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Journal Information

Volume: 5 Num ber/lssue: Year: 1985

Article Information

Article Title: The problem of translation invariance of Gibbs states at low temperatures

Article Author: Dobrushin, R.L. and Pages:

Shlosman, S.B. 52-195

Notes:

Title Requested:

Author: nauk SSSR.

Title: Soviet scientific reviews. Section C, Mathematical physics reviews.

Imprint: c1980-

Link: http://oskicat.berkeley .edu/record=b11994061-S 1

\) \:23 :;15

;-:;,

\ C 'CSS '\,.' S"'f'';('P.r t... ...

____ . __ _

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.0\'J

https://mail.google.com/maillb/300/u/O/?ui=2&ik=bd3e4c3753&vieN=pt&cat=Article%20Reqs&search=cat&th=14f6acfd55e6fe9c&siml=14f6acfd55e6fe9c 1/1

SOVIET SCIENTIFIC REVIEWS CoseTCKIIIe Hay4Hble 063opbl

A series edited by V.I. Gol'danskii (Institute of Chemical Physics, Moscow), R. Z. Sagdeev (Institute for Space Research, Moscow), Maurice Levy (Univer- site Pierre et Marie Curie, Paris), M. Longair (Royal Observatory, Edinburgh), P. Carruthers (Los Alamos National Laboratory, New Mexico), and S. Ichti- taque Rasool (Jet Propulsion Laboratory, Pasadena).

The articles for this series are each written in Russian by a Soviet expert in the field, then rapidly translated into, and published in, the English language. The aim of the series is to make accounts of recent scientific advances in the USSR readily and rapidly available to other scientists. The most recent volumes in each section are listed below.

Section A: Physics Reviews edited by I. M. Khalatnikov Volume 6 (1985)

(ISSN: 0143-0394)

Section B: Chemistry Reviews (ISSN: 0143-0408) edited by N. U. Kochetkov and M. E. Vol'pin Volume 7 (1985)

Section C: Mathematical Physics Reviews (ISSN: 0143-0416) edited by S. P. Novikov

Volume 4 (1984)

Section D: Physicochemical Biology Reviews (ISSN: 0734-9351) edited by V. P. Skulachev

Volume 5 (1984)

Section E: Astrophysics and Space Physics Reviews edited by R. A. Syunyaev

Volume 4 (1985)

(ISSN: 0143-0432)

(

Scientific Reviews, Section C

MATHEMATICAL PHYSICS REVIEWS---1

Volume 5 (1985)

Edited

by

S. P. Novikov

L. D. Landau Institute of Theoretical Physics USSR Academy of Sciences, Moscow

Translated by

MORTON HAMERMESH

University of Minnesota

liT

SOVIET SCIENTIFIC REVIEWS

I I

lr

Vl CONTENTS

1.1. Formulation of the Problem . . . 35

1.2. The Interaction Region . . . 36

1.3. Convergence of Series in the Space of Bounded Func- tions ... 39

2. Existence of the Thermodynamic Limit for Symmetric Systems ... 43

2.1. The Case When the Particles are in a Finite Volume at the Initial Time . . . 43

2.2. The Case Where the Particles Move in a Finite Volume ... 44

3. Global Solutions . . . 46

3.1. Procedure for Extending a Local Solution in Time ... 46

3.2. The Thermodynamic Limit in the CNE ... 47

4. The Thermodynamic Limit for Nonsymmetric Systems . . . 48

4.1. Formulation of the Problem . . . 48

4.2. The Evolution Operator in the Space of Bounded Functions . . . 49

4.3. Existence of the Thermodynamic Limit ... 50

The Problem of Translation Invariance of Gibbs States at Low Temperatures R.L. DOBRUSHIN and S.B. SHLOSMAN 1. Introduction . . . 54

2. Ground States ... ... 60

2.1. States ... 60

2.2. Specifications and Gibbs States .. .. . . .. . . . .. . . 62

2.3. Ground State ... 64

2.4. Periodic Ground States . .. . . .. . . .. . .. .. . . . .. . . .. . . . .. .... 65

2.5. Main Thesis . . . .. .. .. ... . .. . ... . . .. . .. . . .. . . 67

2.6. The CPS-Condition ... 69

2.7. Special Cases-One-Dimensional Models ... 71

2.8. The Symmetric Ferromagnetic Ising Model ... 71

2.9. The Antiferromagnetic Ising Model ... 88

2.10. The Widom-Rowlinson Model ... 91

2.11. Random Ground States-The Antiferromagnetic Case ... 93

2.12. Nonperiodic Random Ground States ... 99

3. Translation-Invariance in Two Dimensions: Basic Ideas ... 114

3.1. Main Result ... 114

CONTENTS 3.2. Basic Reductions ... . 3.3. Contours ... . 3.4. Main Transformation ... . 3.5. Method of Construction of the Mapping TI ... . 4. Proofs-The Ferromagnetic Case ... . 4.1. Introduction ... . 4.2. Partition Functions under Constant Boundary Condi- Vll 115 117 118 120 120 120 tions ... 123

4.3. Estimate of Number of Contours ... 125 4.4. Correlation Functions for Constant Boundary Condi-

tions ... . 4.5. Partition Functions for Given Open Contours ... . 4.6. Correlation Functions for Given Open Contours .. . 4. 7. Estimate of Length of Open Contours ... . 4.8. Construction of the Main Transformation TI ... .

4.9. Geometrical Properties ... . 4.10. Main Estimate ... . 4.11. Check of the Properties of the Transformation ^{TI •.}

5. Proofs-The General Case ... . 5.1. Reduction to the Translation-Invariant Case

5.2. Contours ... . 5.3. Estimate of Number of Contours ... . 5.4. The Pirogov-Sinai Contour Ensemble ... . 5.5. Estimate of the Contour Functional-Symmetric

Case ... . 5.6. Estimate of the Contour Functional-the General

Case ... . 5.7. Partition Functions for Constant Boundary

Conditions ... . 5.8. Correlation Functions for Constant Boundary 5.9.

5.10.

5.11.

5.12.

5.13.

5.14.

5.15.

5.16.

Conditions ... . Open Contours ... . Estimate of the Probability of an Open Contour-The Symmetric Case ... . Open Contours-General Case ... . Partition Functions for a Given Open Contour .... . Correlation Functions for a Given Open Contour ... . Estimate of Length of an Open Contour ... . Estimate of the Intersection of an Open Contour with Layers ... . Main Transformation TI .•...•.•...•...

126 131 133 136 139 141 143 145 146 147 148 153 154 157 160 162 164 167 169 170 172 173 175 176 177

i' lli

ll1 I

·I

viii CONTENTS

5.17. Main Estimate and Check of the Properties of the Transformation 1T • • • • • • • • . • • • • • • • • • • • • • • • • • • • • • • • • . • . • • • . • • . 180 . Appendix I. Ground States of the Model of a One- Dimensional

Lattice Antiferromagnet . . . 183

Appendix II. On Random Ground States of One-Dimensional Antiferromagnetic Model ... _... 188

The Liouville and Sinh-Gordon Equations. Singular Solutions, Dynamics of Singularities and the Inverse Problem Method A.K. POGREBKOV and M.K. POLIVANOV 1. Introduction . . . 198

2. Solution of the Singular Cauchy Problem for the Liouville Equation. Properties of the Solutions . . . 204

2.1. Construction of the Global Solution . . . 204

2.2. Dynamical Variables of the Liouville Field Dynamics of the Singularities ... 209

2.3. Geometrical Properties of Singular Solutions. Topological Charges ... 216

3. The Periodic Problem for the Liouville Equation ... 224

3.1. The Auxiliary Linear Problem ... 224

3.2. The C!k-Matrix for the Liouville Equation ... 234

3.3. The Canonical Transformation to Action-Angle Variables ... 237

3.4. Quasiclassical Quantization of the Liouville Equa- tion. The Possible Role of Singular Solutions . . . 245

4. Singluar Solutions of the Sinh-Gordon Equation . . . 249

4.1. Formulation of the Problem ... 249

4.2. Scheme for Solution of the Direct and Inverse Scattering Problems in the Singular Case . . . 252

4.3. Decomposition of the Set of Solutions of the Sinh- Gordon Equation into Poincare-Invariant Classes ... 256

4.4. Singular Soliton Solutions for the Sinh-Gordon Equa- tion ... 260

5. Conclusion . . . 268

L_

PREFACE TO THE SERIES

Soviet Scientific Reviews presegtly publishes annual volumes in Physics, Chemistry, Mathematical Physics, Physicochemical Biology and Astrophysics and Space Physics. Additional volumes in prepara- tion include Physiology and General Biology, Physicochemical Medi- cine, Microbiology and Biotechnology, Atmosphere, Ocean and Cli- mate, and Physical Methods in Geophysics.

The series is intended to make accounts of recent scientific advances in the USSR more readily and rapidly accessible to the scientist who does not read Russian. Important developments in Soviet science may not receive as much attention as they deserve from the international community because of difficulties with language and distribution. This series, by making available accounts of Russian research in English, will ensure a wider circulation. The articles in these volumes are reviews of recent developments and are written by Soviet experts, most of them in Russian, and translated from the Russian by the publisher.

We are indebted to the volume editors and individual authors for their cooperation in writing and assembling these contributions and getting them to press, usually under considerable time pressure.

The continued success of this series is of course dependent on its continuing to meet the needs and requirements of readers. The distin- guished editorial committee of both Soviet and international scholars has been successful in commissioning the best Soviet contributors for these series and will continue to seek out and publish the best of Soviet science.

In making these volumes available, the publisher hopes to contrib- ute to the further development of international cooperation between scholars and to a greater understanding among scientists.

THE EDITORS

ix

52 D. Ya. PETRINA AND V. I. GERASIMENKO

23. C. Cercignani, Theory and Application of the Boltzmann Equation, Scottish Aca- demic Press, Edinburgh and London, 1975.

0. E. Lanford, Time Evolution of Large Classical Systems, Lee. Notes in Physics, #38, Springer, 1975, pp. 1-111.

H. Spohn, Kinetic Equations from Hamiltonian Dynamics:Markovian Limits, Revs.

Mod. Phys., 53, 569 (1980).

24. A. K. Vidybida, Thermodynamic Limit in Perturbation Theory for Solutions of the Bogoliubov Kinetic Equations, Dokl. Akad. Nauk, Ukr. SSR, ser. A#6, 542 (1975).

25. V. I. Gerasimenko and D. Ya. Petrina, Statistical Mechanics of Quantum-Classical Systems. Nonequilibrium Systems, Kiev, Prepr. 158E, ITP, 1978.

26. V. I. Arnold, Mathematical Methods of Classical Mechanics, Moscow, Nauka, 197 4;

trans!., Springer, New York, 1980.

27. L. A. Pastur, Spectral Theory of the Kirkwood-Salsburg Equations in a Finite Volume, Tear. Mat. Fiz., 18, 233(1974).

28. K. Yosida, Functional Analysis, Springer, Berlin, 1965.

29. T. Kato, Perturbation Theory for Linear Operatbrs, Springer, Berlin, 1966.

30. N. N. Bogoliubov Jr., and B.l. Sadovnikov, Some Problems of Statistical Mechanics, Moscow, Vysh. Shkola, 1975, pp. 352, N. N. Bogoliubov Jr. and B. I. Sadovnikov, Green's Functions and Distribution Functions in the Statistical Mechanics of Classical Systems, JETP 43, 677 (1962).

31. E. G. D. Cohen, The Kinetic Theory of Dense Gases, in Fundamental Problems in Statistical Mechanics, II, 1968, p. 228.

Sov. Sci. Rev. C Math. Phys .. Vol. 5. 1985. pp. 53-196 0143-0416/85/0005-0053 $30.0010

© 1985 harwood academic publishers GmbH and OPA (Amsterdam) B.V.

Printed in the United Kingdom

THE PROBLEM OF TRANSLATION INVARIANCE OF GIBBS STATES AT LOW TEMPERATURES

R. L. DOBRUSHIN AND S. B. SHLOSMAN

Contents

1. Introduction . . . .. . . .. . . .. .. . . .. . 54

2. Ground States ... ... ... 60

2.1. States ... 60

2.2. Specifications and Gibbs States .. ... .. . .. .. ... . . . .. .. .. .. . . .. 62

2.3. Ground State ... ... 64

2.4. Periodic Ground States .. . . . .. .. . . .. .. .. . . .. .. . .. . . . .. .. .. 65

2.5. Main Thesis . . . .. . . .. . . 67

2.6. The CPS-Condition ... ... 69

2.7. Special Cases-One-Dimensional Models ... 71

2.8. The Symmetric Ferromagnetic Ising Model ... 71

2.9. The Antiferromagnetic Ising Model ... 88

2.10. The Widom-Rowlinson Model . .. .. .. .. .. . .... ... . . ... .. ... . . 91

2.11. Random Ground States-the Antiferromagnetic Case .. 93

2.12. Nonperiodic Random Ground States ... ... 99

3. Translation- In variance in Two Dimensions: Basic Ideas .. . 114

3.1. Main Result ... 114

3.2. Basic Reductions ... 115

3.3. Contours ... 117

3.4. Main Transformation ... 118

3.5. Method of Construction of the Mapping 1T ... 120

4. Proofs-the Ferromagnetic Case ... 120

4.1. Introduction ... 120

4.2. Partition Functions under Constant Boundary Conditions ... 123

4.3. Estimate of Number of Contours ... 125

4.4. Correlation Functions for Constant Boundary Conditions ... 126

4.5. Partition Functions for Given Open Contours ... 131

4.6. Correlation Functions for Given Open Contours ... 133

4. 7. Estimate of Length of Open Contours .. . . .. . . .. . .. . . .. . 136

4.8. Construction of the Main Transformation 1T ... 139

4.9. Geometrical Properties ... 141 c

54 ^{R. L.} DOBRUSHIN AND S. B. SHLOSMAN

4.10. Main Estimate ... ..

4.11. Check of the Properties of the Transformation ^{1T •••••••}

5. Proofs-the General Case ... . 5.1. Reduction to the Translation-Invariant Case ... ..

5.2. Contours ... . 5.3. Estimate of Number of Contours ... ..

5.4. The Pirogov-Sinai Contour Ensemble ... ..

5.5. Estimate of the Contour Functional-Symmetric Case .. . 5.6. Estimate of the Contour Functional-the General Case 5. 7. Partition Functions for Constant Boundary Conditions 5.8. Correlation Functions for Constant Boundary

Conditions ... . 5.9. Open Contours ... ..

5.10. Estimate of the Probability of an Open Contour- the Symmetric Case ... . 5.11. Open Contours-General Case ... . 5.12. Partition Functions for a Given Open Contour ... ..

5.13. Correlation Functions for a Given Open Contour .... ..

5.14. Estimate of Length of an Open Contour ... ..

5.15. Estimate of the Intersection of an Open Contour with Layers ... . 5.16. Main Transformation 1T ... ..

5.17. Main Estimate and Check of the Properties of the Transformation 1T ••••••••••••••••••••••••••••••••••••••••••••••••

Appendix I. Ground States of the Model of a

One-Dimensional Lattice Antiferromagnet ... . Appendix II. On Random Ground States of

One-Dimensional Antiferromagnetic Model ... .

1. Introduction

143 145 146 147 148 153 154 157 160 162 164 167 169 170 172 173 175 176 177 180 183 188

In statistical mechanics situations are known in which translation- invariant potentials give rise to non-translation-invariant, and even nonperiodic Gibbs states. Statistical properties of such states are dif- ferent in different parts of the space. Nonperiodic Gibbs states are used to describe surface phenomena in situations where different phases ,co-exist. To their disappearance there corresponds an abrupt change of the properties of the statistics of the surface of phase separa- tion, which is called a roughening transition. Recently additional interest has grown in such phase transitions, related to the possibility

!11

l

INV ARIANCE-GIBBS STATES 55

of interpreting them as phase transitions in lattice gauge models of quantum field theory (cf. Ref. 1).

The existence of a nonperiodic state for the three-dimensional ferromagnetic Ising model was first established in the papers of Dobrushin (Refs. 2,3). There the method used was the contour method of Peierls, applicable at sufficiently low temperatures. This method was later applied with some simplifications by Bricmont, Lebowitz, Pfister and Olivieri to the Widom-Rowlinson model (Ref. 4). In the work of van Beijeren (Ref. 5) the existence of such states was proved by a completely different method, based on the use of correlation inequalities. This method is essentially simpler, and makes it possible to estimate the temperature at which the roughening transition occurs. On the other hand, it does not enable one to study the finer properties of such states (for example, their properties relat- ing to falloff of correlations) and in principle is restricted to the ferromagnetic models.

Already, in Refs. 6, 7 the hypothesis was put forward that in 1 and 2 dimensions, in the case of short range potentials, all Gibbs states are periodic (translation-invariant, in the ferromagnetic case). In dimen- sion 1 this is a consequence of well-known results about the uniqueness of Gibbs states (cf. Refs. 7, 8). Recently in the papers of Aizenman (Ref. 9) and Higuchi (Ref. 10), culminating a series of investigations by other authors ( cf. also Refs. 11-15) this hypothesis was proved for the ferromagnetic Ising model for all temperatures. The method used by them is based on the application of correlation inequalities and consequently also cannot be extended to the nonferromagnetic case.

From this enumeration we see that almost all ofthe investigations have been limited to the ferromagnetic case. The basic aim of this paper is to examine the conditions for existence of nonperiodic states for a more general class of models.

It is well known that the qualitative structure of Gibbs states at low temperatures is disclosed in the study of the ground state of the system, i.e., its state at zero temperature. This approach ( cf. Refs. 16-22) has made it possible to understand quite completely the structures of the periodic Gibbs states arising at low temperatures in terms of the corresponding periodic ground states. The main thesis developed in this paper, is that, in an analogous way, the structure of nonperiod Gibbs states is closely related to the structure of nonperiodic ground states. Section 2 of the paper is devoted to this. There we introduce the general concept of ground states and discuss the conditions under which ground states give rise to nearby Gibbs states at nonzero tern-

56 ^R. L. DOBRUSHIN AND S. B. SHLOSMAN

perature. An important point here is the use of a more general than usual interpretation of the concept of a ground state, in which the ground state is, generally speaking, not a single configuration but rather a random field. It turns out that many of the unsolved problems related to the phase structure of Gibbs states (including periodic ones) reduce to the study of such random fields. In particular, using such an approach one succeeds in constructing an example of a finite potential (an antiferromagnet with particles of several types), in which the Gibbs state is unique both at sufficiently low, and also at sufficiently high temperatures, while in some intermediate range of temperatures there is a nonuniqueness, associated with the breaking of translation invariance of the state. As an illustration we enumerate the ground states of the ferromagnetic and antiferromagnetic Ising model, and also the Widom-Rowlinson model, and discuss from a common point of view the results and hypotheses about the existence of the corre- sponding Gibbs states. The most important and difficult problems here are those related to the concept of the "random ground state" dis- cussed above. The presentation in this Section is in the nature of a sketch and survey, and should rather be regarded as a program for future investigations, in which it is hoped that the authors of this paper will take part. In an appendix written by Burkov and Sinai, still another natural example is given with a nontrivial structure of the ground states.

The rest of this review is of a different character, giving a detailed proof of a new result about the absence of nonperiodic Gibbs states of two-dimensional systems in the low temperature region, under the assumption that the Gertsik-Pirogov-Sinai condition (the Peierls con- dition) is satisfied, i.e., for that class of models for which periodic Gibbs states were studied in the papers of Gertsik, Dobrushin, Pirogov and Sinai (cf., Refs. 16, 18, 19). Our results are formulated in Sec. 3.

In the case of the ferromagnetic Ising model, the result reached by the new method we have introduced is weaker than the Aizenman- Higuchi result. It succeeds in covering not all, but only sufficiently low temperatures. From methodological considerations we nevertheless start our presentation with a detailed analysis of the ferromagnetic case (Sec. 4), where the basic idea of our method is not obscured by technical details. We consider the general case in Sec. 5.

Let u,s briefly describe here, without any pretense of mathematical rigor, the basic ideas of the method used in Sees. 3-5 and its origins.

The physical basis of the specifics of certain low-dimensional systems (primarily one- and two-dimensional ones) consists in the fact that

:,

__ _

INVARIANCE-GIBBS STATES 57

large-scale fluctuations are possible, requiring only a bounded energy.

A simple and beautiful method for the mathematical realization of this idea was proposed by Bricmont, Lebowitz and Pfister (Ref. 23).

Choosingasequenceofincreasingvolumes Vn C 7!_V, they introduce two different sequences of boundary conditions, and the Hamiltonians and

H/;

corresponding to these boundary conditions and a common potential. If it turns out that

- :s; const, (1.1)

then for any inverse temperature

13

the corresponding states in Vn have a ratio of densities that is uniformly bounded inn, and therefore the limiting Gibbs states given by these Hamiltonians are mutually absolutely continuous. On the other hand, from the general results about the uniqueness .of the decomposition of Gibbs states into ex- tremal ones (cf. Refs. 24, 25) it follows that if one of two mutually absolutely continuous Gibbs states is an extremal, then these states must coincide. Using this method, Bricmont, Lebowitz and Pfister gave a very simple proof of the result of Dobrushin (Ref. 7) and Ruelle (Ref. 8) on the uniqueness of the Gibbs state in one-dimensional systems with sufficiently rapid falloff of the potential. Moreover, they proved that, in the two-dimensional ferromagnetic Ising model, the boundary conditions corresponding to plus spins in the upper half- plane and minus spins in the lower do not generate an extremal non-translation-invariant Gibbs state (which is not the case for dimen- sion 3, cf. Ref. 3). The result described above is a consequence of an earlier theorem of Gallavotti (Ref. 12), in which it was proven that the limiting state considered is a halfsum of the limit states generated by constant boundary conditions. However, the new proof is much simpler.

The further development of this idea is related to its application to the proof of the absence in certain models of a breakdown of continuous symmetry. The particular feature of the case of continuous symmetry is that large-scale fluctuations can occur not only with a finite energy, but even with an arbitrarily small energy. Thus Pfister (Ref. 26) and Klein, Landau and Shucker (Ref. 27) used this method for two-dimensional models with continuous symmetry, while Simon (Ref. 28) applied it to one-dimensional models with a long-range potential. Recently, Frohlich and Pfister (Ref. 29) proved by this method the translational invariance of two-dimensional continuous (i.e. non-lattice) models with superstable interaction without hard

58 R. L. DOBRUSHIN AND S. B. SHLOSMAN

core, smooth outside of the origin. Finally, very recently Burkov and Sinai (Ref. 30) used these ideas for the proof of the translational invariance of all Gibbs states in one-dimensional models with a long range interaction, which falls off more slowly than is needed to assure uniqueness of the state.

We shall single out some of the auxiliary arguments used in the papers cited above.

Let us consider a lattice system with configuration space

sr.

^Sup-

pose that a group G acts on S, leaving invariant the Hamiltonian of the system. For anyfunctiong: V G, V C

zv,

there is defined in a natural way its action on configurations CJ" e S:gCJ" = (g1CJ"1 , t e V). Suppose that for each of the volumes Vn

czv,

^Vn

ⁱ zv,

there is given a sub- volume

c

Vn , j

zv,

and a function gCn):Vn G, such that, for some he G,

g{n) = h = const for t e (1.2)

We set CJ"¹ = g<n)CJ", and

H;(CJ") = (1.3)

Making a change of variables in the corresponding configuration inte- grals, it is not difficult to show that for any local observable with support in its expected average value, given by the Hamiltonian H^2, will coincide with the expected average value of the shift by h of this observable given by the Hamiltonian H^1. Thus, if condition (1.1) holds for each hand the limiting state, given by the Hamiltonian H¹, is extremal, it follows that it is invariance under the group G. As ob- served in Ref. 29, instead of the condition (1.1) one can verify the weaker condition

- H;(CJ")> const, (1.4)

where the average value is taken with respect to the Gibbs state with Hamiltonian

The ideas developed earlier cannot be applied directly to the ques- tion of the translational invariance of the two-dimensional limiting state corresponding to a general sequence of boundary conditions. In contrast to the special class of boundary conditions considered in Ref. '

12, for a general boundary condition and its shift neither condition (1.1) nor condition (1.4) are satisfied; one also does not succeed in

I_

INV ARIANCE-GIBBS STATES 59 defining mappings analogous to the mapping CJ" CJ"¹ considered above. We therefore decompose the configuration space Om corre- sponding to volume Vn into disjoint classes T:!ln = U T and define

n'!J

the mapping T T', which takes'classes of configurations one into another. The analog ofthe condition (1.1) is the assertion that the free energies calculated for the classes T and T,, differ only by a constant:

lin

2:

exp(-!3 H(CJ"))- In

2:

^exp

^(-13

H(CJ"))I const (1.5)

creTe <TET'

uniformly inn. An essential point is that, in contrast to condition (1.1), the verification of condition (1.5) requiFes the use of quite delicate information about the statistics of configurations. We shall reduce it to a comparison of the partition functions for constant boundary condi- tions in volumes of different shape. However a sufficiently complete control over the properties of such partition functions is possible only for sufficiently large [3, which limits the method to the case of low temperatures. We note that we verify condition (1.5) not for all T, but only forT e g" C 21, where the set fin = Ll_ Tis such that its measure,

-re5"

given by the Hamiltonian, is bounded from below by a positive con- stant uniformly in n. This can be regarded as a further weakening of condition (1.4). Finally, condition (1.2) corresponds to the assumption that in the restriction to the mapping T T' is reduced to a shift of

these classes of configurations. .

If for the ferromagnetic case the estimates of the dependence of the partition function on the shape of the volume, which we need, have been known for a long time ( cf. Ref. 2), to get them in the general case requires the use of the method of the equivalent contour ensemble of Pirogov-Sinai (Refs. 18-20). For us it also proves to be very useful to use the modification of the Pirogov-Sinai method proposed by M.

Zahradnik (Ref. 21). For the more special, but still wide enough case, where the phase transitions are associated with a breakdown of sym- metry of the ground states, the result we need can be obtained by more elementary methods, without using the cumbersome constructions of Pirogov-Sinai and Zahradnik. Because of this, and also because of the fact that the paper of Zahradnik (Ref. 21) is not easily available as yet, we also present in Sec. 5 alternative contructions applicable to this symmetric case.

The authors are grateful to M. Zahradnik, who presented his work at the Moscow seminar on mathematical methods of statistical physics,

60 ^R. L. DOBRUSHIN AND S. B. SHLOSMAN

and provided us with the possibility of studying the preliminary version of this paper prior to its publication.

2. Ground States

This section is in the nature of a review. In Sees. 2.1 and 2.2 we introduce the notation and recall some well-known concepts. Section 2.3 is devoted to the general definition of a ground state. In Section 2.4 we discuss the relation between the general definition introduced by us and the usual definition of a ground periodic configuration. In Sec. 2.5 we formulate the main hypothesis of this paper, which selects among all ground (not necessarily periodic) configurations those that are stable, i.e., those to which there correspond Gibbs states at low temperatures. (The content of Sees. 3-5 may be interpreted as a proof of this hypothesis for two-dimensional systems). Section 2.6 is devoted to the discussion of the relation of the hypothesis introduced and the Peierls condition (the Gertsik-Pirogov-Sinai condition). In Sec. 2.7 we illustrate the validity of the hypothesis on the example of the one-dimensional case. In Sec. 2.8 we describe the ground configu- rations of the ferromagnetic Ising model for v = 1, 2, 3, and discuss their stability. We present new examples of ground configurations, stable for v ;;,: 4. The similar class of questions for the Ising antiferro- magnet with external magnetic field is discussed in Sec. 2. 9, and for the Widom-Rowlinson model in Sec. 2.10.

The last Sections, 2.11, 2.12 of Section 2 are devoted to the descrip- tion of random ground states for the models listed above. We present a listing of open questions. We discuss possible consequences for the phase diagrams of the models studied. The solution of the problems that arise here for the models traditionally considered is apparently beyond the limits of capacity of present mathematical technique.

Therefore, in Sec. 2.11 we consider one modification of the Ising antiferromagnet for which it is not difficult to prove the nonuniqueness of the random ground state, and derive the resulting consequences about the phase diagram for this model.

2.1. States

Suppose that 71..v is a v-dimensional simple cubic lattice with points t = (t¹, . . . ,tv), where t; are integers. Two types of norm on 71..v are convenient: _I

lltll

⁼ max

lt;l

and

ltl

⁼

(lt1

¹

2+ ... +ltvl2)1/2

i=l, ... ,o

L_

INV ARIANCE-GIBBS STATES 61 We shall interpret 71..v as a graph whose edges join vertices s ,t such that

Is -

tl = 1. We shall also denote by Vn the "cube" in 71..v with sides 2n:

Vn= \t E 71..v :

lltll n\.

Let S be a finite or countable set of states of the particle. (Every- where, except in this section and in Sees. 2.2 and 2.12, unless the contrary is specifically stated, the set S is assumed to be finite. For example, S = [-1, +1\ for the Ising model, S = [-1, 0, +1\ for the Widom-Rowlinson model.) The set of all mappings S, t a1

will be denoted by

n

⁼

sr

and called the set of configurations. For

V C 71..v the set of mappings a: V S will be denoted by Dv. For

V C W C 71..v and a ^E llw, we shall denote by av the restriction aiV.

For V¹' V² 7l_V' V¹

n

V²=

0,

Uv;E llv;, i

=

1 ,2, we shall denote by av,U avz a configuration from

nV'u

^V2 such that ITVJU ITv21 v•=

ap, i = 1,2. For W C 71..v we shall denote by 0'3w the a-algebra of subsets of the space

n,

generated by cylindrical sets of the form

[a En: av = const, V C W,

lVI

< ooj.

From here on

lVI

is the power of the set V, vc= 7l..v\V. We shall de- note the a-algebra 0'3z, by 01. We use T1, t E 71..v to denote the shift transformation on f!:(T₁a)s = a(s-t)' s ^E 7l..v. Similarly, for a E f!v, we denote by T1a a configuration of Dv+t such that (T1a)s = IT(s-t)' S E V

+

t.

By a state we mean a probability measure< >on (f!, 01). For any measurable function 'P( · ) on f!, we mean by <'P> its integral with respect to this measure. A sequence of states< >n is said to be con- vergent to a state < > if, for any V C 7l..v,IVI < oo and any 0'3v- measurable function 'P,

<'P> for n oo.

A state < > is said to be deterministic if there is a configuration a E

n

such that <[a\> = 1. Indeterministic states will be called random. A state is said to be translation-invariant if for all t E 71..v

<'P> = < Tt'P> ' ^(2.1)

where 'P is a 01-measurable bounded function, while (T1'P)(a) =

'P( T -tiT). Similarly, a state < >is said to be 71..' -periodic, if (2.1) is valid

62 ^{R. L.} DOBRUSHIN AND S. B. SHLOSMAN

for all t E 7L', where 7L' C 7Lv is some subgroup of the group 7Lv with finite index.

2.2. Specifications and Gibbs States

A translation-invariant interaction of radius r < oo (or simply an inter- action)isasetoffunctions u

=

tUA(o"), a E n,A

c

^7l_V, IAI < oo]with values in the extended real line IR¹U \+oo], such that

1) UA

=

0 if diam A > r, where diam A = max lis -

s,t ^E A

2) UA(ai) = UA(az) if (ai)A = (<rz)A·

Everywhere except in this paragraph and in 2.12, we shall assume that the potential U is translation-invariant, i.e., that it satisfies the condition:

3) UA+t

=

T1UA·

In the set 2W of all interactions we introduce a metric q, given by the formula*

q(U, U') = sup

I

exp j-UA(a)]- exp \-UA(a)]j A E 7l_V, a En

Suppose that V C 7Lv,

jVJ

< oo (we call the set V a volume), ayE flv,

a

En. We define the relative Hamiltonian Hv(+) by the formula Hv(avl a)

= 2:

UA(aA)

A:A

n

V =/= 0

where

aAI v n A = aAI vn A' aAI AW = ai A\V·

The configuration

a

is called a boundary condition.

By a specification we mean a system of functions Q = \Qv(avl a), ayE

n

y, a ^E Ov,

v c

^7l_V,

lVI

< oo], where Ov ^E ellvc, such that

1) For any a E Ov, Qv(·

I

a) is a probability measure on flv.

2) The function Qv( a vi ·) is ellvc-measurable for any configuration avE flv.

*From here on a + oo = +oo for any a E IR¹U j+ool, e-oo= 0.

INVARIANCE-GIBBS STATES

3) For any V

c

W

c

7Lv,

IWJ

< oo and a e Ow, Qw(!aw:aw

I

w ^\V

u

^a

I

(W \li)<E

Ovli

a)

=

^1'

and for any aw ^E flw

63

Qw(aw

I

a) = Qv(av

I

aw ,:; a(w we) X

2:

^Qw(aw

I

a).

awe

^Ow:

afv,v= aww

Condition 3)-the consistency condition-means that the condi- tional distribution in the volume V, given by the measure Qw(·

I

a) under the condition aww, coincides with Qv(·j aW\v U a (W\V)c).

A state < > is said to _!?e consistent with the specificatioQ_ Q if, for any volume V C 7Lv, a ^E flv and <Tv ^E flv, the probability <flv> = 1, and the conditional probability

<[a

^E O:av=

<Tv] I

a> ⁼ Qv(<Tv

I

a).

The specification Q = Q₁₃.v and the state < > consistent with it are said to be Gibbsian with interaction U and temperature

J3,

if for any volume V C 7Lv, avE flv and a ^E flv,

Qv(avl a) ^exp [-J3Hv(avl av)]

z(v

I

a)

where the partition function Z(V

I

a) is defined by Z(V

I

a)

=

Z¹³(V

I

a)

= 2:

exp [-J3Hv(av

I a)]

<Tj.AE.flv

and the domain of definition flv is defined by Ov

= [a

^E 0:0 < z (V

I

a) < oo].

(2.2)

(2.3)

In the case of

J3 =

1 we shall simply speak of a specification and a Gibbs state with interaction U.

It is known ( cf. Refs. 6, 24, 25) that the set of states consistent with a given specification is convex, and that such a state is representable as a weighted average of extremal states. In the case where Sis finite, this set is definitely not empty.

64 R. L. DOBRUSHIN AND S. B. SHLOSMAN

Sometimes it will be convenient for us to associate with the bound- ary condition <T E ilv and volume V the state < >Q,V,iT• concen- trated on the set /a:avc

=

<Tvc), such that, for any volume W C

z.v

^and

bounded function 'f',

<qJ>Q,v,,:-=

f

'P (av U <Tvc) Qv(davl <T).

Any extremal state with a given specification Q can be obtained as the limit of states < >Q,v",iT" for an increasing sequence of cubes Vn and an appropriately chosen sequence of boundary conditions <Tn.

Such a limit is always a state with specification Q. Referring to a state

< >which is the limit of states< >Q,v",iT' we shall say that it is gen- erated by the boundary condition

cr.

2.3. Ground State

Suppose now that U is some interaction, and <T E ilv. By Mu(<T) we denote the Set of configurations U v E ilv SUCh that

H v( &vi <T) = min H v( a vi <T) ayE ilv

(2.4)

The specification Q = /Qv) is called a ground state specification for the interaction U (cf. Refs. 16, 31) if

{

IMv(rr)l-1, ayE Mv(<T) ,

Qv(avl <T) = 0 , avi, Mv(<T) . ^(2.5)

The specification Q = /Qv) is said to be w-ground (or weakly ground) for the interaction U if Qv( a vi <T) = 0 for av M v(<T).

A state < > will be called a ground ( w-ground) state for the inter- action U if it is consistent with the ground specification (or with one of the w-ground specifications) for the interaction U.

In accordance with what was stated in Sec. 2.2 about states with a given specification, the ground states always exist, form a convex set and can be constructed as the limits of states< > Q, vn,a n for the corre- sponding boundary conditions <T n.

The following simple proposition serves as a jusitification of the above definitions.

Proposition 2.1. Suppose that the sequence of states< > n converges to the state < >.

i

I·;

I

INV ARIANCE-GIBBS STATES 65

1) If the state < > n is Gibbsian with the interaction U and reciprocal temperature where oo, then the state< >is a ground state for the interaction U.

2) If the state < >

n

is Gibbsian with interaction

u<n)

and reciprocal temperature where oo and U, then the state < > is a w-ground state for the interaction U.

Proof: Suppose that Q(n) is a Gibbsian specification, consistent with the state < >no Then Q(n)(avl <T) QJ0)(avl <T), where Q(O) is some specification which, as it is easy to see, is ground in case 1) and w-ground in case 2). From this and from the definition of ground and w-ground states, the assertion of the proposition follows easily.

The proposition just proved shows that in searching for Gibbs states with low temperature and with given (or close to the given) interac- tion, we must consider among the candidates states close to ground (or close to w-ground) states. We say that a w-ground state is stable (w-stable) if there exists a sequence of Gibbs states converging to it, and satisfying the conditions of assertion 1) (or assertion 2) of proposi- tion 2.1. The main purpose of the rest of this Section is to study the ground states and discuss the conditions for their stability.

If a deterministic state < >, concentrated on the configuration a is ground (w-ground), then we shall also say that the configuration a is ground (w-ground). Furthermore, a w-ground configuration will be said to be stable ( w-stable) if the deterministic w-ground state that defines it is such a state. It is easy to see that a configura- tion a is w-ground if and only if for any V C

z.v, I VI

< oo the configura- tion, ayE Mv(a). A w-ground configuration is ground if and only if IMv(a)l = 1 for all V. A ground state is concentrated on w-ground configurations. Therefore from the existence of ground states the existence of w-ground configurations follows. It is possible also that there are no ground configurations, as is seen from the examples considered in Sees. 2.11, 2.12. In constructing ground states as limits of states< > Q,

vn:-n,

we shall restrict ourselves to boundary conditions <T n

which are w-ground configurations.

2.4. Periodic Ground States

In this section we shall assume, in order to simplify the situation, that the interaction takes on only finite values. We recall that a configura- tion a is said to be periodic with respect to a subgroup Z.'

c z.v

^{of finite}

index if T₁a = a for all t E Z.'. The period d of the configuration a is the lowest of the numbers

d

such that

d z.v c

Z.'. Frequently the ground

66 ^R. L. DOBRUSHIN AND S. B. SHLOSMAN

periodic configurations are singled out by minimizing the specific energy. We set

Hv(cr) =

L

UA(cr) , hm= IV ml-¹ inf Hvjcr) .

crEfl

V C 7l.v'

lVI

< ^{00 '}

(2.6)

For any configuration cr E 0 we define its specific energy to be the limit

h(cr) = lim IVml-¹ Hvjcr), (2.7)

if it exists. It is easy to see that this is the case for periodic configu- rations.

Proposition 2.2. (cf. Ref. 20, Sec. 2.2).

1) The limit h = lim hm

exists.

2) For any w-ground configuration cr the specific energy h( cr) exists, and h(cr) =

h.

3) The equality

h

= inf h(cr) crEfl,

holds, where inf is taken over the set of all periodic configurations.

4) If cr is a periodic configuration, and h(cr)

= h,

then cr is a w- ground configuration.

Proof: We note that for some C < oo, for any m and cr E 0, IHv _m (crv ) -_m Hv _m (crv Jcr) _m

I :;;; c

^{mv-1 (2.8)}

(see property 1) of the interaction).

Let us prove 1) Suppose that cr E 0 is such that hk= Hvk(cr), and let crk be the periodic extension of the configuration crv E flv . From A k k

(2.8) it follows that lim lh(crk) - hkl

=

0. Because of the peri-

00

INV ARIANCE-GIBBS STATES 67 odicity of the configuration crk, for any E > 0 there is an m(k,c) such that, form ;3 m(k,E),

IVmi-¹HvjcrtJ:;;; h(crk)

+

€,

/

and thus, for sufficiently large m, hm:;;; hk+ ^€.

Therefore lim sup hm :;;; lim infhm, which proves 1).

2) Suppose that assertion 2) is false, and that for some w-ground configuration cr, lim supiV

mi-

¹Hv (cr) >

h.

We choose configurations

m

crm such that lim

IVmi-

¹Hv (crm) =h. From (2.8) we see that the

m

difference - = o (IV mJ), as m oo. There-

fore, for sufficiently large m, Hv (crv lcr) > Hv lcr), and this contradicts the assumption that cr i; a ;-ground

3). It is clear that for any periodic configuration cr the specific energy h(cr) ^;3

h.

In deriving assertion 1) we constructed a sequence of periodic configurations crk such that h(crk) h for k This proves 3).

4). Suppose, finally, that a 71.'-periodic configuration cr is not w- ground.Supposethatnissolargethatthevectort = (n,n, ... ,n) E 71.^1. Then for some V C 7l.v,

lVI

< oo and cr{;E flv, the energy Hv(crvlo-) >

Hv(crirlcr). Now we construct a new periodic configuration rr. Namely, we consider the configuration crvc, restrict it to V mm choosing the integer m so that dist (V, Vmn) > r, and take the periodic exten- sion. Then, using the n-periodicity of cr, it is not difficult to verify that h(cr) > h(rr) ^;3

h,

and this proves assertion 4).

As the example considered in Ref. 32 shows, for the case of an unbounded space S, when the interaction can also be unbounded, assertion 2) of the theorem just proved may be false even for a stable ground configuration. We also mention that without the condition of periodicity of the configuration cr, assertion 3) of the theorem is false, since a change of the configuration cr on a finite set does not affect h(cr). We note, finally, that the problem remains open whether peri- odic w-ground configurations always exist.

2.5. Main Thesis

In investigating the stability of periodic ground configurations, one frequently uses the following symmetry property. Suppose that G(S) is

68 ^{R. L.} DOBRUSHIN AND S. B. SHLOSMAN

thegroupofpermutationsofthe(finite)setS, G = E17G(S1), i.e., G is t ^E 7l..v

the group of functions g:7l..v----'> G(S) with pointwise product. The group G acts naturally on the set O:(gcr)1= g1cr1• We denote by 2r the sub- group of elements of the group G that leave invariant the interaction U:g E 2r uA(cr) = UA(gcr), for all (TEn, A

c

7l..v. We say that the symmetry condition of the w-ground periodic configurations holds if the set Lu of those configurations is finite, Lu= jcr<l), ... , cr(kl), and the group 2r E!7 ]shifts of 7!..vl acts transitively on Lu· In other words, any configuration cr E Lu can be taken into any other by the com- position of some transformation of 2r and some shift.

Suppose that cr is an arbitrary w-ground configuration. A local perturbation of it is any configuration 0', for which the set jt E 7l..v, 0'₁

-f erA

is finite. We call this set the support of the perturbation 0'

and we denote it by supp (O'jcr). The perturbation is said to be n- connectedif, for any two pointss,t E supp (O'jcr) and there is a sequence t₁ = s, ... ,tk = t, such that t;E supp (O'jcr) and jt;+1 - t;l :s:: n,i =

1, . . . ,k - 1. We denote the set of all local n-conected perturbations of the w-ground configuration cr by Ln(cr). We call the quantity

E(O'jcr) =

L

[UA(O') - UA(cr)]

A ^E 7l..v

(2.9)

the energy of the perturbation 0' (only a finite number of terms in (2.9) are non-vanishing). Because of the definition of the w-ground configuration, E(O'jcr) 0. We say that w-ground configuration cr is nondegenerate (for the interaction U) if, "for any n > 0, E > 0,

max (diam supp (O'jcr)) < oo;

0' E Ln(cr):E(O'jcr) < E

in other words, this means that only bounded n-connected perturba- tions can give rise to a bounded increase in energy. We say that a w-ground configuration is w-nondegenerate, if, for any £ we can find an interaction UE E )lli,suchthatq (U, UE) < c,andtheconfigurationcr is w-ground and nondegenerate for the interaction UE.

Now we are able to formulate our main hypothesis.

Thesis:

1). A w-ground configuration cr is w-stable if and only if it is w- nondegenerate.

2). Suppose that the condition of symmetry of ground periodic configurations is satisfied. Then a ground (not necessarily periodic) configuration cr is stable if and only if it is nondegenerate.

INVARIANCE-GIBBS STATES 69

The main (and not very strong) support in favor of the stated hypothesis is that the authors at present know of no examples that contradict it. It is therefore natural to expect that further investigations will show that there is a need to modify it and to restrict its domain of applicability.

2.6. The CPS-Condition

For the study of the stability of periodic ground configurations, Gertsik (Ref. 17) and Pirogov and Sinai (Refs. 18, 19) introduced the following additional condition (GPS-condition or Peierls condition).

Suppose that there are only a finite number of periodic ground config- urations cr<¹l, ... ,cr<kl, and d is their smallest common period. Let the cubes T1V m• Rm(cr)

=

]t E 7l..v:crv,

f

crVl for all i

=

1, ...

,k),

and Nm(cr) = IRm(cr)i be the powe; of the'"set Rm(cr). We shall say that the GPS-condition is satisfied if, for some

E

> 0, for any ground periodic configuration cr and any local perturbation 0' of it, the energy

E(O'jcr)

E N[ d;

^{1 ] (0') (2.10)}

h [ ^d + ¹

j .

h 1 . . d + ¹ I . d"ff. 1 h w ere -

2- IS t e argest mteger m - ₂^{- .} t ^IS not ¹ tcu t to s ow

[ d + l

l

^A

thatforanys - ₂^- and some E, it follows from (2.10) that

E(O'jcr)

E

^{s Ns} (0') . (2.11)

It is obvious that if 0' E Ln(cr), then for diam supp (O'jcr)----'> oo, also N(O') ----'> oo. Therefore if the GPS-condition is satisfied it follows that all the ground periodic configurations are nondegenerate. Gertsik (Ref. 17) and Pirogov and Sinai (Refs. 18, 19) proved, respectively, the stability and w-stability of all periodic ground configurations if the GPS-condition is satisfied (and, in Gertsik's case, under the additional condition of symmetry of the ground periodic configurations), which is a partial confirmation of the formulated hypothesis. Sinai (Ref. 20) conjectured that the GPS-condition follows from the finiteness of the number of periodic ground configurations. Counterexamples were recently constructed by Pecherski (Ref. 33) and Preiss (private com- munication). In Pecherski's example, the ground periodic configura-

70 R. L. DOBRUSHIN AND S. B. SHLOSMAN

tions are degenerate if the dimension v

=

2 and nondegenerate if v 3. The question of their stability has not been studied.

In the paper of Holsztynski and Slawny (Ref. 34), they single out a class of interactions for which the GPS-condition holds. It is defined as follows:

A configuration (J E

n

that is ground for the interaction

u

is said to be s-ground, if for all A

c zv,IAI

< ^oo,

UA(rr) = min UA(rr) .

CTED

One of the results of Ref. 34 is that if the number of s-ground configu- rations for the interaction U is finite it follows that the GPS-condition is satisfied for it.

Remark In the example of Pecherski the number of (nonperiodic) s-ground configurations is infinite.

It is possible that the investigation of non periodic ground states will be made easier if one introduces a condition analogous to the GPS- condition, but it is not clear how such a general condition should be formulated. It is even less clear how to formulate an analog of such a condition for random ground states.

2. 7. Special Cases. One-Dimensional Models

It is known ( cf. Ref. 7, 8) that for the class of interactions m5 considered, in the case of v = 1 the Gibbs state is always unique. The validity of our thesis must then mean that, if in some one-dimensional model there are two ground configurations, then at least one of them must be degenerate. Assuming, for simplicity, that the interaction takes on only finite values, we can prove the stronger assertion:

Proposition 2.3. Suppose that v = 1, and that rr is a ground configura- tion. Assume that there exists a w-ground configuration rr, not co- inciding with CT. Then rr is a degenerate ground configuration.

The proof of this assertion is immediate in the case where for all t E £'¹, crt=f CTt: then the sequence of local perturbatiOnS U(n)=

IT[-n,n]U CT[-n,n]' is 1-connected, has an increasing diameter and (cf. below for more details) a finite energy.

The same idea is applicable in the general case. We note first that if, for some t E £'¹, rr_1=f rr1, then for any n₁ < t < n_{2 ,} at least one of the restrictions of the configurations rr, rr to the two segments [ n_{1 -}

r, nd,

[n2 , n2+ r], do not coincide. Otherwise, the set M[n,,n₂J(rr) would contain at least two elements, which would contradict the assumption

INVARIANCE-GIBBS STATES 71 that rr is a ground configuration. From this there follows the existence of an infinite monotonic sequence of points t2 , • . . C £'¹ such that ^lti+l- til .;;; r and rrt,=f rr1,. Supposing, for definiteness that this sequence is increasing, we see that the perturbations (j'(n)=

CT[t,,t"J< form a sequence of r-connected local perturba- tions of increasing diameter.

Let us estimate the quantity

E(O'(n)lrr) = H[vnJCrrr1,,drr) - H[Vnl(rr(t,.drr) .

Using (2.8) and the fact that rr is a w-ground configuration, we see that H[t,,tnJ(rr(t,.drr) .;;; H[t,,drr(t,.drr) + 2 C .;;; (since rr is a w- ground configuration).

.;;; H[1,,drr[1,,drr)

+

2 C.;;;

+

4 C

This means that £((j'{n)lrr) .;;; 4C, which proves Proposition 2.3.

We note finally that from theorem 2 of Ref. 31 it follows that if there is a unique periodic ground configuration satisfying the GPS condi- tion, then there is a unique ground state. Thus, in constructing specific examples of nonperiodic ground configurations it is natural to consider interactions that lead to non unique periodic ground configurations. As already pointed out in Sec. 1, we consider below the simplest classes of such interactions: the symmetric ferromagnetic Ising model, the antiferromagnetic Ising model with external field, and two variants of the Widom-Rawlinson model.

2.8. The Symmetric Ferromagnetic Ising Model

Let us assume that S = /-1, +1), and the interaction is

U ^{A (} rr) = { - rr P t ' if A = /s' t)' is - tl = 1

0 for other A. (2.12)

In this model there are exactly two periodic ground configurations, rr+=

/ut

^{= 1) and rr-= /rr1 =} -1). These configurations satisfy the GPS condition (for any v); it is easy to check that for v = 1 they are degenerate, and are nondegenerate for v 2. The assertion that they are unstable when v = 1 and stable for v 2 is well known ( cf. Refs.

7, 8).