Fortschritte der Physik / Progress of Physics: Band 28, Heft 7 1980 [Reprint 2021 ed.] 9783112522929, 9783112522912

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FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM AUFTRAGE D E R P H Y S I K A L I S C H E N GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN R E P U B L I K VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

H E F T 7 • 1980 • B A N D 28

A K A D E M I E

-

V E R L A G

EVP 1 0 , - M 31728

•

B E R L I N

ISSN 0015-8208

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Zeitschrift „Fortschritte der Physik" Herauageber: Prof; Dr. Frank Ksschhihn, Prof. Dr. i m * U n h , ProL Dr. Rudolf Ritsehl, Prof. Dr. Robart Rompo, im A n f i n g der PhyiikallichcQ Geeellschaft der Deutschen Demokratischen Republik. Verlag! Akademie-Verlag, DDR -1080 Berlin, Leipziger StraBe 3 - 4 ; Fernruf! 22 36231 und 1136 >29; Telex-Nr. 114410; Baak: Staatsbank der DDR, Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr. L a u Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Üniversitlt su Bariin, DDR - 1040 Berlin, Hessische StraBe 2. Veröffentlicht unter der T.i«wiannmmcr 1324 dee Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: VEB Druckhaus „Maxim Gorki", DDR - 7400 Altenburg, Carl-von-Ossietzky-StraBe 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik** erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band: 180,— M zuzOglioh Versandspesen (Preis für die DDR: 120,— M). Preis je Heft IS,— M (Preis für die DDR: 1 0 , - H). Bestellnummer dieses Heftee: 1027/28/7. © 1980 by Akademie-Verlag Berlin. Printed in the German Democratie Republie. AN (EDV) 57618

Fortschritte der Physik 28, 3 5 5 - 4 2 6 (1980)

Electrodynamics of Superconductors as a Consequence of Local Gauge Invariance M . F U S C O - G I R A R D , F . MANCINI a n d M . MABINARO

Scuola di Perfezionamento in Scienze Cibernetiche e Fìsiche Fisiche dell'Università Gruppo Nazionale di Struttura della Materia del CNR

di

Salerno1)

Abstract The electromagnetic properties of superconductors are studied in the framework of a quantum gauge-invariant theory. The formulation is developed in the generalized pair approximation which preserves the WardTakahashi identities. The macroscopic equations which regulate current and electromagnetic fields are derived by means of the boson transformation method. Comparison with previous works is reported.

Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Introduction Invariance Properties and WT-identities Equations for the Green's Functions in Pair Approximation Calculation of the Quantities Pt"(x, y; z) and y) Calculation of the Matrices G(x, y; z, w), X(x, y; z) and J(x, x; y) Invariance Properties for the Approximate Solution The Elementary Excitations The Quasi-Particles • Dynamical Mapping Boson Transformation Macroscopic Equations Ground State Energy Conclusions Appendix A Appendix B Appendix C Appendix D Appendix E References

Postai address: Facoltà di Scienze, Università di Salerno, 84100 Salerno, Italy 23

Zeitschritt „Fortschritte der Physik", Heft 7

356 359 364 368 372 375 377 380 383 388 392 396 399 402 405 412 418 421 425

Fortschritte der Physik 28, 3 5 5 - 4 2 6 (1980)

Electrodynamics of Superconductors as a Consequence of Local Gauge Invariance M . F U S C O - G I R A R D , F . MANCINI a n d M . MABINARO

Scuola di Perfezionamento in Scienze Cibernetiche e Fìsiche Fisiche dell'Università Gruppo Nazionale di Struttura della Materia del CNR

di

Salerno1)

Abstract The electromagnetic properties of superconductors are studied in the framework of a quantum gauge-invariant theory. The formulation is developed in the generalized pair approximation which preserves the WardTakahashi identities. The macroscopic equations which regulate current and electromagnetic fields are derived by means of the boson transformation method. Comparison with previous works is reported.

Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Introduction Invariance Properties and WT-identities Equations for the Green's Functions in Pair Approximation Calculation of the Quantities Pt"(x, y; z) and y) Calculation of the Matrices G(x, y; z, w), X(x, y; z) and J(x, x; y) Invariance Properties for the Approximate Solution The Elementary Excitations The Quasi-Particles • Dynamical Mapping Boson Transformation Macroscopic Equations Ground State Energy Conclusions Appendix A Appendix B Appendix C Appendix D Appendix E References

Postai address: Facoltà di Scienze, Università di Salerno, 84100 Salerno, Italy 23

Zeitschritt „Fortschritte der Physik", Heft 7

356 359 364 368 372 375 377 380 383 388 392 396 399 402 405 412 418 421 425

356

M. Fuscp-GIRABD, F. MANCINI and M . MARINABO

1. Introduction Since the discovery of the phenomenon of superconductivity [1], many experimental and theoretical works [2] have been devoted to the analysis of the laws which govern the macroscopic behaviour of a superconductor in the presence of external electromagnetic fields. Up to the present, the most complete formulation from a theoretical point of view of the superconducting state is given by the Gor'kov equations [3]. These are a set of non-linear integro-differential equations which, once solved under the boundary conditions appropriate to the physical situation, would give the macroscopic behaviour of the superconducting system. However, it is a very difficult task to solve the Gor'kov' equations, and analysis is restricted to situations where the order parameter is small (i.e. near the critical temperature: Ginzburg-Landau equations [4]; for H •—• HCt in the mixed state of t y p e - I I superconductors, H W M T E theory [5], etc.) or when the spatial variations of the order parameter are negligible. Only recently [6], numerical solutions of the Gor'kov equations have been obtained to study an isolated flux line in the mixed state. For this reason part of the theoretical results about the electromagnetic behaviour of superconductors has been obtained by using the linear-response method [7] and by computing the Meissner kernel in some approximation (LONDON [5], BCS [9], RICKAYZEN [10] approximation s). However, on this approach attention must be paid to the introduction of the boundary conditions which describe correctly the effect of the external fields. Some years ago an alternative method of studying a superconducting System was introduced [11—12], This method, the boson transformation method, is based on the physical point of view that the macroscopic properties of a superconductor, which is a system exhibiting a macroscopic ordered state, are in large part controlled by the condensation of collective modes (Goldstone particles). Therefore, to obtain the macroscopic behaviour of a superconductor under specific physical boundary conditions, it is sufficient to operate directly on the condensate phase. A n appropriate phase condensation will correspond to each physical boundary condition. The mathematical tool which allows us to formalize this point of view is the so-called boson transformation [ I I ] . In previous works [13,14] it has been shown that this alternative approach to formulating a theory of superconductivity is particularly useful when one is interested in studying situations in which the ground state is non homogeneous (the order parameter is a function of x and t). The reason for this is connected to the fact that by using the boson transformation, the problem of introducing the boundary conditions is very much simplified; indeed, these conditions are introduced directly at the level of the condensate phase. The aim of this paper is to present, in the framework of the boson transformation method, a complete derivation of the macroscopic equations which govern the e.m. behaviour of a superconductor in a generalized pair approximation, which preserves the gauge invariance properties of the theory. We assume that the effects of external fields are small, so that we can confine ourselves to considering only linear approximation. The superconducting model that we investigate is a charged superconductor in which both the electron field and the photon field are quantized and interact through a gauge invariant Hamiltonian. This paper completes the study performed in two previous works. In the first [12], the macroscopic equations were derived in the case of a neutral superconductor (the e.m. field was considered as a classical field). In the second [15], the structure of the macroscopic equations was found by means of general arguments based on the invariance properties of the Lagrangian. As will be clear from the following, the generalization to

Electrodynamics of Superconductors

357

charged superconductors is not trivial, both for practical and general reasons. Indeed, from a general point of view, it gives a more complete description of the ground state and of the elementary excitations of the system. (In particular we obtain a deeper interpretation of the Higgs phenomenon in superconductivity). With respect to practical computations, the macroscopic equations that we obtain are different from those computed in the neutral case. I n addition, we show that the invariance properties are not sufficient to fix in a unique way the structure of the macroscopic equations, and some assumptions need to be introduced [J5]. These assumptions may not be verified in a specific model. Therefore, we conclude that only by fixing a specific dynamical model and solving in some approximation the equations of motion can one uniquely determine the macroscopic behaviour. This paper is divided into three parts. The first part, from section 2 to section 6, contains all the dynamical calculations which serve as preparation for the remaining part of the work; the reader who is more interested to the physical aspects of the theory can skip this part and go directly to section 7. In section 2 we fix the dynamical model which is described by the gauge-invariant Lagrangian density: Jf(x) = Te(x) + Xea(x) + ^int(z) (1-1) i

where £e(z) is the BCS Lagrangian for the electron field; -fem(x) is the free Lagrangian of the electromagnetic field; and £int(a;) describes the interaction of the electron field with the photon field, realized through the minimal electromagnetic interaction. The properties of invariance of this model are expressed through the Ward-Takahashi (WT) identities [16]. In section 3, the pair approximation is introduced and the equations for the Green functions are given. Section 4 and 5 are devoted to the solutions of these equations. In section 6 by using the W T identities, we show that the approximation preserves the invariance properties of the initial Lagrangian. Once we have prepared all the material, in the second and third parts we proceed with the formulation of the boson method. This method is essentially based on two steps: i) Dynamical map ii) Boson transformation. The first step is realized in the second part, from section 7 to section 9. In section 7 and 8, we compute the elementary excitations (poles of the Green's functions). The results are the following: theré are four elementary excitations (quasi-particles) a) Quasi-electron fields

and

with energy spectrum (1.2)

Ek = 1U k 2 + A2

where h2

ek = —— (le2 — kF2); A = energy gap; hkF — Fermi momentum ÁlYfb

(1.3)

b) Boson field B(x) (Goldstone particle), with a gapless energy spectrum ho>B(k2) which tends to 2A when h -> oc. c) Ghost field b(x) with the same energy spectrum of B(x); it is a ghost since it has a negative norm:

(1.4)

Plasmon field U^x) which is a vector field whose longitudinal and transverse components have energy spectra hwL(k2) and hcoT(k2), respectively; these energy spectra have 23*

M. Fusco-Girard, F. Mancini and M. Mabinabo

358

a gap, given by the plasma energy hcop = hcjXL; in the range of values of k that are of physical interest A>L is constant, while OJT increases as ft2. Section 9 is devoted to the computation of the dynamical map; i.e., to perform the expansion of the original Heisenberg fields y>, A^, J ^ . . . in terms of which the lagrangian is written, in normal products of free fields (quasi-particles). The result is the following:

0(x) = 4

d^B - b), U,,)

A

(1.5)

A,(x) = ftcVH-n WW + zv*(-r*) uv(x) + JM)

(4jt)_1 mle{-V*)

= v ( - n D„(d) [B{x) - b(x)] -

where

(1.6)

U,(x) + ...

9(x) = Vt(x)n{x).

(1.7)

(1.8)

Ap is the vector electromagnetic field and J,, is the current density. The dots in (1.6) and (1.7) denote higher-order normal products of free fields. The derivative operators rj, Z„„ D^ m\v are defined by Eqs. (9.17), (9.30), (2.10a), (9.29), respectively. The dynamical map has to be read as a weak relation, the Hilbert space being chosen as the Foek space of the quasi-particles B, b, U^, 0. In the following, we restrict the Fock space by requiring that for any physical state |«) the following condition holds: (B - &) |o>. (1.9) The symbol (—) means the annihilation part of the operators B and b. The Fock space so obtained is called the "physical space". We note that the quasi-particles B and b appear in the observable quantities always in the combination B—b, and therefore they do not contribute to matrix elements on the physical space; this result can be considered as the counterpart of the Higgs phenomenon in our approach. All the results found in the first and second parts are referred to an ideal infinite superconductor : the order parameter is constant and the ground state expectation values of current and electromagnetic field vanish. The extension to the case in which the superconductor has finite dimension and is in the presence of external e.m. fields is accomplished in part three. In section 10, we introduce the boson transformation; that is we transform the Goldstone and plasmon fields as follows D„(8) B(x) -> D^d) B(x) + (ec) -1 V(-V2)

U,(x)

2

p2

U,(x) + V' (- )

D„(8) f(x)

«>(*)

(1.10)

i1-11)

where f(x) and av(x) are c-number functions which satisfy the following equations: D(8)f(x) 1

A^(8)a,(x) = -^(ec)-

V

2

= 0 2

(-F )

(1.12)

D,(d)f(x).

(1.13)

The differential operators D(8), A^{8) are defined by Eqs. (2.38) and (8.22). The transformation of B and U^ induces through the dynamical map a change of the operators AM and J M which acquire, after the transformation, vacuum expectation values different from zero:

A:„ J*

A/; (A/(x)) = a^x) = - 4 - (8%, -

(1.14) d„d,) a,(x).

(1.15)

359

Electrodynamics of Superconductors

(Apt) and { J J ) are connected b y the Maxwell equation and can be interpreted as the macroscopic vector potential and current in the superconductor. In conclusion, the boson transformation induces a ground-state vector potential defined by the equations:

- =

^

=

^

m

é

-^m^bn

= -

to

( L 1 6 )

Zt( V2) FL(X)

- -

(L17)

^

(L18)

- ~ r j t - v > )

è

In section 11, by using the Maxwell equations, we write all the macroscopic equations which determine thè behaviour of the electromagnetic field and current in a superconducting system and present some considerations on the meaning and role of the function f(x). A method of determining the function f(x) in correspondence of specific boundary conditions is also presented. Section 12 is devoted to the computation of the ground-state energy. Section 13 contains a comparison between our results and the ones previously found. The main difference can be understood b y comparing our expression for the transverse part of vector potential with the one computed by using the method of linear response in BCS approximation. Namely we have : aT

4jt(j

B

( k ) = Ti—i k +

(k))

T

57T57

mT2(k2)

4:7t(JeKt(k))T

(°ur

result

)

(BCS result).

JB is the vacuum expectation value of the boson current and is uniquely expressed in terms of the function f(x) ; J e x t is the external current and has to be fixed according to the boundary conditions. We close this section by noting t h a t the quantization of the electromagnetic field modifies, as illustrated in detail in section 13, the results of Ref. [12], where the vector potential was treated as a classical field. Our results differ, somehow, also from those obtained in Ref. [J5], where, though the vector field was quantized, no dynamical calculations were performed.

2. Invariance Properties and Ward-Takahashi Identities I n this section we introduce a gauge invariant model of non-relativistic electrons coupled to the electromagnetic field and interacting with each other through a two-body nonderivative interaction. We shall derive the Ward-Takahashi (WT) identities which express in a very convenient way the invariance properties of the model. The Lagrangian density for the model can be written as : f[y>(x),

A{x)]

=

*,[?(*)]

-

j L

F^x)

F„,(x)

+

+

Ç

(*))

•

-

{?(*)

M * ) (2.1)

359

Electrodynamics of Superconductors

(Apt) and { J J ) are connected b y the Maxwell equation and can be interpreted as the macroscopic vector potential and current in the superconductor. In conclusion, the boson transformation induces a ground-state vector potential defined by the equations:

- =

^

=

^

m

é

-^m^bn

= -

to

( L 1 6 )

Zt( V2) FL(X)

- -

(L17)

^

(L18)

- ~ r j t - v > )

è

In section 11, by using the Maxwell equations, we write all the macroscopic equations which determine thè behaviour of the electromagnetic field and current in a superconducting system and present some considerations on the meaning and role of the function f(x). A method of determining the function f(x) in correspondence of specific boundary conditions is also presented. Section 12 is devoted to the computation of the ground-state energy. Section 13 contains a comparison between our results and the ones previously found. The main difference can be understood b y comparing our expression for the transverse part of vector potential with the one computed by using the method of linear response in BCS approximation. Namely we have : aT

4jt(j

B

( k ) = Ti—i k +

(k))

T

57T57

mT2(k2)

4:7t(JeKt(k))T

(°ur

result

)

(BCS result).

JB is the vacuum expectation value of the boson current and is uniquely expressed in terms of the function f(x) ; J e x t is the external current and has to be fixed according to the boundary conditions. We close this section by noting t h a t the quantization of the electromagnetic field modifies, as illustrated in detail in section 13, the results of Ref. [12], where the vector potential was treated as a classical field. Our results differ, somehow, also from those obtained in Ref. [J5], where, though the vector field was quantized, no dynamical calculations were performed.

2. Invariance Properties and Ward-Takahashi Identities I n this section we introduce a gauge invariant model of non-relativistic electrons coupled to the electromagnetic field and interacting with each other through a two-body nonderivative interaction. We shall derive the Ward-Takahashi (WT) identities which express in a very convenient way the invariance properties of the model. The Lagrangian density for the model can be written as : f[y>(x),

A{x)]

=

*,[?(*)]

-

j L

F^x)

F„,(x)

+

+

Ç

(*))

•

-

{?(*)

M * ) (2.1)

360

M. Fpsco-Girabd, F. Mancini and M. Marinaro

where: s=M' X / y>,+(x) xpt.{y) V(X - y) FAX) =

-

(y) w{x)

Ws

d3y

(2.2)

M„(®).

(2.3)

Jp and Jd are the paramagnetic and the diamagnetic density currents, respectively, expressed b y :

¿b k ^ ~^ ^

jv[x)=

(2,4)

Jd(x) = -^e(x)A(x).

(2.5)

e(x) = eifs+(x)y,s(x).

(2.6)

q(x) is the charge density: 3= 1

In the previous expressions A^(x) = (A{x), A0(x)) denotes the electromagnetic vector potential; y>3{x) is the electron field, s = 1,2 means spin up, spin down, respectively, (that we shall denote also as y>f, ip^);

h F is the Fermi wave vector, fi is the chemical potential. A„B„ means A

B — 4050;

8 d{ = —; Cxi

80 =

Id —. c ot

I t can be immediately verified that the Lagrangian density (2.1) is invariant under the phase transformation: y>3(x)-+euy>,(x);

y>/(x)

e-uip+(x);

A^x) - > A^x)

(2.7)

6 is a constant phase parameter), and also under the gauge transformation: y>s(x)

eWtorWy^x);

yi+(x)

e~ W i F t(x) y>i(x) |0) 4= 0 .

(2.9)

In (2.9) |0) denotes the physical vacuum state. For our further considerations it is convenient to express the invariance properties of the model through the corresponding TFT-identities. Therefore, we devote the remaining part of this section to the derivation of these identities. The derivation will be accomplished by using canonical formalism [25]. At first we define the conjugate momenta and the Hamiltonian; then, by using the Heisenberg equations of motion and equal time commutation relations, we obtain the WT-identities.

Electrodynamics of Superconductors

361

Since the Lagrangian is gauge invariant, one must fix the gauge condition. Following Ref. [75], we shall fix the gauge by requiring that Dpid) Ap(x) = 0.

(2.10)

Here is a derivative operator which contains time derivatives only at first order and is fixed by the condition that the Heisenberg operators y>(x), A^x) can be realized in a suitable Fock space [12] of free quanta. Explicitly we write: 2)M(a) = (a 0 ,c-«» J ,«(-F»)F)

(2.10a)

where vB2(—F2) is a derivative operator, to be fixed by dynamical calculations. The gauge condition (2.10) is incorporated into the theory by adding to the Lagrangian (2.1) the following gauge-fixing term: =

JTf(x)

- 0 { x ) D„(8)

(2.11)

A,{x)

where ,6(x) is an auxiliary Heisenberg field. Thus, keeping in mind (2.1), the Lagrangian density of the model under consideration becomes: I [ f ( x ) , A{x)\

=

JfeM*)]

-

. W r

+

(

J P

eA°

+

PD»(d)Af

-

•

(2-12)

From (2.12), the conjugate momenta of ip3 and AM (denoted as ps and JtM, respectively) can be immediately computed; we have: ps(x)

^

=

-

¿

'

"

ihWs+(x)

=

^

=

-

¿

n0(x)

(

= 1,2)

(s

-

7

^

-

^

(2.13) ( i = 1

)

'2'

3 )

(2 14)

"

(2.15)

= - - ? ( x ) .

C

The Hamiltonian density is therefore: # { * ) =

i

!».(»)

+

3=1 = Jf e («) + where: Jte(x) =

ip+{x)

r 3 [e(-P 2 ) -

t «(x) +

I f d?yrp+{x)

JT[v(x),

M x ) -

rit.%p(y)

A{»)]

(2.16) V{x

-

y ) y>+(y) txt+

y>{x)

(2.17) a(x)

=

~

^ „ ( z ) Foi(x)

^int(s) = - (j p {x) + ^ Jt>g(x)

=

0{x)

Di(8)

Ai{x)

+

i - Ft,(x)

Ft,(x)

(2.18)

(as)) A{xy + e(x) A0(x)

(2.19)

+

(2.20)

Foi(x)

djA0(x).

M. Fttsco-Girabd, F. Mancini and M. Marinaro

362

In Eqs. (2.17) ... (2.20) we have used the matricial notation to express the electron field v

(

x

)

=

w

+

{

x

)

=

n

{

x

)

^

and we have used the Pauli matrices Tl =

(?o)'

T2 =

(?"o)'

»

(o-i)'

T =

(O?)'

/ =

Here and in the following Latin indices run from 1 to 3, while Greek indices run from 0 to 3. The summation over repeated indices is understood. Hamiltonian (2.16) leads to the Heisenberg equations: 8,Ft/l{x)

=

J^x)

-

A{d) y>(x) =

(2.21)

4jiD„(ö) ß(x)

(2.22)

F[V(x)]

D^d) A„(x) = 0

where A{8)

=

7 TT +

JpL = (Jo,J) F[y>{x)] = (pr3 - ZItj) f(x) +

he

—-A2(x)

~

S ( _ P 7 2 ) T3

with

Te2

(2.23) (2.24)

ATL

J 0 = q;J = J f + J D

+ eA0(x)

1

(2.24')

rsrp{x)

[2A{x) -F + F- A(x)] y,(x)

— Àj ¿Pyfav-ipix) y>+{y) V(x — y) r^+rpty) ~ rMx)

V>+(V) V{x — y) r+y>(y)].

(2.25) The quantities fi and A are defined by A = A(0|

(x) n(x)

|0) = AS(x)

Vt

+ ( * ) 10) = - j

(x) |0> = A+T3 Y>).

(2.26) (2.27)

It is worth while to note that, in order to derive the Heisenberg equations from Eq. (2.16), we made use of the equal time canonical commutation relations : [y,(®, t), Ps.{y, f)] + = ihôss-ô(x - y) [A„(x, t), n,{y, i)] = ihô^x

- y)

[A„(x, t), Av(y, *)] = [»„(as, t), 7i,(y, *)] = 0 [Vt(x, t), fAy, „(3) = c~2vB*(-V2)

(2.37)

pa _ V -

(2.38)

Then, after introducing the operators: x(x) = f(x) =

[ f t ( x ) n(x)

~ Vi+(x) Vt + ( a; )] = --j=y>+(xir2y>(x)

lwt( x ) v>dx) + n + ( x ) v r M l = - - j ^ w + i x ) r M x )

(2.39) (2.40)

we note that Eq. (2.21) and the equal time commutation relations allow us to write: [X{x, t), A,(y, i)] = [*(*, t), + (y) S(x,y;z) where

(3.21)

= (W(x)y>+(y)X(z))

(3.22)

Q{Z,W))

(3.23)

= (y,(x)y>+(y)!S(z))

Q(Z, W) = yi+(z) r3 ip(w)

and x is defined in (2.39). We note that the quantities P**, X, G and J are 2 x 2 matrices, whose elements will be denoted by etc. We start by writing the equation satisfied by the matrix P^ix, y,z). To obtain this equation we apply the operators A(8X) and A(—8y) to left and right side of P*{x, y\ z) and use the eq. (2.22); we have immediately: A(8X)

y; z) A{-dy)

=

368

M . FUSCO-GIBARD, F . MANCINI a n d M . MABINAEO

These equations can also be derived by using the perturbative method illustrated in appendix A. Finally, we derive the equation for the Green's functionDpy(x, y). Keeping in mind (2.3) and (2.21) we have: (8%r

-

Dvi(x

- y )

=

4nihd(x

-

y) g^

-

4n(JM(x)

Ak(y))

-

4nJ)„(dx)

(fi(x)

A>{y))

(3.31)

where the metric tensor g^, is defined by:

(~|T)

®" =

I being the 3 x 3 unit matrix. The equations (2.24') and (3.20) allow us to write: (J0(x) (J{(x)

A,(z))

=

-

A,{z))

J j j ^ {{Vx

-

=

e[P\2{x,

Vy)i (P'22(x,

as; a ) — P^x,

x\ z)} y, Z ))},_,

y; z) — P^x,

(3.32)

-

q0Du(x

-

z).

(3.33)' The last term in the r.h.s. of (3.33) has been obtained by remembering that in our approximation the diamagnetic current has been approximated by — (e/2mc2) g0A, with q0 = (e(x)) defined by (2.53). From (3.31), (3.32) and (3.33) we have: (82g0,

— 808v) Dvl(x

— y) =

4mhd(x

— y) goi

— 4ne{P'22(x,

x;y)

— P'n(x,

x;

8{x - y) (d*git =

— did,) Dy>(x,

47iihd(x

-

y) gu

+ IG

q0Du{x,

y))

(3.34)

y) +

z) -

4 7ich — {(Vx

-

V„) [(.P\ 2 {x, y; z) -

4MH ^

^

D(x -

P^x,

y,

y).

z)]}x=v

(3.35)

The last terms in the r.h.s. of (3.34) and (3.35) have been obtained by using the WTidentity (2.49). In conclusion, we have determined in pair approximation a set of coupled equations (3.26), (3.28), (3.29), (3.34) and (3.35) which are satisfied by the Green's functions. The solution of these equations will be given in the next sections. 4 . C a l c u l a t i o n o f t h e Q u a n t i t i e s P"(x,y;

z) a n d Dvv(x,

y).

In order to solve the Eq. (3.26), it is convenient to introduce the Fourier transforms. That is we write: Jib

—

/ D„(x

- y ) = i J

r

d*k

e^P«^)

(4.1)

e ^ W ^ k ) .

(4.2)

368

M . FUSCO-GIBARD, F . MANCINI a n d M . MABINAEO

These equations can also be derived by using the perturbative method illustrated in appendix A. Finally, we derive the equation for the Green's functionDpy(x, y). Keeping in mind (2.3) and (2.21) we have: (8%r

-

Dvi(x

- y )

=

4nihd(x

-

y) g^

-

4n(JM(x)

Ak(y))

-

4nJ)„(dx)

(fi(x)

A>{y))

(3.31)

where the metric tensor g^, is defined by:

(~|T)

®" =

I being the 3 x 3 unit matrix. The equations (2.24') and (3.20) allow us to write: (J0(x) (J{(x)

A,(z))

=

-

A,{z))

J j j ^ {{Vx

-

=

e[P\2{x,

Vy)i (P'22(x,

as; a ) — P^x,

x\ z)} y, Z ))},_,

y; z) — P^x,

(3.32)

-

q0Du(x

-

z).

(3.33)' The last term in the r.h.s. of (3.33) has been obtained by remembering that in our approximation the diamagnetic current has been approximated by — (e/2mc2) g0A, with q0 = (e(x)) defined by (2.53). From (3.31), (3.32) and (3.33) we have: (82g0,

— 808v) Dvl(x

— y) =

4mhd(x

— y) goi

— 4ne{P'22(x,

x;y)

— P'n(x,

x;

8{x - y) (d*git =

— did,) Dy>(x,

47iihd(x

-

y) gu

+ IG

q0Du{x,

y))

(3.34)

y) +

z) -

4 7ich — {(Vx

-

V„) [(.P\ 2 {x, y; z) -

4MH ^

^

D(x -

P^x,

y,

y).

z)]}x=v

(3.35)

The last terms in the r.h.s. of (3.34) and (3.35) have been obtained by using the WTidentity (2.49). In conclusion, we have determined in pair approximation a set of coupled equations (3.26), (3.28), (3.29), (3.34) and (3.35) which are satisfied by the Green's functions. The solution of these equations will be given in the next sections. 4 . C a l c u l a t i o n o f t h e Q u a n t i t i e s P"(x,y;

z) a n d Dvv(x,

y).

In order to solve the Eq. (3.26), it is convenient to introduce the Fourier transforms. That is we write: Jib

—

/ D„(x

- y ) = i J

r

d*k

e^P«^)

(4.1)

e ^ W ^ k ) .

(4.2)

36»

Electrodynamics of Superconductors Keeping in mind (4.1), (4.2) and (3.12), Equation (3.26) becomes: l 3 0

P"(x.

X

"d kd*q (271)' S(q - k).

P y in (4.3) we obtain :

d*p

— « /

(2^

fte ^"(fc) - er3 D0„(k) D^{k) (4.4) mc + — PiS&-)

where (4.5) In addition, by using Eq. (4.3), we can compute the quantity

Wx - Vv)i P"(x, y\ Thus, we have :

d*k iJ {2nf

d*p he (4.6) PjDj^k) PiS(P+)P°*(k) - er3D0li(k) mc + —-S(p_)

P^ik)

(2n ?

It may be immediately seen that the equation (4.4) can be put in the form:

Pim = N%(k) - j P%(k) QxySp(k)

(4.7)

where we have indicated the non-homogeneous term by: W )

=

and we have set :

T

D»»{k) (t%)*Qayif (k) - — D^k) Q{y.Jk) 7TIC

Q«y6fs(k) m j= d*p iXhS«y{P+) SSfl(P-) Qi-Mk) = m J -^¡L Pisay(yip-)-. P+) s,

(4.8) (4.9)

B y solving the system (4.7) (see Appendix C), we obtain the matrix elements of P 0f< in terms of photon propagator D^,. We report the combination of the matrix elements of which are directly connected with the coefficients of the dynamical mapping : 4eA {-hk0R(k) D0fl(k) + (kjimc) Djfl(k) PIW - P°£{k) = -ii2{A,x)FTW(k) (4.10) X [*(*) + /¿(k)]}

nm

-

p°m = (A^)ft

Explicit expressions for

(4.11) î a , , « - (K/M

X

The quantities Q and Q'j are defined in appendix B. In the previous equations, the symbol {0-fi^px denotes the Fourier transform of the vacuum expectation value of the time ordered product of the operators O^x) 02(y): (OAW

= - t / d*x(f)1(x) 02(y))

(4.12')

Equations (4.10), (4.11) and (4.12) give the matrix elements of Pil'(k) in terms of the photon propagator Dp,(k). Therefore, we must use the equation (3.34) and (3.35) to complete our computation. Let us write Eqs. (3.34) and (3.35) in momentum space: (—k2g0y + k0kt) Dtl(k) = 4nhg0i - Am[PH(k) - Pft(fc)] - 4nih (~k2giy

¿Ljlih 6 + hkr) DvX(k) = 4nhg u - — [P»(4) + P » ( * ) ] - —

D (k\ k, * (4.13) D(k) Dix{k)

Qo

where D^k) = [ic~2vB2(k2) k, 16-%]

(4.15)

D(k) = c" 2 [¿o2 - vB2(k2) k2] = c-2[k02 - mB2(k2)]

(4.16)

mB2{k2) = k2vB2(k2).

(4.17)

Keeping in mind Eqs. (4.11) and (4.12), we can solve the system (4.13)—(4.14). The solution is derived in appendix C; here we give only the final results: _ W(k)

Doi{k)

=

^hB2(k2) k0kj c3D2(k) =

u

W

_ m 4 nhk02k2 - oc(k) c2D2{k) ^

"

W

~ c'2k02 - k2 - mT2(k) ~ V - n(k)

, "

(4 18b)

^*

C )

'

where the functions