Operator Algebras and Quantum Statistical Mechanics 2: Equilibrium States. Models in Quantum Statistical Mechanics [2 ed.] 3540614435, 9783540614432

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Series Editors: R.

Balian, Gif-sur-Yvette, France

W.

Beiglböck, Heidelberg, Germany

H.

Grosse, Wien, Austria E. H. Lieb, Princeton, NJ, USA N.

H.

Reshetikhin, Berkeley, CA, USA Spohn, München, Germany

W.

Thirring, Wien, Austria

in

Physics

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l ONLINE LIBRARY

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Ola Bratteli Derek W. Robinson

Operator Algebras and Quantum Statistical Mechanics 2 Equilibrium States. Models in Quantum Second Edition

Springer

Statistical Mechanics

Professor Ola Bratteli Universitetet i Oslo Matematisk Institutt Moltke Moes vei 3 1

0316 Oslo, Norway e-mail: [email protected] Home page:

http://www.math.uio.no/~bratteli/

Professor Derek W. Robinson Australian National

University

School of Mathematical Sciences ACT 0200 Canberra, Australia e-mail: [email protected] Home page:

http://wwwmaths.anu.edu.au/~derek/

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Second Edition 1997. Second

Printing

2002

ISSN 0172-5998 ISBN 3-540-61443-5 2nd Edition ISBN 3-540-1038 1-3

Ist Edition

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subject to Copyright. All rights are reserved, whether the whole or part of the material is conceraed, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Vedag. Violations are liable for prosecution under the German Copyright Law. Springer- Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business

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use

of

general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the specific Statement, that such names are exempt from the relevant probreak tective laws and regulations and free for general use.

absence of therefore Cover

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production GmbH, Heidelberg

acid-free paper

SPIN 10885999

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543210

To

Trygve Bratteli, Samuel Robinson, and Harald ROSS

Preface to the Second Edition

Fifteen years have and much has ments

passed since completion of the first edition of this book happened. Any attempt to do justice to the new develop-

would necessitate at least

edition of the current

one

new

volume rather than

a

second

Fortunately

other authors have taken up the challenge of describing these discoveries and our bibliography includes references to a variety of new books that have appeared or are about to one.

appear. We consequently decided to keep the format of this book äs a basic reference for the operator algebraic approach to quantum statistical mechanics and concentrated on correcting, improving, and the

updating

material of the first edition. This in itself has not been easy and changes occur throughout the text. The major changes are a corrected presentation of Bose-Einstein condensation in Theorem 5.2.30, insertion of a general result on the absence of symmetry breaking in Theorem 5. 3. 3 3 A, and an extended

in

The discussion of

in Sects. 6.2.6 and

description of the dynamics of the A^-Fmodel phase transitions in specific models, 6.2.7, has been expanded with the focus shifted from

model to

genuine quantum situations

such

äs

the

Example

6.2.14.

the classical

Heisenberg

Ising

and X-Y

models. In addition the Notes and Remarks to various subsections have been considerably augmented. Since

our

interest in the

subject of equilibrium states and models considerably in the last fifteen years

statistical mechanics has waned

of it

VIII

Preface to the Second Edition

impossible to prepare this second edition without the and encouragement of many of our friends and colleagues. We are Support indebted to Charles Batty, Michiel van den Berg, Tom ter Eist, particularly

would have been

Jürg Fröhlich, Taku Matsui, Andre Verbeure, helpful advice, and we apollatter. We the often for are especially grateful to Aernout ignoring ogize for Werner Reinhard and Enter van counselling us on recent developments and giving detailed suggestions for revisions.

Dai Evans, Mark Fannes,

and Marinus Winnink for information and

Oslo and Canberra 1996

Ola Bratteh Derek W. Robinson

Contents Volume 2

States in

Quantum Statistical Mechanics

5.1. Introduction

5.2. Continuous

3

Quantum Systems.

I

5.2.1. The CAR and CCR Relations

5.2.2. The CAR and CCR 5.2.3.

States and

Algebras Representations

6 6 15

23

5.2.4. The Ideal Fermi Gas

45

5.2.5. The Ideal Böse Gas

57

5.3. KMS-States

76

5.3.1. The KMS Condition

5.4.

l

76

5.3.2. The Set of KMS States

1 12

5.3.3. The Set of Ground States

131

Stability 5.4.1. 5.4.2.

and

Equilibrium

Stability Stability

144

of KMS States

144

and the KMS Condition

176

X

Contents Volume 2

5.4.3.

Gauge Groups and Systems

the Chemical Potential

197

5.4.4. Passive

211

Notes and Remarks

217

Models of

235

Quantum Statistical Mechanics

6.1. Introduction

237

6.2

Quantum Spin Systems

239

6.2.1. Kinematical and

239

Dynamical Descriptions Equilibrium The Maximum Entropy Principle Translationally Invariant States Uniqueness of KMS States Nonuniqueness of KMS States

6.2.2. The Gibbs Condition for

261

6.2.3.

266

6.2.4. 6.2.5. 6.2.6.

286 306 317

6.2.7. Ground States

6.3. Continuous

338

Quantum Systems. II

353

6.3.1. The Local Hamiltonians

355

6.3.2. The Wiener 6.3.3. The 6.3.4. The

Integral Thermodynamic Limit. Thermodynamic Limit.

366 I. The Reduced

Density

Matrices

II. States and Green's Functions

381 395

6.4. Conclusion

422

Notes and Remarks

424

References

463

Books and

Monographs

465

Articles

468

List of

487

Subject

Symbols Index

499

Contents Volume l

Introduction

l

Notes and Remarks

C*-Algebras 2.1.

and

16

von

Neumann

Algebras

C*-Algebras

19

2.1.1. Basic Definitions and Structure

19

2.2. Functional and 2.2.1.

Spectral Analysis

2.2.3.

Approximate

Representations 2.3.1.

25

Resolvents, Spectra, and Spectral Radius Identities and

32

Quotient Algebras

and States

Representations

39

42 42

2.3.2. States 2.3.3. Construction of

25

'

2.2.2. Positive Elements

2.3.

17

48

Representations

54

XII

Contents Volume l

2.3.4. Existence of

Representations

2.3.5. Commutative

2.4.

von

Neumann

2.4.1.

58

C*-Algebras

6l

Algebras

Topologies

on

65

^()

65

2.4.2. Definition and of

von

Elementary Properties Neumann Algebras

75

2.4.4.

79

Quasi-Equivalence

of

Representation

2.5. Tomita-Takesaki Modular of

von

2.5.1.

Neumann cr-Finite

Theory

and Standard Forms

Algebras Neumann

2.5.3.

Algebras Group Integration and Analytic Elements for One-Parameter Groups of Isometries on Banach Spaces

2.5.4.

Self-Dual Cones and Standard Forms

von

2.5.2. The Modular

2.6.

7l

2.4.3. Normal States and the Predual

83 84 86

97

102

Quasi-Local Algebras

118

2.6.1. Cluster

1 18

2.6.2.

129

2.6.3.

Properties Topological Properties Algebraic Properties

2.7. Miscellaneous Results and Structure 2.7.1.

Dynamical Systems and Crossed Products Operator Algebras Weights on Operator Algebras; Self-Dual Cones of General von Neumann Algebras; Duality and Classification of Factors; Classification of C -Algebras

2.7.2. Tensor Products of 2.7.3.

Notes and Remarks

Groups, Semigroups,

3.1. Banacb 3.1.1.

3.1.2.

3.1.3. 3.1.4.

3.1.5.

136 136 142

145

152

and Generators

Space Theory

Uniform

133

Continuity Strong, Weak, and Weak* Continuity Convergence Properties Perturbation Theory Approximation Theory

157

159 161

163 183

189 198

Contents Volume l

3.2.

XIII

205

Algebraic Theory 3.2.1. Positive Linear

205

3.2.2. General

228

3.2.3.

3.2.4. 3.2.5. 3.2.6.

Maps and Jordan Morphisms Properties of Derivations Spectral Theory and Bounded Derivations Derivations and Automorphism Groups Spatial Derivations and Invariant States Approximation Theory for Automorphism Groups

244

259 263 285

Notes and Remarks

298

Decomposition Theory

309

4.1. General

311

Theory

311

4.1.1. Introduction 4.1.2.

4.1.3.

4.2.

315

Barycentric Decompositions Orthogonal Measures

333

4.1.4. Borel Structure of States

344

Extremal, Central, and Subcentral Decompositions

353

4.2.1. Extremal

353

4.2.2.

Decompositions Decompositions

362

Central and Subcentral

4.3. Invariant States 4.3.1.

4.3.2. 4.3.3

4.3.4

4.4.

367 367

Ergodic Decompositions Ergodic States Locally Compact Abelian Groups Broken Symmetry

386 400

416

432

Spatial Decomposition

433

4.4.1. General 4.4.2.

Theory Spatial Decomposition

and

Decomposition

of States

442

Notes and Remarks

451

References

459

Books and

Monographs

461

Articles

464

List of

481

Subject

Symbols Index

487

States in

Quantum Statistical Mechanics

5.1. Introduction

In this

chapter, and the following one, we examine various applications of C*algebras and their states to statistical mechanics. Principally we analyze the structural properties of the equilibrium states of quantum Systems consisting of a large number of particles. In Chapter l we argued that this leads to the study of states of infinite-particle Systems äs an initial approximation. There are two approaches to this study which are to a large extent complementary. The first approach begins with the specific description of finite Systems and their equilibrium states provided by quantum statistical mechanics. One then rephrases this description in an algebraic language which identifies the equili brium states äs states over a quasi-local C*-algebra generated by subalgebras corresponding to the observables of spatial Subsystems. Finally, one attempts to calculate an approximation of these states by taking their limit äs the volume of the System tends to infinity, the so-called thermodynamic limit. The infinitevolume equilibrium states obtained in this manner provide the data for the calculation of bulk properties of the matter under consideration äs functions of the thermodynamic variables. By this we mean properties such äs the particle density, or specific heat, äs functions of the temperature and chemical potential, etc. In fact, the infinite-volume data provides a much more detailed, even microscopic, description of the equilibrium phenomena although one is only generally interested in the bulk properties and their fluctuations. Examination of the thermodynamic limit also provides a test of the scope of the usual statistical mechanical formalism. If this formalism is rieh enough to describe phase transitions, then at certain critical values of the thermodynamic parameters there should be a multiplicity of infinite-volume limit states arising from slight variations of the external interactions or boundary conditions. These states would correspond to various phases and mixtures of these phases. In such a Situation it should be possible to arrange the limits such that phase Separation takes place and then the equilibrium states would also provide information concerning interface phenomena such äs surface tension. The second approach to algebraic statistical mechanics avoids discussion of the thermodynamic limit and attempts to characterize and classify the equili brium states of the infinite System äs states over an appropriate C*-algebra. The elements of the C*-algebra represent kinematic observables, i.e., observables at a given time, and the states describe the instantaneous states of the System. For a complete physical description it is necessary to specify the dynamical law

4

States in

Quantum Statistical Mechanics

governing the change with time of the observables, or the states, and the equilibrium states are determined by their properties with respect to this dynamics. The general nature of the dynamical law can be inferred from the usual quantum-mechanical formalism and it appears that there are various possibiUties. Recall that for finite quantum Systems the dynamics is given parameter group of *-automorphisms of the algebra of observables,

A^Tt(Ä)^e^^^Ae-^^^

by

a one-

,

where H is the

selfadjoint Hamiltonian operator of the System. Thus it appears dynamics of the infinite System should be determined by a continuous one-parameter group of *-automorphisms T of the C*-algebra of observables. This type of dynamics is certainly the simplest possible and it occurs in various specific models, e.g., the noninteracting Fermi gas, some of which we examine in the sequel. Nevertheless, it is not the general Situation. The difRculty is that a group of this kind automatically defines a continuous development of every state of the System. But this is not to be expected for general infinite Systems in which compHcated phenomena involving the local accumulation of an infinite number of particles and energy can occur for natural that the

certain initial states. Thus it is necessary to examine weaker forms of evolution. For example, one could assume the dynamics to be specified äs a group of of the von Neumann algebras corresponding to a subclass of C*-algebra. Alternatively one could adopt an infinitesimal description and assume that the evolution is determined by a derivation which generates an automorphism group only in certain representations. Fach of these possible structures could in principle be verified in a particular model by a thermodynamic limiting process and each such structure provides a framework for characterizing equilibrium phenomena. To understand the type of characterization which is possible it is useful to refer to the finite-volume description of equilibrium. There are various possible descriptions of equilibrium states, which all stem from the early work of Boltzmann and Gibbs on classical statistical mechanics, and which differ only in their initial specification. The three most common possibilities are the microcanonical ensemble, the canonical ensemble, and the grand canonical ensemble. In the first, the energy and particle number are held fixed; in the second, states of various energy are allowed for fixed particle number; and in the third, both the energy and the particle number vary. Fach of these descriptions can be rephrased algebraically but the grand canonical be the Hubert space of description is in several ways more convenient. Let states for all possible energies and particle numbers of the finite System, and H and N, the selfadjoint Hamiltonian and number operators, respectively. The Gibbs grand canonical equihbrium state is defined äs a state over ^(), or ^^(), by

automorphisms states

over

the

Tr^(.-/^^^)

^^-^(^)^Tr,M^)

'

Introduction

where K erator.

for all

H

5

// G [R, and it is assumed that e~^^ is a trace-class opH is lower semi-bounded and the trace-class property is valid

iiN,

Typically ß > Q. The parameters ß

and /i

to the inverse

correspond

temperature of

the System, in suitable units, and the chemical potential, respectively, and therefore this description is well-suited to a given type of material at a fixed

temperature. Now if the generalized evolution A G

^()

^

then the trace-class property of f

are

T,(^)

-

is defined

e^^^Ae-^^^

e~^^ allows ^

T

one

G

^()

by ,

to deduce that the functions

cDß^^(Ait(B]]

analytic in the open strip 0 < Im ^ < jS and continuous Strip. Moreover, the cyclicity of the trace gives

on

the boundaries

of the

^ß.M'^t(B]]\t^iß This is the KM S condition which will

role

=

o}ß^^(BA)

brieüy described throughout this chapter. we

.

in

Chapter l and which significance of this over ^^(), i.e., the

One

important uniquely determines the Gibbs state only State over ^^() which satisfies the KMS condition with respect to T at the value ß is the Gibbs grand canonical equilibrium state. This can be proved by explicit calculation but it will in fact follow from the characterization of extremal KMS states occurring in Section 5.3. It also follows under quite general conditions that the KMS condition is stable under limits. Thus for a System whose kinematic observables form a C*-algebra ^ and whose dynamics is supposed to be given by a continuous group of *-automorphisms T of ^, it is natural to take the KMS condition äs an empirical definition of an equilibrium play

an

condition is that it

state.

Prior to the analysis of KMS states we introduce the specific quasi-local C*algebras which provide the quantum-mechanical description of Systems of point particles and examine various properties of their states and representations. In particular we discuss the equilibrium states of Systems of non-interacting particles. This analysis illustrates the thermodynamic limiting process, utilizes the KMS condition äs a calculational device, and also provides a testing ground for the general formalism which we subsequently develop. In the latter half of the chapter we discuss attempts to derive the KMS condition from first principles.

5.2. Continuous

Quantum Systems.

I

5.2.1. The CAR and CCR Relations There are two approaches to the algebraic structure associated with Systems of point particles in quantum mechanics. The first is quite concrete and physical. One begins with the Hubert space of vector states of the particles and subsequently introduces algebras of operators corresponding to certain particle observables. The second approach is more abstract and consists of postulating certain structural features of a C*-algebra of observables and then proving uniqueness of the algebra. One recovers the first point of view by passing to a particular representation. We discuss the first concrete approach in this subsection and then in Section 5.2.2

we

examine the abstract formulation.

The

quantum-mechanical states of n identical point particles in the configuration space U^' are given by vectors of the Hubert space L~(R"^''). If the number of particles is not fixed, the states are described by vectors of the direct sum

space

^=@L\K"-) 77

i.e.,

\l/ {jA^''^}>o, where given by

sequences

of

norm

i/^

is

=

ll'Af

=

l'/'^'P

+

E

,

>0

\l/^^^

G

C,

!/^^''^

G

L'([R'^')

for

fdxr--dx\il^^"\x,,...,x)\^

/?

>

l, and the

.

n>\'^ There If

is, however, a further restriction imposed by quantum statistics. G 5 is normalized, then

i/^

dp(xi,...,Xn)

=

\il/^''\xi,...,Xn)\^dxi

-

dx^

is the quantum-mechanical probability density for \l/ to describe n particles at the infinitesimal neighborhood of the points ;ci x^ The normalization of \l/ , corresponds to the normalization of the total probability to unity. But in .

.

.

,

.

microscopic physics identical particles are indistinguishable and this is refiected by the symmetry of the probability density under interchange of the particle coordinates. This interchange defines a unitary representation of the permutation group and the symmetry is assured if the ij/ transform under a suitable subrepresentation. There are two cases of paramount importance.

Continuous

Quantum Systems.

I

7

\l/^"^

of each ij/ are Symmetrie under The first arises when the components Particles whose transform in this manner are of coordinates. states interchange called bosons and

said to

are

satisfy

Böse

statistics. The second

(-Einstein)

ease

anti-symmetry of the i/^^"^ under interchange of each pair of particles are called /ermzö/?^ and are said to satisfy Fermi (-Dirac} statistics. Thus to discuss these two types of particle one must examine the Hubert subspaces 5.^, of 5, formed by the ij/ {^ }n>Q whose sign). These components are Symmetrie (the + sign) or anti-symmetric (the subspaces are usually called Fock spaces but we will also use the term for more general direct sum spaces. To describe particles which have internal structure, e.g., an intrinsic angular momentum, or spin, it is necessary to generalize the above construction of corresponds

to

coordinates. The associated

=

-

Fock space. Assume that the states of each

particle form a complex Hubert space l) and f) denote the /7-fold tensor product of i^ with itself. Fur ther introduce the Fock space g(l)) by

let

l)''

=

t)

0

I)

0

(g)

5(1))

©

=

t)"

,

n>0

if

where

i/^^"^

C. Thus

a

\\j

vector

G

is

5(f))

a

{iA''"''}/2>o

sequence

^^ vectors

subspace of g(l)) formed by the vectors with all components except the th equal to zero. In Order to introduce the subspaces relevant to the description of bosons and fermions we first define operators P on (5(^) by G

t)'^ and l)"

can

^+(/i

for all

be identified

äs

/.)

=

P-(fl^f2^"'^ fn]

=

/2

/l, ...,/

(TII, 712,

,

7r)

G

^

^

f). The

sum

the closed

(n

!)"^ V /.,

(n !)

is

of the indices and n is

/.,

0

Z-^^

^

over

one

^ Bnfn, all

^

even

/.

TI;

(1.

,

fn.

permutations

if TT is

0

0

and minus

2,

one

Extension

...,)

F-^

if TC is odd.

l and by linearity yields two densely defined operators with ||P|| the P extend by continuity to bounded operators of norm one. The P+ and P_ restricted to i^", are the projections onto the subspaces of 1^" corresponding to the one-dimensional unitary representations n \-^ l and TC 8;^ of the permutation group ofn elements, respectively. The Bose-Fock space g^(t)) and the Fermi-Fock space (5_({)) are then defined by =

H-

S(^)=/'5(l)) and the

corresponding -particle subspaces I)'^ by I)^ on g(^) by

==

{)"

P

.

We also define

number operator N

D(N)

=

(lA; l

^

=

{^^"^}.>o, E^'ll^^'^^ll' '^>o

and

7v,A

=

{.AW}>o


0

The

simplest example

one

then has

of this second

quantization

jr(i)=7v If u is

and

unitary, [/

is defined

Un(P(fl

/2

0

0

extending by continuity.

by L/o

^

fn)]

=

The second

given by choosing

//

=

H ,

.

H and

P(Ufl

=

is

by setting 0

^/2

quantization

0

0

Ufn)

of U is denoted

by r(f/),

where

r(u)

@u

=

.

n>0

Note that r (U) is

is

a

strongly

unitary. The

notation dY and P is chosen because if Ut unitary group, then

r(t/,) Next

we

=

e'^^

continuous one-parameter

wish to describe two

bosons and fermions,

respectively.

-

e'"^''(^)

.

C*-algebras of observables associated with Both algebras are defined with the aid of

particle "annihilaüon" and "creation" operators which are introduced äs follows. For each / e 1) we define operators a(f), and *(/), on 5(1)) by initially setting a(/).A(0) 0,a*(/)^() /, / e ^, and =

a(/)(/i f2---fn)

*(/)(/! /2---/)

=

=

n^'^(f, /i)/2

/3

(+l)'^V

/l-

=

/

,

Continuous

Extension

i/^^"^

l)\

G

by linearity again yields easily calculates that

I

9

if

densely defined operators and

two

one

||a*(/),AW||

||a(/)^Wi|