermi surface, in the Landau theory backscattering processes have negligible effects. We will see that in 1D (where there is no curvature) they play a key role. R

R

L

R

L

L

R

R

R

L

R

R

(a)

(b)

(c)

Figure 6.5: a) forward scattering, b) backscattering, c) umklapp scattering Precisely at half-filling in addition to the back-scattering and forward scattering interactions (Fig.6.5a and 6.5b), an umpklapp interaction must †

The notation has a historical origin and it is by now traditional.

14

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

also be considered: this is a scattering process in which momentum conservation is conserved up to a reciprocal lattice vector G = 2π (Fig.6.5c). An umpklapp process has the form   Humklapp = gu lim ψR† (x)ψR† (y)ψL(x)ψL (y) + R ↔ L y→x

(6.47)

where gu = Ve (4pF ≃ 2π). This coupling cannot be expressed in terms of densities of right and left movers. Since the Hamiltonian of the Luttinger model is written in terms of ρR (x) and ρL (x), it is invariant under a continuous chiral transformation, i.e. it is invariant under an arbitrary continuous translation. An umpklapp term reduces this continuous symmetry to the (discrete) symmetry of lattice displacements. We will now see that the Luttinger model can be solved exactly by a mapping (or transformation) known as Bosonization.

6.5

Bosonization

We are now going to discuss some subtle but very important properties of one-dimensional Fermi systems. To date, these properties are known not to generalize to higher dimensions. Some superficially similar ideas have been recently discussed in the context of “anyon superfluids” (which we will discuss later). The physics is quite different, though. A very important tool for the understanding of one-dimensional Fermi systems is the bosonization transformation. This transformation was first discussed by Bloch and Tomonaga. It was rediscovered (and better understood) by Lieb and Mattis in the 1960’s, and by Coleman, Luther, and Mandelstam in the 1970’s. Witten solved the non-Abelian version of bosonization in 1984. We will only consider the Abelian case. Let us consider first a theory of non-interacting (spinless) fermions with Hamiltonian H0 given by (in units in which the Fermi velocity is vF = 1) H0 =

Z

dxψ † iα∂x ψ

(6.48)

where α = γ5 (defined in the previous section), with canonically quantized

6.5.

15

BOSONIZATION

Fermi fields, i.e. {ψα† (x), ψα′ (x′ )} = δαα′ δ(x − x′ ), {ψα (x), ψα′ (x′ )} = {ψα† (x), ψα† ′ (x′ )},

(6.49)

at equal times.

6.5.1

Anomalous Commutators

Consider now the “vacuum states” |0i and |Gi, where |0i is the empty state and |Gi is the filled Fermi sea obtained by having occupied all the negative energy one-particle eigenstates of the Hamiltonian H0 . The Hamiltonian H0 relative to both vacua differs by normal ordering terms. Indeed, for any eigenstate |F i of H0 one can write H0 =: H0 : +EF |F ihF |

(6.50)

where : H0 : is the Hamiltonian normal ordered with respect to |F i, i.e. : H0 : |F i = hF | : H0 := 0

(6.51)

and EF is the energy of |F i H0 |F i = EF |F i.

(6.52)

Clearly, if we choose |0i or |Gi as the reference state, EF will be different. The currents and densities also need to be normal ordered. This is equivalent to the subtraction of the (infinite) background charge of the reference state, say of the filled Fermi sea. We will see that these apparently “formal” manipulations have a profound effect on the physics. Let us compute the commutator of the charge density and current operators at equal times [j0 (x), j1 (x′ )]. Relative to the empty state |0i, both operators are already normal ordered since a state with no fermions has neither charge nor current, i.e. j0 (x)|0i = 0,

j1 (x)|0i = 0.

(6.53)

Let us consider the right and left components of the current ρR,L defined by

1 ρR,L = (j0 ± j1 ). 2

(6.54)

16

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

where

ρR = ψR† ψR

(6.55)

ρL = ψL† ψL

(6.56)

1 X † ρR (p) = √ ψR (k)ψR (k + p) L k

(6.57)

[ρR,L (p), ρR,L (p′ )]|φi = 0.

(6.58)

and In Fourier components, we find

which annihilates the empty state |0i. In fact, for any state |φi with a finite number of particles, the result is

Consider now the filled Fermi sea, |Gi. Explicitly we can write Y † Y † |Gi = ψR (p) ψL (q)|0i. p0

In other words, in |Gi all right moving states with negative momentum and all left moving states with positive momentum are filled (see Fig.6.6). ε

p

Figure 6.6: Vacuum |Gi is obtained by filling the right moving states with negative momentum (filled circles) and filling the left moving states with positive momentum (empty circles). Let us compute the commutator [ρR (x), ρR (x′ )] at equal times. The operator ρR (x) is formally equal to a product of fermion operators at the same point. Since we anticipate divergencies, we should “point-split” the product ρR (x) = ψR† (x)ψR (x) = lim ψR† (x + ǫ)ψR (x − ǫ) ǫ→0

(6.60)

6.5.

17

BOSONIZATION

and write ρR in terms of a normal ordered operator : ρR : and a vacuum expectation value ρR (x) =: ρR (x) : + limhG|ψR† (x + ǫ)ψR (x − ǫ)|Gi. ǫ→0

(6.61)

The singularities are absorbed in the expectation value. Consider a system on a segment of length ℓ with periodic boundary conditions and expand ψR (x) in Fourier series +∞ 2πxp 1 X ψR (p)ei ℓ . ψR (x) = √ L p=−∞

(6.62)

The vacuum expectation value to be computed is hG|ψR† (x

1 + ǫ)ψR (x − ǫ)|Gi = ℓ

+∞ X

ei

2π [(x−ǫ)p′ −(x+ǫ)p] ℓ

p,p′ =−∞

hG|ψR (p)† ψR (p)|Gi. (6.63)

Using the definition of the filled Fermi sea, we get hG|ψR† (p)ψR (p′ )|Gi = δp,p′ θ(−p), Hence

(6.64)

hG|ψL† (p)ψL (p′ )|Gi = δp,p′ θ(+p).

hG|ψR† (x

(6.65)

0 1 X −i 2πp (2ǫ) . e ℓ + ǫ)ψR (x − ǫ)|Gi = ℓ p=−∞

(6.66)

This is a conditionally convergent series. In order to make it convergent, we will regulate this series by damping out the contributions due to states deep below the Fermi energy. We can achieve this if we analytically continue ǫ to the upper half of the complex plane (i.e. ǫ → ǫ + iη) to get the convergent expression ∞ 1 X i 4πp (ǫ+iη) † hG|ψR (x + ǫ)ψR (x − ǫ)|Gi = lim e ℓ η→0 ℓ p=0 1  = lim  4π η→0 ℓ 1 − ei ℓ (ǫ+iη)

= lim

η→0

=

1

ℓ

i . 4πǫ

(ǫ −i 4π ℓ

+ iη)




(6.67)

18

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

Thus, the result is hG|ψR† (x + ǫ)ψR (x − ǫ)|Gi =

i 4πǫ

(6.68)

Similarly, the expectation value hG|ψL† (x + ǫ)ψL (x − ǫ)|Gi, is found to be given by −i hG|ψL† (x + ǫ)ψL (x − ǫ)|Gi = . (6.69) 4πǫ The current commutator can now be readily evaluated h i † † ′ ′ ′ ′ ′ ψR (x + ǫ)ψR (x − ǫ), ψR (x + ǫ )ψR (x − ǫ ) [ρR (x), ρR (x )] = lim ǫ,ǫ′ →0 n δ(x′ − x + ǫ′ + ǫ)ψR† (x + ǫ)ψR (x′ − ǫ′ ) = lim ′ ǫ,ǫ →0 o −δ(x − x′ + ǫ′ + ǫ)ψR† (x′ + ǫ′ )ψR (x − ǫ) .

(6.70)

The contributions from normal ordered products cancel (since they are regular). The only non-zero terms are, using Eq.(6.66),   iδ(x − x′ + ǫ + ǫ′ ) iδ(x′ − x + ǫ′ + ǫ) ′ . (6.71) − [ρR (x), ρR (x )] = lim ǫ,ǫ′ →0 2π(x − x′ + ǫ + ǫ′ ) 2π(x′ − x + ǫ + ǫ′ )

Thus, in the limit we find

i ∂x δ(x − x′ ) 2π

(6.72)

i ∂x δ(x − x′ ). 2π

(6.73)

[ρR (x), ρR (x′ )] = − and [ρL (x), ρL (x′ )] = + In terms of Lorentz components, we get

i [j0 (x), j1 (x′ )] = − ∂x δ(x − x′ ), π

(6.74)

[j0 (x), j0 (x′ )] = [j1 (x), j1 (x′ )] = 0.

(6.75)

whereas The commutator [j0 (x), j1 (x′ )] has a non-vanishing right-hand side which is a c-number. These terms are generally known as Schwinger terms. They are pervasive in theories of relativistic fermions. But terms of this sort are also found in non-relativistic systems of fermions at finite densities. In fact, these terms are the key to the derivation of the f -sum rule.

6.5.

19

BOSONIZATION

6.5.2

The Bosonization Rules

We thus notice that the equal-time current commutator [j0 (x), j1 (x′ )] acquires a Schwinger term if the currents and densities are normal ordered relative to the filled Fermi sea. The identity of Eq. (6.74) suggests that there should be a connection between a canonical Fermi field ψ with a filled Fermi sea and canonical Bose field φ. Let Π(x) be the canonical momentum conjugate to φ, i.e. at equal times [φ(x), Π(x′ )] = iδ(x − x′ ).

(6.76)

If we identify the normal ordered operators 1 j0 (x) = √ ∂x φ(x) π

(6.77)

and

1 1 j1 (x) = − √ ∂t φ(x) ≡ − √ Π(x), π π we see that Eq. ((6.76) implies

(6.78)

i 1 [∂x φ(x), Π(x′ )] = − δ ′ (x − x′ ) (6.79) π π which is consistent with the Schwinger term. These equations can be written in the more compact form 1 jµ = √ ǫµν ∂ ν φ π

(6.80)

where ǫµν is the (antisymmetric) Levi-Civita tensor and we are using from now on the notation t → x0 , x → x1 and x ≡ (x0 , x1 ). We then arrive at the conclusion that the current commutator with a Schwinger term, Eq. (6.74), is equivalent to the statement that there exists a canonical Bose field φ whose topological current, Eq.(6.80), coincides with the normal ordered fermion current. The fermion current jµ is conserved, i.e. ∂µ j µ = 0

(6.81)

which is automatically satisfied by Eq.(6.80). In the case of the free theory, the number of left and right movers are separately conserved. This means that not only should jµ be conserved, but jµ5 , defined by ¯ µγ 5ψ jµ5 = ψγ

(6.82)

20

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

should also be conserved. Using the identity γµ γ5 = ǫµν γ ν

(6.83)

we see that jµ and jµ5 are in fact related by jµ5 = ǫµν j ν .

(6.84)

The divergence of jµ5 can be computed in terms of the Bose field φ as follows 1 1 ∂µ j 5µ = ǫµν ∂µ jν = √ ǫµν ǫνλ ∂µ ∂ λ φ = √ ∂ 2 φ. π π

(6.85)

Thus, the conservation of the axial current jµ5 implies that φ should be a free canonical Bose field ∂µ j 5µ = 0 ⇒ ∂ 2 φ = 0 (6.86) where ∂ 2 ≡ ∂02 − ∂12 .

(6.87)

The Lagrangian for these bosons is simply given by∗ 1 (∂µ φ)2 . 2

(6.88)

 1 2 Π + (∂1 φ)2 2

(6.89)

L0 = and the Hamiltonian is H0 =

Conversely, if φ is not free jµ5 should not be conserved. We will see below that this is indeed what happens in the Thirring-Luttinger model. Before doing that, let us consider a set of identities, originally derived by Mandelstam. We should expect that these identities should be highly nonlocal, although they should have local anti-commutation relations. These identities, like all others derived within the Bosonization approach, only make sense within the Operator Product Expansion: the operators so identified give rise to the same leading singular behavior when arbitrary matrix elements are computed. Also, from the Jordan-Wigner analogy, we should expect that the fermion operators, as seen from their representation in terms of bosons, should act like operators which create solitons. ∗

Here we have set, for the moment only, vF = 1.

6.5.

21

BOSONIZATION

The free Bose field φ can be written in terms of creation and annihilation operators. Let φ+ (x) (φ− (x)) denote the piece of φ(x) which depends on the creation (annihilation) operators only φ(x) = φ+ (x) + φ− (x)

(6.90)

where φ(x) is a Heisenberg operator (x ≡ (x0 , x1 ), see Eq.(6.80)). Obviously, φ− annihilates the vacuum of the Bose theory. The operators φ+ and φ− obey the commutation relations [φ+ (x0 , x1 ), φ− (x′0 , x′1 )] = lim ∆+ (x0 − x′0 , x1 − x′1 ) ǫ→0

(6.91)

where ∆+ is given by ∆+ (x0 − x′0 , x1 − x′1 ) = −

  1 ln (cµ)2 ((x1 − x′1 )2 − (x0 − x′0 + iǫ)2 ) . (6.92) 4π

The arbitrary constant µ has dimensions of mass, i.e. (length)−1 , and it is necessary to make the argument of the logarithm dimensionless. It is customary to do this calculation by adding a small mass µ and to consider the limit |x1 − x′1 | ≪ µ−1 . In this case the numerical constant c is related to Catalan’s constant. Consider now the operators Oα (x) and Qβ (x) defined by Oα (x) = eiαφ(x) and Qβ (x) = eiβ

R x1

−∞

dx′1 ∂0 φ(x0 ,x′1 )

≡ eiβ

(6.93) R x1

−∞

dx′1 Π(x0 ,x′1 )

.

(6.94)

When acting on a state |{φ(x′ )}i, Oα (x) simply multiplies the state by eiαφ(x) . The operator Qβ (x) has quite a different effect. Since Π(x) and φ(x) are conjugate pairs, Qβ (x) will shift the value of φ(x0 , x′1 ) to φ(x0 , x′1 ) + β for all x′1 < x1 . Thus, Qβ (x) creates a coherent state which we can call a soliton Qβ (x)|{φ(x0 , x′1 )}i = |{φ(x0 , x′1 ) + βθ(x1 − x′1 )}i.

(6.95)

Consider now the operator ψα,β (x) of the form ψα,β (x) = Oα (x)Qβ (x) = eiαφ(x)+iβ

R x1

−∞

dx′1 ∂0 φ(x0 ,x′1 )

(6.96)

22

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

and compute the product ψα,β (x)ψα,β (x′ ) at equal times (x′0 = x0 ). Using the Baker-Hausdorff formula ˆ

ˆ

ˆ ˆ

ˆ

ˆ ˆ

ˆ

1

ˆ ˆ

eA eB = eB eA e−[A,B] = eA+B− 2 [A,B]

(6.97)

ˆ B] ˆ is a complex valued distribution, we get where [A, ′

ψα,β (x)ψα,β (x′ ) = ψα,β (x′ )ψα,β (x)e−iΦ(x,x )

(6.98)

where Φ(x, x′ ) is given by (all the commutators are understood to be at equal times and x0 = x′0 but x′1 6= x1 ) ′

2

′

iΦ(x, x ) = −α [φ(x), φ(x )] − β −αβ

Z

x′1

−∞

2

Z

x1

dy1

−∞

Z

dy1′ [φ(x), Π(y ′)] − αβ

= −iαβ.

x′1

dy1′ [Π(y), Π(y ′)] +

−∞ Z x1

dy1[Π(y), φ(x′ )]

−∞

(6.99)

For the operators ψα,β (x) to have fermion commutation relations we need to choose αβ = ±π. It is useful to write left and right components of the Fermi field in the form  cµ 1/2 µ R β 2π x1 ′ ′ e 8ǫ : e−i β −∞ dx1 Π(x0 ,x1 )+i 2 φ(x) : (6.100) ψR (x) = 2π  cµ 1/2 µ R iβ 2π x1 ′ ′ ψL (x) = e 8ǫ : e−i β −∞ dx1 Π(x0 ,x1 )− 2 φ(x) : . (6.101) 2π The constant β is arbitrary and it can be chosen by demanding that the currents satisfy the operator identity 1 jµ = √ ǫµν ∂ ν φ. π

(6.102)

From Eqs.(6.100) and (6.101), it follows that the free fermionic current is identified with the bosonic operator jµ = Thus, we must choose β =

√

β ǫµν ∂ ν φ. 2π

4π for the free fermion problem.

(6.103)

6.5.

23

BOSONIZATION

The free scalar field operator φ(x) and the canonical momentum Π(x) have the mode expansions Z ∞  dk 1  a(k) ei(|k|x0 −kx1 ) + a† (k) e−i(|k|x0−kx1 ) φ(x) = 2π 2|k| Z−∞  ∞ dk 1  Π(x) = i|k| a(k) ei(|k|x0−kx1 ) − i|k| a† (k) e−i(|k|x0 −kx1) −∞ 2π 2|k| (6.104) where the creation and annihilation operators obey standard commutation relations, i.e. [a(k), a† (k ′ )] = (2π) 2|k| δ(k − k ′ ). The field operator φ(x) and the canonical momentum Π(x) admit a decomposition in terms of right and left moving chiral bosonic fields, φR (x) ≡ φR (x0 − x1 ) and φL (x) ≡ φL (x0 + x1 ), which are given by Z ∞  dk 1  a(k) eik(x0 −x1 ) + a† (k) e−ik(x0 −x1 ) φR (x0 − x1 ) = 2π 2k 0 Z 0  dk −1  −ik(x0 +x1 ) † ik(x0 +x1 ) φL (x0 + x1 ) = a(k) e + a (k) e −∞ 2π 2k (6.105) It is convenient to introduce the dual field ϑ(x), defined by Π(x) = ∂1 ϑ(x) or, equivalently (up to a suitably defined boundary condition), Z x1 ϑ(x) ≡ dx′1 Π(x0 , x′1 )

(6.106)

(6.107)

−∞

The field operator φ(x) and the dual field operator ϑ(x) obey the CauchyRiemann equations ∂0 φ = ∂1 ϑ, ∂1 φ = −∂0 ϑ (6.108) as operator identities. The chiral decomposition reads φ(x0 , x1 ) = φR (x0 − x1 ) + φL (x0 + x1 ) ϑ(x0 , x1 ) = −φR (x0 − x1 ) + φL (x0 + x1 )

(6.109)

24

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

In this subsection we will work primarily with the free fermion problem. In this case the Mandelstam identities, Eq.(6.100) and Eq.(6.101), take the simpler form √ 1 : ei2 πφR (x) : 2πa0 √ 1 ψL (x) ∼ √ : e−i2 πφL (x) : 2πa0

ψR (x) ∼ √

(6.110)

where a0 is a short distance cutoff. It is interesting to consider products of the form limy1 →x1 ψR† (x)ψL (y) and limy1 →x1 ψL† (y)ψR (x) at equal times. We will use Mandelstam’s formulas, Eq.(6.110), to derive an operator product expansion for ψR† ψL and ψL† ψR , both to leading order. We find lim ψR† (x)ψL (y) =

y1 →x1

√ √ 1 : e−i2 πφR (x) :: e−i2 πφL (y) : . 2πa0

(6.111)

We can make use of the Baker-Hausdorff formula once again, now in the form ˆ

ˆ

ˆ+ ,B ˆ−]

: eA :: eB := e[A

ˆ

ˆ

: eA+B :

(6.112)

and write down a bosonic expression for ψR† ψL . The normal ordered operator is, by definition, regular. Thus we can take the limit readily to find ˆ

ˆ

lim : eA+B :=: e−iβφ(x) : .

y→x

(6.113)

This operator is multiplied by a singular coefficient that compensates for the fact that ψR† ψL and e−iβφ have superficially different scaling dimensions. An explicit calculation gives the asymptotic result √ 1 : e−i2 πφ(x) : y1 →x1 2πa0

lim ψR† (x)ψL (y) = lim

y1 →x1

(6.114)

Similarly, one finds the identification lim ψL† (x0 , y1 )ψR (x0 , x1 ) =

y1 →x1

√ 1 : e+i2 πφ(x) : . 2πa0

(6.115)

¯ i.e. the CDW order paTo sum up, the Dirac mass bilinear operator ψψ, rameter, is identified with √ ¯ ¯ 0 , x1 )ψ(x0 , y1) = 1 : cos( 4πφ(x)) : ψ(x)ψ(x) ≡ lim ψ(x (6.116) y1 →x1 πa0

6.5.

25

BOSONIZATION

and, similarly,

√ 1 5 ¯ : sin( 4πφ(x)) : . (6.117) ψ(x)iγ ψ(x) = πa0 As we saw above, at half filling the Umklapp operators play a crucial role at half-filling. These operators enter in the interaction Hamiltonian, through terms of the form Z n o † † 2 2 (6.118) Hu ∼ dx1 (ψR ψL ) + (ψL ψR ) .

These terms can be bosonized using the Mandelstam identities Eq.(6.110). Indeed, we get the equal-time operator expansion lim (ψR† (x)ψL (y))2 =

y1 →x1

√ √ 1 −i 4πφ(x) −i 4πφ(y) : e :: e : (2πa0 )2 √ 1 −4π[φ+ (x),φ− (y)] :: e−i2 4πφ(x) : = 2 : e (2πa0 ) √ 1 −4π∆+ (0+ ,x1 −y1 ) : e−i2 4πφ(x) : = 2 e (2πa0 ) (6.119)

where Eqs.(6.97) and (6.91) have been used. In short, the bosonized version of the Umklapp terms is √ 1 −i4 πφ(x) : : e (2πa0 )2

(6.120)

√ 1 +i4 πφ(x) :. : e (2πa0 )2

(6.121)

lim (ψR† (x)ψL (y))2 =

y1 →x1

and likewise lim (ψL† (y)ψR (x))2 =

y1 →x1

Finally let us consider the bosonized form of the pairing field which for spin-less fermions can only be OP (x) = ψR† (x)ψL† (x) (and its adjoint). It is straightforward to see that it maps onto √

OP (x) ∼ ψR† (x)ψL† (x) ∼ const. e−i2

π(φR −φL )

√

∼ const ei2

πϑ(x)

(6.122)

Hence while charge fluctuations (and hence CDW order) are related to the fluctuations of the field φ, superconducting fluctuations are related to fluctuations of the dual field ϑ, and hence to fluctuations of the canonical momentum

26

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

Π. In this sense charge and superconducting (phase) fluctuations are complementary with each other. Thus, typically (but not always), if one orders and has long range correlations, the other cannot not order and must have short range correlations.

6.6

The Bosonized form of the Luttinger Model

We will now use the identities we just derived to find the bosonized form of the Luttinger model. As we saw the free fermion system maps onto the free boson system (with the same velocity vF ). Hence the free fermion Hamiltonian density (the Dirac Hamiltonian density) becomes  vF  2 (6.123) H0 = Π + (∂x φ)2 2

which, in terms of the field φ and the dual ϑ has the symmetric (self-dual) form  vF  (6.124) (∂x ϑ)2 + (∂x φ)2 H0 = 2 The right and left moving (fermion) densities ρR and ρL map onto ρR ρL

 1  = √ ∂x φ − Π ≡ 2 π  1  = √ ∂x φ + Π ≡ 2 π

1 √ ∂x (φ − ϑ) 2 π 1 √ ∂x (φ + ϑ) 2 π

(6.125) (6.126)

In terms of the right and left moving densities the Hamiltonian is∗
H = (πvF + g4 ) ρ2R + ρ2L + 2g2 ρR ρL

(6.127)

Similarly, the backscattering term becomes  g2  (∂x φ)2 − Π2 2g2 ρR ρL → 2π

(6.129)

Hence, the forward scattering term of the Luttinger Hamiltonian becomes    g4  2 Π + (∂x φ)2 (6.128) g4 ρ2R + ρ2L → 2π

∗

In high energy physics this is known as the Sugawara form.

6.6. THE BOSONIZED FORM OF THE LUTTINGER MODEL

27

Thus, we see that the Hamiltonian of the Luttinger model can be represented by an effective bosonized theory, which includes the total effects of forward and backscattering interactions, and which has the (seemingly) free bosonic Hamiltonian of the form  v 1 2 2 (6.130) Π + K(∂x φ) H≡ 2 K

with an effective velocity v and stiffness K (also known as the Luttinger parameter) given by r g4  2  g2  2 v = − (6.131) vF + π π s vF + gπ4 + gπ2 K = (6.132) vF + gπ4 − gπ2 Thus, we see that the Luttinger model, which describes the density fluctuations of a 1D interacting fermion system, is effectively equivalent to a free Bose field with (in addition to the renormalized stiffness K) an effective speed v for the propagation of the bosons (the density fluctuations). We see immediately two effects • The only effect of the forward scattering interactions, parametrized by the coupling g4 , is only to renormalize the velocity. • The backscattering interactions, with coupling g2 , renormalize the velocity and the stiffness. Furthermore, for repulsive interactions g2 > 0, the stiffness is renormalized upwards, K > 1, while for attractive interactions, g2 , it is renormalized downwards. We will see that these effects are very important. • The bosonized form of the Luttinger model has the obvious invariance under φ → φ + θ, where θ is arbitrary. This is the bosonized version of the continuous chiral symmetry of the Luttinger model or, equivalently, the invariance of the original fermionic system under a rigid displacement of the density profile. Due to this invariance the system has long lived long wavelength density (particle-hole) fluctuations that propagate with speed v. In other words, the system has long lived (undamped) sound modes (i.e. phonons) much as a 1D quantum elastic solid would.

28

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID • We saw that in higher dimensions there are similar collective modes, zero sound, which eventually become (Landau) damped. In 1D for a system with a strictly linearized dispersion these modes are never damped.

• This feature of the Luttinger model is, naturally, spoiled by microscopic effects we have ignored, such a band curvature that can be shown to contribute non-quadratic terms to the bosonized Hamiltonian of the form (∂x φ)3 and similar. These non linear terms have two main effects: a) they break the inherent particle-hole symmetry of the Luttinger model, and b) they cause the the boson (the sound modes) to interact with each other and decay, which leads to damping.

At half-filling (obviously on a lattice) we have to consider also the Umklapp term, which becomes √ Hu ∼ gu cos(4 πφ) (6.133) This term formally breaks the continuous U(1) chiral symmetry φ → φ + θ to √ n π a discrete symmetry subgroup φ → φ + 4 , where n ∈ Z. We will see that when the effects of this operator are important (“relevant”) there is a density modulation (a CDW) which is commensurate with the underlying lattice and there is a gap in the fermionic spectrum. In its absence, the fermions remain gapless and the CDW correlations are incommensurate. Finally, we note that the total charge of the system is

Q = −e

Z

e dxj0 (x) = − √ π

Z

e dx∂x φ(x) = − √ ∆φ π

(6.134)

where ∆φ = φ(+∞)−φ(−∞). Hence, in the charge neutral sector the system must obey periodic boundary conditions, ∆φ = 0. Conversely, boundary √ conditions involving the winding of the boson by ∆φ = N π, where N ∈ Z, amount to the sector with charge Q = −Ne. We will now summarize our main operator identifications:

29

6.7. SPIN AND THE LUTTINGER MODEL

j0 j1 ψR ψL ¯ ψψ ¯ 5ψ iψγ ψR† ψL† ψR† ψL† ψR ψL

6.7

√1 ∂x φ → π → − √1π ∂x ϑ √ → ei2 √πφR → e−i2 √πφL → cos(2√ πφ) → sin(2√ πφ) → ei2√πϑ → ei4 πφ

(6.135)

Spin and the Luttinger model

We will now consider the case of the Luttinger model for spin-1/2 fermions. We will use the same bosonization approach as before. In this context it is known as Abelian Bosonization as the SU(2) symmetry of spin is not treated in full. A more correct (and more sophisticated) approach involves Non-Abelian Bosonization that we will not do here. We Hamiltonian density for the Luttinger model for spin-1/2 fermions with both chiralities, denoted below by s = +1 (for R )and s = −1 (for L), is X X † H = −ivF sψs,σ ∂x ψs,σ σ=↑,↓ s=±1

+g4

X

† † ψs,σ ψs,−σ ψs,−σ ψs,σ

(forward, same branch)

σ,s

+g2

X

† † ψ1,σ ψ−1,σ ′ ψ−1,σ ′ ψ1,σ

(forward, both branches)

σ,σ′

+g1,k

X

† † ψ1,σ ψ−1,σ ψ1,σ ψ−1,σ

(backscattering, no spin flip)

σ

+g1,⊥

X

† † ψ1,σ ψ−1,−σ ψ1,−σ ψ−1,σ

(backscattering, with spin flip)

σ

(6.136)

For the system to be invariant under SU(2) spin rotations the couplings must satisfy g1,k = g1,⊥ ≡ g1 .

30

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID The umklapp scattering term now is † † Hu = g3 ei(4pF −G)x ψ−1,↑ ψ−1,↓ ψ1,↓ ψ1,↑ + h.c.

(6.137)

where G is a reciprocal lattice vector. As before, we will ignore umklapp processes unless we are at half-filling. Once again we begin with the free fermion. We then introduce two Bose fields φ↑ and φ↓ , and their respective canonical momenta, Π↑ and Π↓ . The corresponding free boson Hamiltonian is  vF X  2 (6.138) Πσ + (∂x φσ )2 H0 = 2 σ We now define the charge and spin Bose fields φc and φs ,  1  φc = √ φ↑ + φ↓ 2  1  φs = √ φ↑ + φ↓ 2

(6.139) (6.140)

in terms of which H0 becomes a sum over the charge and spin sectors  v   vF  2 F (6.141) Πc + (∂x φc )2 + Π2s + (∂x φs )2 H0 = 2 2

where Πc and Πs are the momenta canonically conjugate to φc and φs . We will now see that the interactions will lead to a finite renormalization of these parameters, leading to the introduction of a charge and a spin velocity, vc and vs , and of the charge and spin Luttinger parameters Kc and Ks . The charge and spin densities and currents are r 1 2 j0c = j0↑ + j0↓ = √ ∂x (φ↑ + φ↓ ) = ∂x φc (6.142) π π  1 1 ↑ 1 ↓ s j0 − j0 = √ ∂x (φ↑ − φ↓ ) = √ ∂x φs (6.143) j0 = 2 π 2π

Using the bosonization identities we can write the Luttinger Hamiltonian in the form  v  1  vc  1 2 s 2 2 2 H = Π + Kc (∂x φc ) + Π + Kc (∂x φs ) 2 Kc c 2 Ks s √ √ (6.144) +Vc cos(2 2πφc ) + Vs cos(2 2πφs )

6.7. SPIN AND THE LUTTINGER MODEL where vc and vs are the charge and spin velocities, q
2 1 vc = (2πvF + g4 )2 − g1,k − 2g2 2π q
2 1 vs = (2πvF − g4 )2 − g1,k 2π

Kc and Ks are the charge and spin Luttinger parameters, s 2πvF + g4 + 2g2 − g1,k Kc = 2πvF + g4 − 2g2 + g1,k s 2πvF − g4 − g1,k Ks = 2πvF − g4 + g1,k

31

(6.145) (6.146)

(6.147) (6.148)

The couplings Vc and Vs , due to umklapp and backscattering with spin flip respectively, are given by Vc =

g3 , 2(πa)2

Vs =

g1,⊥ 2(πa)2

(6.149)

In what follows we will neglect umklapp processes and hence set Vc = 0. In the absence of backscattering, g1 = 0, this model is known as the TomonagaLuttinger model. • We now see that this model describes a system with charge and spin bosons, the charge and spin collective modes of the fermionic system. In general the charge and spin velocities are different. • There is no mixing between charge and spin bosons: spin-charge separation • We also see that for repulsive interactions the charge mode propagates faster than the spin mode, vc > vs . • In the same regime, Kc > 1 while Ks < 1. This will have important consequences. • The fermion operators, with chirality s and spin σ become ψs,σ = √

1 Fs,σ e−iΦs,σ (x) 2πa

(6.150)

32

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID where Φs,σ =

r

i π h (ϑc − sφc ) + σ (ϑs − sφs ) 2

(6.151)

and Fs,σ are Klein factors that ensure that fermions with different labels anti-commute with each other, {Fs,σ , Fs′ σ′ } = δs,s′ δσ,σ′ ,

(6.152)

and a is a short distance cutoff. We can now express all the operators we are interested in, in terms of bosons. • Charge-Density-Wave: OCDW = e−i2pF x

X σ

† ψ1,σ (x)ψ−1,σ →

• Spin-Density-wave: (3)

OSDW =e−i2pF x

(±)

OSDW

X

√  √ 1 −i2pF x 2πφs e−i 2πφc (x) e cos πa (6.153)

† 3 ψ1,σ (x)τσ,σ ′ ψ−1,σ ′

σ,σ′

√  √ 1 −i2pF x 2i sin →− e 2πφs e−i 2πφc (x) πaX † ± −i2pF x ψ1,σ (x)τσ,σ =e ′ ψ−1,σ ′

(6.154)

σ,σ′

→

1 −i2pF x −i√2πφc ±i√2πϑs (x) e e e πa

• Singlet Superconductivity:

√

† † OSS = ψR,↑ ψL,↓ → ei

2πϑc

• Triplet Superconductivity

√

(1)

† † OT S = ψR,↑ ψL,↑ → ei

(−1) OT S

=

† † ψR,↓ ψL,↓

√

e−i

(6.155)

2πφs

√ 2πϑc i 2πϑs

e

√ √ i 2πϑc −i 2πϑs

→ e

e

(6.156)

(6.157) (6.158)

33

6.8. CORRELATION FUNCTIONS

6.8

Correlation functions

We will now compute the correlation functions of the Luttinger model. We will do first the spinless case.

6.8.1

The spinless case

The bosonized Luttinger Hamiltonian density for spinless fermions is
H = (πvF + g4 ) ρ2L + ρ2R + 2g2 ρR ρL (6.159) We will diagonalize this Hamiltonian by means of a Bogoliubov transformation (which is canonical): ρR = cosh λ ρ˜R + sinh λ ρ˜L ρL = sinh λ ρ˜R + cosh λ ρ˜L where

1 ρ˜R = √ ∂x φ˜R , π

With the choice tanh 2λ = −

1 ρ˜L = √ ∂x φ˜L π g2 πvF + g4

(6.160) (6.161) (6.162) (6.163)

the Hamiltonian becomes

where, as before, and


v ˜ 2 + (∂x φ) ˜2 (∂x ϑ) H = πv ρ˜2R + ρ˜2L = 2 q πv = (πvF + g4 )2 − g22

(6.164)

(6.165)

K +1 K−1 √ , sinh λ = √ (6.166) 2 K 2 K and r πvF + g4 + g2 K= (6.167) πvF + g4 − g2 The propagator of the field φ˜ = φ˜R + φ˜L is∗  (x − x′ )2 − v 2 (t − t′ )2 + a2 + iǫ    1 0 ′ ′ ˜ ˜ (6.168) hT φ(x, t)φ(x , t ) i = − ln 2 4π a0 cosh λ =

∗

We have regularized the propagator so that it vanishes as x′ → x and t′ → t.

34

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

from where we get  (x − x′ ) − v(t − t′ ) + iǫ  D  E 1 T φ˜R (x, t)φ˜R (x′ , t′ ) = − ln 4π a0 D  E  ′ 1 (x − x ) + v(t − t′ ) + iǫ  ′ ′ ˜ ˜ T φL (x, t)φL (x , t ) = − ln 4π a0 (6.169) Using these expressions we get (K + 1)2 (K − 1)2 hT (φ˜R (x, t)φ˜R (x′ , t′ )i + hT (φ˜L(x, t)φ˜L (x′ , t′ )i 4K 4K (K + 1)2 (K − 1)2 hT (φ˜R(x, t)φ˜R (x′ , t′ )i + hT (φ˜L(x, t)φ˜L (x′ , t′ )i hT (φL(x, t)φL (x′ , t′ ))i = 4K 4K (6.170)

hT (φR (x, t)φR (x′ , t′ ))i =

The fermion propagator The propagator for right moving fermions is √ √ 1 ′ ′ hT (ei 2πφR (x,t) e−i 2πφR (x ,t ) )i 2πa0 1 ′ ′ = e2πhT (φR (x,t)φR (x ,t ))i 2πa0 2 2    (K+1)  (K−1) a0 1 a0 4K 4K = 2πa0 (x − x′ ) − v(t − t′ ) + iǫ (x − x′ ) + v(t − t′ ) + iǫ (6.171)

hT (ψR (x, t)ψR† (x′ , t′ ))i ∼ √

and for left moving fermions √ √ 1 ′ ′ hT (e−i 2πφL (x,t) ei 2πφL (x ,t ) )i 2πa0 1 ′ ′ e2πhT (φL (x,t)−φL (x ,t ))i = 2πa0 2 2   (K+1)  (K−1) a 4K 4K

hT (ψL (x, t)ψL† (x′ , t′ ))i ∼ √

=

1  a0 ′ 2πa0 (x − x ) + v(t − t′ ) + iǫ

0

(x −

x′ )

− v(t − t′ ) + iǫ

(6.172)

35

6.8. CORRELATION FUNCTIONS In the free fermion case, K = 1, this propagator just becomes hT (ψR (x, t)ψR† (x′ , t′ ))i =

1 a0 ′ 2πa0 (x − x ) − v(t − t′ ) + iǫ

(6.173)

(and a similar expression for left movers). We see that while in th free fermion case the propagator has a simple pole (and hence a finite fermion residue Z = 1, as soon as the interactions are turned on the pole becomes a branch cut. Order parameters 1. The propagator of the CDW order parameter is found to be √ √ 1 1 i2 πφ(x) −i2 πφ(x) hT (e e )i ∼ (2πa0 )2 (2πa0 )2



 K1 a20 (x − x′ )2 − v 2 (t − t′ )2 + iǫ (6.174)

2. The propagator for the superconducting order parameter is, instead, √ √ 1 1 i2 πϑ(x) −i2 πϑ(x) )i ∼ hT (e e 2 (2πa0 ) (2πa0 )2

6.8.2



K a20 (x − x′ )2 − v 2 (t − t′ )2 + iǫ (6.175)

The spin 1/2 case

The behavior of the correlation functions for the case of spin-1/2 fermions can be computed similarly. Since the Hamiltonian of the Luttinger model decomposes into a sum of terms for the charge and spin sectors, respectively, we will find that the correlation functions factorize into a contribution from the charge sector and a contribution from the spin sector. We will not do all possible cases but just the most interesting ones. Since H = Hc + Hs , the propagators factorize. In other terms, the system behaves as if the electrons have fractionalized into two independent excitations: a) a spinless holon with charge −e, and b) a spin-1/2 charge neutral spinon. This feature is known as spin-charge separation. It is a robust feature of these 1D systems in the low energy limit. We will follow the same approach as in the spinless case, although we will do it less explicitly. Here too we define the densities of right and left moving

36

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

fermions with either spin polarization, 1 ρR,σ = √ ∂x φR,σ , π

1 ρL,σ = √ ∂x φL,σ π

(6.176)

and write the Luttinger Hamiltonian in terms of these densities. It reduces to H = Hc + Hs (6.177) where



1 1 2g2 − g1,k ρc,R ρc,L (6.178) (πvF + g4 ) ρ2c,R + ρ2c,L + 2 2
1 1 (πvF − g4 ) ρ2s,R + ρ2s,L − g1,k ρs,R ρs,L (6.179) = 2 2

Hc = Hs

We now perform Bogoliubov transformations (separately) for charge and spin, whose parameters λc and λs are 2g2 − g1,k πvF + g4 g1,k = + πvF − g4

tanh 2λc = −

(6.180)

tanh 2λs

(6.181)

The Luttinger parameters Kc and Ks are s πvF − g4 + 2g2 − g1,k 2λc Kc = e = πvF + g4 − 2g2 + g1,k s πvF − g4 − g1,k Ks = e2λs = πvF − g4 − g1,k and

πvs

q

(πvF + g4 )2 − 2g2 − g1,k q
2 = (πvF − g4 )2 − g1,k

πvc =


2

(6.182) (6.183)

(6.184) (6.185)

The transformed densities and bosons are denoted by 1 ρ˜c,R = √ ∂x φ˜c,R , π 1 ρ˜s,R = √ ∂x φ˜s,R , π

1 ρ˜c,L = √ ∂x φ˜c,L π 1 ρ˜s,L = √ ∂x φ˜s,L π

(6.186) (6.187)

6.8. CORRELATION FUNCTIONS

37

The fermion propagator The operators for right and left moving fermions with spin σ now take the form √ √ 1 (6.188) ψR,σ ∼ √ ei 2πφR,c eiσ 2πφR,s 2πa0 √ √ 1 e−i 2πφL,c e−iσ 2πφL,s ψL,σ ∼ √ (6.189) 2πa0 After some algebra we find † † hT ψR,↑ (x, t)ψR,↑ (0, 0)i = hT ψR,↓ (x, t)ψR,↓ (0, 0)i γc +γs  −1/2  −1/2 a a0 + i(vc t − x) a0 + i(vs t − x) = 0 2π  −γc /2  −γs /2 2 2 2 2 × x + [(a0 + ivc t) ] x + [(a0 + ivs t) ]

(6.190)

and † † hT ψL,↑ (x, t)ψL,↑ (0, 0)i = hT ψL,↓ (x, t)ψL,↓ (0, 0)i γc +γs  −1/2  −1/2 a0 a0 + i(vc t + x) a0 + i(vs t + x) = 2π  −γc /2  −γs /2 × x2 + [(a0 + ivc t)2 ] x2 + [(a0 + ivs t)2 ]

(6.191)

where γc,s =

1  1 1 Kc,s + − 4 Kc,s 2

(6.192)

Order parameters 1. The CDW correlator is † hT OCDW (x, t)OCDW (0, 0)i = √ √ √ √ 1 −i 2πφc (x,t) i 2πφc (0,0) 2πφ (x, t)) cos( 2πφ hT cos( e i (0, 0))ihT e s s (πa0 )2  2K1   2K1 a20 a20 2  c s = (πa0 )2 x2 − vc2 t2 + a20 + iǫ x2 − vs2 t2 + a20 + iǫ (6.193)

38

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID 2. The (transverse) SDW correlator is (±)

(±)

†

hT OSDW (x, t)OSDW (0, 0)i = √ √ √ √ 1 ±i 2πϑs (x,t) ∓i 2πϑs (0,0) −i 2πφc (x,t) i 2πφc (0,0) hT e e ihT e e i (2πa0 )2   2K1   K2s a20 1 a20 c = (2πa0 )2 x2 − vc2 t2 + a20 + iǫ x2 − vs2 t2 + a20 + iǫ (6.194) 3. The singlet superconductor correlator is † hT OSS (x, t)OSS (0, 0)i = √ √ √ √ 1 i 2πϑc (x,t) −i 2πϑc (0,0) −i 2πφs (x,t) i 2πφs (0,0) hT e e ihT e e i (2πa0 )2  K2c   2K1 a20 a20 2  s = 2 2 2 2 2 2 2 2 2 (πa0 ) x − vc t + a0 + iǫ x − vs t + a0 + iǫ (6.195)

6.9

Susceptibilities of the Luttinger Model

6.9.1

The fermion spectral function and ARPES

The fermion (electron) spectral function As,σ (p, ω) (where s = R, L and σ =↑, ↓) is defined by 1 As,σ (p, ω) ≡ − Im Gret s,σ (p, ω) πZ Z ∞   ∞ 1 dx dte−i(px−ωt) G(x, t) + G(−x, −t) = 2π −∞ −∞ (6.196) where Gret s,σ (x, t)

Dn oE = −iθ(t) ψs,σ (x, t), ψs,σ (0, 0)

(6.197)

is the fermion retarded Green function, G(x, t) = GR,↑ (x, t) (since the system is invariant under parity and spin reversal) is the time-ordered propagator we derived before, and p is measured form the Fermi point at pF . The detailed

6.9. SUSCEPTIBILITIES OF THE LUTTINGER MODEL

39

form of the spectral function for the general case is complicated. Explicit expressions are given in the book of A. Gogolin et al.∗ Here we will just quote the main results and analyze its consequences. For a free fermion system the Luttinger parameters Kc = Ks = 1 and hence γc = γs = 0. Similarly, the charge and spin velocities are equal in that case, vc = vs = vF . Hence, in the free fermion case, we see that the spectral function As,σ (p, ω) reduces to the sum of two poles (resulting from the poles in the propagator), for right and left movers respectively, each with a quasi-particle residue Z = 1. The situation changes dramatically for the interacting case no matter how weak the interactions are. For simplicity we will only discuss the case in which the system has a full SU(2) spin invariance, in which case Ks = 1 and γs = 0. We see that instead of poles, the fermion propagator has branch cuts, whose tips are located at ω = ±pvc,s (± here stands for right and left movers). An analysis if the integral shows that close to these singularities the spectral function has the behavior† A(p, ω ≃ pvc ) ∼ θ(ω − pvc ) (ω − pvc )(γc −1)/2 A(p, ω ≃ −pvc ) ∼ θ(−ω − pvc ) (−ω − pvc )γc A(p, ω ≃ pvs ) ∼ θ(ω − pvs ) (ω − pvs )γc −1/2

(6.198) (6.199) (6.200)

where p is the momentum of the incoming fermion (measured from pF .) Thus, the free fermion poles are replaced in the interaction system by powerlaw singularities. These results show clearly the spin-charge separation: an injected electron has decomposed into (soliton-like) excitations, holons and spinons, that disperse at a characteristic (and different) speed. In an angle-resolved photoemission (ARPES) experiment high-energy photons impinge on the surface of a system. If the photons energ is high enough (typically the photons are X-rays from synchrotron radiation of a particle accelerator), there is a finite amplitude for an electron to be ejected from the system (a photo-electron), leaving a hole behind. In an ARPES experiment the energy and momentum (including the direction) of the photo-electron are measured. It turns out that the intensity of the emitted photo-electrons is proportional to the spectral function of the hole left behind at a known momentum and energy. Although it is not technically possible to do an ARPES ∗

A. O. Gogolin, A. A. Nersesyan and A. M. Tsvelik, Bosonization and Strongly Correlated Systems; see Chapter 19. They use the notation ρs,σ (p, ω) for the spectral function. † For the SU (2) symmetric case, γs = 0 and the is no singularity at ω ∼ −pvs .

40

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

p − ω(p)

p

p − ω(p)

p

ω

EF

ω

(b)

(a)

(c)

EF

(d)

Figure 6.7: ARPES spectra: a) MDC in a Fermi liquid, b) EDC in a Fermi liquid, c) MDC in a Luttinger liquid, d) EDC in a Luttinger liquid.

experiment in a literally one-dimensional system, it is possible to do it quasi1D systems, arrays of weakly coupled 1DEGs. Experiments of this type are done in systems of this type, such as the blue bronzes, although their degree of quasi-one-dimensionality is not strong enough to see the effects we discuss here. The data from ARPES experiments is usually presented in terms of cuts of the spectral function: a) as energy distribution curves (EDC’s) in which the spectral function at fixed momentum is plotted as a function of energy, and b) as momentum distribution curves (MDC’s) in which the spectral function at fixed energy is plotted as a function of momentum (see Fig.6.7 a-d.) Even if an ARPES experiment could be done in a Luttinger liquid, it is important

6.9. SUSCEPTIBILITIES OF THE LUTTINGER MODEL

41

to include the effects of thermal fluctuations since all experiments are done at finite temperature. One important effects is that the singularities of the spectral functions will be rounded at finite temperature. For example, the singularity of the EDC near the charge right moving branch for p = 0 (near pF ), which diverges as ω (γc −1)/2 as ω → 0 at T = 0, saturates at finite temperature T with maximum ∼ T (γc −1)/2 (which will grow bigger as T is lowered.) The same holds for the EDC at ω = 0 as a function of momentum p (at pF ), which will saturate at a value ∼ (T /vc )(γc −1)/2 . A more detailed study of the spectral function at finite temperature T (which must be done numerically) shows that the EDCs are much broader than the MDCs and look like what is shown in Fig.6.7.‡

6.9.2

The tunneling density of states and STM

In a scanning tunneling microscopy (STM) experiment, a (very sharp) metallic tip (typically made of a simple metal such as gold) is placed near a very flat (and clean) surface of en electronic system. There a finite voltage difference V between the tip and the system and, depending on its sign, electrons will tunnel from the tip to the system or viceversa. An STM instrument is operated by scanning the system (i.e. by displacing the tip) while keeping the tip at a fixed distance from the surface and at a fixed voltage difference. If the tip is sharp enough the intensity of the measured tunneling current (of electrons), which reflects the local changes of the electronic structure, can be used to map the local environment with an atomic precision.

V Γ

x=0

x

Figure 6.8: Sketch of an STM setup. ‡

Similar behaviors are seen in ARPES experiments in high temperature superconductors.

42

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

We will now see that the local differential conductance measured in STM has direct information on the local density of states. To see how this works let us consider a simple model of the operation of the STM. Let Htip be the Hamiltonian which describes the electronic states in the tip. Let us denote by ψtip (t) the be fermionic operator that removes an electron from the tip in some one-particle state (which we will not need to know) close to the tip Fermi energy. The tunneling process from the tip to the 1DEG at a point x = 0 is described by a term in the Hamiltonian of the form  X X † † Htunnel = δ(x)Γ ψσ† (0)ψtip +h.c. ≡ δ(x)Γ ψR,σ (0)+ψL,σ (0) ψtip +h.c. σ

σ

(6.201)

The tunneling current operator J at the point contact is given by  i X h † † J = ieΓ ψR,σ (0) + ψL,σ (0) ψtip − h.c.

(6.202)

σ

We will assume that the energy of this state is higher than the Fermi energy in the Luttinger liquid by an amount equal to eV , where V is the voltage difference. We will assume that this is a rather uninteresting metal well described by a Fermi liquid with a density of one-particle states ρtip (E) which is essentially constant for the range of voltages V used. Hence we can use the approximation ρtip (EF + eV ) ≃ ρtip (EF ). A fixed voltage V is equivalent to a difference of the chemical potentials of eV between the tip and the 1DEG. The same physics can be described by assigning the following phase factor to the tunneling matrix element Γ e

Γ → Γei ~ V t

(6.203)

both in the tunneling Hamiltonian and in the definition of a tunneling current.§ We now use perturbation theory in powers of the tunneling matrix element Γ to find expectation value of the current operator, which will be denoted by I. To the lowest possible order in Γ, I is given by Z e 2 0 I = 2π |Γ| dE ρLL (E, T ) ρtip (E + eV, T ) (6.204) ~ −eV §

This is equivalent to a time-dependent gauge transformation.

6.9. SUSCEPTIBILITIES OF THE LUTTINGER MODEL

43

and the differential tunneling conductance G(V, T ) is G(V, T ) =

2πe 2 dI ≃ |Γ| ρtip (0)ρLL (E, T ) dV ~

(6.205)

Here ρLL (E, T ) is the one-particle local density of states of the Luttinger liquid 1 ρLL (E, T ) = − ImGret LL (x = 0, ω = E, T ) π Z ∞ 4 dp ret AR,↑ (p, E, T ) = − ImGR,↑ (x = 0, E, T ) = 4 π −∞ 2π (6.206) where AR,↑ (p, E, T ) is the spectral function defined above, and the factor of 4 arises since right and left movers (with both spin orientations) contribute equally at equal positions (denoted by x = 0). Alternatively, Z ∞ E 4 dte−i ~ t Gret (6.207) ρLL (E, T ) = − Im R,↑ (x = 0, t, T ) π −∞ At T = 0, by computing this Fourier transform one finds that ρLL (E) has a power-law behavior ρLL (E) ∝ E 2(γc +γs ) (6.208) Hence, the differential tunneling conductance essentially measures the local density of states of the Luttinger liquid.¶ Therefore, at T = 0, the differential tunneling conductance behaves as GLL (V ) ∝ V 2(γc +γs )

(6.209)

whereas for T > 0 one finds a saturation for V ≪ T : GLL (V, T ) ∝ T 2(γc +γs )

(6.210)

The crossover between the T > 0, V → 0 Ohmic behavior and the T → 0, V > 0 Luttinger behavior occurs for eV ∼ kB T . In contrast, for a free fermion (and for a Landau Fermi liquid) GF L (V ) = const ¶

This is actually a general result

(6.211)

44

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

since in this case γc = γs = 0. Therefore, for a free fermion we find that the point contact is Ohmic, I ∝ V . For a Luttinger liquid there is instead a power-law suppression of the tunneling differential conductance for T ≪ V (see Fig.6.9), and Ohmic behavior for T ≫ V (with a conductance that scales as a power of T .) These behaviors reflect the fact that there are no stable electron-like quasiparticles in the Luttinger liquid: the electron states are orthogonal to the states in the spectrum of the Luttinger liquid leading to a vanishing of the quasiparticle residue and to characteristic power-law behaviors in many quantities. This fact is known as the orthogonality catastrophe. G=

dI dV

V dI Figure 6.9: Differential tunneling conductance G = dV as a function of bias voltage V , in a Fermi liquid (brown), and in a Luttinger liquid at T = 0 (dark blue) and at T > 0 (red).

6.9.3

The fermion momentum distribution function

We will now discuss the fermion momentum distribution functions at zero temperature, T = 0. Since we have right and left movers, with both spin orientations, in principle we have four such functions. However, the Luttinger liquid state is invariant under global spin flips, ↑↔↓, and under parity, R ↔ L. Thus all four momentum distributions are equal to each other. Let us

6.9. SUSCEPTIBILITIES OF THE LUTTINGER MODEL

45

compute, say, nR,↑ (p) which is given by the equal time correlator † lim hψR,↑ (p, t)ψR,↑ (p, t′ )i Z Z +L/2 1 +L/2 ′ † dx (x, t)ψR,↑ (x′ , t′ )i dx′ e−ip(x−x ) ′ lim + hT ψR,↑ = lim L→∞ L −L/2 t →t+0 −L/2 Z ∞ dω = AR,↑ (p, ω) −∞ 2π (6.212)

nR,↑ (p) =

t′ →t+0+

(here T means time-ordering!). n(p) 1

pF

p

Figure 6.10: The fermion momentum distribution function in a Luttinger liquid. A lengthy computation of the Fourier transforms leads to the result at T = 0 (here p is measured from pF ) nR,↑ (p) ∼ const + # |p|2(γc +γs ) sign(p)

(6.213)

At finite temperature T > 0 this singularity is rounded by thermal fluctuations which dominate for momenta |p| . kB T /vc , which lead to a smooth momentum dependence in this regime. This is why Luttinger behavior is difficult to detect in the momentum distribution function. Thus, instead of a jump (or discontinuity) of Z (the quasiparticle residue) at pF (the Fermi liquid result), in a Luttinger liquid there is no jump (since Z = 0!). Instead we find that the momentum distribution function has a weak singularity at pF . This is what replaces the “Fermi surface” in a Luttinger

46

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

liquid. we will show below that this happens since the Luttinger liquid is a (quantum) critical system and the fermions have an anomalous dimension given by 2(γc + γs ).

6.9.4

Dynamical susceptibilities at Finite Temperature

Finally, we will discuss the behavior of dynamical susceptibilities at finite T > 0. In the preceding sections we gave explicit expressions for the correlators (time-ordered) of various physical quantities (order parameters and currents) at T = 0 in real space and time. Here we will need the dynamical susceptibilities, which are the associated retarded (instead of time-ordered) correlators at finite T > 0 in real momentum and frequency. We saw earlier in the class that we can determine all of these properties from the temperature correlators, i.e. in imaginary time τ , restricted the the interval 0 ≤ τ < 1/T (with kB = 1). We accomplish this by first making the analytic continuation vt → −ivt, (6.214) which implies to introduce the complex coordinates x − vt → z = x + ivt and x + vt → z¯ = x − ivt

(6.215)

Next we perform the conformal mapping from the complex plane labelled by the coordinates z to the cylinder, labelled by the coordinates w = x + ivτ (see Fig.6.11) T

x + ivt → e2π v (x+ivτ )

(6.216)

Thus, the long axis of this cylinder is space (labelled by −∞ ≤ x ≤ ∞), and the circumference is the imaginary time τ , 0 ≤ τ ≤ 1/T . Under the conformal mapping the boson propagator (in imaginary time t) tuns out to transform as πT 1
ln (6.217) hφ(x, t)φ(x′ , t′ )i → T ′ 2π sinh π v (w − w ) where w = x + ivτ . The computation of the correlators of the observables we are interested in is the same than at T = 0, except that the boson propagator changes as

replacemen 47

6.9. SUSCEPTIBILITIES OF THE LUTTINGER MODEL z

vt

w z

P

P′ O 1

w

τ

x 0

−∞

x

Figure 6.11: The conformal mapping z = e2πT w/v which maps the complex plane z = x + ivt to the cylinder w = x + ivτ . Under this mapping the origin O on the plane maps onto −∞ on the cylinder. shown above.k Thus, to compute the temperature propagators we perform (a) the analytic continuation followed by (b) the conformal mapping. This leads to the following identification for the power-law factors in the correlators∗∗ γ γ   1 1 → (x − x′ ) ∓ v(t − t′ ) + iǫ (x − x′ ) ± iv(t − t′ )  γ → 

sinh



πT v



πT v

(x − x′ ) ± iv(τ − τ ′ )

 

(6.218)

where γ is an exponent. Notice that the temperature changes the behavior of the boson propagator on distances long compared with the thermal wavelength v/T to ′

′

hφ(x, t)φ(x , t )i → ln k



πT 2



−

πT x − x′ + . . . v

(6.219)

The form of the boson propagator on the cylinder insures that the correlators are translation invariant and periodic (or anti-periodic for fermions). This result can also be derived by an explicit calculation of the propagator (without using conformal mappings). ∗∗ To restore proper units we must set T /v → kB T /~v. Here we use kB = ~ = 1.

48

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

The long distance behavior of the boson propagator at finite T changes the behavior of the other correlators as well. In this regime they exhibit exponential decay of correlations over distances long compared with the thermal wavelength. Contrary to what we did in perturbation theory, where the correlators are given in momentum and frequency space, the bosonization approach yields the exact correlators in real space and time. Thus to compute spectral functions and other quantities of interest we now must perform Fourier transforms on the analytic continuation of these expressions (some of which have a somewhat involved analytic structure). But the expressions we have are not perturbative, they are exact! The CDW susceptibility The thermal CDW correlation function, i.e. the CDW propagator in imaginary time at finite temperature T , is D  E † DCDW (x, τ ; T ) = Tτ OCDW (x, τ )OCDW (0, 0)  2K1   2  ∼ 



 × 

sinh

sinh

 

c

πT vc

πT (x vc

πT (x vs



+ ivc τ ) sinh 

πT vs

2



πT (x vc

− ivc τ )

 

  + ivs τ ) sinh πT (x − ivs τ ) vs

 2K1

 

s

(6.220)

The CDW dynamical susceptibility at finite temperature χCDW (p, ω; T ) is the Fourier transform of this expression in x and τ (after an analytic continuation to real time t). The Fourier transform has a complex analytic structure due to the branch cuts and to the difference in the charge and spin velocities. Of direct physical interest is the imaginary time of the dynamical susceptibil′′ ity, χCDW (p, ω; T ) at finite temperature which is measured by inelastic X-ray ′′ scattering.†† Although the general form of χCDW (p, ω; T ) can be determined ††

′′

Up to a Bose factor χCDW (p, ω) is proportional to the inelastic cross section.

6.9. SUSCEPTIBILITIES OF THE LUTTINGER MODEL

49

ω

ω = pv

pv ω = −pv

Figure 6.12: numerically, a simple expression (which captures the main physics) can be obtained by setting vc = vs = v:      ω + pv ω − pv sin(πγ) ′′ f (6.221) χCDW (ω, p > 0; T ) ∼ − 2(1−γ) Im f T 4πT 4πT Here p is measured from 2pF , and 1 γ= 2



1 1 + Kc K s



(6.222)

The complex function f (x) is given by f (x) =

Γ( γ2 − ix) Γ(1 − γ2 − ix)

(6.223)

where Γ(z) is the Euler Gamma function. At very low temperatures, |ω ± ′′ pv| ≪ T , χCDW (p, ω; T ) converges to the T = 0 result:

1 1 2 ω − p2 v 2 − 2 (1− Kc )
θ(ω−pv)+θ(−(ω+pv) (6.224) χCDW (p, ω; T = 0) ∝ 4π 2 ′′

where I have set Ks = 1 (we will see below that this is required by the SU(2) spin rotational invariance). Since Kc > 1 (for repulsive interactions)

50

CHAPTER 6. THEORY OF THE LUTTINGER LIQUID

we see that the spectral function is largest near ω = ±pv (“on-shell”) where it diverges as a power law. Notice that this is a one-sided singularity as the spectral function vanishes on the other side of the mass shell condition (the edge of the shaded area in Fig.6.12.) This divergence is cutoff at finite temperature T , where it takes the maximum value determined by T . The same behavior is found for the full static susceptibility at the ordering wave vector (which here this means p = 0). Although it can be determined ′′ from the general expression for χCDW (p, ω; T ), it is instructive to determine it more directly by the following simple argument. At finite temperature, the long distance behavior of the correlators in real space on distances long compared to the thermal wave length v/T is an exponential decay. This behavior effectively cuts off the infrared singularities in many quantities such as the static susceptibility at the ordering wavevector Q = 2pF , χCDW (ω = 0, p = 0) (again, the momentum p is measured from the ordering wavevector). This quantity can be computed directly from a Fourier transform of the thermal CDW correlation function in imaginary time at T = 0 with a long distance cutoff of v/T . χCDW (0, 0; T ) ∼

Z

dx

|x| 1 (repulsive interactions) but it does for Kc < 1 (attractive interactions). The different behavior of the superconducting and CDW susceptibilities follows directly from duality.

Physics 561, Fall Semester 2015 Professor Eduardo Fradkin Problem Set No. 1: Quantization of Non-Relativistic Fermi Systems Due Date: September 14, 2015 1

Second Quantization of an Elastic Solid

Consider a three-dimensional elastic solid in the continuum harmonic approximation. The classical Lagrangian for this system is   Z 3 3 X  2  X ρ Γ K ~ · ~u(~x, t) L = d3 x ∂i uj (~x, t) ∂i uj (~x, t) − ∇ u˙ 2i (~x, t) −   2 2 2 i=1 i,j=1 (1) where u(~x, t) is the displacement field, i.e. the local distortion of the continuous solid at a point ~x at time t. 1. Find the Classical Hamiltonian. 2. Quantize the system in (i) position and (ii) momentum space. 3. Derive expressions for the creation and annihilation operators of the normal modes. 4. Construct the following states: (a) The ground state |0i. (b) A state with one longitudinal phonon with momentum ~k. (c) A state with one longitudinal phonon of momentum ~k and one transverse phonon of momentum ~q.

2

Creation and Annihilation Operators 1. Let |ϕi be a one-particle state and |ψ1 , . . . , ψn i an n-particle state (properly symmetrized or antisymmetrized). Show that the destruction operators a(ϕ) satisfies a(ϕ)|ψ1 , . . . , ψn i =

n X

ζ k−1 hϕ|ψk i|ψ1 , . . . , ψk−1 , ψk+1 , . . . , ψn i

(2)

k=1

where ζ = ±1 (+1 for Bosons and −1 for Fermions). Give an interpretation for this formula in the case in which , in a system of free fermions, the 1

n-particle state is the ground state |Gi and |ϕi is a single particle state localized at ~x. 2. Prove the formula where

3

  a(ϕ1 ), a† (ϕ2 ) −ζ = hϕ1 |ϕ2 i

(3)

[A, B]−ζ = AB − ζBA

(4)

The Free Electron Gas

Consider an electron gas with one particle wave functions ϕp~,σ (~x) for states with momentum ~ p and z-component of the spin σ =↑, ↓. The single particle energy of the state ϕp~,σ (~x) is E(~ p) ≡ E(p2 ) (independent of the spin and isotropic). Assume that the system has N particles of mass M , and that the energy function is p~ 2 E(p2 ) = (5) 2M The creation and annihilation operators for a fermion of spin σ =↑, ↓ at a point ~x are denoted by ψσ† (~x) and ψσ (~x) and obey canonical anticommutation relations: n o {ψσ (~x), ψσ′ (~x ′ )} = ψσ† (~x), ψσ† ′ (~x ′ ) = 0, n o ψσ (~x), ψσ† ′ (~x ′ ) = δσ,σ′ δ (3) (~x − ~x ′ ) (6)

and the second-quantized Hamiltonian is Z X ~2 ~ σ† (~x) · ∇ψ ~ σ (~x) H = d3 x ∇ψ 2M

(7)

σ=↑,↓

1. Construct the ground state |Gi for a system with N particles. Find the values of the Fermi energy EF and the Fermi momentum pF . Calculate the ground state energy EG . 2. Construct an excited state with one electron with spin ↑ and momentum p~ and one hole with spin ↓ and momentum ~q, and find their energies. 3. The current carried by a one-particle state |ϕ, σi is Z i X h ~ σ − ∇ϕ ~ ∗ ϕσ ~j = −i e~ d3 x ϕ∗σ ∇ϕ σ 2m

(8)

σ=↑,↓

The corresponding second quantized current operator is Z i X h e~ ~ σ (~x) − ∇ψ ~ † (~x) ψσ (~x) d3 x ψσ† (~x)∇ψ J~ = −i σ 2m

(9)

σ=↑,↓

Check that this operator reproduces Eq.(8) by computing the matrix elements of the second-quantized operator of Eq.(9) between the one-particle states |ϕσ i = ψσ† (ϕ)|0i and |χσ i = ψσ† (χ)|0i. 2

4. Find the Heisenberg equation of motion of the fermion operators ψσ (~x) and ψσ† (~x). Show that it has the same form as the Schr´odinger equation for the wave function of a free particle of mass M . Note: recall that the Heisenberg equation of motion of an operator A(t) is ∂A(t) i = [H, A(t)] (10) ∂t ~ where A(t) = eiHt/~ Ae−iHt/~ (11) 5. Show that the particle density operator X ρ(~x, t) = ψσ† (~x, t)ψσ (~x, t)

(12)

σ

and the current density operator ~ x, t) = −i e~ J(~ 2m

X h

σ=↑,↓

i ~ σ (~x, t) − ∇ψ ~ σ† (~x, t) ψσ (~x, t) ψσ† (~x, t)∇ψ

(13)

obey the Heisenberg equation of motion ∂ρ ~ ~ +∇·J =0 ∂t

(14)

i.e. the continuity equation.

4

Free fermions in one dimension

Consider a system of particles of mass M , charge e and spin spin- 21 . The particles are restricted to move on a line of length L and do not interact with each other. we will assume that the one-particle wave functions hx, σ|ψi = ψσ (x) (with σ =↑, ↓) obey periodic boundary conditions, ψ(x) = ψ(x + L)

(15)

Consider the problem in general, without specifying the number of particles at first. 1. Write down the one-particle states ψσ (x) which obey the boundary conditions given above. 2. Use fermion creation and annihilation operators a† (x) and a(x) in position space to write an expression for the Hamiltonian of this free fermion system in Fock space. Write the same Hamiltonian in momentum space. 3. Compute the anticommutators {a(p), a(p′ )}, {a† (p), a† (p′ )} and {a(p), a† (p′ )}.

3

4. Construct the ground state |gndi for a system of N fermions with spin spin- 21 . Assume that N is an even number and that N/2 is odd. Compute the Fermi energy EF , namely the energy of the topmost occupied state. How many single particle states with this energy are present?. 5. Define a new set of creation and annihilation operators that annihilate the ground state |gndi. Write the Hamiltonian in terms of these new operators. Construct the spectrum of excitations in terms of particle and hole states with a given spin. Compute the excitation energies as a function of momentum p determine and the degeneracies of the single particle and single hole states with arbitrary spin. 6. Construct excited states with two particles, two holes and one particle and one hole. In each case assume that the excitations have momenta p and p′ and arbitrary spin. Compute the energy and the total number of fermions for each state.

5

Thermodynamics of the Ideal Fermi Gas

In this problem you will work out the thermodynamic properties of an ideal spinless non-relativistic Fermi gas at finite temperature T and density 1/v, where v is the specific volume. 1. Use the Grand Canonical Ensemble to show that the equation of state is given by 1 P = 3 f5/2 (z) kB T λT 1 1 = 3 f3/2 (z) v λT where 2 f5/2 (z) = √ π

Z

∞

∞ X √ zn −x (−1)n+1 5/2 dx x log(1 + z e ) = n n=1

Z

∞

0

2 f3/2 (z) = √ π

√ dx x

0

and λT =

∞ n X ze−x n+1 z = (−1) 1 + z e−x n=1 n3/2



2π~2 mkT

3/2

is the thermal wavelength. 2. Show that in the low density (or high temperature) limit the equation of state free Fermi gas also has the form of a virial expansion. Compute the second virial coefficient for this free Fermi gas. 4

3. Show that the density ρ = 1/v and the energy density u = U/V can be written in the form  3/2 Z ∞ √ 1 2m ǫ ρ = dǫ 4π 2 ~2 eβ(ǫ−µ) + 1 0  3/2 Z ∞ 2m ǫ3/2 1 dǫ u = 4π 2 ~2 eβ(ǫ−µ) + 1 0 4. You will now use the integrals you just derived to investigate the low temperature limit of this gas. You will need to use the Sommerfeld expansion of these integrals: Z ∞ g(ǫ) I = dǫ β(ǫ−µ) e +1 0 Z βµ Z ∞ Z µ g(µ − x/β) dx g(µ + x/β) dx − g(ǫ) dǫ + = x+1 e β ex + 1 β 0 0 0 Z µ 2 π = g(ǫ) dǫ + 2 g ′ (µ) + . . . 6β 0 (16) Use this approximation to compute the specific heat of a Fermi gas at low temperatures. Write your answers in terms of the Fermi energy ǫF , the limiting value of the chemical potential at T = 0. Find the relation between ǫF and the specific volume v. 5. Use the same approximation to calculate the pressure in a Fermi gas at very low temperatures. Find the limiting value P0 of the pressure of the Fermi gas at T = 0 and explain its physical meaning.

5

Solution 1 September 10, 2015

1

Second quantization of an elastic solid

1.1 For the classical Lagrangian ( 3 ) Z 3 Xρ X K Γ ~ · ~u(x, t))2 L = d3 x u˙ 2i (x, t) − ∂i uj (x, t)∂i uj (x, t) − (∇ 2 2 2 i=1 i,j=1 (1) The canonical momentum is Πi (x, t) =

δL = ρu˙ i (x, t) δ u˙ i

(2)

Hence the classical Hamiltonian is Z H=

3

dx

3 X i=1

Z Πi u˙ i − L =

d3 x

3 X Π2 i

i=1

2ρ

+

3 K X Γ ~ ∂i uj (x, t)∂i uj (x, t) + (∇ · ~u(x, t))2 2 i,j=1 2

(3)

1.2 For the quantum theory, it obeys the equal-time commutation relation [ui (x), Πj (x0 )] = iδ(~x − ~x0 )δij

1

(4)

Since ui (x) and Πi (x) are real, they satisfy ui (x) = u†i (x),

Πi (x) = Π†i (x)

(5)

After performing Fourier transformation, we can quantize the theory in momentum space. Define Z √ ui (p) = ρ d3 xe−i~x·~p ui (x) Z 1 Πi (p) = √ d3 xe−i~x·~p Πi (x) (6) ρ We have [ui (p), Πj (q)] = i(2π)3 δ 3 (~p + ~q)δij

(7)

where we used Z

3 3

(2π) δ (~p + ~q) =

d3 xe−i~x·(~p+~q)

(8)

The Hamiltonian in the momentum space is   Z d3 p 1 1 2 H= Πi (−p)Πi (p) + ωij (p)ui (−p)uj (p) (2π)3 2 2

(9)

where ωij2 (p) =

Γ K 2 p δij + pi pj ρ ρ

with p2 = p21 + p22 + p23 . This matrix has two eigenvalues   K +Γ 2 2 ωL (p) = p, ρ

ωT2 (p)

(10)

 =

K ρ



p2

(11)

For ωL2 (p), it corresponds to the longitudinal mode, with eigenvector parallel to the p~. For ωT2 (p), it corresponds to the transverse mode. In three dimensional space, there are two transverse modes perpendicular to the longitudinal mode. The Hamiltonian can be split into two terms H = HL + HT , Z 1 d3 p HL = ΠL (−p)ΠL (p) + ωL2 (p)uL (−p)uL (p) 2 (2π)3 Z 1 d3 p X a HT = Π (−p)ΠaT (p) + ωT2 (p)uaT (−p)uaT (p) (12) 2 (2π)3 a=1,2 T 2

1.3 The creation and annihilation operators for the normal modes are ! p 1 i ΠL (p) aL (p) = √ ωL (p)uL (p) + p 2 ωL (p) ! p 1 i aL (p)† = √ ΠL (−p) ωL (p)uL (−p) − p 2 ωL (p) ! p 1 i aaT (p) = √ ωT (p)uaT (p) + p ΠaT (p) 2 ωT (p) ! p 1 i aaT (p)† = √ ωT (p)uaT (−p) − p ΠaT (−p) 2 ωT (p) The Hamiltonian in terms of a(p) and a† (p) is Z d3 p ωL (p)aL (p)† aL (p) HL = (2π)3 Z d3 p X HT = ωT (p)aaT (p)† aaL (p) (2π)3 a=1,2

(13)

(14)

1.4 (a) The ground state is defined in this way aaT (p)|0i = 0

aL (p)|0i = 0,

(15)

(b) A state with one longitudinal phonon with momentum ~k is aL (k)† |0i

(16)

(c) A state with one longitudinal phonon of momentum ~k and one transverse phonon of momentum ~q is aL (k)† aaT (q)† |0i.

3

(17)

2

Creation and Annihilation Operators

2.1 hχ1 , · · · , χN −1 |a(ϕ)|ψ, · · · , ψN i = hϕ, χ1 , · · · , χN −1 |ψ1 , · · · , ψN i hϕ|ψ1 i hχ1 |ψ1 i · · · hχN −1 |ψ1 i . . . . . . . . = . . . . hϕ|ψN i hχ1 |ψN i · · · hχN −1 |ψN i ζ hχ1 |ψ1 i · · · hχN −1 |ψ1 i N X .. .. k−1 .. = ζ hϕ|ψk i . . . k=1 hχ1 |ψN i · · · hχN −1 |ψN i =

N X

(18) Hence we have a(ϕ)|ψ1 , · · · , ψN i =

ζ k−1 hϕ|ψk i|ψ1 , · · · , (no ψk ), ψN i

(19)

k=1

2.2 By using the above equation, we can get a(ϕ1 )a† (ϕ2 ) − ζa† (ϕ2 )a(ϕ1 ) = hϕ1 |ϕ2 i

(20)

[a(ϕ1 ), a† (ϕ2 )]−ζ = hϕ1 |ϕ2 i

(21)

Hence

3

ζ

ζ k−1 hϕ|ψk ihϕ, χ1 , · · · , χN −1 |ψ1 , · · · , (no ψk ), ψN i

k=1

N X



The Free Electron Gas

3.1 The Hamiltonian in the momentum space is X H= Ei,σ a†i,σ ai,σ i,σ

4

(22)

where Ei = p2i /2M . Since the electron has spin, each energy state can be filled up with two electrons with spin up and spin down. If N is even number, the ground state has the lowest N/2 single-particle energies filled with E1 ≤ E2 ≤ · · · ≤ EN/2 ,

p1 ≤ p2 ≤ · · · ≤ pN/2

(23)

The Fermi energy is EN/2 and Fermi momentum is pF = pN/2 . The ground state energy is EGS = 2(E1 + E2 + · · · + EN/2 )

(24)

If N is odd number, the ground state has the lowest (N − 1)/2 single-particle energies filled with two electrons. The energy level E(N +1)/2 has only electron. The Fermi energy is E(N +1)/2 and Fermi momentum is pF = p(N +1)/2 . The ground state energy is EGS = 2(E1 + E2 + · · · + E(N −1)/2 ) + E(N +1)/2

(25)

3.2 The excited state with one electron with spin ↑ and momentum p~ and one hole with spin ↓ and momentum ~q is |Ψi = a†↑,p a↓,q |GSi

(26)

E = EGS + p2 /2M − q 2 /2M

(27)

The energy is

3.3 The current operator is Z X  e~ d3 x ψσ† (x)∇ψσ (x) − ∇ψσ† ψσ (x) J = −i 2m σ=↑↓

(28)

The expectation value for this operator is Z Z 3 hϕσ |J(x)|ϕσ i = d y d3 zhϕσ |yih0|ψσ (y)J(x)ψσ† (z)|0ihz|ϕσ i Z X  e~ = −i d3 x ϕ†σ (x)∇ϕσ (x) − ∇ϕ†σ ϕσ (x) 2m σ=↑↓ =j

(29) 5

3.4 The equation of motion for ψσ (y) satisfies ∂ψσ (y) = i[H, ψσ (y, t)] = ieiHt [H, ψσ (y)]e−iHt ∂t Z X 1 iHt [∇ψσ† (x) · ∇ψσ (x), ψσ (y)]e−iHt = ie d3 x 2M σ=↑↓ =

i ∇2 ψ(y, t) 2M

(30)

3.5 Using the equation of motion Eq.(30) and its Hermitian conjugate, we have  ∂ρ X i  = −(∇2 ψ † )ψ + ψ † ∇2 ψ ∂t 2M σ X i   = −∇ · (∇ψ † ψ) + ∇ψ † · ∇ψ + ∇ · (ψ † ∇ψ) − ∇ψ † · ∇ψ 2M σ =∇·J

(31)

Hence we have ∂ρ +∇·J =0 ∂t

4

(32)

Free fermions in one dimension

4.1 The one particle state with periodic boundary condition is ψσ (x) = ei

2πn x L

(33)

where n ∈ Z.

4.2 The Hamiltonian in position space for a general free fermion model is Z ∇2 H = dxa† (x)[− + V (x)]a(x) (34) 2M 6

The above Hamiltonian in momentum space can be written as Z Z Z dp p2 † dp dq H= a (p)a(p) + V (q)a† (p + q)a(p) 2π 2M 2π 2π

(35)

where Z V (q) =

dxV (x)e−iqx

(36)

4.3 The anticommutators {a(p), a(p0 )} = {a† (p), a† (p0 )} = 0

(37)

{a(p), a† (p0 )} = δp,p0

(38)

The anticommutator

4.4 Assume that the above Hamiltonian can be diagonalized and written in this form X H= E(p)b†σ (p)bσ (p) (39) p,σ

As shown in Fig.1, the ground state has the lowest N/2 single-particle energyis filled with Ep1 ≤ Ep2 ≤ · · · ≤ Ep(N/2−1)/2

(40)

The Fermi energy EF = Ep(N/2−1)/2 . Since particle has 1/2 spin, each level is double degenerate. There are in total four single particle state with EF .

4.5 The Hamiltonian is H=

X

E(p)b†σ (p)bσ (p)

p,σ

7

(41)

p(N/2-1)/2

-p(N/2-1)/2

-p2

p1=0

p2

Figure 1: (Color online) The ground state for N particles The excited state (with a given spin) is 0 |ψi = b†σ (q1 )b†σ (q2 ) · · · b†σ (qm )bσ (q10 )bσ (q20 ) · · · bσ (qm )|GSi

(42)

It has m pairs of particle-hole qi ≥ pN/2 and qi0 ≤ pN/2 . Pm where Pm excitations, The excitation energy is i=1 E(qi ) − i=1 E(qi0 ). The degeneracy of the single particle and single hole states with arbitrary spin is 8.

4.6 The excited state with two particles and two holes is |ψi = b†σ1 (q1 )b†σ2 (q2 )bσ3 (q3 )bσ4 (q4 )|GSi

(43)

This state has N particle and the excitation energy is E(q1 )+E(q2 )−E(q3 )− E(q4 ). Since the excitation have momenta p, q1 + q2 − q3 − q4 = p. The excited state with one particles and one holes is |ψi = b†σ1 (q1 )bσ2 (q2 )|GSi

(44)

This state has N particle and the excitation energy is E(q1 ) − E(q2 ). Since the excitation have momenta p0 , q1 − q2 = p0 .

8

5

Thermodynamics of the Ideal Fermi Gas

5.1 The thermodynamic potential for ideal fermi gas is   Z p2 d3 p −µ) −β( 2m Ω = −kB T V log 1 + e (2π)3 Z   = −kB T V dN0 () log 1 + ze−β

(45)

where z = eβµ and the one-particle density of state is 2π(2m)3/2 √ N0 () =  (2π)3

(46)

Hence we have 1 ∂Ω P =− = kB T kB T ∂V 1 = 3 f5/2 (z) λT

Z

  dN0 () log 1 + ze−β (47)

where 2 f5/2 (z) = √ π

∞

Z

√ dx x log(1 + ze−x )

(48)

0

and  λT =

2π mkT

1/2 (49)

The specific volume satisfies 1 hN i 1 ∂Ω =ρ= =− = v V V ∂µ 1 = 3 f3/2 (z) λT

Z dN0 ()

ze−β 1 + ze−β (50)

where 2 f5/2 (z) = √ π

∞

Z 0

9

√ ze−x dx x 1 + ze−x

(51)

5.2 Since f5/2 (z) =

∞ X

(−1)n+1

zn n5/2

(−1)n+1

zn n3/2

n=1

f3/2 (z) =

∞ X n=1

(52)

In the high temperature limit, z → 1, the equation of state of free Fermi gas also has the form of a virial expansion. The second coefficent is −1/25 /2 for P and −1/23/2 for 1/v.

5.3 From Eq.(50), we have 1 ρ= 2 4π



2m ~2

3/2 Z

∞

d 0

√  eβ(−µ) + 1

(53)

The energy density satisfies Z  U = dN0 () u= V 1 + z −1 eβ  3/2 Z ∞ 1 2m 3/2 = 2 d 4π ~2 eβ(−µ) + 1 0

(54)

5.4 Using the Sommerfeld expansion, we have Z  U = dV N0 () 1 + z −1 eβ Z ∞ π2 ≈ g()d + 2 g 0 (µ) 6β 0 2 3 π (kB T )2 = N F + N 5 4 F where g() = V N0 (). 10

(55)

Hence the specific heat is Cv =

2 π 2 kB T 2 F

(56)

5.5 Using the same approximation, we can show that for Eq.(47), P =

π 2 N (kB T )2 2 N F + 5 V 6 F V

(57)

At T = 0, P = 25 NVF , which is known as Fermi pressure. It is nonzero due to the effects of the Pauli principle which keeps fermions from occupying the same single-particle state.

11

Physics 561, Fall Semester 2015 Professor Eduardo Fradkin Problem Set No. 2: Green Functions and Perturbation Theory Due Date: October 12, 2015 1

Antiferromagnetic Spin Waves

In this problem you will consider the Heisenberg model of a one-dimensional quantum antiferromagnet. I first give you a brief summary on the Heisenberg model. You do not need to have any previous knowledge on magnetism (or the Heisenberg model) to do this problem. You will be able to solve this problem only with the methods that were discussed in class. The one-dimensional Heisenberg model is defined on a linear chain ( a onedimensional lattice) with N sites. The lattice spacing will be taken to be equal to one (i.e., it is the unit of length). The quantum mechanical Hamiltonian for this system is N/2 X ˆ Sˆk (n) · Sˆk (n + 1) (1) H =J n=−N/2+1

where the exchange constant J > 0 ( i.e., an antiferromagnet) and the operators Sˆk (k = 1, 2, 3) are the three angular momentum operators in the spin-S representation ( S is integer or half-integer) which satisfy the commutation relations (2) [Sˆj , Sˆk ] = iǫjkl Sˆl For simplicity we will assume periodic boundary conditions, Sˆk (n) ≡ Sˆk (n+N ). In the semi-classical limit, S → ∞, the operators act like real numbers since the commutators vanish. In this limit, the state with lowest energy has nearby spins which point in opposite ( but arbitrary!) directions in spin space. This is the classical N´eel state. In this state the spins on one sub-lattice ( say the even sites) point up along some direction in space while the spins on the other sub-lattice ( the odd sites) point down. At finite values of S, the spins can only have a definite projection along one axis but not along all three at the same time. Thus we should expect to see some zero-point motion precessional effect that will depress the net projection of the spin along any axis but, if the state is stable, even sites will have predominantly up spins while odd sites will have predominantly down spins. This observations motivate the following definition of a set of basis states for the full Hilbert space of this system. The states |Ψi of the Hilbert space of this chain are spanned by Q the tensor product of the Hilbert spaces of each individual j th spin |Ψj i, |Ψi = j ⊗|Ψj i. The latter are simply the 2S + 1 degenerate multiplet of states with angular 1

momentum S of the form {|S, M (j)i} (|M (j)| ≤ S) which satisfy ~ 2 (j)|S, M (j)i S S3 (j)|S, M (j)i

= =

S(S + 1)|S, M (j)i M (j)|S, M (j)i

(3)

The states in this multiplet can be obtained from the highest weight state |S, Si by using the lowering operator Sˆ− = Sˆ1 − iSˆ2 . Its adjoint is the raising operator Sˆ+ (j) = Sˆ1 (j)+iSˆ2 (j). For reasons that will become clear below, it is convenient to define for j even ( even site) the spin-deviation operator n ˆ (j) ≡ S − Sˆ3 (j). For an odd site ( j odd) the spin deviation operator is n ˆ (j) ≡ S + Sˆ3 (j). For j even, the highest weight state |S, Si is an eigenstate of n ˆ (j) with eigenvalue zero while the state |S, −Si has eigenvalue 2S n ˆ (j)|S, Si = n ˆ (j)|S, −Si =

(S − Sˆ3 (j))|S, Si = 0 (S − Sˆ3 (j))|S, −Si = 2S |S, −Si

(4)

whereas for j odd the state |S, −Si has zero eigenvalue while the state |S, Si has eigenvalue 2S. In terms of the operators n ˆ (j), the basis states are {|S, M (j)i} ≡ {|n(j)i}, ˆ ± where M (j) = S ∓n(j). For even sites, the raising and lowering operators S(j) act on the states of this basis like     21 n−1 + ˆ S |ni = 2S 1 − n |n − 1i 2S h  n i 21 Sˆ− |ni = 2S(n + 1) 1 − |n + 1i (5) 2S For odd sites the action of the above two operators is interchanged. The action of the operators Sˆ± is somewhat similar to that of annihilation and creation operators in harmonic oscillator states. For this reason we define a set of creation and annihilation operators a ˆ† and a ˆ such that √ a ˆ† |ni = n + 1|n + 1i √ a ˆ|ni = n|n − 1i (6) which satisfy the conventional algebra [ˆ a, a ˆ† ] = 1. Since we have two sub-lattices ± and the operators Sˆ are different on each sub-lattice, it is useful to introduce two types of creation and annihilation operators: the operators a ˆ† (j) and a ˆ(j) which act on even sites, and ˆb† (j) and ˆb(j) which act on odd sites. They obey the commutation relations h i   a ˆ(j), a ˆ† (k) = ˆb(j), ˆb† (k) = δjk i h i h ˆ(j), ˆb(k) = 0 [ˆ a(j), a ˆ(k)] = ˆb(j), ˆb(k) = a (7)

and similar equations for their hermitian conjugates. It is easy to check that the action of raising and lowering operators on the states {|ni} is the same as the action of the following operators on the same states 2

1. On even sites: Sˆ+ (j) Sˆ− (j)

 1 n ˆ (j) 2 2S 1 − a ˆ(j) 2S  1 √ n ˆ (j) 2 = 2Sˆ a† (j) 1 − 2S

=

√

Sˆ3 (j)

= S −n ˆ (j)

n ˆ (j)

= a ˆ† (j)ˆ a(j)

(8)

2. On odd sites: Sˆ− (j)

=

Sˆ+ (j)

=

Sˆ3 (j)

=

n ˆ (j)

 1 √ n ˆ (j) 2 ˆ b(j) 2S 1 − 2S 1  √ n ˆ (j) 2 2Sˆb† (j) 1 − 2S

−S + n ˆ (j) † ˆ ˆ = b (j)b(j)

(9)

Notice that although the integers n can now range from 0 to infinity, the Hilbert space is still finite since (for even sites) Sˆ− |n = 2Si = 0. Similarly, for odd sites, the state |n = 2Si is anihilated by the operator Sˆ+ . 1. Derive the quantum mechanical equations of motion obeyed by the the spin operators Sˆ± (j), Sˆ3 (j) in the Heisenberg representation, for both j even and j odd. Are these equations linear? Explain your result. 2. Verify that the definition for the operators S ± and S3 of equations 8 and 9 are consistent with those of equation 5. 3. Use the definitions given above to show that the Heisenberg Hamiltonian can be written in terms of two sets of creation and annihilation operators a ˆ† (j) and a ˆ(j) (which act on even sites), and ˆb† (j) and ˆb(j) which act on odd sites. 4. Find an approximate form for the Hamiltonian which is valid in the semiclassical limit S → ∞ ( or S1 → 0). Include terms which are of order S1 (relative to the leading order term). Show that the approximate Hamiltonian is quadratic in the operators a and b. 5. Make the approximations of section (4) on the equations of motion of section (1). Show that the equations of motion are now linear. Of what order in S1 are the terms that have been neglected?

3

6. Show that the Fourier transform r a ˆ(q) ˆb(q)

=

r

=

2 X iqj e a ˆ(j) N j even 2 X −iqj ˆ b(j) e N

(10)

j odd

followed by the canonical (Bogoliubov) transformation cˆ(q) ˆ d(q)

= cosh(θ(q)) a ˆ(q) + sinh(θ(q)) ˆb† (q) = cosh(θ(q)) ˆb(q) + sinh(θ(q)) a ˆ† (q)

(11)

yields a diagonal Hamiltonian HSW of the form HSW = E0 +

Z

+π 2

−π 2

dq ω(q)(ˆ nc (q) + n ˆ d (q)) 2π

(12)

ˆ where n ˆ c (q) = cˆ† (q)ˆ c(q), n ˆ d (q) = dˆ† (q)d(q), provided that the angle θ(q) ˆ and their hermitian conis chosen properly. The operators cˆ(q) and d(q) jugates obey the algebra of eq (6). Derive an explicit expression for the angle θ(q) and for the frequency ω(q). 7. Find the ground state for this system in this approximation ( usually called the spin-wave approximation). 8. Find the single particle eigenstates within this approximation. Determine the quantum numbers of the excitations. Find their dispersion (or energymomentum) relations. Find a set of values of the momentum q for which the energy of the excited states goes to zero. Show that the energy of these states vanish linearly as the momentum approaches the special points and determine the spin-wave velocity vs at these points. Note: This is the semi-classical or spin-wave approximation. The identities of eq (8) and eq (9) are known as the Holstein-Primakoff identities. 9. Derive an expression for the following propagators in terms of time-ordered expectation values of the bosonic a and b operators introduced above (a) (b)

D33 (nt, n′ t′ ) = −ihgnd|T Sˆ3 (n, t)Sˆ3 (n′ , t′ )|gndi

(13)

D+− (nt, n′ t′ ) = −ihgnd|T Sˆ+ (n, t)Sˆ− (n′ , t′ )|gndi

(14)

in momentum and frequency space. Be very careful and very explicit in the way you treat the poles of these propagators. Show that your choice of frequency integration contour yields a propagator which satisfies the correct boundary conditions. 4

10. Use Wick’s theorem to find an expression for the corresponding timeordered functions in the spin-wave approximation in momentum and frequency space. 11. Use the results of the prvious sections to show that D+− (p, ω) has, in the limit ω → 0, a pole at p = π. Calculate the residue of this pole. The residue is the square of the order parameter of the system in this approximation.

2

The Electron Gas

In this problem you will consider the weakly interacting electron gas we discussed in class. 1. Show that the Feynman propagator for the non-interacting system ′

† ′ ′ Gσσ 0 (x, x ) = −i0 hG|T ψσ (x)ψσ′ (x )|Gi0

(15)

where |Gi0 is the ground state of the non-interacting system, has the Fourier transform δσσ′

′

p, ω) = Gσσ 0 (~

ω−

E0 (~ p) h ¯

+ isign(ω) δ

(16)

2

p − EF . Show that this expression is consistent with the where E0 (~ p) = 2m propagator being time ordered.

2. The one-particle density matrix is defined by the equal time ground state expectation value X hG|ψσ† (~x)ψσ (~y )|Gi (17) σ

(a) Find an exact relation between the one-particle density matrix and the Feynman propagator (b) Compute the one-particle density matrix for non interacting fermions. Discuss it behavior at both short and long distances R =| ~x −~y | compared with the Fermi wavelength λF = ¯h/pF . 3. (a) Draw all the Feynman diagrams in momentum space that contribute to the electron propagator to second order in the electron-electron interaction potential (b) Find the contribution of each diagram and express your result a suitable momentum integral(s). Make sure you indicate the multiplicity of each diagram (i. e. how many diagrams have the same weight) and the fermionic sign of each diagram. Do not do the integrals. (c) Show that the vacuum diagrams cancel out to this order.

5

(d) Classify your connected diagrams into one-particle reducible and one particle irreducible diagrams. Indicate the contributions to the elec′ tron self-energy Σσσ (~ p, ω) at second order in the interaction potential. 4. We will imagine that the electron system interacts with an external potential Vext (~x) which we will take to be static and to correspond to a point charge of strength Q at the origin, Vext (~x) = Qδ (3) (~x). The change of the local density δρ(~x) caused by the perturbation Vext (~x) is obtained by the expression δρ(~x) = hG|ψσ† (~x)ψσ (~x)|Gi Vext − hG|ψσ† (~x)ψσ (~x)|Gi Vext =0 (18)

(a) Use perturbation theory in the external potential to find an expression for δρ(~x) to linear order in the external potential Vext (~x) in terms of the density propagator of the electronic system that we discussed in class. Which regime of the density propagator is of interest in this case?

(b) Use the results discussed in class to draw conclusions on the spacial behavior of the density. You may want to phrase your answer in momentum space using spacial Fourier transforms. Note: you may quote results on any integrals that you may need from my notes or from textbooks.

6

Solution 2 October 14, 2015

1

Antiferromagnetic Spin Waves

1.1 The Heisenberg Hamiltonian is N/2 X

H=J

Sˆk (n) · Sˆk (n + 1)

n=−N/2+1

=

 J X  ˆ+ S (n) · Sˆ− + Sˆ− (n) · Sˆ+ (n + 1) + J Sˆ3 (n) · Sˆ3 (n + 1) 2 j

(1)

where Sˆ− = Sˆ1 − iSˆ2 and Sˆ+ = Sˆ1 + iSˆ2 . To calculate the quantum mechanical equations of motion obeyed by ± ˆ S (j) and Sˆ3 (j), we first need to calculate the commutators [Sˆ+ (j), H] =

Jˆ J S3 (j)(Sˆ+ (j − 1) + Sˆ+ (j + 1)) − Sˆ+ (j)(Sˆ3 (j − 1) + Sˆ3 (j + 1)) 2 2 (2)

where we used [Sˆ3 , Sˆ± ] = ±Sˆ± [Sˆ+ , Sˆ− ] = 2Sˆ3 Similarly, we can calculate [Sˆ− (j), H] and [Sˆ3 (j), H].

1

(3)

The Heisenberg equations of motion are i Jh ∂0 Sˆ+ (j) = −i Sˆ3 (j)(Sˆ+ (j − 1) + Sˆ+ (j + 1)) − Sˆ+ (j)(Sˆ3 (j − 1) + Sˆ3 (j + 1)) 2 h i J ∂0 Sˆ− (j) = −i −Sˆ3 (j)(Sˆ− (j − 1) + Sˆ− (j + 1)) − Sˆ− (j)(Sˆ3 (j − 1) + Sˆ3 (j + 1)) 2 h i J ∂0 Sˆ3 (j) = −i Sˆ+ (j)(Sˆ− (j − 1) + Sˆ− (j + 1) + Sˆ− (j)(Sˆ+ (j − 1) + Sˆ+ (j + 1)) 4 (4) They are non-linear equations. This suggests that the Heisenberg model in general is not a free bosonic system.

1.2 For the even sites, by using Eq.(8), we have     12   12 √ n − 1 n ˆ (j) + a ˆ(j)|ni = 2S 1 − Sˆ |ni = 2S 1 − n |n − 1i 2S 2S h n i 21 − ˆ S |ni = 2S(n + 1)(1 − ) |n + 1i (5) 2S From the above equation, we have (for the even sites) n ˆ (j)|S, Si = (S − Sˆ3 (j))|S, Si = 0 n ˆ (j)|S, −Si = 2S|S, −Si

(6)

Hence Eq.(7) and Eq.(8) are consistent with Eq.(4).

1.3 The Heisenberg Hamiltonian is H = Heven + Hodd h i X † † ˆ ˆ = JS a ˆ (j)ˆ a(j) + b (j + 1)b(j + 1) j∈even   n ˆ (j + 1) 1/2 n ˆ (j) 1/2 n ˆ (j + 1) 1/2 n ˆ (j) 1/2 † † ˆ ˆ ) (1 − ) a ˆ(j)b(j + 1) + a ˆ (j)b (j + 1)(1 − ) (1 − ) + JS (1 − 2S 2S 2S 2S X + Ja ˆ† (j)ˆ a(j)ˆb† (j + 1)ˆb(j + 1) − S 2 + (ˆ a ↔ ˆb) (7) j∈odd 2

1.4 In the semiclassical limit S → ∞, for the above Hamiltonian, if we only include terms which are of order 1/S, the Hamiltonian is i X h † † † † ˆ ˆ ˆ ˆ Heven = JS a ˆ(j)b(j + 1) + a ˆ (j)b (j + 1) + a ˆ (j)ˆ a(j) + b (j + 1)b(j + 1) − S j∈even i X h ˆb(j)ˆ Hodd = JS a(j + 1) + ˆb† (j)ˆ a† (j + 1) + ˆb† (j)ˆb(j) + a ˆ† (j + 1)ˆ a(j + 1) − S j∈odd (8) The above Hamiltonian takes a quadratic form.

1.5 In the semiclassical limit, the equations of motion becomes i JS hˆ† b (j − 1) + ˆb† (j + 1) − 2ˆ a(j) ∂0 a ˆ(j) = −i 2 h i JS ˆb(j − 1) + ˆb(j + 1) − 2ˆ a† (j) (9) ∂0 a ˆ† (j) = −i 2 The above result is for the even j. For the odd j, we only need to switch a ↔ b to get the similar result. The equations of motion are now linear. The term n ˆ (j)ˆ n(j + 1) is neglected.

1.6 By using the Fourier transformation r 2 X iqj a ˆ(q) = e a ˆ(j) N j∈even r 2 X −iqj ˆ ˆb(q) = e b(j) N j∈odd

(10)

The Hamiltonian can be written as  X N † † † † H = 2SJ a ˆ(q)ˆb(q) cos(q) + a ˆ (q)ˆb (q) cos(q) + a ˆa ˆ(q) + ˆb (q)ˆb(q) − S 2 q (11) 3

By performing the canonical transformation cˆ(q) = cosh(θ)ˆ a(q) + sinh(θ)ˆb† (q) ˆ = cosh(θ)ˆb(q) + sinh(θ)ˆ d(q) a† (q)

(12)

ˆ − The Hamiltonian has off-diagonal term cos(q)(cosh2 (θ) + sinh2 (θ))ˆ c(q)d(q) ˆ c(q). To diagonalize the Hamiltonian, this term is equal 2 cosh(θ) sinh(θ)d(q)ˆ to zero and this requires that cos(q) = tanh(2θ) The Hamiltonian after diagonalization takes the following form Z π 2 dq HSW = E0 + ω(q) (ˆ nc (q) + n ˆ d (q)) − π2 2π

(13)

(14)

where ω(q) = | sin(q)|.

1.7 The ground state satisfies ˆ cˆ(q)|GSi = d(q)|GSi =0

(15)

1.8 The single particle eigenstate can be generated by cˆ† (q) or dˆ† (q). cˆ† (q)|GSi = |q, 1i, dˆ† (q)|GSi = |q, 2i

(16)

E(q) = 2N JS| sin(q)|

(17)

The dispersion relation is

It equals to zero when q = nπ. Since q ∈ [− π2 , π2 ], when q is around zero, the energy of excited state goes to zero. Around q = 0, E(q) ≈ 2N JS|q| The energy vanishes linearly as the momentum q approaches to zero. 4

(18)

1.9 In the spin wave approximation, X √ ˆ c† (q) − sinh θd(q)) Sˆ+ (j) = e−iqj 2S(cosh θˆ q

ˆ−

S (j) =

X

√ c(q) − sinh θdˆ† (q)) eiqj 2S(cosh θˆ

(19)

q

When both n and n0 are even or odd numbers, D+− (jt, j 0 t0 ) = −ihGS|T Sˆ+ (j, t)Sˆ− (j 0 , t0 )|GSi X 0 ˆ dˆ† (q)|GSi = −4Si e−iEq (t−t ) θ(t − t0 )hGS| cosh2 θˆ c† (q)ˆ c(q) + sinh2 θd(q) q

− 4Si

X

0

ˆ dˆ† (q)|GSi eiEq (t−t ) θ(t0 − t)hGS| cosh2 θˆ c† (q)ˆ c(q) + sinh2 θd(q)

q

(20) The propagator in the momentum space is Z   D+− (q) = 4Si dt θ(∆t)e−i(Eq +ω)∆t cosh2 (θ) + θ(−∆t)ei(Eq −ω)∆t sinh2 (θ)   4S 1 − | sin(q)| 1 + | sin(q)| = + (21) | sin(q)| ω − Eq + i ω + Eq − i When one is odd and one is even, the calculation is similar, D+− (jt, j 0 t0 ) = −ihGS|T Sˆ+ (j, t)Sˆ− (j 0 , t0 )|GSi X 0 ˆ dˆ† (q)|GSi e−iEq (t−t ) θ(t − t0 )hGS| cosh θ sinh θˆ c† (q)ˆ c(q) + cosh θ sinh θθd(q) = −4Si q

− 4Si

X

0 ˆ dˆ† (q)|GSi eiEq (t−t ) θ(t0 − t)hGS| cosh θ sinh θθˆ c† (q)ˆ c(q) + cosh θ sinh θθd(q)

q

(22) The propagator in the momentum space is   4S cos(q) 1 1 D+− (q) = + | sin(q)| ω − Eq − i ω + Eq + i 5

(23)

For D33 , D33 (jt, j 0 t0 ) = −ihGS|T (S − n ˆ (j, t))(S − n ˆ (j 0 , t0 ))|GSi

(24)

It includes term hGS|ˆ a† (j)ˆ a(j)|GSi and term hGS|ˆ a† (j)ˆ a(j)ˆ a† (j 0 )ˆ a(j 0 )|GSi. The propagator in the momentum space is   1 1 δq+q0 ,0 X 2 D33 = N S δq,π δq0 ,π − + + ω − iδ ω + iδ N
k = cosh θk−q/2 sinh θk+q/2 − cosh θk+q/2 sinh θk−q/2   1 1 × − (25) ω − Ek−q/2 − Ek+q/2 + i ω − Ek−q/2 − Ek+q/2 − i

1.10 The propogator is already calculated in the above problem.

1.11 From the result on D+− (p, ω), there is a pole when cos2 (q/2) → 0 . This requires that q = π.

2

The electron gas

2.1 The Feynman propagator for the non-interacting system 0

G0σ,σ = −i0 hG|T ψσ (x)ψσ† 0 (x0 )|Gi0 X = −iσσ,σ0 ϕ∗α (r)ϕα (r0 )× α

h

θ(α − G)e

i(Eα −EG )(t−t0 ))

i(Eα −EG )(t−t0 ))

0

i θ(t − t) 0

θ(t − t ) − θ(G − α)e   X θ(α − G) θ(G − α) ∗ 0 + = sigmaσ,σ0 ϕα (r)ϕα (r ) ω − (Eα − EG ) + iδ ω − (Eα − EG − iδ) α (26)

6

For free fermion, we have σ,σ 0

G0

Z = δσ,σ0

"

θ(|p| − pF )

ip(r−r0 )

d3 pe

ω−

p2 2m

+ iδ

+

θ(pF − |p|) ω−

p2 2m

− iδ

# (27)

Hence after Fourier transformation, we have GF (p, ω) =

θ(|p| − pF ) p2 2m

+

θ(pF − |p|) 2

p ω− + iδ ω − 2m − iδ 1 1 = = ω − E(p) + i(|p| − pF ) ω − E(p) + isign ω

(28)

2.2 (a) The density operator is hρ(x, y)i = −i lim GF (x, t, σ, y, t0 , σ 0 ) 0 t →t

(b) Using the result in Eq.(28), we have Z Z π dp p2 i cos θp|x−y| e θ(pF − |p|) dθ sin θ ρ(x, y) ∼ (2π)3 0 Z pF 1 sin(p|x − y|) ∼ dp p 3 (2π) 0 |x − y| 1 1 ∼ 2 (−pF cos(pF r) + sin(pF r)) r r

(29)

(30)

where r = |x − y|. In the short distance limit when r > 1/pF , the second term is negligible, we have ρ(r) ≈ −

pF cos(rpF ) π 2 r2

It decays as 1/r2 and is also oscillatory. 7

(32)

3 (a.). f~r)t

t~

R

or-cter ~

k-k_ (

~ k)

-t\

~

--7

)

P..

-r

-t \

-l

@

1_8

lb .

~0~

If CoY +he I

Gr, ~ C:-2)

~

f!rft

s

or-oter- . Grocfz 1 )

l. b(C j!._j ~

D

~

j~Cr/-k-1

J

q,cJ,

qock) C:t;, lj,)

~r

l C(%))

l 'L)

j2J~'-k Gt k > q ~~-v . t(J ~'l

( C:r,C~J 2 G-o~~

J

kr

to) ([9 CF-,

~ (" +k }e (x0 , t)

(35)

where O(x, t) = A(x, t). Hence we have J1A (x0 , ω) = −

2 e R (x0 , ω) ImD A −β~ω 1−e

6

(36)

2.4 The correlation function 0 † 0 G> A (x0 , t − t ) = hA(x0 , t)A (x0 , t )i 0

0

= hei(Htip +H)t/~ A(x0 )e−i(Htip +H)t/~ ei(Htip +H)t /~ A† (x0 )e−i(Htip +H)t /~ i 0

0

= hei(Htip +H)t/~ ψ † (x0 )ϕ(x0 )e−i(Htip +H)t/~ ei(Htip +H)t /~ ϕ† (x0 )ψ(x0 )e−i(Htip +H)t /~ i 0

0

0

0

= heiHt/~ ψ † (x0 )eiH(t −t)/~ ψ(x0 )e−iHt /~ eiHtip t/~ ϕ(x0 )eiHtip (t −t)/~ ϕ† (x0 )e−iHtip t /~ i (37) When Γ = 0, there is no tunneling and so we can separate the ground state into the tip and the system ground state |Gi = |Gsys i|Gtip i

(38)

Hence 0 † 0 ∗ † 0 G> A (x0 , t − t ) = hGsys |ψ (x0 , t )ψ(x0 , t)|Gsys i hGtip |ϕ(x0 , t)ϕ (x0 , t )|Gtip i 0 ∗ > 0 = G< (39) sys (x0 , t − t ) Gtip (x0 , t − t )

2.5 Notice that J1A (x0 , ω)

Z

∞

=

dteiωt G> A (x0 , t)

Z−∞ ∞

∗ > dteiωt G< sys (x0 , t) Gtip (x0 , t) Z Z Z−∞ ∞ dω 00 t −iω00 t tip dω 0 iω0 t sys 0 iωt e J2 (x0 , ω ) e J1 (x0 , ω 00 ) = dte 2π 2π −∞ Z ∞ Z Z dω 0 iω0 t −β~ω0 sys dω 00 t −iω00 t tip iωt 0 = dte e e J1 (x0 , ω ) e J1 (x0 , ω 00 ) 2π 2π Z−∞ 0 dω sys 0 = J1 (x0 , ω 0 )J1tip (x0 , ω + ω 0 )e−β~ω (40) 2π

=

Hence we have J1A (x0 , ω)

Z =

dΩ sys J (x0 , Ω)J1tip (x0 , ω + Ω)e−β~Ω 2π 1

where Ω = ω 0 . 7

(41)

2.6 According to question 2.2, we have eV 2eΓ2 R eA ImD (x0 , − ) ~ ~  eV eV eΓ2  =− 1 − e−β~(− ~ ) J1A (x0 , − ) ~ ~ Z 2 dΩ −β~Ω 2 eΓ βeV 2 eV eR (x0 , x0 , Ω) eR (e − 1) e ) = ImG eV ImGtip (x0 , x0 , Ω − sys −β~Ω −β~(Ω− ) ~ 2π 1+e ~ ~ 1+e (42)

It =

Notice that 1 eR (x, x, ω) ρtip (x, ω) = − G π tip 1 eR ρsys (x, ω) = − G (x, x, ω) π sys

(43) (44)

We have Z eΓ2 βeV 2 eV dΩ −β~Ω 2 It = (e − 1) e π 2 ρsys (x0 , Ω) ) eV ρtip (x0 , Ω − −β~Ω −β~(Ω− ) ~ 2π 1+e ~ ~ 1+e   Z 2πeΓ2 βeV eV 1 1 = (e − 1) dΩρsys (x0 , Ω)ρtip (x0 , Ω − ) 1− eV ~ ~ eβ~Ω + 1 1+ eβ~(Ω− ~ )   Z eV eV 2πeΓ2 βeV (e − 1) dΩρsys (x0 , Ω)ρtip (x0 , Ω − )f (β~Ω) 1 − f β~(Ω − ) = ~ ~ ~ (45) where f (z) =

ez

1 +1

(46)

2.7 Notice that    Z 2πeΓ2 βeV eV eV (e − 1) dΩρsys (x0 , Ω)ρtip (x0 , Ω − )f (β~Ω) 1 − f β~(Ω − ) It = ~ ~ ~   Z 2πeΓ2 eV 1 1 = dΩρsys (x0 , Ω)ρtip (x0 , Ω − ) − eV ~ ~ 1 + eβ~(Ω− ~ ) eβ~Ω + 1 (47) 8

At T = 0, we have   Z eV 2πeΓ2 eV ) θ( − Ω) − θ(−Ω) It = dΩρsys (x0 , Ω)ρtip (x0 , Ω − ~ ~ ~ Z eV ~ 2πeΓ2 eV = dΩρsys (x0 , Ω)ρtip (x0 , Ω − ) (48) ~ ~ 0

2.8 Assuming that ρtip (x0 , ω) = ρtip (0), we have dIt dV d = dV

Gt (V ) =

2

2πeΓ ~

eV ~

Z

dΩρsys (x0 , Ω)ρtip (x0 , Ω −

0

2πeΓ2 eV d = ρtip (0)ρsys (x0 , ) ~ ~ dV 2πe2 Γ2 eV = ρtip (0)ρsys (x0 , ) 2 ~ ~



eV ~

eV ) ~

!



(49)

2.9 2.9.1

T =0

(a) If ρsys (x0 , ω) = ρsys (0), according to the result in Eq.(49), we have Gt (V ) =

2πe2 Γ2 ρtip (0)ρsys (0) ~2

(50)

(b) If ρsys (x0 , ω) ∝ |ω|α , according to the result in Eq.(49), we have eV 2πe2 Γ2 ρtip (0)ρsys (x0 , ) 2 ~ ~ eV α 2πe2 Γ2 ∝ ρtip (0) ~2 ~ 2 2+α 2πΓ e = ρtip (0)V α ~2+α

Gt (V ) =

9

(51)

(c) If ρsys (x0 , ω) ∝ θ(|ω| − ∆/~)(|ω| − ∆/~)ν , we have 2πe2 Γ2 eV ) ρ (0)ρ (x , tip sys 0 ~2 ~ eV eV 2πe2 Γ2 − ∆/~)ν ∝ ρ (0)θ( − ∆/~)( tip ~ ~ ~2 2πe2+ν Γ2 = ρtip (0)θ(V − ∆/e)(V − ∆/e)ν ~2+ν

Gt (V ) =

2.9.2

(52)

T 6= 0

Using the result in Eq.(45),    Z eV eV 2πeΓ2 βeV (e − 1) dΩρsys (x0 , Ω)ρtip (x0 , Ω − )f (β~Ω) 1 − f β~(Ω − ) It = ~ ~ ~ (53) We can derive that   Z dIt 2πeΓ2 ∂ 1 Gt (V ) = = ρtip (0) dΩρsys (x0 , Ω) dV ~ ∂V 1 + eβ~(Ω− eV~ ) Z eV 2πe2 Γ2 β eβ~(Ω− ~ ) = ρtip (0) dΩρsys (x0 , Ω)  (54) 2 ~ β~(Ω− eV ) ~ 1+e (a) If ρsys (x0 , ω) = ρsys (0), we have 2πe2 Γ2 β ρtip (0) Gt (V ) = ~

Z dΩρsys (x0 , Ω) 

2πe2 Γ2 β = ρtip (0)ρsys (0) ~ =

2πe2 Γ2 ρtip (0)ρsys (0) ~2

10

Z dΩ 

eβ~(Ω−

eV ~

)

β~(Ω− eV ) ~

1+e

eβ~(Ω−

eV ~

1 + eβ~(Ω−

2

)

eV ~

)

2 (55)

In this case, the nonzero temperature has no effect on the conductance. (b) If ρsys (x0 , ω) ∝ |ω|α , we have eβ~(Ω−

2πe2 Γ2 β Gt (V ) = ρtip (0) ~

Z

2πe2 Γ2 β ρtip (0) ∝ ~

Z

eV ~

)

dΩρsys (x0 , Ω)  2 eV 1 + eβ~(Ω− ~ )

2πe2 Γ2 β ρtip (0) = ~

eβ~(Ω−

α

dΩ|Ω| 

)

β~(Ω− eV ) ~

1+e

∞

Z

eV ~

dΩΩα

0

eβ~(−Ω−

eV ~

1 + eβ~(−Ω−

2

)

∞

Z

eV ~

)

dΩΩα

+ 0

eβ~(−Ω+

eV ~

(1 + eβ~(−Ω+ (56)

!

)

eV ~

) 2 )

At finite temperature, the simple power law disappears. (c) If ρsys (x0 , ω) ∝ θ(|ω| − ∆/~)(|ω| − ∆/~)ν , we have 2πe2 Γ2 β ρtip (0) Gt (V ) = ~

Z

2πe2 Γ2 β ∝ ρtip (0) ~

Z

dΩρsys (x0 , Ω) 

eβ~(Ω−

eV ~

1 + eβ~(Ω−

)

eV ~

)

2 eβ~(Ω−

ν

dΩθ(|Ω| − ∆/~)(|Ω| − ∆/~) 

1+e

eV ~

)

β~(Ω− eV ) ~

2

(57) We can split the above integral into two parts (1) Ω ∈ [−∞, −∆/~] and (2) Ω ∈ [∆/~, ∞]. The final result is 2πe2 Γ2 β ρtip (0) Gt (V ) ∝ ~

Z 0

∞

dΩΩν

eβ~(−Ω− (1 +

eV ~

−∆/~)

eV eβ~(−Ω− ~ −∆/~) )2

Z + 0

∞

dΩΩν

eβ~(Ω−

(1 + eβ~(Ω− (58)

From the above result, we can see that at finite temperature, we do not have a simple power law anymore. We have an infinite sum over positive frequencies. Furthermore, it is also determined by the temperature and the gap ∆.

11

eV ~

+∆/~)

eV ~

+∆/~) 2 )

!

Physics 561, Fall Semester 2015 Professor Eduardo Fradkin Problem Set No. 4: One-dimensional conductors Due Date: Wednesday December 2, 2015 Polyacetylene is a long polymer chain of the type (CH)n . The motion of the conduction electrons in polyacetylene can be described by the following model, due to W.P. Su, J.R. Schrieffer and A. Heeger. [Some general references for this problem are, W. P. Su, J. R. Schrieffer and A. Heeger, Phys. Rev. B 22, 2099 (1980), R. Jackiw and J. R. Schrieffer, Nucl. Phys. B 190[FS3], 253 (1981), and E. Fradkin and J. E. Hirsch, Phys. Rev. B 27, 1680 (1983).] In this model, one considers a linear chain of carbon atoms, C, with classical equilibrium positions at the regularly spaced sites {x(n)|x(n) = n a0 } (where a0 is the lattice constant). The carbon atoms share their electrons in their π-orbitals, and there is one such electron per carbon atom. These electrons are allowed to hop from site to site. This hopping process is modulated by the lattice vibrations. Since the mass M of the atoms is much larger than the mass of the electrons (or, what is the same, the tunneling hopping (kinetic) energy t of the electrons is much larger than the kinetic energy of the atoms), we can give an approximate description by treating the atoms classically while treating the electrons quantum mechanically. In this problem set we will consider a simplified version of this problem in which the electrons are considered to be spinless (i.e., their spins have been fully polarized by a magnetic field). The Hamiltonian for this system on a one-dimensional lattice (a chain) with N (even) sites, is N

H

= −

2 X

n=− N 2 +1 N

+

2 X

n=− N 2 +1



  [t¯ − α (un − un+1 )] c† (n)c(n + 1) + h.c.

Pn2 K 2 + (un − un+1 ) 2M 2



(1)

where c† (n) and c(n) are fermion operators which create and destroy a π (spinless) electron at the nth site of the chain, un ≡ u[x(n)] are the displacements of the coordinates of the carbon atoms measured from their classical equilibrium positions x(n), P (n) are their momenta, M is the carbon mass, K is the elastic constant, t¯ is the electron hopping matrix element for the undistorted lattice, and α is the electron-phonon coupling constant. Polyacetylene has one electron per carbon atom and, hence, it is a half-filled band system: there are N electrons in a chain with N atoms. The study of this problem is greatly simplified by considering a continuum version of the model. If the coupling constant α is not too large, the only 1

physical processes which are important are those which mix nearly degenerate states, i.e., the only electronic states that will matter are those within a narrow band of width 2Ec centered at the Fermi energy EF = 0, where Ec is a high energy cutoff. In this limit, the single particle dispersion law can be linearized, E(p) ≃ vF (p ± pF ). These electronic states can be regarded as right moving electrons ( with p ≃ pF ) and left moving electrons ( with p ≃ −pF ). Here vF is the Fermi velocity. These considerations motivate the following way of writing the electron operators i 1 h ipF x(n) c[x(n)] = √ e R[x(n)] + e−ipF x(n) L[x(n)] (2) 2a0

Likewise, since the only processes in which phonons mix electrons near ±pF have momentum q ≃ 0 (forward scattering) or q ≃ 2pF = π/a0 (backward scattering), it is also natural to split the phonon fields into two terms u[x(n)] =u0 [x(n)] + e2ipF x(n) ∆+ [x(n)] + e−2ipF x(n) ∆− [x(n)] ≡u0 [x(n)] + cos(πn)∆[x(n)]

(3)

where u0 [x(n)] represents phonons with wave-vectors close to q = 0, while ∆± (n) represent phonons with wave-vectors close to q = ±2pF . For a half-filled band (i.e., one electron per carbon site), pF = 2aπ0 , there is only one slow phonon field ∆± [x(n)] ≡ ∆[x(n)] since in this case q ≃ 2pF = π/a0 and it differs from −q by a reciprocal lattice vector, 2π/a0 . Within these approximations, the effective continuum Hamiltonian only involves the left and right moving fermions and the phonons with q ≃ aπ0 :   Z X −ivF ∂x g∆(x) H= ψ ′ (x) = R, L dx ψα† (x) g∆(x) +ivF ∂x α,α′ α α,α′   Z 1 2 Π2 (x) + dx + ∆ (x) (4) 2(M/4K) 2 where the two-component spinors ψ(x) and ψ † (x) are  
R(x) ψ(x) ≡ , ψ † (x) = R† (x), L† (x) L(x)

(5)

is a Fermi field which represents the right and left moving electrons close to the Fermi energy, which obey equal-time canonical anticommutation relations n o ψα (x), ψα† ′ (x′ ) = δαα′ δ(x − x′ ), {ψα (x), ψα′ (x′ )} = 0 (6)

Here, α = R is the right moving component of the fermion and α = L is the left moving component. The Bose field ∆(x) represents lattice vibrations with momentum close to 2pF , and it represents small fluctuations around a staggered distortion of the position of the atoms. Π(x) is the canonical momentum conjugate to the 2pF phonon field, and obey equal-time canonical commutation relations [∆(x), Π(y)] = iδ(x − y) (7) 2

The effective electron-phonon coupling constant is g ∝

√α . Dt¯

1. Consider first the case of zero electron-phonon coupling, g = 0. Compute in this limit the following propagators at T = 0 (a) The propagator for right-moving electrons Z Z ′ ′ SRR (p, ω) = dx dtei(p(x−x )−ω(t−t ) hT R(x, t)R† (x′ , t′ )i (b) The propagator for left-moving electrons Z Z ′ ′ SLL (p, ω) = dx dtei(p(x−x )−ω(t−t ) hT L(x, t)L† (x′ , t′ )i (c) The propagator for the 2pF phonon field ∆, Z Z ′ ′ G∆ (p, ω) = dx dtei(p(x−x )−ω(t−t ) hT ∆(x, t)∆(x′ , t′ )i

(8)

(9)

(10)

2. Determine the form of the electron-phonon vertex shown in Fig.1

L

R ∆

∆

L

R

Figure 1: The electron-phonon vertex . 3. Consider the limit in which the mass of the carbon atoms is very heavy, i.e., M → ∞. In this adiabatic limit, the phonon kinetic energy term of the effective Hamiltonian of Eq.(4) can be neglected compared to the other terms and the phonon fields (displacements) ∆(x) become classical variables. Find the ground state vector |gndi and energy of the system in this limit by calculating the (constant) value ±∆0 for which the total ground state energy (i.e. fermions plus vibrations) is minimized. You will have to cutoff some of the integrals at a relative momenta ±Ec /pF . Note: This spontaneous staggered distortion of the lattice is called dimerization. This phenomenon is known as the Peierls Instability. This dimerized ground state is a bond charge-density wave (CDW). 3

4. Compute the fermion propagator Sαα′ (p, ω) in momentum and frequency space within this approximation. Show that for g 6= 0 right and left moving fermions are coupled and that as a result there is non-vanishing component for SRL (p, ω). Find explicit expressions for all the four components of Sαα′ (p, ω) in the adiabatic approximation. 5. Determine the energy spectrum and quantum numbers of the single-particle electronic states in this approximation. Show that the spectrum of single particle excitations (quasiparticles) has a gap. Find a formula for the gap in terms of the effective coupling constant g and ∆0 . What is the total functional dependence of the gap on the coupling constant g? 6. Show that the Hamiltonian of Eq.(4) is invariant under the global discrete symmetry transformation     R(x) R(x) → , ∆(x) → −∆(x) (11) L(x) −L(x) Show that, in terms of the lattice model, this transformation is equivalent to a translation of all the fields by one lattice spacing, i.e., half of the period of the dimerization. Show that the local operator OCDW (x) = R† (x)L(x) + L† (x)R(x)

(12)

is odd (i.e., changes sign) under this global discrete symmetry and hence it is the bond CDW order parameter. 7. Compute the ground state expectation value for the charge density wave order parameter OCDW we just defined in the M → ∞ approximation. Show that this order parameter has non-vanishing expectation value only if ∆0 6= 0 and establish a connection between both quantities. 8. Use the bosonization methods discussed in class to find an expression for the bond CDW order parameter in terms of the (charge) Bose field φ of the bosonized fermions. 9. Use the result of item 8 (and the results derived in class for spinless Luttinger liquids) to find the bosonized Hamiltonian for the SSH Hamiltonian of Eq.(4). Show that this bosonized Hamiltonian √ is invariant under the symmetry ∆(x) 7→ −∆(x) and φ(x) 7→ φ(x) + π/2. Interpret this symmetry in terms of the dimerization pattern. 10. In this last question we will consider a classical configuration of the phonon field ∆(x) that satisfies the boundary conditions limx→∞ ∆(x) = ∆0 limx→−∞ ∆(x) = −∆0 and interpolates smoothly and monotonically between these two limits. This state is known as a soliton. Let us consider the one-dimensional Dirac Hamiltonian that we used in Eq.(4)   −ivF ∂x g∆(x) (13) hDirac = g∆(x) +ivF ∂x 4

(a) Show that if ∆(x) is a soliton configuration, then the Dirac Hamiltonian of Eq.(13) has a square-integrable (normalizable) eigenvector with vanishing eigenvalue (i.e., a fermion zero-mode) of the form   R −f (x) ψ(x) = e (14) L where R and L are constants. Find explicit expressions for the constants R, L and the function f (x) in terms of the coupling constant g, the Fermi velocity vF and the soliton configuration ∆(x). In order to obtain simple expressions it is useful to make the “thin-soliton” approximation, ∆(x) = ∆0 sign(x) (15) (b) Use the expression for the total charge in terms of the boson φ(x) obtained by bosonization to show that the charge of the soliton is 1/2 (in units of the electric charge e).

5

Solution 4 December 6, 2015

1

One-dimensional conductors

1.1 In the g = 0 limit, the Hamiltonian is   Z Z  †  Π2 (x) 1 † H = dx R (x)(−ivF ∂x )R(x) + L (x)(ivF ∂x )L(x) + dx + 2(M/4K) 2∆2 (x) (1) (a) Suppose Z R(x, t) =

dq R(q, t)ei(qx−E(q)t) 2π

(2)

where E(q) can be obtained from diagonalizing the Hamiltonian in momentum space and E(q) = qvF . The propagator for the right-moving electron is hT R(x, t)R† (x0 , t0 )i = θ(t − t0 )hR(x, t)Rdag (x0 , t0 )i − θ(t0 − t)hR† (x0 , t)R(x, t)i Z  dqdq 0 iq(x−vF t)−iq0 (x0 −vF t0 )  0 † 0 0 0 † 0 0 e θ(t − t )hR(q, t)R (q , t )i − θ(t − t)hR (q , t )R(q, t)i = (2π)2 Z dq iq(x−vF t−x0 +vF t0 ) = e [θ(t − t0 )θ(q) − θ(t0 − t)θ(−q)] (3) 2π

1

Performing Fourier transformation for the above result, we have Z Z 0 0 SRR (p, ω) = dx dte−i[p(x−x )−ω(t−t )] hT R(x, t)R† (x0 , t0 )i Z Z Z dq iq(x−x0 )−iqvF (t−t0 ) −i[p(x−x0 )−ω(t−t0 )] = dx dte e [θ(t − t0 )θ(q) − θ(t0 − t)θ(−q)] 2π Z ∞ Z 0 = dtei(ω−pvF +i) tθ(p) − dtei(ω−pvF −i)t θ(−p) −∞

0

=

iθ(p) iθ(−p) + ω − pvF + i ω − pvF − i

(4)

(b) Similarly, we have Z L(x, t) =

dq R(q, t)ei(qx−E(q)t) 2π

(5)

where E(q) = −qvF . Therefore we can just take vF → −vF and calculate the propagator for the left-moving electrons similarly Z dq iq(x+vF t−x0 −vF t0 ) † 0 0 hT L(x, t)L (x , t )i = e [θ(t − t0 )θ(−q) − θ(t0 − t)θ(q)] 2π (6) Then we have SLL (p, ω) =

iθ(p) iθ(−p) + ω + pvF + i ω + pvF − i

(c) The Hamiltonian for phonon is Z 1 4K H∆ (x) = dx(Π2 (x) + ω02 ∆2 (x)) 2M p where ω0 = M/4K. Define Z dq a(x, t) = a(q, t)eiqx−t/ω0 2π Z dq 0 † 0 0 −i(q0 x0 −t0 /ω0 ) a† (x0 , t0 ) = a (q , t )e 2π 2

(7)

(8)

(9)

The above Hamiltonian can be written as Z 1 1 H∆ (x) = dx (a† (x)a(x) + ) ω0 2

(10)

The propagator for the phonon field is 1 hT ∆(x, t)∆(x0 , t0 )i = hT (a(x, t) + a† (x, t))(a(x0 , t0 ) + a† (x0 , t0 ))i ω0   Z Z dq −iq(x−x0 )+i(t−t0 )/ω0 1 dq iq(x−x0 )−i(t−t0 )/ω0 0−t 0 = + θ(t ) θ(t − t ) e e 2ω0 2π 2π (11) Hence we have Z

Z

0

0

G∆ (p, ω) = dx dte−i[p(x−x )−ω(t−t )] hT ∆(x, t)∆(x0 , t0 )i Z ∞  Z 0 1 iωt −it/ω0 −t iωt it/ω0 t = dte e e − dte e e 2ω0 0 −∞   1 1 i − = 2ω0 ω − 1/ω0 + i ω + 1/ω0 − i i = M 2 ω − 1 + i 4K

(12)

1.2 For the electron-phonon vertex shown in Fig.1, it represents the interaction between electron and phonon, the corresponding interaction terms are g∆(x)L† R g∆(x)R† L

(13)

1.3 In the limit M → ∞, the Hamiltonian is   Z Z 1 −ivF ∂x g∆(x) † H = dxψ (x) ψ(x) + dx ∆2 g∆(x) ivF ∂x 2

3

(14)

Use the Fourier expansion Z ψ(x) =

dp ψ(p)ipx 2π

The Hamitonian can then be written as   X Z dp 1 pvF g∆ † H= ψp,α ψp,α0 + N a0 ∆2 g∆ −pvF α,α0 2π 2 α,α0 =R,L This Hamiltonian can be diagonalized by defining r r E + pvF E − pvF Rp + Lp ap = 2E 2E r E − pvF E + pvF bp = − Rp + Lp 2E 2E

(15)

(16)

(17)

Therefore, we have Z H=

   dp † †
E 0 1 ap ap b p + N a0 ∆2 0 −E b 2π 2 p

(18)

where E=

q

p2 vF2 + g 2 ∆2

(19)

After diagonalizing the Hamiltonian, we see that there are both positive and negative energy excitations. The negative energy excitation looks like excitation from holes and we can perform a particle-hole transformation by taking bp = d†p . Then we have Z dp 1 H= E(p)(a†p ap + d†p dp − 1) + N a0 ∆2 (20) 2π 2 The ground state is defined by ap |GSi = 0 dp |GSi = 0 The ground state energy is Z Ec /pF q 1 dp p2 vF2 + g 2 ∆2 + N a0 ∆2 Egnd = − 2 −Ec /pF 2π 4

(21)

(22)

To minimize the energy E0 , we need to find ∆, which satisfies dEgnd =0 d∆

(23)

This leads to g2∆ N a0 ∆ − arcsinh πvF



Ec vF g|∆|pF

 =0

(24)

Hence we have 1 Ec vF   gpF sinh N a0 πvF g2   2Ec vF N a0 πvF ≈± exp − gpF g2

∆0 = ±

(25)

1.4 The fermion propagator Z dqdq 0 iqx−iq0 x0 † 0 0 e hT R(q, t)L† (q 0 , t0 )i hT R(x, t)L (x , t )i = (2π)2 Z  dqdq 0 iqx−iq0 x0  0 † 0 0 0 † 0 0 e θ(t − t )hR(q, t)L (q , t )i − θ(t − t)hL (q , t )R(q, t)i = (2π)2 (26) According to Eq.(17), we have r r E + pvF E − pvF Rp = ap − bp 2E 2E r r E − pvF E + pvF Lp = ap + bp 2E 2E

(27)

Using the above equations, we can show that hR(q, t)L† (q 0 , t0 )i =

g∆ 0 δ(q − q 0 )2πe−iE(q)(t−t ) 2E(q)

(28)

g∆ 0 δ(q − q 0 )2πeiE(q)(t−t ) 2E(q)

(29)

Simiarly, we have hL† (q 0 , t0 )R(q, t)i = −

5

Therefore we have  dqdq 0 iqx−iq0 x0  0 † 0 0 0 † 0 0 hT R(x, t)L (x , t )i = e θ(t − t )hR(q, t)L (q , t )i − θ(t − t)hL (q , t )R(q, t)i (2π)2 Z h i dq iq(x−x0 ) g∆ 0 0 e θ(t − t0 )e−iE(q)(t−t ) + θ(t0 − t)eiE(q)(t−t ) (30) = 2π 2E †

0

0

Z

In momentum space, it becomes Z Z Z i dq iq(x−x0 ) g∆ h 0 0 −i[p(x−x0 )−ω(t−t0 )] SRL (p, ω) = dx dte e θ(t − t0 )e−iE(q)(t−t ) + θ(t0 − t)eiE(q)(t−t ) 2π 2E  Z ∞ Z 0 g∆ dteiωt e−t e−iE(p)t + dteiωt et eiE(p)t = 2E(p) 0 −∞   ig∆ 1 1 = − (31) 2E(p) ω − E(p) + i ω + E(p) − i Thus when g 6= 0, the left and right movers are coupled together and there is non-zero component for SRL (p, ω). Similarly, we can calculate the fermion propagator for the right mover hT L(x, t)R† (x0 , t0 )i = hT R(x, t)L† (x0 , t0 )i

(32)

This leads to ig∆ SLR (p, ω) = 2E(p)



1 1 − ω − E(p) + i ω + E(p) − i

 (33)

Using the same method, we can show that hT R(x, t)R† (x0 , t0 )   Z dq iq(x−x0 ) E − qvF iE(q)(t−t0 ) 0 E + qvF −iE(q)(t−t0 ) 0 e θ(t − t ) e − θ(t − t) e = 2π 2E 2E (34) In momentum space, it equals to SRR (p, ω) =

E(p) + pvF i E(p) − pvF i + 2E(p) ω − E(p) + i 2E(p) ω + E(p) − i

6

(35)

The fermion propagator for the left mover is hT L(x, t)L† (x0 , t0 )   Z dq iq(x−x0 ) E + qvF iE(q)(t−t0 ) 0 E − qvF −iE(q)(t−t0 ) 0 = e θ(t − t ) e − θ(t − t) e 2π 2E 2E (36) In momentum space, it equals to SLL (p, ω) =

i E(p) + pvF i E(p) − pvF + 2E(p) ω − E(p) + i 2E(p) ω + E(p) − i

(37)

1.5 The energy spectrum is p E(p)± = ± (vF p)2 + (g∆)2

(38)

There is an energy gap because   N a0 πvF 4Ec vF exp − E(p = 0)+ − E(p = 0)− = 2g∆ ≈ pF g2

(39)

The single particle state can be represented as a†p |GSi,

d†p |GSi

(40)

1.6 Under the global discrete transformation, R(x) → R(x),

L(x) → −L(x),

∆(x) → −∆(x),

(41)

The Hamiltonian becomes    X Z dp
−ivF ∂x −g∆ R(x) † † R (x) −L (x) + ··· H= −g∆ ivF ∂x −L(x) 2π α,α0 =R,L (42) It is identical to the original Hamiltonian and thus the Hamiltonian is invariant under the global discrete symmetry transformation. 7

In terms of the lattice model,  1  ipF x(n) e R + e−ipF x(n) L c[x(n)] = √ 2a0

(43)

under the global Z2 transformation,  1  ipF x(n) e R − e−ipF x(n) L 2a0 = (−i)c[x(n + 1)]

(44)

u[x(n)] = u0 + cos(πn)∆

(45)

c0 [x(n)] = √

where we use pF = π/2. For u[x(n)],

under the Z2 transformation, u0 [x(n)] = u0 − cos(πn)∆ = u[x(n + 1)]

(46)

Therefore this transformation is equivalent to a translation of all the fields by one lattice spacing. For the CDW order parameter, under the Z2 transformation, we have 0 OCDW (x) = −R† (x)L(x) − L† (x)R(x) = −OCDW (x)

(47)

It is odd under the global discrete symmetry.

1.7 The ground state expectation value for OCDW in the M → ∞ limit is hGS|OCDW |GSi = hGS|R† (x)L(x) + L† (x)R(x)|GSi Z Ec /pF Z Ec /pF dp g∆0 dp g∆0 =− − Ec /pF 2π 2E(p) Ec /pF 2π 2E(p) Z Ec /pF dp g∆ N a0 ∆0 =− =− 2 2 2 2 g −Ec /pF 2π g ∆0 + p vF

(48)

This suggests that the order parameter has non-vanishing expectation value only if ∆ 6= 0. 8

1.8 Using the bosonization method, we have √ 1 : ei2 πφR (x) : R(x) ∼ √ 2πa0 √ 1 L(x) ∼ √ : e−i2 πφL (x) : 2πa0 where

√ Z x1 π β φR (x) = √ φ(x) − dx01 Π(x0 , x01 ) β 4 π √ Z −∞ β π x1 0 dx Π(x0 , x01 ) φL (x) = √ φ(x) + β −∞ 1 4 π

(49)

(50)

We can derive that 1 −i2√πφ(x) 1 i2√πφ(x) e + e 2πa0 2πa0 √ 1 = cos(2 πφ(x)) πa0

R† (x)L(x) + L† (x)R(x) =

(51)

1.9 According to the rule in Eq.(49), the bosonized Hamiltonian is     Z  X Z dp
−ivF ∂x −g∆ 1 2 Π2 R(x) † † R (x) −L (x) + ∆ H= + dx −g∆ ivF ∂x −L(x) 2π 2M/4K 2 0 α,α =R,L   Z Z
g∆(x) √ vF Π2 1 2 2 2 = dx cos(2 πφ) + dx + ∆ Πφ + (∂x φ) + 2 πa0 2M/4K 2 (52) This √ Hamiltonian is invariant under the symmetry ∆ → −∆ and φ → φ + π/2. This symmetry is equivalent to the chiral symmetry defined in the fermion model and corresponds to two dimerization pattern.

1.10 (a) For the Dirac Hamiltonian  HDirac =

 −ivF ∂x g∆(x) g∆(x) ivF ∂x 9

(53)

When ∆(x) takes a soliton configuration, there is a fermionic zero mode of the form   R −f (x) ψ(x) = e (54) L To find the explicit form of f (x) function, we need to solve the following equation    −ivF ∂x g∆(x) R −f (x) e =0 (55) g∆(x) ivF ∂x L This leads to f 0 (x) =

ig∆(x)L ig∆(x)R =− vF R vF L

(56)

Solving this equation gives R/L = ±i. When R/L = i, we have f 0 (x) =

g∆(x) vF

In the thin-soliton approximation ∆(x) = ∆0 sign(x), we have r g g∆0 f (x) = ∆0 |x|, L = vF 2vF Therefore the zero mode wavefunction is normalizable and equals to  r g∆0 − vg ∆0 |x| i e F ψ(x) = 1 2vF

(57)

(58)

(59)

When R/L = −i, f (x) = − vgF ∆0 |x|, the solution is not normalizable and is non-physical. (b) The charge in terms of boson is Z Z e e Q = −e dxj0 (x) = − √ dx∂x φ(x) = − √ (φ(∞) − φ(−∞)) (60) π π √ We know the Hamiltonian is invariant under ∆ → −∆ and φ → φ + π/2. For the soliton configuration, ∆(x = ∞) = −∆(x = −∞), thus we have √ φ(−∞) = φ(∞) + π/2. The charge Q is equal to e Q= (61) 2 This demonstrates that the solition carries fractional charge. 10