Fortschritte der Physik / Progress of Physics: Band 26, Heft 7/8 1978 [Reprint 2021 ed.] 9783112519141, 9783112519134

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BEZUGS MÖGLICH R E I T E N Bestellungen sind zu richten — in der DDR an eine Buchhandlung oder an den Akademie-Verlag, DDR - 108 Berlin, Leipziger Straße 3 - 4 — im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der BRD und Westberlin an eine Buchhandlung oder an die Auslieferungsstelle KUNST UND WISSEN, Erich Bieber, 7 Stuttgart 1, Wilhelmstraße 4—6 — in Österreich an den Globus-Buchvertrieb, 1201 Wien, Höchstädtplatz 3 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, DDR - 701 Leipzig, Postfach 160, oder an den Akademie-Verlag, DDR - 108 Berlin, Leipziger Straße 3 - 4

Zeitschrift „Fortschritte der Physik" Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Artur L6sche, Prof. Dr. Rudolf Ritsehl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R -108 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 22 36 221 und 22 36229; Telex-Nr. 114420; B a n k : Staatsbank der D D R , Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R - 104 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: V E B Druckhaus „Maxim Gorki", D D R - 74 Altenburg, Carl-von-Ossietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik 44 erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je B a n d : 180,— M zuzüglich Versandspesen (Preis für die D D R : 120,— M). Preis je H e f t 15,— M (Preis für die D D R : 2 0 , - M). Bestellnummer dieses Heftes: 1027/26/7/8. © 1978 by Akademie-Verlag Berlin. Printed in the German Democratio Republio. AN (EDV) 57618

ISSN 0 0 1 5 - 8 2 0 8 Fortschritte der Physik 26, 3 9 7 - 4 3 9 (1978)

Linear Fermion Systems, Molecular Field Models, and the KMS Condition L E O VAN H E M M E N *

Institut des Hautes Etudes Scientifiques, 91440 Bures-sur- Yvette, France

Abstract We take advantage of the simplifications made possible by assuming the system to be infinite from the beginning, to study some exactly solvable models in statistical mechanics. (1)- For fermi lattice systems with quadratic hamiltonians the unifying concept of our treatment is a precise formulation of the idea of a linear system. This concept makes it possible to define a dynamical matrix which incorporates all the information one needs for equilibrium and non-equilibrium problems. As an example, we consider the alternating XY chain. Furthermore we pay some attention to ergodicity, pointing out the similarity to the infinite harmonic crystal. (2) For molecular field models, including the BCS model, the method is to consider an arbitrary extremal homogeneous state, to construct the dynamics in the representation determined by that state, and then to use the KMS condition to single out the equilibrium state of the infinite system. A synthesis of these methods — (1) and (2) — enables us to give a new simple treatment of the BCS model and a direct analysis of its mathematical structure. In addition we show the equivalence of Kac and van der Waals (spin) systems in the so-called van der Waals limit. Finally we indicate briefly how to deal with molecular field models with polynomial and more general interactions.

Contents I.

Mathematical preliminaries 1.1. The algebra of observables 1.2. Time evolution and KMS condition 1.3. Hilbert space techniques . 1.4. Space averages and their properties 1.5. Convolution algebras

398 399 399 401 402 404

II.

Linear fermion systems 2.1. Fermi lattice systems with bilinear hamiltonians 2.2. A nonprimitive fermi lattice 2.3. General theory: the interplay between dynamics and KMS 2.4. Ergodicity 2.5. High- and low-temperature limits

404 405 409 411 41§ 416

III.

Molecular field models 3.1. A preview of the method 3.2. van der Waals spin systems

417 417 419

*) Supported by the Netherlands Organization for the Advancement of Pure Research (Z. W. 0.). 30

Zeitschrift „Fortschritte der Physik", Heft 7/8

398

LEO VAN HEMMEH 3.3. 3.4. 3.5. 3.6.

The BCS model General theory: BCS and similar models Kac and van der Waals models . . . . A molecular field model a la Thompson .

Appendix: Non-equilibrium molecular field theory

422 426 431 432 433

Introduction

The main intention of this review is to offer the reader a suitable formalism and some useful methods for efficiently analyzing linear fermion systems and molecular field models. Not all of the results are new, some of them — like BCS — are even respectable by their age and input, but nevertheless the methods of reaching them contain some novelty. The main intention of this introduction is to describe the organization of the paper. The first two sections of Chapter I I . treat fermi lattice systems with bilinear hamiltonians (e.g. various one-dimensional XY models belong to this class) in a rather concrete and simply way. Dynamics (one-sentence proof) and thermodynamics are solved explicitly. We extend our formalism to nonprimitive fermi lattices and then apply it to the alternating XY chain. The reader who is mainly interested in the BCS and similar models can now proceed directly to Chapter I I I . A general and more abstract analysis of linear fermion systems is presented in section 2.3. Section 2.4. is devoted to the ergodicity problem; its moral is that linear fermion systems in equilibrium can be ergodic only if they are infinite. Molecular field models are exactly solvable only after the thermodynamic limit: section 1. of Chapter I I I previews the method; sections 2. and 3., on van der Waals spin systems and the BCS model respectively, are illustrations of how the method works in practice. The general and more abstract considerations do not appear before section 3.4. The final two sections are again of a more practical nature. In this chapter we develop a version of the BOGOLIUBOV-HAAG method [49, 50~\ as proposed originally by DE VEIES [42, S3]. An appendix is devoted to non-equilibrium problems. In Chapter I. we have collected some mathematical preliminaries. Only section 1.5. and the KMS condition in section 1.2. are needed in Chapter II. The other sections, including 1.2., will be indispensible for the reading of Chapter I I I .

I. Mathematical Preliminaries

In this chapter we introduce some mathematical notions we need later on, together with some notation. The reader who is not interested in technical details can skip all these sections as we shall refer to them whenever we use their results. In section 1. we describe in general terms the algebra of observables and their quasilocal structure. Next we outline, in section 2., the construction of a dynamics and for•mulate (a version of) the KMS condition; moreover some words are devoted to analytic elements. Then, in section 3., we quickly show how to associate a hilbert space with a given state and how to prove the existence of averages (if any) in that space (section 4.); within this context we also pay attention to ergodicity. Finally, in section 5., convolution algebras are discussed.

Molecular Field Models and the KMS Condition

399

1.1. The algebra of observables Because measurements are made in finite subsets of the space and can be done simultaneously for disjoint regions, one assigns to each bounded [2] region A an algebra of observables sé (A), more specifically a C*-algebra, in such a way that [2]

A} n A2

0 => [¿¿{Ai), j/(/1 2 )] = 0, i.e. causality

Ax C A¡t

jé(A¡) C sé(A2),

i.e. isotony.

(1.1.1) (1.1.2)

By the isotony séL = U sé (A) is again an algebra, in fact a normed algebra, of strictly A

local observables; for mathematical convenience one adds its limit points, obtaining sé = séL. The algebra of quasi-local observables sé is a C*-algebra. Natural characteristics of any C*-algebra include its completeness and its norm satisfying \\A*A| = \\A\\ 2, while in addition H^H ^ |[4|| \\B\\. An important consequence of causality is asymptotic abelianness with respect to space translations: lim ||[axA, B]|] = 0

(1.1.3)

for all A and B from sé, where a , means translation by x. Another useful property of quasi-local algebras should also be mentioned [3—7]: they are simple. As a consequence any non-zero representation n of is one-to-one (faithful). So we are permitted to identify sé and n(sé), as we shall do sometimes by dropping the n from the notation. A state co is a positive linear functional on sé, characterized by A>(AA -f- ¡UB) = XOJ (A) + ¡UCO(B), A>(A*A) 2; 0 and co( 1) = 1; these requirements are quite reasonable for w (A) to represent an expectation .value. 1.2. Time evolution and KMS condition Since sé describes the observables of an infinite system, the construction of a dynamics [2, 3] is non-trivial. We sketch the procedure for lattice systems, which is, after all, quite simple. Suppose we are given a hamiltonian HA for any finite region A of a lattice system. Pick an A from sé(A0) and consider for all A contraining A0 i.e.

ott A{A)

= exp ( f f l A t ) A exp ( - i H j )

(1.2.1)

oo /.yin *T A{A)

= Z - R CAM), n=0 n\

CM)

= [HA, A].

(1) In order to prove that the limit ctt(A) = lim txt A(A) exists for A reasonable properties, one usually shows first \\CÁ n(A)\\ ^ C slM nn\

(1.2.2)

oo, and has some (1.2.3)

where C st depends on A, so that the series (1.2.2) converges absolutely (in norm) and uniformly (in A) when |i| < M~ l, and one can interchange the summation and the limit procedure A oo. 30*

400

L E O VAN HEMMEN

(2) Secondly, one exhibits lim Ca»(A) =a„

in si\

n = 1, 2, ...

(1.2.4)

A—>co

with convergence in the norm of si. Thus one is able to calculate xt(A) by *t{A)

=

(itY*

00

M - l i m Ca*{A) = Z =0 n - A — n = 0

E

(

li'tV - j - a

n

-

n

(1-2.5)

,

and ot {A) is in si provided < M' . (3) The *-algebraic properties [•§] of D(k) = I I v ; ; rv~'), y-f(k) we have is given by its two-point correlation functions (2.3.8) only. Because dynamics and thermodynamics are fully determined by the infinitesimal generator 0 of the time evolution exp \it&] — i.e. more precisely, (2.3.4) — it seems plausible that one can also find a simple criterion for the ergodicity of co with respect to at in terms of 0. This is indeed the case. Theorem: The KMS state a> of a linear fermion system is ergodic with respect to the dynamical evolution a t if and only if the spectrum of 0 is continuous. Proof: a(0), the spectrum of 0 considered as an operator in H — cf. (2.3.2) — , is continuous if and only if its point spectrum ap(0) is empty, and in that case surely H2 = ker 0 = (0). We have to show T

A,

B

€

lim ^ f r->oo J- J

dtco[xt(A)

B)

=

w{A)

a(B)

(2.4.1)

o

i.e. (1.4.8), if and only if a(0) is continuous. Thereto we divide the proof into three parts. Gluing them together gives the desired result. T

(1) Define

rj(J)

i r = lim — I dtf(t), T^oo

-L

J 0

\v(f)\

provided the limit exists. One has 11/11»,

^V(\f\)

where H/H«, is the supremum of / on R; / 2; 0 implies a n d VON N E U M A N N

rj(f)

22 0. According to

(2.4.2) KOOPMAN

[36]

qfl/l) = 0 o lim f{t) = 0 t—>oo

if

t $ I = set of "density zero", (2.4.3)

(2) Suppose a(0) is continuous. Let A and B in (2.4.1) be products of jB(£)'S; the consideration of this situation suffices. Evaluating w(oit (^4) B) by means of the Wick formula and observing w is a ( -invariant, we have to show that the average rj of all "pairings" of atA with B are zero in order to conclude that (2.4.1) holds. Pick a B1 from A and a B2 from B. The function f(t) = « ( « ¡ ( i ^ ) _B2) can be represented as /(i) = f dv(H) exp (it?.) where v is the distribution function of an (absolutely) finite measure — combine (2.3.4) 31*

416

L E O VAN H E M M E N

and (2.3.8). But rj(\f\2) is zero if and only if v(X) is continuous [37]; the latter condition is implied by a(0) being continuous. Cf. also [21, p. 396]. (3) The condition that op{&) is empty is also necessary. Suppose on the contrary that it were not. Zero is clearly not in ] / f t — exp [ — itA] / f t ; say ft =• / f t . Study w{BlB2B3Bi) with A = BtB2 («¡-invariant) and B = BiBl (arbitrary); then (2.4.1) gives a contradiction. The details are left to the reader. Anyhow, there is something very amusing in this result, especially when one is looking at fermi lattice systems. In our study of the infinite harmonic crystal, a classical linear lattice system, we have shown [31] — cf. also [38] — that the equilibrium state ¡i^, which we determined by means of the classical KMS condition, is ergodic with respect to the dynamical evolution if and only if, indeed, the spectrum of the dynamical matrix (unitarily equivalent to the interaction matrix!) is continuous. This is clearly never the case if the system is finite. Remark: A statement similar to our theorem also appears in [