Lecture Notes in Physics Edited by H. Araki, Kyoto, J. Ehlers, MQnchen,K. Hepp, Z0rich R. Kippenhahn,MLinchen,H. A. Weidenmeller, Heidelberg and J. Zittartz, Kbln Managing Editor. W. BeiglbSck
257 Statistical Mechanics and Field Theory: Mathematical Aspects Proceedings of the International Conference on the Mathematical Aspects of Statistical Mechanics and Field Theory Held in Groningen, The Netherlands, August 26-30, 1985
Edited by T.C. Dorlas, N.M. Hugenholtz and M. Winnink
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo
Editors
T.C. Dorlas N.M. Hugenholtz M. Winnink Institute for Theoretical P~ysics, University of Groningen P.O. Box 800, Groningen, The Netherlands
ISBN 3-540-16777-3 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-16777-3 Springer-Verlag NewYork Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2153/3140-543210
PREFACE
This volume of LECTURENOTES IN PHYSICS contains the text of the lectures presented during the International Conference on the Mathematical Aspects of S t a t i s t i c a l Mechanics and Field Theory, held in Groningen, the Netherlands, 26-30 August 1985. Some of the participants have provided us with an abstract of the poster(s) they presented during the poster sessions. We have incorporated these abstracts as we]l in this volume. As is unavoidable, in those abstracts claims are presented without proof. Interested readers are encouraged to contact the authors. The organizers of this conference thought i t a good idea to have a conference on mathematical physics in the Netherlands where this subject is receiving more attention nowadays. We hope, among other things, to have given an impetus to mathematical physics in the Netherlands. We want to express our sincere gratitude to the s c i e n t i f i c advisory board, the lecturers, the participants, the Congress Bureau of Groningen University, the secretary of the I n s t i t u t e for Theoretical Physics Ms. M. Boering, and last but not least to the sponsors of this conference, Philips Research Laboratories, Shell Nederland BV, IAMP, Groninger Universiteitsfonds, Koninklijke Nederlandse Akademie van Wetenschappen, College van Bestuur RUG, Fakulteit der Wiskunde en Natuurwetenschappen RUG and Subfakulteit Natuurkunde.
T.C. Dorlas N.M. Hugenholtz M. Winnink
Scientifc Advisory Committee: H. Araki, J. FrOhlich, R. Haag, N.M. Hugenholtz, J.L. Lebowitz
CONTENTS
A MODEL FOR CRYSTALLIZATION: A VARIATION ON THE HUBBARD MODEL T. Kennedy, E.H. Lieb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FIRST ORDER PHASE TRANSITIONS AND PERTURBATION THEORY J. Bricmont, J. Slawny . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
LOW TEMPERATURE CONTINUOUS SPIN GIBBS STATES ON A LATTICE AND THE INTERFACES BETWEEN THEM
-
A PIROGOV-SINAI TYPE APPROACH
M. Zahradnik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
SPIN GLASSES, EFFECTIVE DECREASE OF LONG-RANGE INTERACTIONS A.C.D. van Enter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
ANALYTIClTY IN SOME MODELS OF QUANTUM STATISTICAL MECHANICS H. A r a k i
......
~. ..........................................................
89
K-THEORY OF C*-ALGEBRAS IN SOLID STATE PHYSICS .J. B e l l i s s a r d
..............................................................
99
QUANTUM FIELD THEORY WITHOUT CUTOFFS: RENORMALIZABLE AND NONRENORMALIZABLE A. Kupiainen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
157
RENORMALIZATION GROUP METHODS IN RIGOROUS QUANTUM FIELD THEORY K. Gaw~dzki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17]
RENORMALIZATION GROUP METHODS FOR CIRCLE MAPPINGS O.E. Lanford I I I
..........................................................
]76
CORRELATIONS AND FLUCTUATIONS IN CHARGED FLUIDS Ph.A. M a r t i n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
190
MODELS OF STATISTICAL MECHANICS IN ONE DIMENSION ORIGINATING FROM QUANTUM GROUND STATES H. Spohn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
209
WHY DO BOSONS CONDENSE? J.T.
Lewis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
234
Vl
BLACK HOLES AND QUANTUM MECHANICS G. ' t Hooft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
257
QUANTUM FIELD THEORY AND GRAVITATION R. Haag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
258
A REMARK ON ANTIPARTICLES H.J. Borchers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
269
NOTES ON THE CANONICAL ANTICOMMUTATION RELATIONS R.V. Kadison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
281
A DIFFERENTIAL-GEOMETRIC SETTING FOR BRS TRANSFORMATIONS AND ANOMALIES D. K a s t l e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
295
POSTER ABSTRACTS RENORMALIZATION OVA HIERARCHICAL FERMION MODEL IN TWO DIMENSIONS T.C. Dorlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
313
GRIBOV COPIES AND ABSENCE OF SPONTANEOUS SYMMETRY BREAKING IN COMPACT U(1) LATTICE HIGGS MODELS Ch. Borgs, F. N i l l
........................................................
319
RENORMALIZATION GROUP APPROACH FOR THE ISING MODEL AS AN APPROXIMATE SOLUTION OF THE DIAPHANTIAN SYSTEM OF EQUATIONS S. Rabinovich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
320
TRANSFORMATION OF THE TRANSFER MATRIX FOR THE ISING MODEL UNDER DECIMATION S. Rabinovich . . . . . . . . . . . . . .
...............................................
321
SOME RESULTS CONCERNING THE LOCALIZATION PROBLEM IN ONE-DIMENSIONAL QUASlPERIODIC SYSTEMS D. P e t r i t i s
...............................................................
322
RANDOM WALKS ON RANDOM LATTICES W.Th.F. den H o l l a n d e r , P.W. K a s t e l e y n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
323
THE ROLE OF TRANSVERSE FLUCTUATIONS IN MULTIDIMENSIONAL TUNNELING A. Auerbach, P. van Baal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
323
Vll
SELF-SIMILAR TEMPORAL BEHAVIOR OF P4~NDOMWALKS IN RANDOM MEDIA J. Bernasconi, W.R. Schneider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
324
LOOP EXPANSIONS IN THE PRESENCE OF GRIBOV COPIES F. N i I l
....................................................................
325
NON-EXISTENCE OF LONG-P~ANGE-ORDER FOR A CERTAIN ID MODEL AND THE SOLITON PICTURE H. Grosse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
325
QUANTUM FIELDS OUT OF THERMAL EQUILIBRIUM W. Schoenmaker, R. Horsley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
326
STATISTICAL MECHANICAL APPROACH TO THE CONSTRUCTION OF FOUR DIMENSIONAL BOSON FIELD THEORIES G.A. Baker, J.R. Johnson, J.D. Johnson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
326
CORRELATION LENGTHS AT ZERO TEMPERATURE IN ONE DIMENSION J.-J.
Loeffel
..............................................................
327
HORIZONTAL WILSON LOOPS IN FINITE TEMPEPJ~TURE LATTICE GAUGE THEORIES C. Borgs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
327
THE RANDOM BOND POTTS AND ASHKIN-TELLER MODELS P.L. C h r i s t i a n o , S. Goulart Rosa J r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
328
SPECTRAL LINE SHAPES IN QUANTUM MARKOV SCATTERING H. Maassen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
328
EXACT BOGOLIUBOV LIMITS FOR THE BASSICHIS-FOLDY MODEL AND CONTINUED FRACTIONS D. Masson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
328
A M o d e l for C r y s t a l l i z a t i o n : A Variation on t h e H u b b a r d M o d e l
Tom Kennedy and Elliott H. Lieb Departments of Mathematics and Physics Princeton University Jadwin Hall, P.O. Box 708 Princeton, NJ 08544
Much attention has been paid to the question of proving the existence of long range order in model statistical mechanical systems in which the basic atomic constituents interact with short range forces. An important example is a lattice spin system in which the spin at each site represents the localized spin of an atom located at that site and where the short range, pairwise interaction (Ising or Heisenberg) reputedly comes from an interatomic exchange energy. Another problem - so far unsolved - is the existence of periodic crystals which are supposed to come from short range (e.g. Lennard- Jones) interatomic potentials. In the real world, however, these interactions are not given a-priori; it is ultimately itinerant electrons and their correlations that give rise to the long range ordering. In other words, a deep unsolved problem is to derive magnetism or crystallization from the SchrSdinger equation - or some caricature of it. The construction of a simple model based on itinerant electrons, and the rigorous derivation of ordering from it, is a challenge for mathematical physicists. A lattice model of itinerant electrons that is believed to display ferro and antiferromagnetism - if it could be solved - is the Gutzwiller-Hubbard-Kanamori model [1-3]. We are also unable to solve it, but we have succeeded in proving that a simplified version of it does display crystallization. It is a toy model but it is, to our knowledge, the first example of this genre. Roughly, it has the same relation to the Hubbard model as the Ising model has to the quantum Heisenberg model. Here we shall give a brief report of our results, the full details of which will appear elsewhere [4]. The Hubbard model, which is the motivation for our model, is defined by the second-quantized Hamiltonian
_
x c:oc o + 2 u a
x,y•h
xTn ,
(1)
x~.A
with the following notation: a --- +1 denotes the 2 spin states of the electrons; A is a finite lattice; cxa is a fermion annihilation operator for a spin a electron at x E A; n~a = c ~ c ~ a is the number operator for spin a at x. Electrons interact
only at the same site with an energy 2U, and tx~ = ty~ is the hopping amplitude from x to y. The crucial assumption will be made that A is the union of two sublattices A U B such that tx~ = 0 unless x E A,y E B or x E B , y E A. The number of sites in A,A, and B are denoted by IAI, IAI and IS]. The two sublattices need not be isomorphic. Thus, for example, a face-centered cubic lattice is allowed with A=face centers and B = c u b e corners. A is said to be connected if every x, y E A can be joined by a "path" through nonzero t's. In our model we assume that one kind of electron (say a = - 1 ) does not hop. One can say that these electrons are infinitely massive. The Hamiltonian is then x,yEh
xEA
with nx = c~cx (the subscript a is omitted since the dynamic electrons have a = +1) and with W ( x ) = +1 if a fixed electron (a = - i ) is at x and W(x) = 0 otherwise. It will be recognized immediately that (2) is just a fancy way to say that the movable electrons are independent, with a single particle Hamiltonian
h=T+V,
(3)
with T being the IAt-square matrix t~y and Vxu = 2UW(x)4~. It is convenient to write h = h + U with h--T+US (4) with S ~ = sx~xy, and sx = 1 (resp. -1) if x is occupied (resp. unoccupied). The {sx) are like Ising spins. We shall henceforth call the movable particles "electrons" and the fixed particles "nuclei". This terminology is most appropriate if U < 0, for then H does represent a lattice system of electrons and nuclei in which all Coulomb interactions except the on-site electron-nucleus attraction and the on-site infinite nuclear repulsion are regarded as "screened out". This conforms with the spirit of the original Hubbard model. The electron number, the nuclear number and the total particle number, all of which commute with H, are, respectively Ne=Enx, xEA
1
Nn=~E[sx+ll=EW(x), xEh
JC=N~+Nn.
(5)
xEh
It is to be emphasized that W does not represent a disordered potential. We take the "annealed", not the "quenched" system. The ground state for fixed N~ and N,t is defined by taking the ground state of H (with respect to the electrons) for each W and then minimizing the result with respect to the location of the nuclei. The ground state energy will be denoted by E(Ne,N,~). Likewise, for
positive temperature, we take Tre-~H with respect to both the electron variables and the nuclear locations. Since txv = 0 for x, y on the same sublattice, the spectra of H and H H are invariant under t~y --* - t x v (all x, y). There is also a hole-particle symmetry. If c~ ---* c~,cx ---* c?=,n~ --* 1 - nx, t=y --* - t x y , then H(U) --~ H ( - U ) + 2UNn. If W(x) ---* 1 - W ( x ) , then H(U) --~ H ( - U ) ÷ 2UNe. A similar s y m m e t r y holds for H H. Thus, the U > 0 and U < 0 cases are similar - - from the mathematical point of view. Our results are of two kinds. The first concerns the ground state which we prove always has perfect crystalline ordering and an energy gap (defined later). T h e second concerns the positive temperature ( 1 / k T = / 3 < oo) grand canonical state. For large/3 and dimension d > 2, the long range order persists. For small /3 it disappears and there is exponential clustering of the nuclear correlation functions.
The Ground State T h e o r e m 1: (a) Let U < O. Under the condition )4 = Ne + N,~ < 2JAJ, the ground state (i.e. we minimize E(N~,N,~) over the set N~ + Nn <_ 2JAJ) occurs for Ne = Nn = JA I and a minimizing nuclear configuration is W(x) = WA(X) =-- l(x e A),O(x • A). Under the condition J4 < 2IBJ, the ground state is Ne = Nn = ]SJ and the B sublattice is occupied (W = WB). If A is connected, these are the only groundstates, i.e. if ]AJ > JBJ the ground state is unique; if ]AJ = JB I it is doubly degenerate. No assumption is made about the sign or magnitude or periodicity of the txv other than try = 0 for x, y E A or x, y E B. (b) Let U > O. Under the condition )4 >_ JAJ + JB], there are two ground states: Ne = IAI,N,~ = JBJ,W = W s and Ne = JBJ,Nn = JAJ,W = WA. If A is connected, these are the only ground states. The condition N = JAJ is called the half-filled band. If JAJ = JBJ = JAm/2, the crystal occurs at the half-filled band. If A is a cubic lattice, for example, this means that the ground state is a cubic lattice of period v/2 oriented at 45 ° with respect to A. Theorem 1 relies heavily on the fact t h a t the electrons are fermions. The ground state would be completely different if they were bosons. For bosons and for A a cubic lattice, the nuclei would all be clumped together in the ground state instead of being spread out into a crystal. By using rearrangement inequalities it is possible to describe this clumping quantitatively. Next, we define the energy gap. Actually two different definitions are of interest. First, let
E()¢) -
min{E(N~,N,~)lNe + Nn = M}.
(6)
T h e chemical potential is defined by ~()4) _= E()¢ + 1 ) - E(~/).
(7)
We say there is a gap of the first kind at A{ if ~ ( , ~ / ) - ~ ( J ~ / - 1) __~ E1 > 0
(s)
with el being independent of the size of the system. We say there is a gap of the second kind at Ne, Nn if E ( N e + 1, Nn) + E(Ne - 1, Nn) - 2E(Ne, Nn) >_ 62 > O.
0)
In other words, the nuclear number is fixed in the second definition. A gap is one indication t h a t the system is an insulator, for it implies that it costs more energy to put a particle into the system than is gained by removing one. The first kind of gap is relevant if one views our model as an approximation to the Hubbard model; the second is relevant from the "electrons and nuclei" point of view. T h e o r e m ~. Assume that h is not only connected but that every x, y E lk can be connected by a chain with Itabl >_ 6 for every a,b on the chain. Also, assume that IITI[ <_ r. The following energy gaps exist with e2 >_ ea > 0 and depending only on 6, r and U: U
O: First kind at )4 -- IAI + ISl. Second kind at Ne = IAI,Nn = IBI and at Ne = ]B],Nn = [A I. In order to give the flavor of our methods, the proof of Theorem 1 will be given here. T h e proof of Theorem 2 is more complicated. Proof of Theorem 1: Let Ax _< A2 -< .-. be the eigenvalues of h in (4). They depend on the nuclei. For Ne electrons the ground state energy, E, of H satisfies
1%
(a0) j-----1
,~" < 0
But T r h = UY]~sx = 2UNn - UIAI and Ihl = ( T 2 + U 2 + u g ) 1/2 with Jxy = t~y[sx + su]. Since the function 0 < x --* z 1/2 is concave, f ( y ) = T r { T 2 + U2 + y U J ) 1/2 is concave in y E [-1,1]. But f ( - 1 ) = f(1) (since spec(h) is invariant under T -* - T ) , so f(1) _< f(0), with equality if and only if g - 0. Thus E _> UXt - ~1 U]A] - ~ T r ( T 2 + U 2) a/~
(11)
If A is connected, the only ~ays to have J = 0 are either W = WA or WB. Consider U < 0 and N <_ 21A ], whence, from (11), 1 E >_U(21A ]- ~lhl) - ~ Vr(T 24-
U2) 1/2"
(12)
If W = WA then, as is easily seen, h has precisely IAI negative and ]B I positive eigenvalues. Thus, if W = WA and Ne -- IAI, then (12) is an equality. The other cases are similar. []
Grand Canonical Ensemble
First, we define the partition function E. A nuclear configuration is denoted by S = {sx},s~ = +1, and the As- are the eigenvalues of h in (4). If #,~,pe are the nuclear and electronic chemical potentials, I^1 E= Z
1 exp[~3pn( Z sz + IAI)] r I {14- exp[-~(A s 4- u - ~,)]).
S
x
(13)
j=l
The product in (13) is just the well known Fermi-Dirac grand canonical partition function for the electrons, We want to choose #e,#n so that (N~) = ½1AI and 1 (N~) = ~IAI, or ( ~ sz) = 0. From the fact that if T -~ - T , spec(h) --~ spec(h), one has that when S -+ - S , spec(h) -~ -spec(h). It is then easy to see that the desired chemical potentials are #e = #n = U. Since ~ ,~j = U ~ sz, (13) becomes in this case (after dropping an irrelevant factor 2iAle3vhAI/2) E= ~
exp[-flF(S)]
(14)
S
with
- 3F(S)=
I^1 ~ gncosh(~3)~s')= Tr£ncosh[~3h] jml
1 2 q- U 2 4- U J)1~2]. = Trgncosh[~3(T
(15)
Thus, (14) is like an Ising model partition function but with a complicated, temperature dependent "spin-spin" interaction, F(S), given by (15) in terms of the eigenvalues of h. With respect to this "spin measure" we can talk about the presence or absence of long range nuclear order in the thermodynamic limit.
6 •
In order to discuss this limit we henceforth restrict ourselves to a translation invariant nearest neighbor hopping on a cubic lattice in d dimensions. What we are able to prove is summarized in the schematic figure below and in T h e o r e m 3: For all U and su~eiently large fl there is long range order
for d >_ 2 (the same kind as in the ground state). For all U and sul~ciently small/~ there is none; indeed there is exponential decay of all nuclear correlation functions.
13 Presumably there is no intermediate phase, but we cannot prove this. For large U,/~e is clearly linear in U. For small U, we have the bound on the lower /~c ~ U -1/(2+d) and the bound on the upper/~c ~ I~nUIU -2. Our guess is that the true state of affairs is/~c ~ U --2. For large/3 we use a Peierls argument; for small/~ we use Dobrushin's uniqueness theorem. A sketch of our proof omitting many important details - is the following. For simplicity, we here consider only large U > 0. Define, for x > 0, P(x) = ~ncoshx 1/2. (16) -
[We note in passing that P is concave, and we see from the last expression in (15), using the proof in Theorem 1, that F(S) has its minima at precisely the
same values of W (or S) as in Theorem 1.] P is a Pick (or Herglotz) function with the representation
1,2 ~ = + x]-' x - 1 / 2 t a n h x 1/2 = E [ C k + ~)
P'(x) =
(17)
k=O
1 2 (T 2 + U 2 ~i U J) with gz~ = tx~(sx + sy). We are interested in x = ~/~ L o n g R a n g e O r d e r (large/~): Choose S, and then define antiferromagnetic contours in the usual way. If y is a connected contour component and r + is the whole contour, we want to prove that A -- F(F + y) - f ( r ) > C[ l for a suitable constant C = C(U). Obviously, Jr+~ = J r + Jr. To remove % we change sx to - s x inside ~. For 0 < t < 1, define J(t) = Jr + tJ~. Then, assuming for simplicity that r lies entirely outside ~, we have, by differentiating (15) and using (17), that
-/~A=
/~2U k~0
dtTr(GkJ~),
(18)
o
with Gk = [(k+ ~) 1,2 1r2 + ~ /~2(T2 + U 2 + U Jr + tug~)] -1
(19)
Integrating by parts, this becomes --~/k = - ( ) ~ 4 U 2 / 1 6 )
dr(1 - t)TrGkJ~GkJ~ + . . .
(20)
[The ... terms in (20) come from t = 0 in the partial integration. They are small and easily bounded for large U, but they have to be treated more judiciously when U is small.] If A, B, D are matrices with A > B >__0, then T r A D ? A D >_ T r B D ? B D . Also, A _> B > 0 =~ 0 < A -1 <_ B -1. Moreover, T 2 < (2d) 2 and U J ~_ T 2 + U 2. Using this, we have the matrix inequality
ak > [(k + --
:
+
2((2d)2 + V2)]-,.
(21)
2 /
Summing on k, A > (const.) U2(C2d) 2 + U2)-3/2Tr(J~)2. Clearly Tr(J~) 2 = (const.)]~]. Thus, for U large, C(U) > (const.)U -1, and thus long range order exists in d >__2 if/~/U is large enough. A b s e n c e of Long R a n g e O r d e r (small j3): Dobrushin's uniqueness theorem [5-7], together with the modification in [8], gives the following criterion for
exponential clustering. We have to bound the change in F when we change the spins at x and y (taking the worst case with respect Vo the other spins). Call this f~u" T h e requirement is that for some m > 0 and all x,
Z fzvexp[mlx - Y]] < 1.
(22)
Y
By an argument similar to the preceding (large U)
A~ z (Z3u2/16)~ T~a~J~a~J~ k
(~aU2/16)
~ la~(~,y)l~
(23)
k
Here, Jz = ifzT + Tifz and Gk(x, y) is the x, y matrix element of (19). To implement (22) we now require an upper bound on Gk(x,y) that has exponential decay. For this purpose the Combes-Thomas argument is ideal. Let Q be the matrix with elements Q~u = 6~ve n~ and with ]n I = 1. Then
QGklQ -1 = G-~1 + Rk - Lk.
(24)
The "remainder" Rk can be bounded for large U: IIRkH < C~2U for some constant C. Similar to (21), for large U
a ; ~ _> [(~ + 1 ~ 2 , + 8 ~ U ~ ] = ~
(25)
Thus,
IIL~-'II < [ ~ k - c~2u] -1.
(26)
Since I(L;1)=~l _ IIi;'ll, we have from (24), I(QCkQ-~)~I < IlL;Ill, and thus I(ak)~l _< exp[~. (y - x)]{~k - cz2u} -1.
(27)
This holds for all n, so summing on k, fx~ < (const.) U-le -21x-yl. Hence, (22) holds with m = 1 if ~/U is small enough. The support of both authors by the U.S. National Science Foundation, grant PHY 8116101 A03, is gratefully acknowledged.
References
1. M. C. Gutzwiller, Phys. Rev. Letters 10, 159-162 (1963), and Phys. Rev. 134, A923-941 (1964), and 137, A1726-1735 (1965). 2. J. Hubbard, Proc. Roy. Soc. (London), Ser. A276, 238-257 (1963), and 277, 237-259 (1964). 3. J. Kanamori, Prog. Theor. Phys. 30, 275-289 (1963). 4. T. Kennedy and E. H. Lieb, An itinerant electron model with crystalline or magnetic long range order (in preparation). 5. R. L. Dobrushin, Theory Probab: and Its. Appl. 13, 197-224 (1968). 6. L. Gross, Commun. Math. Phys. 68, 9-27 (1979). 7. H. FSllmer, J. Funct. Anal. 46, 387-395 (1982). 8. B. Simon, Commun. Math. Phys. 68, 183-185 (1979). 9. J. M. Combes and L. Thomas, Commun. Math. Phys. 34, 251-270 (1973).
FIRST ORDER PHASE TRANSITIONS AND PERTURBATION THEORY
J. Bricmont and J. Slawny I n s t i t u t de Physique T h e o r i q u e , 2, ch. du Cyclotron Universite Catholique de Louvain B - 1348 L o u v a i n - l a - N e u v e , Belgium Center for T r a n s p o r t T h e o r y and Mathematical Physics V i r g i n i a Polytechnic I n s t i t u t e and State U n i v e r s i t y Blacksbur9, V i r g i n i a 24061, U . S . A .
I. INTRODUCTION While in most situations the state of
a system changes
smoothly when e x t e r n a l
parameters such as pressure, magnetic field, or temperature are varied, t h e r e are also sometimes sudden jumps in the d e n s i t y , occur
at
Statistical
some values of these external Mechanics
is to
understand
the magnetization or the e n e r g y t h a t
parameters. these
One of the basic goals of
phase
transitions
starting
from
a
microscopic description. Since the invention of the Peierls' argument [41, 3 0 ] , have been shown to e x h i b i t f i r s t - o r d e r
phase transitions
some basic questions remain unanswered.
For example:
a great v a r i e t y of models [21, 26,
22].
However
- We do not have a proof of the occurrence of f i r s t - o r d e r phase transitions f o r a reasonable model of a simple f l u i d .
For a simple l i q u i d - v a p o u r phase t r a n s i t i o n ,
we cannot go much beyond van der Waals t h e o r y [61, 39]. - We do not have a complete u n d e r s t a n d i n g of the Gibbs phase rule, even for classical lattice systems (for partial results see [34, 4 5 ] ) . that, t y p i c a l l y ,
One would like to show
given a set of parameters, n phases will coexist on a manifold of
codimension n-1 in that space. -
From
a more
practical
point
of
view,
one
would
like
to
have
reliable
techniques to compute the phase diagram of alloys with interactions of interest. tn these lectures,
we shall discuss how the Pirogov-Sinai t h e o r y [42, 47, 48]
and some of its extensions [33, 8,
10, 13, 18, 14, 60]
p r o v i d e at least partial
answers to these questions. The main emphasis will be on the use and justification of p e r t u r b a t i o n theory.
11
To
develop
a perturbation
theory,
we
need
a
reference
system
which
is
"completely" known. At low temperatures such a reference system may consist of zero temperature states, or ground states, y i e l d i n g low temperature expansions. The f i r s t proof of convergence of a low temperature expansion was given by Minlos and Sinai [40] for the Ising model. Minlos and Sinai reformulated this model in terms ground
of a gas of contours separating states
occur,
t h e r e f o r e one could
At
regions of the lattice where d i f f e r e n t
low temperatures,
activity
of
use methods developed to control
this
gas
is
small
and
low a c t i v i t y expansion to
prove convergence here. We review this in Sect. II. However, it is often useful, and sometimes imperative, to have a p e r t u r b a t i o n t h e o r y around more general objects than ground states,
which we call restricted
ensembles. Such an ensemble consist of a subset of the phase space, t o g e t h e r with a measure (usually a Gibbs state) on it, which gives a better approximation to the true
phase than
a ground
state.
A
restricted
ensemble often
consists
fluctuations around a ground state as in the case of continuous
of
small
spin models or
quantum field t h e o r y [26, 33, 18]. However, this notion is useful in other contexts as well, as we shall t r y to demonstrate on the following examples: -
Lattice spin
related
by
Pirogov and
systems
symmetries
Sinai
of
with the
a finite
number
This
Hamiltonian.
(when the
of
number of ground
ground
states
is the
situation
state
is f i n i t e ) .
which
are
not
considered by We give an
introduction to t h e i r t h e o r y by discussing a simple example in Sect. 111. Lattice
spin
systems
with
an
infinite
number
of
ground
states.
In
this
situation, the t h e o r y is far from complete. The notion of a restricted ensemble is very
useful
in many
cases where an
infinite
number
of
ground
states
occurs
because it allows us to reduce this i n f i n i t e set to the consideration of a f i n i t e set of 9round states called dominant ground states (Sect. IV). Fluids.
T h e r e are v e r y few results on phase transitions for one component
fluids (see however [34]),
but several results are available f o r phase separation in
mixtures [44, 37, 8]. Restricted ensembles are essential in this case since t h e r e is no notion of a ground state for continuum fluids. The reference system here is an ideal or almost ideal gas (Sect. V ) . We discuss an extension of the Pirogov-Sinai t h e o r y to
- L a r g e e n t r o p y models.
the
q-state
Ports model for
q - l a r g e and for
the temperature where q÷l
phases
coexist, [8, 14, 60]. q of these phases are small perturbations of the corresponding ground
states
but
there
is
an additional
high-temperature
phase which
is
in
equilibrium with the other phases only because of its large e n t r o p y . This phase is a small p e r t u r b a t i o n of the restricted ensemble which is maximally disordered (Sect.
Vi).
12
II. THE ISING MODEL: LOW TEMPERATURE EXPANSION One of the basic open problems in the t h e o r y of f i r s t - o r d e r phase t r a n s i t i o n s is to understand
the phase diagram of
simple f l u i d s .
T h e r e are deep open
problems
concerning c r y s t a l l i s a t i o n but even t h e l i q u i d - g a s t r a n s i t i o n is understood the f o l l o w i n g lattice-gas approximation (see, intermolecular
potential
usually
long-range attractive part. Zd
(d
is the
potential,
us
of
instance,
a
[49,
short-range
only in
Sect. 5 - 2 ] ) :
repulsion
The
and
of
a
Let us c o v e r t h e continuum w i t h the cells of the lattice
dimension of
let
consists
for
require
space)
that
each
and,
in
cell
place of
the
repulsive
be occupied
by
at most
part
of
the
one p a r t i c l e .
F u r t h e r m o r e , one assumes t h a t the a t t r a c t i v e potential is insensitive to the position of the p a r t i c l e in the cell.
Then one obtains a lattice gas model, with a v a r i a b l e
p(a) at each lattice point a, which is equal 1 if the cell is occupied, and 0 if it is empty.
A
change
f e r r o m a g n e t i s m ; the
of
variables,
p(a)
=
a t t r a c t i v e potential
½(l+o(a)),
of the lattice
leads
us
to
gas yields
a
mode]
of
a ferromagnetic
interaction of the magnetic system. The simplest version of
such a f e r r o m a g n e t i c model is the
nearest n e i g h b o r
Ising model: H = -
Y. oCa)~Cb) - h~: oCa) , a
oCa) = ±1 ,
(2.1)
w h e r e means t h a t a and b are nearest n e i g h b o r s , nearest model,
neighbor
(n.n.)
pairs,
called
especially at low t e m p e r a t u r e s ,
bonds. but
Many
before
and the sum is o v e r all
results
are
discussing
known
them,
for
this
we want
to
stress t h e simplifications introduced by t h i s lattice a p p r o x i m a t i o n . Since the
lattice is given
a priori,
we do not have the problems (massless
e x c i t a t i o n s ) associated with the formation of c r y s t a l s . -
We have also i n d i r e c t l y i n t r o d u c e d a symmetry into the problem: H in (2.1) is
i n v a r i a n t under the change of h into -h and a(a) into - o ( a ) f o r all lattice sites a. This
symmetry
transition,
yields
if any,
a very
drastic
simplification:
is expected to occur at h=0, i.e.
because
of
it,
the
phase
at a temperature independent
value of h. This is c e r t a i n l y not the case when t h e r e is no symmetry between the phases
e.g.
in
a
liquid-gas
transition,
where
the
pressure
for
which
phase
coexistence takes place depends in a n o n - t r i v i a l way on the t e m p e r a t u r e . - Finally, t h e r e is a w e l l - d e f i n e d reference system about which one can t r y
to
c o n s t r u c t a low t e m p e r a t u r e expansion, namely the g r o u n d states. For h > 0 (resp h < 0)
t h e r e is a unique g r o u n d state w i t h all spins equal to +1 ( r e s p .
h=0 one has both of these g r o u n d states.
-1).
For
13
We shall recall now a d e r i v a t i o n of the low temperature expansion for the free energy of the Ising model, [19], to converge.
an expansion which,
in this case, can be proved
It will be convenient to define the free energy with the g r o u n d - s t a t e
energy substracted,
like
in ~(BI +) below: the
+ after the vertical
bar indicates
which g r o u n d state is considered. Thus ~(BI ÷) = - B-11im IA1-1 log ZCAI ÷) , A where
IAI
is the number of
sequence of
finite
subsets
points in A. The limit is taken over an expanding of
the
lattice
with
not
too
large
boundaries,
like
NxN-squares with N*~, or, more generally over Van Hove sequences, [43, Chapter 2]. Furhermore, Z(EI +) =
~: exp -13HA(oI+)
,
(2.2)
(5
and HACoI÷) = Z ( a C a ) a ( b ) - l ) In
(2.2)
the
configurations
s~Jm is
(2.3)
.
over all
are extended
complement of A,
configurations
to all
the
and the summation
Hamiltonian H^(-I ÷) is as in
(2.1),
o=(o(a))a~ A in A;
lattice by
is o v e r all the but with
more
involved
than
necessary for
when we discuss the
in
a(a)=l
bonds
(2.3)
for
these
a in
the
intersecting A.
The
h=O and the energy of the ground
state subtracted: o ( a ) o ( b ) replaced by o ( a ) o ( b ) - l . a manner
setting
The expansion will be derived in
this
model.
however,
be useful
simplicity
and in o r d e r to make the calculations e x p l i c i t ,
This
Pirogov-Sinai t h e o r y
derivation in
Sect.
3.
will, For
we shall work in two
dimensions. Let us start by computing the f i r s t o r d e r term in the expansion of ~(151*). 13 large, flips
the leading term in the p a r t i t i o n function (2.2)
i.e.
occupied
from
configurations
by plus-spins.
where all the
Restricting the
sites
For
comes from isolated spin
adjacent to a minus-spin
sum in the partition
are
function to these
configurations one obtains approximately, by adding the contributions of O, 1, 2 . . . . spin flips: Z ( A I + ) = I + I A l e - 4 B + ½ 1 A I ( I A I - 1 ) ( e - 4 B ) 2 + . . . = ( I + e -4~ ) I^1,- e x p ( I A l e-4B) .
(2.4)
This yields 13~(61+) = - exp-41~, to leading o r d e r . One could obtain in this fashion higher o r d e r terms but convergence of this expansion is best proved after an introduction of contours - the basic notion of
14
Peierls' argument: Consider a c o n f i g u r a t i o n o=(o(a))a~Z2 equal +1 e v e r y w h e r e , w i t h a possible exception of a finite set. Define the b o u n d o r y B(o) of o by B(o) = {: o(a) # o ( b ) } . Now decompose B(o) into connected components in the usual way: associate to each bond the bond of the dual lattice which is p e r p e n d i c u l a r to it, and decompose the r e s u l t i n g set of lines into connected pieces, called contours.
For each c o n t o u r ~"
set [~'] = {a = z d : t h e r e exists b such t h a t ~ ~'} . We mention t h a t this definition of contours which is convenient f o r the Ising model needs to be modified f o r more general models. In the special case of the Ising model this general notion would be defined as follows.
Cover t h e lattice w i t h squares of
side R (where R is chosen to be l a r g e r than the
range of the
here is 1) and define a square C to be r e g u l a r if a l c
i n t e r a c t i o n , which
(:the r e s t r i c t i o n of o to C) is
equal to the r e s t r i c t i o n to C of one of the g r o u n d states ( i . e .
o(a)=+l f o r all a¢C
or o(a)=-I
a connected set of
f o r all asC).
Then
a contour
is a pair made of
i r r e g u l a r squares t o g e t h e r with the r e s t r i c t i o n of o to these squares. One i m p o r t a n t p r o p e r t y of this general definition is t h a t a c o n t o u r is no l o n g e r a p u r e geometrical object b u t
r a t h e r it is a subset of the lattice t o g e t h e r With a c o n f i g u r a t i o n on it
from which one can read off the b o r d e r i n g g r o u n d states. With
any
of
the
two
above
definitions,
contours
enjoy
the
following
two
i m p o r t a n t geometric p r o p e r t i e s :
P r o p e r t y 1.
They are closed in the sense t h a t f o r any c o n t o u r ~', z d \ [ ~ "] ( o r
the complement of the s u p p o r t of ~' in the general case) can be decomposed into several connected components o n l y one of which is i n f i n i t e .
The l a t t e r is called t h e
e x t e r i o r of ~' while the f i n i t e components form the i n t e r i o r of ~;, Int~'. there
is a unique g r o u n d
state
(+ o r
-)
along
the
b o u n d a r y of
Moreover,
each of
these
components. This p r o p e r t y is obvious in the case of the Ising model; it also holds w h e n e v e r the number of g r o u n d states is f i n i t e b u t does not,
in general, when t h e
number of g r o u n d states is i n f i n i t e (see Sec. 4).
Property
2.
The
number of
contours
of
length
n
(having
n bonds)
and
containing a f i x e d bond is bounded b y c n w h e r e c is a constant ( d e p e n d i n g only on the lattice and the number R e n t e r i n g the general definition of c o n t o u r s ) . Using
the f i r s t
property
subfamily of outer contours, the c o n t o u r s .
we can
define,
for
any
family
of contours,
the
namely those t h a t do no lie in the i n t e r i o r of any of
15
Let us consider the p a r t i t i o n f u n c t i o n Z(~') of all the c o n f i g u r a t i o n s h a v i n g o n l y one o u t e r c o n t o u r ~" and consider the Hamiltonian ( 2 . 3 ) . Then
Z(~') = e x p ( - 2 B l ~ l ) ]I Z ( I n t i E I E ( i ) ) i
where
,
(2.5)
I EI is the number of bonds in ~', t h e p r o d u c t runs o v e r all the connected
components Int.~" of the i n t e r i o r of ~', and
~(i) = ÷ o r - d e p e n d i n g on t h e g r o u n d
state w h i c h , b y P r o p e r t y 1, b o r d e r s I n t . L
Finally, Z(AI ÷) is defined b y (2.2) and
1
I
Z(A I - ) is defined s i m i l a r l y , b u t w i t h o ( b ) = - I f o r b in the complement of A. Now f o r any A, we can w r i t e Z(AI +) = where
Z H Z(~), w ~sw
the
sum
(2.6)
is over oli compatible
families u of outer contours
product over contours of the family; compatiblity means
in A and the
here that the supports of
the contours are disjoint and that no contour lies in the interior of another contour of the family. The
{important) factorization property implicit in (2.5) derives here from the
fact that w e have a fixed ground state in the exterior of all outer contours. Now
to get our final expansion w e
use the symmetry of the Hamiltonian under
the global spin-flip relating the two ground states, which implies that for any A,
Z(AI ÷) = Z ( A I - )
•
(2.7)
Using (2.7) and ( 2 . 6 ) , one t r a n s f o r m s (2.5) i n t o : Z ( k l +) =
~
][
w ~sw
exp(-2Bl~l)]I Z(Intk~l+) k
,
(2.8)
with the s u m extended over the same set of w as in (2.6). This expansion is in a form which is suitable for iteration; iterating it one gets ZCAI*) =
~. ]T z(~') ,
zC~')=exp-2f31~'J
,
(2.9)
w h e r e the sum now runs o v e r all families u of contours t h a t are n o n - i n t e r s e c t i n g , w i t h o u t the c o n s t r a i n t t h a t the
contours are o u t e r c o n t o u r s .
The
identity
(2.9)
shows t h a t Z(A1 +) is equal to the p a r t i t i o n f u n c t i o n of a gas of an i n f i n i t e n u m b e r of "species of hard-core
partic|es"
exclusion
(all c o n t o u r s ' - m o d u l o t r a n s l a t i o n s )
(non-intersecting
contours),
interacting through
and h a v i n g
activity
z(~).
a
This
implies immediately t h a t f o r any c o n t o u r ~ t h e p r o b a b i l i t y t h a t i t is a c o n t o u r of a c o n f i g u r a t i o n is bounded b y exp-2t~l~rl ( P e i e H s ~ b o u n d ) . The basic estimate t h a t allows us to c o n t r o l this system is:
16 Z
z(~') + 0 (exponentially fast) as IB ~ - ,
(in the case considered here the sum is bounded b y
where b is any f i x e d bond
exp-4B,
for
(2.9a)
B large enough).
This
follows
easily
from
the fact
that
z(;O
is
decreasing v e r y fast when IZl is increased, here z(~)=exp-2t~'l, and that there are not too many contours with given IZl ( P r o p e r t y 2). The convergence of the low temperature expansion follows now from the results on low a c t i v i t y expansion for the gas of contours:
Define a m u l t i p l i c i t y f u n c t i o n as
a map from the set of contours into non-negative integers, with a f i n i t e support. A complex-valued function ¢ defined on the multiplicity functions can be identified with the formal power series
X ~(O)z O ,
(2.10)
where the sum is over all possible m u l t i p l i c i t y functions, and
O = ]I Z~,~(~)
Z
Z~, being a variable associated to ~. Define now ¢ as follows:
¢(~)=1
if ~(~')<1
for all ~" and the contours in the
support of 0 are pairwise disjoint, and ¢(~)=0 otherwise.
Then,
(2.0) is the value
of the formal power series (2.10) for z~,=exp(-2Bl~l) if ~ck, and z~,=0 otherwise. Let cT be the function corresponding to the logarithm, in the sense of formal power series, of <~.
Proposition ([23, 40, 4 6 ] ) . X
I~T(o)z 0'1 ~ 0
I f Iz~l < exp(-2Bl~'l) Call 7;) then f o r any c o n t o u r ~'
as
B~ ~
(2.11)
~:~(~')#0 F u r t h e r m o r e , setting in
X ~T(~)zO z~,=O f o r series
(2.12) ~ not
from
contained in it, one obtains f o r each
which
log
Z(it[ ÷) is
obtained
by
A an absolutely
substitution
expansion implies in t u r n that
log ZCAI +) = -
I A I B ~ ( B I ÷) ÷ ACA,6)
,
with B~(81 +) being an analytic f u n c t i o n of z=exp-2B, while
convergent
z~,=exp-281~q.
This
17
la(A,~)l
s 8(S)IaAI
(2.13)
,
with 6 (i~)=0(exp(-213)) • Given the expansion ( 2 . 1 2 ) , the p r o o f of the rest of the Proposition is easy. Using this phases
expansion,
[40,
invariant
23]:
Gibbs
one obtains a detailed d e s c r i p t i o n
There
are e x a c t l y t w o
states)
related
spontaneous magnetization. of
(+)-spins,
by
pure
the
phases
spin-flip
of the
(i.e.
low t e m p e r a t u r e
extremal
symmetry,
translation
with
opposite
In each phase, t y p i c a l c o n f i g u r a t i o n s consist of a "sea"
f o r example, w i t h
small islands of the opposite phase.
f u n c t i o n s are e x p o n e n t i a l l y c l u s t e r i n g and analytic in e x p ( - 2 ~ ) . a complete d e s c r i p t i o n of the l o w - t e m p e r a t u r e phases.
Correlation
Thus one has q u i t e
However, in o r d e r to obtain
these results one has to use many of the simplifications of the Ising model t h a t are not available in more realistic systems. Not much needs to be changed if we consider h i g h e r spins ( i . e . ,
c(a) assuming
the values - n , - n + l . . . . . n, w i t h the same Hamiltonian o r f e r r o m a g n e t i c system w i t h a f i n i t e number of g r o u n d states.
However, if we deal with a continuous d i s t r i b u t i o n
of a ( a ) , o r w i t h quantum f i e l d theories [ 2 6 ] , then complications arise. Let
us consider,
for
example,
an
Ising
model w i t h
continuous
spin:
the
Hamiltonian H is as before, (2.1) w i t h h=0, b u t a(a) is now u n i f o r m l y d i s t r i b u t e d in the interval
[-1,+1],
instead of t a k i n g t h e values +1 and -1 o n l y . As in the Ising
model, t h e r e are two g r o u n d states , t h e ( + ) - g r o u n d state which is +1 e v e r y w h e r e and t h e
(-)-ground
symmetry. the g r o u n d
state.
The g r o u n d
states
are again
related b y the s p i n - f l i p
However, one has now the f o l l o w i n g problem: even in the f i n i t e volume, state c o n f i g u r a t i o n s have p r o b a b i l i t y zero and t h e r e are excitations of
the g r o u n d states with ( r e l a t i v e ) e n e r g y which is a r b i t r a r i l y small. To deal with this problem one can i n t r o d u c e two r e s t r i c t e d ensembles: +
X R = {o: o(a) > 0, all a ~ Z d } , X R = {0: o(a) < 0, all a ~ Z d } . Then
one can
o(a)o(b)<0,
as in the
complement of ensembles.
define
contours Ising
a contour,
If one t r i e s
as connected
model. the
Thus,
families in
configuration
to expands t h e
of
bonds
each connected belongs
to
partition function
one
for
which
component of of
the
the
restricted
into a sum o v e r o u t e r
c o n t o u r s , as in the ( 2 . 6 ) , one does not have any more a f a c t o r i z a t i o n into p a r t i t i o n f u n c t i o n s of o u t e r contours. However, it t u r n s out t h a t the r e s t r i c t e d ensembles are v e r y d i l u t e at low t e m p e r a t u r e s . Namely, one has a c o n v e r g e n t expansion f o r these ensembles
one
derives
h i g h - t e m p e r a t u r e expansion,
from
which
o r from
all
the
properties
low f u g a c i t y
obtained
expansion
usually
of a d i l u t e
from
gas.
In
18
particular the factorization in (2.6)
holds in the sense that one can construct a
low-temperature expansion for
complete model which
the
takes
the
form
of
a
combined expansion, fi r s t into contours and then a "high-temperature expansion" in each restricted ensemble; we refer to [9] for detailes. Such a combined expansion has appeared f i rs t in other contexts, particularly in lattice and continuum models of Quantum Field Theory, see [26] and references therein. For other extensions of the Pirogov-Sinai theory to continuous-spin systems we refer to [18] and to Zahradnik's contribution to the present volume. One can avoid this combined expansion and obtain directly the Peierls' bound on the probability of a contour,
as done originally in [55].
many examples where chessboard estimates,
based on the
This is one of the Reflection
Positivity
property of the model, [26, 21], proved to be very useful. To
clarify
the
importance of
the
symmetry
in
the
preceding arguments
consider the Ising model, but with an external field. Now the weight e x p ( - 2 B l ~ l ) of a contour is modified slightly, which for a small field is not essential. However, (2.7) would no longer hold and the iteration procedure in (2.8, converge. Recall that (2.7)
comes from the symmetry.
2.3) would not
Since such symmetry is
expected for a realistic liquid-gas model, a fi r s t step may be to extend the analysis above to lattice models without symmetry.
This is quite non t r iv ia l and is the
content of the Pirogov-Sinai theory.
III. MODELS WITHOUT SYMMETRY This section is divided into four subsections. First we introduce a simple model on which the main features of the Pirogov-Sinai "theory can be most easily explained, and we discuss its phase diagram from the point of view of perturbation theory. Then we explain some of the ideas of the Pirogov-Sinai theory. In Sect. 3, we state the
general
result
of
Pirogov and
Sinai
for
this
model
perturbation theory for the phase diagram in more details.
and
we discuss
the
Finally (Sect. 4) we
generalize our model and explain some recent applications to the theory of random surfaces [5]. 1. Phase diagram of the Blume-Capel mode/. This model, [4, 11], is defined by the Hamiltonian H =
X
( s (a ) - s (b ) ) =
,
s(a) = 0,±1 .
(3.1)
19
It has three ground states, {+), (-) and (0), which are equal everywhere to +1, -1 and O, respectively. The model has still the s(~) -~ -s{a) symmetry but only the (+) and (-) ground states are related by this symmetry; the (0) ground state is in a class by itself. One could expect that at temperatures low enough the model has three pure phases which
are small perturbations of the three ground states.
We will
now give a
perturbation-theoretic argument showing that this is not true. More precisely, we will argue that for the original Hamiltonian (3.1) there is no low temperature phase which
is a small perturbation of the (+)-ground
perturbation
of
the
Hamiltonian
such
a
state,
phase may
and that for a suitable
appear;
similarly
for
the
( - ) - g r o u n d state. Suppose that in a large but finite region A the (+)-boundary conditions produce such a low temperature phase, i.e. that a typical configuration of such a system is +1 almost everywhere, with small excitations distributed rarely but homogeneously in A.
At very low temperatures the main contribution to the partition function Z(AI +)
+
+
+
0
+
+
0
+
0
+
0
+
0
+
+
0
0
+
+
+
0
0
0
+
0
+0 ÷0
+
0
+
0
+
000
0
+
+0
0 0 0
0 + 0
0
+
+0
0 - 0
0 0 0
0
+
+0
0 0 0
0
+
+0
0 + 0
0 0 0
0
+
+0
0 0 0
0 - 0
O+
+0
0 0 0
+0 + +
A' 0 +
0 +
0 +
0 +
0 +
0 +
0 +
0 +
+
0
0
0
0
+
+
+
+
0
+
0
+ +
Fig. 1. Typical configurations of the low-temperature "(+)-phase" of the model (3.1): tunneling to the {O)-phase.
of this system would come f i r s t from the ground state and then from the excitations of lowest energy, for which s(a)=O at a number of isolated points of A. Thus, see (2.4), Z(AI +) = 1 + IAle -4~ + ½1Al(IAl-1)(e-4~) 2 + . .. = e x p ( I A l e -4B) .
(3.2)
20
Similar a r g u m e n t yields f o r the p a r t i t i o n f u n c t i o n of the l o w - t e m p e r a t u r e ( 0 ) - p h a s e Z(A}0) = 1 * 21Ale -413 + ½(21al)(21Al-1)(e-4!~) = * . . . since here
l o w e s t - e n e r g y excitations
= exp(21AI e-4B) ,
can be obtained in
(3.3)
two ways:
s(a)
can
be
changed e i t h e r to *1 o r to -1. Now, consider the c o n t r i b u t i o n to Z(AI+) of c o n f i g u r a t i o n s of Fig. inside
A one has a s l i g h t l y
NxN-square
and
/,'
is
smaller
an
region A' with
(N-2) x ( N - 2 ) - s q u a r e ,
the
(0)-phase.
the
1.
If
contribution
Here
A is an of
such
c o n f i g u r a t i o n s to the p a r t i t i o n f u n c t i o n is e x p ( - 4 ( N - 2 ) 4 g ) • Z(A'I0) = e x p ( - 4 ( N - 2 ) 4 B + 2 ( N - 2 ) 2 e - 4 ~ ) which f o r large N is l a r g e r than Z ( A I * ) . is unstable
against insertions of
the
,
(3.4)
This shows t h a t the conjectured ( * ) - p h a s e
(O)-phase,
and
t h a t the assumption
of the
existence of a l o w - t e m p e r a t u r e ( * ) - p h a s e leads to a c o n t r a d i c t i o n . This p e r t u r b a t i o n - t h e o r e t i c a r g u m e n t can be cast into a form which hints at a more systematic development (see the end of Sect. 4). i n t e r p r e t e d as a p a r t i t i o n f u n c t i o n
The sum in
(3.2) can be
of a "gas" of lowest e n e r g y excitations of the
( + ) - g r o u n d state. The free e n e r g y of this
gas is denoted
~R(131+) (R
stands here
f o r " r e s t r i c t e d ensemble", as explained in t h e i n t r o d u c t i o n ) . Thus
13~R(131* ) = and
_
SR(131-)=$R(I~I*).
I~$R(1310) =
(3.5)
e-4B , o(e -41~) ,
I n t e r p r e t i n g (3.3) in a similiar manner:
_ 2e-413 + o(e -41~) .
(3.6)
We stress the p o i n t t h a t here, in c o n t r a d i s t i n c t i o n to the Ising model, the p a r t i t i o n function depends on the b o u n d a r y conditions in an essential way. Since the RHS of that
one can
potentials
infer
(3.4)
is essentially
exp(-B(laA'l*lA'l~R(~10))),
the l o w - t e m p e r a t u r e phases
~R(BI*), ~R(131-),
and
SR(~I0)
by
comparing the
it appears
thermodynamic
of the c o r r e s p o n d i n g r e s t r i c t e d ensembles.
Namely, we will say t h a t the ( 0 ) - g r o u n d state is dominant since it has more lowest e n e r g y excitations than the o t h e r g r o u n d
states, o r e q u i v a l e n t l y since the leading
term of the l o w - t e m p e r a t u r e expansion of t h e free e n e r g y of the r e s t r i c t e d ensemble of the gas of the low e n e r g y excitations is smaller f o r (0) than f o r (+) o r ( - ) . above
argument
suggests
that
only
dominant
ground
states
l o w - t e m p e r a t u r e phases. One can in fact p r o v e (see A p p e n d i x ) :
give
rise
The to
21
Theorem.
For B large,
model (3.1).
there is only one translation i n v a r i a n t Gibbs state f o r the
In typical
configurations
of this state the d e n s i t y of sites a where
s(a)=0 tends to 1 as B -~ ~. This theorem, and its proof, are included here f o r pedagogical reasons.
The
method of the proof that we use is applicable in a much broader context (see [10] and Sect.
IV).
The result
itself is a special case of the uniqueness
results of
Preiss (unpublished) and Zahradnik [54]. Remarks: (11 Using correlation inequalities one can show as in [38] that there is actually a unique Gibbs state for this model. inherent
in
our
method.
However,
in
this
(2) The restriction
particular
model,
to large B is
we do not expect
several phases at higher temperatures. The p e r t u r b a t i o n - t h e o r e t i c argument indicates how to obtain the thermodynamic phases corresponding to the g r o u n d states (+) and (-) of ( 3 . 1 / .
Namely, p e r t u r b
BH by replacing it with BHg = BH
-
gY. s(a) = a
(3.71
Now f o r any g > 0 we have only two g r o u n d states,
(+) and ( - ) . T h e y are related
by symmetry of the Hamiltonian and, f o r temperatures low enough (depending on g) a Peierls
argument,
similar
coexistence of two phases.
to the
one used
for
the
Ising
model,
proves
the
However, one expects that there will be a line g(B) on
which three phases, (+), (-) and (0) will coexist. The
perturbation-theoretic
argument which
excluded
the
(+)-phases
for
g=0
yields now that g(B) is given to o r d e r exp-4B by the condition that B~R(B,gl +) - g = B~R(B,910) , -g on the LHS being the energy of the ( + ) - g r o u n d state. (see (3.5),
The f r e e energies are
(3.6)):
B~R(B,gl0) = _ (eg+e-g)e -413 + o(e -4B) ,
B~R(B,gl ÷) = _ e-ge-4B + o(e -4B) .
Thus one expects the line of coexistence to be given by: g(B)=exp-4B+higher o r d e r terms. restricted
One
can
obtain
a better
approximations
for
g(B)
by
considering
the
ensembles which include h i g h e r - o r d e r excitations of the ground states,
excitations with energies not exceeding some cutoff e n e r g y E (two adjacent 0 spins in the + ground state or one - spin in this ground state, etc.1. However this would g i v e only the asymptotics of the line g(B) of coexistence; to prove the existence of g(B), and the fact that the obove computations yield a curve asymptotic to g(l~) we
22
need more sophisticated arguments, since t h e r e is no evidence t h a t the p r o c e d u r e o u t l i n e d above will converge when E is increased. The existence of 9(B)
is one of
the main results of the Pirogov-Sinai t h e o r y , specialized to the present model:
2.
The Pirogov-Sinai theory
We will sketch now the proof, a c c o r d i n g to Pirogov and Sinai [ 4 2 , 4 7 ] ,
that there
exists a line 9(B) on which the t h r e e l o w - t e m p e r a t u r e phases coexist. Define: Z(kl*)
= X exp-BHk(sl*)
,
(3.8)
S
with H =
X ( s ( a ) - s ( b ) ) = - 9X s(a) = • a
We let eo, e+ and e
(3.8a)
denote the e n e r g y p e r lattice site of the c o r r e s p o n d i n g g r o u n d
states:
eo(g)
e,(g) = - g .
= 0 ,
\
We i n t r o d u c e also the
partition
functions
with
the
same b o u n d a r y conditions
as
before, b u t with no s u b s t r u c t i o n of the g r o u n d state e n e r g y : ZS(A) = ( e x p - l A I B e ) Z ( A l ~ ) Define
contours,
,
almost
~ = O, ± . as
in
the
Ising
model,
as
subsets
of
the
lattice
composed of connected families of bonds f o r which sCa)~s(b), t o g e t h e r w i t h a c o n f i g u r a t i o n on it.
Unlike in t h e case of the Ising model, here tlle e n e r g y of a
contou r , E(~)
=
(sCa)-s(b)) = ,
E
w h e r e t h e summation is o v e r all t h e bonds of ~', depends not o n l y on its length b u t also on t h e n e i g h b o r i n g c o n f i g u r a t i o n . With these d e f i n i t i o n s one has the expansion: Z(AI*) = E
]I exp(-BE(~')) ]I ( e x p l l n t i ~ ' l B e *) z ~ ( i ) ( I n t i ~') , ~'E~
(3.9)
I
w h e r e t h e sum is o v e r all families of o u t e r contours in A, and the second p r o d u c t runs o v e r all the connected Components of t h e i n t e r i o r of the c o n t o u r ~', indexed b y i. E(i)=O, * o r - according to t h e value of t h e spins along t h e b o u n d a r y of Int.~. I
23
Now, we would like to iterate this expansion, as we did in the case of the tsing model, in o r d e r to obtain e v e n t u a l l y a sum over families of pairwise nonintersecting, but
not
necessarily
outer,
expansion for Z ( A I * ) ,
contours.
This
would
yield
a
convergent
polymer
p r o v i d e d the resulting activities of the contours are small
enough. Such an expansion would allow us to d e r i v e all the desired properties of the
(*)-phase,
like
Peierls'
properties of pure phases. (0)- and the (-)-phases,
bound on
probability
of
contours,
and
clustering
We would like also to have a similar expansion f o r the and the condition of coexistence of such a three phases
should determine 9(1~). To obtain this, we w r i t e (3.10)
z ~ ( i ) ( I n t i ~) = ( z ~ ( i ) ( I n t i ~ ' ) / Z + ( I n t i ~ ' ) ) . Z + ( I n t i ~) , as we did in the
Ising model, except that
the ratio was equal to 1 t h e r e ,
by
symmetry. Inserting (3.10) into (3.9) and i t e r a t i n g , we obtain the expansion Z(AI +) = ~
~
where
sum
the
z+(~)
is
,
(3.11)
over
all
families
¢0 of
non-overlapping
contours
in
A,
not
necessarily o u t e r ones, and the activities are z+(~') = exp(-BE(~')) ]t ( z ¢ ( i ) ( I n t i 2 r ) / Z + ( I n t i ~ ' ) ) i
.
(3.12)
Note that the families of contours o v e r which we sum in (3.11) are not, in general, families of
contours
of configurations
of our model
(unlike
in the
Isin9
case).
Indeed, because of o u r induction method, all contours have the ( * ) - g r o u n d state in their
exterior,
even those t h a t
may be in the
(-)-
or
(0)-interior
of another
contour. The convergence of the polymer expansion depends on an estimate like (2.3a) and would clearly
hold here
if,
in
(3.12),
we had only the factor
exp-13E(~),
because E(~)->cl~rl, where I~1 is the number of bonds in ~'. So we would like to show that the factors Z ° ( I n t i ~ ) / Z * ( I n t i ~') are not too large. From general arguments we know o n l y that
(3.13)
Ilog Z°(A) - log Z+(A)I s const, laAI Since the b o u n d a r y of Int~" is just
~" we have to f i n d out how does the const,
in
(3.13) depend on IS. From the i n t u i t i v e p i c t u r e described in Sect. 1, we expect that it should grow like l~ if we are not on the coexistence line. Indeed, if the (O)-phase is
the
dominant one,
for
example,
then,
in
a system
with
the
(*)-boundary
24
conditions, a contour " t u n n e l l i n g " to the (O)-ground state will develop itself and its length
will
be
approximately equal
to
[~AI.
Its
weight
will
be of
exp(-[5]aA]) which will give a constant approximately equal to [5. if the const,
the
order
And vice versa,
in (3.13) grows like [5 then the polymer expansions (2.2a) about the
ground states are out of control and there is no reason to expect that we are on the coexistence line. Now, what is the expected behaviour of the const, of (3.13) on t h e c o e x i s t e n c e line?
From the
results
obtained
for
the
Isin9
model
(the
convergent
polymer
expansion of the Proposition), we expect t h a t log ZS(A) = - I ^ I B ~ ( B ) + a~(A,B) , where in the f i r s t
(3.14)
~=*,0,-,
(volume) term $ is the bulk free energy
(independent of the
b o u n d a r y conditions); A~(^,[5) is the b o u n d a r y term which we expect to satisfy the bound (2.13): la~CA,[5)l
~ ~([5)I~AI
(3.15)
,
where 6([5)~0 e x p o n e n t i a l l y fast as [5~. Now, the common volume terms cancel out in (log Z°(A)-Iog Z+(A)),
and t h e r e f o r e the bound
(3.15) implies that the const,
in
(3.13) is of o r d e r 6([5). Thus, we have two extreme situations: either the const, in (3.13) grows like [5 f o r large enough ~, (away from the coexistence line), or it tends e x p o n e n t i a l l y to zero with [5 (if we are on the coexistence l i n e ) . T h e r e f o r e we can expect to be able to construct 9([5) by imposing the condition that
(3.15)
holds f o r ~=+, - and 0
simultaneously. How does one show that such a line 9([5) exists? If we had only to deal with the contours of lowest energy (isolated zeros in the + ground state) then, as our calculation above shows, one could set g([5)=exp-4[5*higher o r d e r terms, and one would get equality of the b u l k free energies for all the ground states, and also the bound (3.15)
to o r d e r exp-4B.
It is f a i r l y obvious that we could still find a
line 9([5) if we had only a f i n i t e number of contours
(up to lattice t r a n s l a t i o n s ) .
But we have an i n f i n i t e number of them. However, 9([5) can be constructed via an i n f i n i t e sequence of approximations for the following reasons: when the size I~l of the contour Condition
If increases,
of [24]
E(~') grows
and [42]).
p r o p o r t i o n a l l y to it:
So, r o u g h l y speaking,
E(~')_>c[~'l
(the P e i e r l s
if we h a v e a line gn([5) such
that we would have coexistence if we considered only contours of size less than or equal to n, then, to include contours of size n*l we shall only have to modify gn by
an
amount
like
exp(-[5(n+l)).
This
gives
approximations to the t r u e phase coexistence line g([5).
a
convergent
sequence
of
25
We stress the importance of the Peierls Condition in the above arguments. Once a notion of a contour can be introduced in a model in such a way t h a t the Peierls Condition holds the most essential elements of the Pirogov-Sinai t h e o r y can be taken o v e r
( f o r instance, see [57, 59] f o r an extension of the theory to some
models with an i n f i n i t e number of g r o u n d states, and the model of Sect. 4). When the
Peierls
Condition
does
not
hold,
one
can
sometimes
substitute
for
"averaged" version - the Peierls' bound, f o r a suitably defined contours,
it
its
as done
in [8] and the work described in Sect. IV.
3. Genera/ phase diagrams and perturbation theory We start with the general result of Pirogov-Sinai, specialized to this model. Let us introduce an additional p e r t u r b a t i o n of the Hamiltonian (3.7): ~ H ( g , h ) = I~H - g~: s(a) = - h~: s(a) . a a
(3.16)
Then we have the following zero-temperature phase diagram in the ( g , h ) - p l a n e . The ground state is unique except along t h r e e half-lines: there are two ground states, (*)
and ( - ) ,
h=-g;
on the half-line g>0 h=0; two ground states,
and (-)
(*)
and (0), f o r g<0
and (0) for g<0 h=g. Of course, for g=h=0 one has three ground
states (see Fig. 1 in [48]). A set of perturbations such as gZs(a)=*h~:s(a) above "removes completely the degeneracy of the ground states" since e v e r y possible combination of the original ground states appear on this z e r o - t e m p e r a t u r e phase diagram on manifolds of the correct dimension (n ground states coexist on manifolds of codimension n - l ) . Pirogov-Sinai t h e o r y then says that f o r small Igl
and
Ihl
(here
The
in fact f o r all
(g,h)),
and for low temperatures, the phase diagram is a small deformation of the
one at
T=0.
Thus
we
have one
point,
g(l~),
with
three
phases,
three
lines
emanating from that point with two phases, etc.. Moreover, this t h e o r y allows one to j u s t i f y the p e r t u r b a t i o n expansion, which was our
starting
point.
Let us explain
how does one obtain
the
perturbation
expansion of the phase diagram in a more general situation [48]: Define an excitation of a ground state as a configuration which coincides with this g r o u n d
state everywhere,
with a possible exception of a finite set of sites.
into connected components, called elementary excitations, in such a way that the ( r e l a t i v e ) energy of an excitation with respect Decompose
every
excitation
to the g r o u n d state is just the sum of the (relative) energies of the components. Note t h a t this decomposition may d i f f e r from that into contours. For example, in the Ising model, a large bubble of minus-spins in the ( * ) - g r o u n d
state with a plus-spin
in its middle is an elementary excitation g i v i n g rise to two contours: one separating
26 the bubble from the surrounding plus-spins, and another one separating the central spin from the bubble. To compute the asymptotics of the lines and surfaces of phase coexistence up to an
err o r of
order
exp-BE,
we consider for
ensemble consisting of its excitations with
each
ground
state the
restricted
elementary components of energy not
exceeding E. This we call the gas of excitations of the ground state G with energy not exceeding E. For example, in the Blume-Capel model considered above, the gas of f i r s t - o r d e r excitations of the ( * ) - g r o u n d state, i.e. excitations with energy not exceeding 4,
consists
surrounded by sites
of all
configurations
b where s(b)=+l.
with
isolated
sites
where
The free energy ~R(BIG) of
s(a)=0, this
gas
(denoted PE G in [48]) has a convergent (polymer) expansion, similar to the one in the Proposition, if B is large enough (depending on E). Indeed, convergence of this expansion rests on an estimate of the form (2.9a) which clearly holds here: we have a finite number of elementary excitations (modulo translations) and therefore in a suitable analogon of (2.9a) one has to sum over a finite set. The
definition of
~R'
and the
construction of
the
phase diagram which
follows, makes sense for a much larger class of systems, than those to which the Pirogov-Sinai theory applies, many of them with an infinite number of ground states (see Sect.
IV).
(regularity) support.
Namely, it is enough that the Hamiltonian satisfies the following
condition:
The
energy
of an
excitation
tends
to
infinity
with
its
Note that this condition is not satisfied by the one-dimensional Ising
model (see Sect. 4 for more interesting examples). To obtain the line of coexistence of all the phases, we set equal the free energies of the restricted ensembles corresponding to all the ground states, and solve for g,
h (in our example h will always be zero, by symmetry). This is what
we did in our f i r s t - o r d e r calculations.
To obtain the surface of the coexistence of
the (+)- and (0)-phases, for example, we again set the corresponding free energies equal, and impose the condition that the restricted (-)-ensemble has a higher free energy. This yields an asymptotic expansion for the line and for the surfaces of phase coexistence [48]. We do not expect this expansion to be convergent. However the coefficients of the terms in this expansion are easy to compute. Also,
it often
happens that knowing the f i r s t few terms in the expansion gives the topological structure of the phase diagram (i.e.
the incidence properties of the lines and
surfaces of the diagram). This structure is not affected by higher o r d e r which only give a better numerical approximation to the true phase diagram.
terms
27
4. A p p l i c a t i o n s to s u r f a c e models
Let us consider t h e same Hamiltonian s(a)
take
2n'1
values:
(3.1) as in the Blume-Capel model, b u t
s ( a ) = - n , - n * l . . . . . n.
Blume-Capel model implies t h a t t h e r e will
Then
the
same analysis
as
in
let the
be a unique phase at low t e m p e r a t u r e s
(see b e l o w ) . Also, we can add to ~H 2n p e r t u r b a t i o n s which l i f t the d e g e n e r a c y of the g r o u n d states, f o r example, ~: k: k#0, and
~ ( k ) ~: 6 ( s ( a ) - k )
,
I kl<-n
obtain
a
complete
phase
diagram
for
small
temperatures
and
small
1~.
C o n s i d e r i n g the lowest e n e r g y e x c i t a t i o n s , we deduce t h a t the coexistence between the
phase
corresponding
to
s(a)=n
]~(n)-iJ(n-1)=exp(-413)+o(exp-413),
and
while
the
one
for
coexistence
s(a)=n-1
between
will
the
(0)-
occur
for
and
the
(-*)-phases will o c c u r f o r ~(*1) = exp(-4n213) * h i g h e r o r d e r terms . Indeed the ( n - 1 ) - g r o u n d the ( n ) - g r o u n d
(3.17)
state has more excitations of e n e r g y equal to 4 than
state ( l i k e in the Blume-Capel model w h e r e n=l) w h i l e t h e 0 and -'1
g r o u n d states are identical in t h e i r low e n e r g y excitations up to e n e r g y 4n z which are g i v e n , in the 0 g r o u n d state, b y s(a)=-*n, s(b)=O f o r l a - b l = l . It t u r n s out t h a t this discussion has an i n t e r e s t i n g application to the t h e o r y of random
surfaces
Is(a)-s(b)l replaced
by
(see
[5]).
First,
observe
that
if
we
replace
(s(a)-s(b)) = by
in the Hamiltonian (3.1) t h e r e are no dramatic changes: n = in (3.17) is n but
particular there
the
phase diagram
is a unique
has the
same q u a l i t a t i v e
phase at low t e m p e r a t u r e s .
properties,
However,
this
in
model is
isomorphic to the SOS model of a surface which f l u c t u a t e s between two hard walls (s(a)
is the h e i g h t of the surface above site a and the walls are at height -n and
+n). Now, c o n s i d e r a d i f f e r e n t model, w h e r e the interface is constrained to be above one
fixed
wall,
say
at
height
0:
the
model
H=F.Is(a)-s(b) I and the c o n s t r a i n t s(a)>0 (all a).
is
defined
by
the
Hamiltonian
Let us tie the surface to t h e wall
(s(a)=0) on the b o u n d a r y of a box A and ask f o r the value of l i m < s(0) >A .
(3.18)
It is well known [16, 23, ,56] t h a t if t h e r e is no wall this limit is f i n i t e at low t e m p e r a t u r e s if the dimension of the lattice is l a r g e r than o r equal to 2 ( a c t u a l l y , f o r at t e m p e r a t u r e s if d_>3, [6, 2 9 ] ) . However, the presence of the wall induces an effective repulsion of the surface so t h a t
the
limit
(3.18)
is i n f i n i t e f o r
all
dimensions at low t e m p e r a t u r e s
(and,
28
presumably,
at
all
temperatures).
This
fact
is
an
easy
consequence
of
the
Pirogov-Sinai t h e o r y in the form discussed above. Indeed,
< s('0) >A > < s(0) >A,2n '
(3.19)
w h e r e on the RHS we have added the c o n s t r a i n t s(a)<_2n f o r all a. i n t u i t i v e and actually follows from FKG-inequalities.
This is f a i r l y
Now, change the variables on
t h e RHS of (3.19) to s ( a ) ' = s ( a ) + n . The s ( a ) ' variables are e x a c t l y like in t h e model discussed above
(]s(a)'l<_n).
Since t h e r e
is a unique
phase ( a c t u a l l y , a unique
Gibbs state) f o r this model at low t e m p e r a t u r e s which moreover is concentrated on small p e r t u r b a t i o n s of the ( 0 ) - g r o u n d state, < s(0)'> A ( ] s ( a ) ' l < n )
will go to zero,
as A-~®, i n d e p e n d e n t l y of the b o u n d a r y c o n d i t i o n s . I n s e r t i n g this into (3.19) we get
lim
< s(0)
>A ~ n .
But since n is a r b i t r a r y ,
our claim is p r o v e d . For more d e t a i l s , we r e f e r the reader
to [ 5 ] .
IV. INFINITE NUMBER OF GROUND STATES; OPEN PROBLEMS We will
now discuss on a simple example some of the e x i s t i n g extensions of the
Pirogov-Sinai
theory,
and
some of the
open
problems r e g a r d i n g
low-temperature
b e h a v i o r of classical lattice systems. The model is a version of the f a m i l i a r spin-½ a n t i f e r r o m a g n e t on a square lattice.
It is chosen here because it i l l u s t r a t e s most of
t h e points we want to make. A t t h e expense of complicating t h e notation we could have discussed models of alloys which are of some importance f o r applications (see Fig.
5 of [ 4 8 ] ) .
Let a(b)=-+l be a spin-½ v a r i a b l e at a point b of the simple square lattice Z 2 and let
H = Hnn - =Hnn n
hM
( * H v) ,
(4.1)
where H
nn
= E o(a)o(b) ,
H
nnn
= E o(a)a(b) ,
M = E o(a) ,
w i t h t h e sum e x t e n d e d over all pairs of nearest neighbors of the lattice in Hnn, o v e r all pairs of n e x t nearest n e i g h b o r s in Hnn n, and o v e r all points of t h e lattice
29
in M;
the term H v,
see
[4.3),
will be included
later,
when we will
consider a
t h r e e - d i m e n s i o n a l " s t a b i l i z a t i o n " of this model. The
antiferromagnetic
Hamiltonian
c h e s s b o a r d - l i k e g r o u n d states.
Hnn
is
well
known
to
have
two
To obtain t h e z e r o - t e m p e r a t u r e phase diagram of H
one can g r o u p its terms as follows: H = E [ ½ I; o ( a ) o ( b ) - ¢ ~ oCa)o(b) - ¼hi: oCa)] = i: i ° , [] nn nnn [] w h e r e t h e f i r s t sum is o v e r all elementary squares [] ( " p l a q u e t t e s " ) of the lattice, the first
sum in
the square
bracket over
(all f o u r )
p l a q u e t t e ; the second o v e r the t w o pairs of n . n . n . ' s
pairs of
n.n.'s
within
the
w i t h i n the p l a q u e t t e , and the
last o v e r the f o u r points of the p l a q u e t t e . Now, minimization of i o yields Fig. 2:
g=2
[7g=3 /
(+;) o
g=l
(::)-+,
,o
(':> N Q=4 s o\ Fig. 2.
\
C o n f i g u r a t i o n s of the s q u a r e minimizing i ° are obtained by
indicated on the f i g u r e ,
r o t a t i n g those
g is t h e number of ( p e r i o d i c ) g r o u n d states (obtained by
p a t c h i n g the indicated c o n f i g u r a t i o n s of s q u a r e s ) . e n t r o p y per lattice site of the set of all g r o u n d
If g is i n f i n i t e s indicates t h e states. The
Pirogov-Sinai t h e o r y
applies w h e n e v e r g is f i n i t e .
For
points
of t h e
lines of
this
phase diagram minimizing
plaquettes are those of each of t h e adjacent open regions. in t h r e e g r o u n d
states f o r the points of
the
configurations
of
the
For example, this results
h a l f - l i n e h=4,
~>0 and a n o n - z e r o
30 entropy
for
the
half-line
h*4¢=4,
~<0.
In
the
later case,
any
configuration
obtained from the ( * ) - g r o u n d states by flipping spins on a set of points no two elements of which are n.n.'s or n . n . n , s is a ground state again. In each of the
(three) open regions with
a finite g the ground states are
equiva/ent in the sense that they are related by translations and rotations which are symmetries of the Hamiltonian. For the (9=2) - and (g=4)-regions the Peierls' argument of Dobrushin [15] and Gerzik [24] yields two and four low-temperature pure phases, respectively. The Pirogov-Sinai theory applies both in these regions In the last region
and in the neighborhood of each point of the half-line h=4, ~>0.
the phase diagram is best computed using the perturbation expansion, which is justified
with the
help of
the
Pirogov-Sinai theory:
Let SR(B,~,h[G=nnn)
and
~R(B,¢,hlG°) be the free energies of the restricted ensembles of excitations of the corresponding ground states, with energies not exceeding E.
For h=4 the lowest
order excitations are obtained by flipping one of the spins in such a way that the only non ground states plaquettes are those with PZ-configurations. This can be done in n l ( G ° ) = l ways per lattice site in the case of the G o ground state, and in nl(G=nnn)=½ ways per lattice site for G=nnn. Hence G O dominates in order 1 and only
the
G°-phase
SR(B,¢,h(B) IG°) G°-phase
and
is
present
at
low
= SR(B,¢,h(B) lG=nnn) the
O=nnn-phases,
temperatures.
for the line
yields
Indeed,
h(B)
the
equation
of coexistence of the
h([B)=4-(1/2l~)exp-8¢B*o(exp-8¢B),
in
accordance with the argument based on domination. The rest of the plane: the interval DB and the strip with base BC, including the points of its boundary, is not covered by the Pirogov-Sinai theory.
In fact
there are only a few rigorous results about the system in this region, not even a formal
perturbation
Iow-temperatu re
theory
behavior
which (see,
would however,
three-dimensional stabilization of this model).
lead
to
below,
plausible the
conjectures
discussion
of
on a
For, for the values of the parameters
on the boundary of the strip the system admits local changes of the ground states without a change in the energy. Hence, the number of ground states in a square of edge-length L is of order c L2, and therefore the ground-state entropy is non-zero. Consider, for instance, for h+4¢=0 and ~<-½ (i.e.
on the half-line emanating from
the point B of Fig. 2) the ground state G=nn consisting of alternating rows of plus- and minus-spins. Flipping any number of minus-spins which are not nearest neighbors produces PZ-plaquettes only,
which are of the same energy as those
occurring in G=nn. Hence one will have an infinite number of kinds of excitations with
the same energy and the standard perturbation expansion does not make
sense. Following [63], points of the zero-temperature phase diagram for which the ground-state entropy is non-zero will be called super-degenerate. This perturbation expansion makes also no sense for (¢,h) on the interval DB or in the strip, though here local change of a ground state does not yield a ground
31
state any more:
now some of the excited
configurations can be changed locally
without changing their energy. We will discuss this in detail for the interval DB, after describing the ground states and low-energy excitations. All the periodic ground states of the system are among those which can be obtained from G=nn by flipping all spins on a number of lines parallel to one of the coordinate
axes.
For
instance
flipping
spins
on
every
second
line
yields
a
G2nnn-ground state. The plaquette P~ obtained from the ground state plaquettes P= by flipping one of the (-)-spins has the lowest energy
= {o(P 1) - Cn(p 2) . Hence the f i rs t excitation energy is 4s. An excitation with this energy is obtained when one (-~)-spin is fli.pped in any of the ground states. However flipping any interval of spins which begins and ends with a minus, like -*- or - * - + - * - ,
yields
again an excitation with four Pl-plaquettes and energy 4~. Thus if A is a square of side L, the coefficient of exp-134¢ of the expansion of (1/IAI)log Z(AI G2) is of order at least L, as L -~ ®, and hence has no thermodynamic limit. However if we ignore the fact that the free energy of the gas of low energy excitations does not exist, but just compare the coefficients of the low-temperature expansion of log Z(AIG=nn) and log Z(AIGSnnn) we would say that the ground state G=nnn is dominant since G= admits twice as many excitation with, say, (-+-)-intervals flipped (both nnn horizontal and vertical) than G=nn (only vertical or only horizontal); in a similar way G2nnn wins not only over G2nn but over any other periodic ground state. It is not clear at all whether these considerations lead to right conclusions about low-temperature phases
of the system (see,
however,
concerning a three-dimensional stabilization of the model).
below, for
results
For, most likely, the
low-temperature expansion of the free energy of the system starts not with exp-41~¢ but with exp-21~z, due to presence of nonlocalized excitations: flipping a half-line of alternating spins of a ground state Which starts
with
a minus-spin one gets a
configuration with only two Pl-plaquettes. Indeed such non-localized excitations do contribute to low-temperature expansion in one-dimensional models, or ferromagnetic models in any dimension, [32, 48]. Though some models with similiar features have been treated in a number of works
(see [50, 51], for example), the prob/em of
computation of the low-temperature expansion for general models of this type seems to be open. We return now to a discussion of the super-degenerate points (the boundary of
the
strip;
non-zero
low-temperature
expansion
zero-temperature
limits
of
ground-state does
not
equilibrium
entropy). make
states
sense zero
Although at order
the
these of
standard
points, a
the
perturbation
32 expansion, in a sense, are of interest here. A discussion of these limits clarifies the situation conceptually and yields n o n - t r i v i a l information on the low-temperature phase diagram [62, 64, 17, 63]. We explain f i r s t the situation on the example of the antiferromagnet in external field [62, 64, 17]. Consider the model (4.1) with ¢=0. We let h-~0 as T-K) (see Fig. 3).
More
precisely we approach the point h=0, T=0 along one of the inclines, i.e. h(T)=4*h'T,
we set
where h' is the inverse slope of the line; for h'=0 one has the usual
zero-temperature limit. Substituting this into ( 4 . 1 ) , with 13=T-1, we obtain (4.2)
I3H = 6(Hnn - h ( T ) M ) = ~H° - h'M ,
where Ho=Hnn-4M. Now one would expect that as 13-*® the term ISH° will suppress the contribution to equilibrium states of all but the g r o u n d state configurations of H° [62].
Thus in
the limit one obtains a state which is supported by the ground state configurations of H
O
with
interaction defined by -h'M.
This
is indeed the case (Slawny,
1980,
unpublished; [64, 17]). The g r o u n d states of H° are supported by configurations minimizing {n for each
plaquette
D.
Inspection
of
Fig.
2
shows
that
these
are
exactly
the
configurations in which no minus-spin is a nearest neighbor of another minus-spin.
T
i / // I
I I
/
I l
~ I I I I I
h I I I I
,Jl
I
Z Fig. 3.
Blowing-up the super-degenerate points:
approaching
the
point
(T=O,h=4)
from
different
the limiting states obtained by directions
are
identified
equilibrium states of a lattice gas with d i f f e r e n t values of the a c t i v i t y z.
with
33 In an isomorphic lattice gas language in which plus-spins are identified with empty
sites and minus-spins with occupied, one obtains a model in which no two particles are allowed to occupy the same or n e i g h b o r i n g sites and the a c t i v i t y is z=exp-2h'. This
picture suggests
that close to a super-degenerate point one should use a
p e r t u r b a t i o n expansion about the limiting state of the hard core lattice gas and to i n t e r p r e t the properties of the system u n d e r consideration in terms of the later model. We refer to [62, 63, 64, 17] for an implementation of these ideas. More generally,
for
any
H° with
non-zero g r o u n d - s t a t e
entropy
limits of
equilibrium states of 13Ho+H', as 6-~, are equilibrium states, in the sense of [65], of the Hamiltonian H' restricted to the set of ground states of Ho, and the limits of the pressure and entropy are equal to those of the limiting system (Slawny, 1980, unpublished).
The
reader may want
to work
out
the
limiting
systems
for
the
super-degenerate points of Fig. 2 d i f f e r e n t from C, and the two-parameter families of interactions obtained by approaching these points from d i f f e r e n t directions. We now pass to a three-dimensional "stabilizat'on' are
able
to
develop
an
extension
of
the
of the model f o r which we
Pirogov-Sinai
theory
describing
its
low-temperature behavior. The model is on a simple cubic lattice Z 3, with the Hamiltonian (4.1), but with the
term
Hv
included
("stacking"
of
[12]):
the two-dimensional
layers,
with
i n t r a l a y e r interaction H2=Hnn-~Hnnn-hM, are coupled by the term H
v
= -J~' o(a)o(a+v) , a
J > 0 ,
(4.3)
where v is the unit vector in the vertical direction. Since the coupling constant J is positive, the ground states of the f u l l Hamiltonian are obtained by repeating in each horizontal plane the same g r o u n d
state of
H =. Hence the zero-temperature
phase diagram remains as on Fig. 2, with the change that the g r o u n d - s t a t e e n t r o p y is now zero everywhere: the number of g r o u n d states in a cube of edge-length L is of o r d e r c L (as L -~ =) for the points of the strip and the interval L= order c on the lines where the e n t r o p y was non-zero p r e v i o u s l y .
DB, and of
Now, in the later case, there is still no perturbation t h e o r y since there is an i n f i n i t e number of kinds of excitations with given energy, as in the (g==., s=0)-case in two dimensions.
For,
the two-dimensional model admits a local change of the
g r o u n d state w i t h o u t a change in the energy. number
of
neighboring
planes
one obtaines
Repeating this change in a f i n i t e a configuration
in
which
only
the
vertical bonds going up from the uppermost plane and down from the lowest plane are " e x c i t e d " ,
independently of the
Thus,
"cL-case '' in two dimensions,
as the
number of planes affected the
"cL=-case ''
by the
in three
change.
dimensions
admits no standard perturbation expansion and the problem of the n a t u r e of the
34
low-temperature phase (phases?) is open (see [12] f o r a discussion of a simi[iar situation). sense
However for all o t h e r (~,h) the standard p e r t u r b a t i o n expansion makes
and
conclusions
derived
from
it
are
justified
by
an
extension
of
the
Pirogov-Sinai t h e o r y of [10]. We will now discuss in some detail the example of the i n t e r v a l DB. First of all we note that for the points of the i n t e r i o r of the interval the r e g u l a r i t y condition of Sect. 111.3 is satisfied. states are dominant there.
Furthermore, the two G=nnn-ground
For, as in the two-dimensional version, the lowest o r d e r
excitations are obtained by f l i p p i n g one minus-spin. and nl(G)=½
for any ground
obtained
flipping
by
a
state G.
This
has now e n e r g y 4~+2J,
If J is not too large,
(-+-)-interval
(energy
4E+6J),
with
the next o r d e r is n2(G=nnn)=t
and
n2(G2nn)=½, and intermediate values of n 2 for other periodic ground states. Hence, G=nnn
is
dominant
in
second
order,
and
therefore
one
would
expect
two
low-temperature phases. T h a t this conclusions are correct follows from proof ("Peierls' bound for large-scale contours")
[10].
The main points of the
are explained in the A p p e n d i x ,
especially in the proof of Lemma 1.
V. CONTINUUM FLUIDS The f l u i d models f o r which most results are known about phase transitions, and the Widom-Rowlison model [52] in p a r t i c u l a r ,
concern phase separation in mixtures.
In
the simplest version of this model, one considers two species of particles A and B, with
no interaction between particles of the same species and with a hard-core
exclusion between A- and B-particles.
Then Ruelle proved,
[44], that the A - r i c h
and the B-rich phases coexist if the activities z(A) and z(B) of the two species are equal and large enough. This
result holds also if t h e r e is a
suitable interaction
among the A particles and among the B-particles p r o v i d e d t h a t t h e y are identical, i.e. t h a t the A-B symmetry of the model still holds [44, 39, 58]. Comparing = this to the Isin9 model, we could say that we have kept the symmetry between the two phases b u t we do not have a lattice any more, perturb important.
around.
In
fact
this
is
an
nor do we have ground states to
example
where
A natural problem is an extension of this
situation (the interaction between the A-particles between
the
B-particles).
This
is
done
in
[8]
restricted result
ensembles
are
to a non-symmetric
is d i f f e r e n t from the interaction but
under
rather
restrictive
assumptions on these interactions (see below). The
restricted
ensembles
considered
by
Ruelle
(in
a
somewhat
different
terminology) are defined as follows: Cover the space with non-overlapping cubes so that if we have an A particle in some cube then the h a r d - c o r e forbids B particles
35
to be in any of t h e adjacent cubes.
Define one r e s t r i c t e d ensemble b y
t h a t t h e r e be at least one A - p a r t i c l e in each cell, B - p a r t i c l e in each cell.
requiring
and another w i t h at least one
Now, if we have a c o n f i g u r a t i o n o f one of the r e s t r i c t e d
ensembles in one region of space and of the o t h e r in another region, t h e h a r d - c o r e condition implies t h a t some of the cells must be empty. Define then a c o n t o u r ~ as a connected
set of
empty c e l l s and
a c t i v i t i e s empty cells are u n l i k e l y ,
I~1 as the number of cells in ~r. For large and we can a p p l y the Peierls a r g u m e n t .
t h e A - B s y m m e t r y , one shows t h a t f o r l a r g e z ( A ) = z ( B ) ~" is bounded b y e x p ( - 0 ( z ( A ) ) l ~ ' l ) . separation is s t a n d a r d .
Using
the p r o b a b i l i t y of a c o n t o u r
Going from this estimate to the p r o o f of phase
The new f e a t u r e of this
p r o o f is the i n t r o d u c t i o n
of t h e
r e s t r i c t e d ensembles in which outside of a c o n t o u r we have a gas and not j u s t one c o n f i g u r a t i o n . Note t h a t in the A - r e s t r i c t e d ensemble only A particles occur while in t h e actual A - r i c h phase t h e r e will be some (small) "islands" of the B p a r t i c l e s . Using an extension of the Pirogov-Sinai t h e o r y in which the g r o u n d states are replaced b y r e s t r i c t e d ensembles, A-A
and the
B-B interactions
[8] p r o v e an A - B phase separation, even if the
b r e a k the A - B
symmetry.
However,
one must then
r e q u i r e t h a t each of the r e s t r i c t e d ensembles be d i l u t e in the sense t h a t t h e y are both w i t h i n the region of convergence of t h e i r Mayer expansion. Such a limitation is not necessary in t h e A - B symmetric case. So, at p r e s e n t , we need e i t h e r s y m m e t r y or
diluteness
which
one-component f l u i d :
prevents
us from
one could i n v e n t
dealing
with
a liquid-gas
transition
in
a
r e s t r i c t e d ensembles f o r t h e l i q u i d and the
gas phases b y locally f i x i n g the f l u i d d e n s i t y at a p p r o p r i a t e values b u t t h e r e is no s y m m e t r y between these ensembles and i t is not clear what kind of diluteness would hold f o r the l i q u i d r e s t r i c t e d ensemble. Of course, one can obtain a one-component fluid
model
particles.
from
the
Widom-Rowlison
Then t h e A-B
A-B
phase separation
model translates
by
"integrating
out"
the
B
into a l i q u i d - g a s t r a n s i t i o n .
T h e e f f e c t i v e interactions between the A - p a r t i c l e s induced b y the B - p a r t i c l e s can b e computed e x p l i c i t l y when t h e r e is no i n t e r a c t i o n among the B - p a r t i c l e s since, f o r a fixed
configuration
of
the
A-particles,
the
B-particles
form
a free
gas
[52].
U n f o r t u n a t e l y these interactions are q u i t e u n r e a l i s t i c f o r a model of a one component fluid.
V I . THE POTTS MODEL A t each lattice site we have a v a r i a b l e i n t e g e r ; the Hamiltonian is H = -
Y. 6 ( s C a ) , s ( b ) ) ,
s(a) t a k i n 9 values 1 . . . . . . q, w h e r e q is an
36 where 6 is the Kronecker delta-function and the summation is over pairs of nearest neighbors of the lattice. We shall consider q large and follows [53]:
d>2.
The phase diagram is expected to be as
There are q ground states s(a)=m (all aCzd), m=l . . . . . q, and, for
temperatures low enough, there are q phases which are small perturbations of these q ground states.
As the temperature is raised, we do not reach a critical point, as
one does in the Ising model (which correspond s to q=2), but rather a f i r s t - o r d e r transition. <6(s(a),s(b))> discontinuous.
At
this
transition
temperature
T(q)=l~(q)
the
m e a n energy
(a,b nearest neighbors) and the order parameter <6(s(a),m)> are For temperatures above T ( q ) ,
there is only one phase; at T ( q ) ,
there are q * l phases, one high-temperature phase and q low-temperature phases. These latter phases are referred to as the "ordered phases" because, in each of them, the mean energy is close to 1 (for large q) i.e. typical configurations have most of their bonds "unbroken" (s(x)=s(y) if x and y are nearest neighbors). high-temperature phase is called "disordered" have most of their bonds "broken" (s(x)~s(y)
The
because its typical configurations if x and y are nearest neighbors),
and the average energy tends to zero as q-~®. There are many configurations with broken bonds for large q, and thus the disordered phase has large entropy which compensates for its smaller energy than that of the ordered phases, which explains why all these phases can coexist. This model has been analyzed using the reflection positivity property of the interaction by Kotecky and Shlosman [35].
Here we want to consider it from the
point of view of the Pirogov-Sinai theory because it illustrates nicely the use of restricted ensembles
[8,
14, 60].
Also, there is a vague analogy between this
model and a liquid-gas transition: the ordered phases have a lot of energy (like the liquid phase) while the disordered one is favored by entropy (like the gaseous phase). There is a more precise analogy between this model and the Blume-Capel model
(3.7) with small positive g: at low temperatures, there are two (ordered) phases and, at some higher temperature T(g) we reach a f i r s t - o r d e r transition point where three phases coexist. g=0 Hamiltonian,
The third phase corresponding to the ground state G o of the
has larger
entropy
(of
low energy
excitations) than
the two
ordered phase and becomes the unique phase for temperatures above T ( 9 ) . Returning to the Potts model, we define q *l (the ordered ones) are trivial; configuration: s(x)=m (all x),
they consist of the corresponding ground state
m=l . . . . . . q.
The last ensemble is made of all the
configurations for which all bonds are broken: XR D = (s: s(x) ~ s(y)
restricted ensembles: q of them
if x and y are n . n . ' s } ,
37
together
with the
Gibbs
state
induced by
H on
it:
the conditional p r o b a b i l i t y
d i s t r i b u t i o n s of this Gibbs state in finite volumes are just the normalized counting measures, since H=0 on X D R" The free energy of these ensembles is easy to compute.
For the ordered
ensembles, it is just the ground state e n e r g y , equal to -d (d bonds per lattice site in d dimensions).
For the
disordered ensemble, the free energy equals,
for q
large enough h, _B-l(iog q , f(q-1))
,
(6.1)
where f is an analytic function, and f(0)=0
(see [8, proof of Theorem 5 ] ) .
One expects t h e r e f o r e phase coexistence between the ordered phases to occur f o r B(q) "- Clog q ) / d
(to leading o r d e r ) .
Let us write B=B'log q.
Now we should
t h i n k of log q as playing the role of B in the Blume-Capel model while B' plays the role
of g.
restricted
For Io9 q-~ we expect the following: ensemble is
Blume-Capel model. G ÷, G-.
And f o r
similar
to the
For B'>l/d,
unique
for
if B'
state
G o of
the
the ordered ensembles replace the g r o u n d states
t~'=l/d we have coexistence of the ordered and the disordered
ensembles (just like at g=0 we had three g r o u n d states). more precise if one restricts
A c t u a l l y , the analogy is
oneself to quantities which are i n v a r i a n t u n d e r the
symmetry of the Hamiltonians, i.e.
under the global spin f l i p for the Blume-Capel
model or under global permutations of the spins in the Potts model.
Then one has
only one ground state for g=0 and one restricted ensemble for 13'#l/d. For the Blume-Capel model, the Pirogov-Sinai t h e o r y proves the existence of a line g(B)
on which
all three
p e r t u r b a t i v e l y (see Sect. 3). gives
similar results
phases coexist.
Moreover, one can compute g(B)
An extension of the Pirogov-Sinai t h e o r y [8, 14, 60]
f o r the
Potts model:
f o r any q large enough t h e r e
B'(q),
B ' ( q ) - ~ l / d as q-~,., for which q+l phases coexist. To f i n d
the
perturbation
expansion f o r
B'(q)
one has to compute the free
energies of the gases of excitations of the restricted ensembles (which replace the ground
states),
then
set
them
equal
and
solve for
B'(q).
The
lowest
order
excitations of the disordered ensemble are given by one unbroken bond < x , y > (i.e. s(x)=s(y)), site.
surrounded by broken bonds.
T h e r e are d such excitations per lattice
One looses a factor of q (to leading o r d e r )
in the sum over configurations
( e n t r o p y factor) but one gains a factor exp 13 on the energy side. Combining these two c o n t r i b u t i o n s ,
one obtains the free e n e r g y of this
new ensemble where all
bonds, adjacent to unbroken bonds are broken; this is similar to the gas of isolated spin flips in Sect. 2. To leading o r d e r , it is equal . B-l(iog q+f(q-1)) _ (d'expB)/Bq ,
(6.2)
38 where
If(q-1)l
~ 0(l/q)
,
(6.3)
since f is analytic and zero at the origin. For
the
ordered
ensemble,
the
lowest
order
excitations
corresponding ground state e v e r y w h e r e apart form one point.
are
equal
to
the
This yields (6.4)
- d - (q-1)/B.exp(-2dB)
f o r the leading o r d e r terms of the free e n e r g y of the restricted ensemble. Setting equal (6.2) and (6.4), term of o r d e r zero.
w r i t i n g B=B'Iog q and solving f o r B'one obtains 1/d as the
Inserting B'=l/d in the second term of (6.2) and using (6.3)
one obtains -d - 0 ( ( q log q ) - l )
= _ B,-1 _ d/(B'lo9 q ) q ( 1 - 1 / d )
, 0 ( ( q log q ) - l )
,
which, to leading o r d e r , yields B'(q) = 1/d * 1 / ( q ( l " l / d ) l o g
q) .
(6.5)
This coincides with the leading o r d e r terms of the exact answer [53] for d=2: B(q) = B'(q)log q = ½log q * log ( l * ( 1 / V q ) ) = I o g ( l * 4 q ) . However, (6.5)
holds for all dimensions, and higher o r d e r corrections can be
obtained by including higher o r d e r excitations (see also [25] and [28]).
Acknowledgements. J. B. thanks A. El Mellouki, J. Frohlich, K. Kuroda and J. L. Lebowitz for discussions and collaboration on some of the topics of the lectures, and Center
for
hospitality.
Transport J.
S.
Theory
thanks Joel
and
Mathematical
Lebowitz
and
Physics
Daniel Styer
at
Virginia
for
Tech
for
discussions,
and
Insitut de Physique Theorique de Universite Catholique de Louvain for hospitality; his w o r k on this paper has been supported in p a r t by NSF g r a n t MCS 8301709.
APPENDIX: PROOF OF THE THEOREM OF SECT. II1.1 We explain
first
the main
ideas of the
proof.
We want
to
show that,
at low
temperatures, there is a unique phase which is a small p e r t u r b a t i o n of the ground
39 state G o .
If G o was the unique g r o u n d
state, we would define t h e contours of a
c o n f i g u r a t i o n s as connected regions w h e r e s does not coincide w i t h G o . Then we would t r y
to p r o v e a Peierls'
(0)-boundary conditions.
bound f o r
From this
such
contours
in the Gibbs
estimate one would deduce,
state w i t h
using an idea of
Gallavotti and Miracle-Sole [23] which is recalled below (see Lemmas 2 and 3 ) , typical
configurations
of
any
translation
invariant
Gibbs
state
are
that small
p e r t u r b a t i o n s of t h e g r o u n d state G = and t h a t the l o w - t e m p e r a t u r e phase is unique. However, such a simple scheme of p r o o f has to be modified here, because t h e r e are t h r e e g r o u n d states, G , G , G o , not one. C l e a r l y , t h e r e is no damping f a c t o r m
the
Hamiltonian,
and
hence
no
Peierls'
bound,
for
the
regions
where
a
+
c o n f i g u r a t i o n coincides w i t h e i t h e r G
or G
and which would be r e g a r d e d as p a r t
of a c o n t o u r according to the scheme o u t l i n e d above. We know,
h o w e v e r , from the discussion of Sect.
III t h a t if we associate w i t h
each g r o u n d state the r e s t r i c t e d ensemble consisting of its lowest e n e r g y e x c i t a t i o n s then
the
(0)-restricted
r e s t r i c t e d ensemble. ensembles
replacing
ensemble has the
lowest free
energy
among these t h r e e
This suggests t h a t we use o u r f i r s t idea b u t w i t h r e s t r i c t e d ground
states:
the
contours
of
a configuration
s
are
now
connected subsets of the lattice on which t h e r e s t r i c t i o n of s does not belong to the (0)-restricted there will
ensemble. If we t r y
to estimate t h e p r o b a b i l i t y
be some damping f a c t o r
coming from
the
inclusion
of these c o n t o u r s , of the
low e n e r g y
e x c i t a t i o n s . The p r o b a b i l i t y t h a t a c o n f i g u r a t i o n s r e s t r i c t e d to a region A belongs to t h e ( * ) - r e s t r i c t e d
ensemble, is bounded b y the r a t i o of the p a r t i t i o n f u n c t i o n s ,
in /~, of t h e gas of low e n e r g y excitations of ( * ) to the p a r t i t i o n f u n c t i o n of the gas of l o w - e n e r g y excitations of (0). This ratio is a p p r o x i m a t e l y equal exp(-IAlexp-4~)
,
(A.1)
(see (3.5), (3.6)). However, at low t e m p e r a t u r e s this damping f a c t o r becomes v e r y small and it is not obvious
how to
control
the
sum o v e r
all the
contours.
To
deal w i t h
this
problem, we shall define contours on a large ( t e m p e r a t u r e - d e p e n d e n t ) scale. Cover the lattice with boxes B(i) Then,
from
(A.1)
we
r e s t r i c t e d a given box B ( i ) This
whose size is chosen so t h a t
can
deduce
that
the
belongs to the ( * ) - r e s t r i c t e d
c o a r s e - g r a i n e d d e s c r i p t i o n causes
e x c i t a t i o n s than the lowest o r d e r ones and,
I B ( i ) l e x p ( - 4 ~ ) ~- as l~®.
probability
an obvious
that
the
configuration
ensemble goes to zero as problem: t h e r e
are o t h e r
if we define o u r contours on such a
large scale, we loose some localization of these e x c i t a t i o n s . Namely an excitation w i l l have t o be in a box B ( i )
b u t i t may be a n y w h e r e in t h a t box. Moreover, we may
have many d i f f e r e n t e x c i t a t i o n s in a g i v e n b o x . So i t seems t h a t we are back to t h e
40 problem of controlling the sum over the contours. Observe however that the energy of the lowest excitations above the lowest ones (whose energy is equal to 4) is 6 ÷
(corresponding e.g. to two adjacent sites with s(a)=0 in the ground state G ). The p r o b a b i l i t y that such an excitation (or any higher energy excitation) is found in a region k can be bounded, by the usual Peierls' argument, by Iklexp(-6B) Therefore,
if we choose the size of the boxes in such away t h a t
with c between 4 and 6, say 5, then, for any box B ( i ) ,
]B(i)l=exp(cB)
the following two events
will be v e r y unlikely when B -~ =: the
configuration
restricted
to
B(i)
belongs
to
the
(+)-
or
(-)-restricted
ensemble. there an excitation (of any of the three ground states) of energy greater that 4 in B ( i ) . From this one deduces rather easily that a typical configuration belong to the ( 0 ) - r e s t r i c t e d ,ensemble in most of these
large
scale boxes,
and from this
the
uniqueness of the phase is deduced using the ideas of Gallavotti and Miracle-Sole mentioned at the beginning of this a p p e n d i x . Of course, the numbers 4 and 6 used here are accidental; what matters is the discreteness
of
the
set of
excitation
energies.
We may
summarize
the
above
discussion as follows: one associates with each energy E of elementary excitations a distance scale ¢ = exp(½BE) in the following way: consider a temperature dependent family of boxes k(B) of volume exp(BE').
If E' > E,
IA(I~)I
>> g2 and there will be
many excitations of energy E in k(B), f o r large B. "Many" means that we are almost in the thermodynamic limit, i.e. the free energy of the gas of excitations of energy <_ E is equal to its bulk contribution plus a ( r e l a t i v e l y ) small b o u n d a r y term. On the o t h e r hand, if E' < E, then, as 13 -~ ®, it becomes v e r y u n l i k e l y to find one
excitation of energy E in A(B).
even
It is the presence of these d i f f e r e n t distance
scales associated with the d i f f e r e n t excitation energies that makes the proof outlined above work. Now we define the restricted ensembles more precisely and discuss t h e i r main properties. Let X = (-1, 0, *1} z= be the configuration space of the model (we set d=2 for simplicity) and let X R = (sEX: s(a)=0,+l for all a, and if s(a)=0 then s ( b ) = * l if I b - a ] = l }
41 be the restricted ensemble of configurations of the gas of lowest energy excitations of the ( * ) - g r o u n d state. X R is defined similarly, and XRO = {s s X: if s(a)=-I or *1 then s(b)=0 if I b - a l = l } The
subscript
R at
X
("restricted
ensemble")
indicates
that
only
elementary
excitations up to some o r d e r (here up to f i r s t order) are considered. E A (¢=+,0,-) is the set of restrictions to A of the configurations For A c Z 2, XR, of X R. We define the p a r t i t i o n function of the restricted ensemble with boundary conditions s'¢XR,Ac as Z R ( ^ I s ' ) = ~: exp-BHA(s) , where the sum is over szX R which are equal to s' on Ac, and HA(s)
= ~; (sCa)-s(b)) = ,
where the sum is over pairs of n . n . ' s
with at least one point in A. These
p a r t i t i o n functions depend on the restriction
of s' to a~. only, and t h e r e f o r e the
same notation will be used f o r any s'¢XR, M with ~t~ c M. It is easy to realize that ZRCAI*) is just the partition function of a hard-square lattice gas: at each lattice site we can have a particle (corresponding to spin zero) and no two particles can be adjacent. The a c t i v i t y of a particle is equal to the Boltzmann weight of a zero spin,
z=exp-413. This
identification helps to realize that,
small, we have a convergent low a c t i v i t y
for
expansion f o r this
13 large,
The free e n e r g y is analytic in z and we have good clustering correlation functions.
i.e.
for z
restricted ensemble. properties of the
In p a r t i c u l a r , we may write, f o r all A
log ZR(AIs) = - IAI~$R(I~I*) * o(e-41~)l~AI , where B~R([~I+)
= - e -4~ * O(e -81~) ;
see (3.5).
Similar formulas hold f o r Z R ( A I - )
(A.2)
and for ZR(AIO),
except that
in the
latter case we have B~R(B]O) = - 2e -4~ * 0(e -8B) .
(A.3)
42
Now we introduce the contours.
T h e y will be defined on two d i f f e r e n t scales.
On the f i r s t scale, we just have the o r d i n a r y contours with e n e r g y large enough. The second, large, scale will depend on I~ and will be used to take advantage of the difference in the number of low e n e r g y excitations of the (0)- and the (+)-
and
( - ) - g r o u n d states. Let s be an excitation (Sect. Define its retouch, r e t ( s ) , that
Ib-al=l,
lowest
excitations
especially 111.3) of the ( * ) - g r o u n d
by setting r e t ( s ) ( a ) = * l
and r e t ( s ) ( a ) = s ( a )
energy
III,
of
s,
otherwise. but
not
state.
if s(a)=0 and s ( b ) = * l for b such
Thus,
in
ret(s)
the excitations
we remove all the
of
higher o r d e r .
For
÷
example, if ssX R then ret(s)
is the ( * ) - g r o u n d
state. The retouch of excitations
of the o t h e r ground states is defined similarly. A configuration s is retouched if s=ret(s).
Let s be such a configuration.
boundary is the set of pairs where s(a)#s(b). configuration s is a pair ~'=([~'],s[~])
(the support of ~'),
where [~']
Its
A small scale contour ~; of a is a maximal
connected subset of the boundary of ret(s) and s[~,] is the restriction of s to [~']; this definition of [~'] is d i f f e r e n t from t h a t of Sect.
II, but closely related to it.
The e n e r g y of ~ is (A.4)
E(~) = ~ ( s ( a ) - s ( b ) ) = , where the sum is over pairs of n . n . ' s contained in [~'].
Note that since we defined
contours by starting with a retouched configuration the lowest e n e r g y excitations (s(a)=0,
s(b)=*t,
for la-bl=l)
appear in the restricted ensembles but do not give
rise to contours, as they would with the usual definitions. In p a r t i c u l a r , we have min~, E(~') : 6 ,
(A.5)
not 4, and it follows immediately from ( A . t ) The
large
scale contours
that E(~)->I~'I = number of bonds in [~r].
are defined as follows:
Let
L(13)=exp(513/2)
(the
reason f o r this choice will become clear l a t e r ) . Cover Z 2 with large boxes B(i) = B(O) * ½Li ,
i ~- Z = ,
where B(O) is a square of side L(B) centered at the o r i g i n ;
a regular
box
of a configuration
s if
SlB(i)
t
O
XR, B ( i ) ,
IB(i)i=exp(5B). and
it
B(i) is
is irregulor
otherwise. Thus we may distinguish between two types of i r r e g u l a r boxes B(i) of a configuration s: ÷
t y p e 1: SlB(i ) s XR, B(i) U X R , B ( i ) t y p e 2: The support of a small scale contour of s intersects B ( i ) .
43
A l a r g e scale c o n t o u r r is a connected family of i r r e g u l a r boxes.
I trll
= number of boxes in r ,
Thus,
in o u r case,
Irl
We set
= number of sites in r .
Irl is bounded from above and from below by a constant times
exp(SB) I I r l l Now we state t h r e e
Lemmas the
last two of which
are f a i r l y
Lemma 1. The p r o o f of the theorem follows also in a b y these lemmas, [ 2 3 ] ,
standard,
given
now s t a n d a r d way from
so that we shall c o n c e n t r a t e on the p r o o f of the new i n g r e d i e n t
(Lemma 1) and sketch only the rest of the p r o o f s . kemma 1. Assume t h a t 6 is l a r g e e n o u g h . T h e r e e x i s t s a c o n s t a n t c such t h a t , o f o r all f i n i t e A c Z 2, all b o u n d a r y c o n d i t i o n s s z XR,AC and all c o n t o u r s r c A,
PA(rJs) <- exp(-cl311rl
I)
,
w h e r e PA(-Js) is t h e Gibbs measure i n A w i t h b o u n d a r y c o n d i t i o n s s.
If f is a function on X M, M c Z 2, we i d e n t i f y it w i t h the c o r r e s p o n d i n g f u n c t i o n on X w h i c h depends on the r e s t r i c t i o n of the c o n f i g u r a t i o n to M o n l y , and we w r i t e f C(M).
For A contained in M and s~X we define
A(S) =
T. f ( s ' ) P A ( s ' i s )
,
w h e r e the sum is o v e r c o n f i g u r a t i o n s s' in A. Lemma 2.
o
For a n y f i n i t e s u b s e t M of the l a t t i c e , a n y f z C(M) and a n y s ~- XR,
the thermodynamic limit
0 = l i m A(S) A e x i s t s and does n o t d e p e n d on s z X oR.
Moreover,
IA(s) - < f > ° l
<-I I l l l® e x p ( - c l 3 d ( M , Z Z \ A ) ) ,
where
positive
c is
large-scale
a
units:
constant
and
d(
d(A,B)=L(13)-linf{la-bl:
E u c l i d e a n d i s t a n c e of a and b, f o r example.
,
)
is
acA,
the
b=B}
distance
,
between
where
la-bl
sets
in
is
the
44
For any configuration s ~ X and f i n i t e subset A of the lattice consider the set of the large-scale contours of s which
intersect both A and its complement.
Let
rA(S) be the union of intersections of these contours with A - an analogon of the "open contours" of the IM of [23]. Lemma 3 [23].
There are
two positive constants, c and c', such that
PA( I I r A I l > cl~AI is) -< e x p ( - c ' ~ l a A I ) f o r ony f i n i t e A, s s X, and B Iorge enough.
Given the lemmas, we can prove the theorem: As in [23], we have to show that for any f i n i t e subset M of the lattice, a n y f¢C(M), and any s~X, l i m l A l - 1 ]~ <~(a)f>A(S ) = o , A a
(A.6)
where ~(a) is the translation by a (acting on functions) and the sum is over all a such t h a t M + a c A. Now, conditioning on rA:
<.>A(S) = ~: PA(rAIs)<.>(rA)
-
By Lemma 3 we may assume that r A covers a small p a r t of A; indeed, IrAI <- O(expS•)l I r A I I <- O(expSB)laAI , which i s much smaller than
IAI,
f o r IAI large.
Decompose the complement of r A in A into connected components: For most a of the LHS of ( A . 6 ) , definition of contours, restricted
ensemble
the configuration on the b o u n d a r y of A(i)
X oR.
A\rA=UiA(i).
$(a)f will be in C ( A ( i ) ) for some i.
Thus,
for
most
a,
<$(a)f>([' A)
But by our
belongs to the
will
be
equal
to
<$(a)f>A(i)(s) for some A(i) and some s~X~. Now we can use Lemma 2, which says that <$(a)f>A(i)(s) is almost equal to o provided a*A° is far from the b o u n d a r y of A(i).
But
again, since
IrAI
is much smaller than
covered by the A(i)'s with IA(i)I_>IAI l - s ,
IAI
the fraction
of A that is
z=O, goes to 1 as A-~ ®. So we can assume
A(i) to be large and to have a small b o u n d a r y (since the latter is contained in rA); thus,
f o r most a, we are far from the b o u n d a r y of A(i)
and this concludes the
p roof. Now, given Lemma 1, it is rather easy to prove Lemmas 2 and 3. we
observe
that
in
(O)-restricted-ensemble.
the
complement
However,
for
this
of
the
contours,
ensemble
we
have
For Lemma 2, we a
have
the
convergent
45
Iow-fugacity expansion. Now, combine this expansion with a (large-scale) contour expansion.
Convergence of the combined expansion then
follows from
Lemma 1,
which gives a suitable damping factor f o r the contours, and the following remarks. When we
sum over all contours
containing a given
contours having a given (large scale) size I l r l l = n
box,
the
number of
grows like c n where c does not
depend on L([}). - When we consider the combined expansion, we have objects defined on two d i f f e r e n t scales: the contours and the graphs of the low f u g a c i t y expansion in the restricted ensemble. However, one shows t h a t the sum over all graphs connected to a given box B in a contour is at most e x p ( O ( e x p - 4 ~ ) l a B I ) , where the boundary aB of B has a size L(ib)=exp(5B/2) which is much smaller than exp(4~) and, t h e r e f o r e , this c o n t r i b u t i o n is a small correction to the weight of the contour. Once
we have
a convergent
expansion for
the Gibbs
state with
boundary
conditions in the (O)-restricted-ensemble, Lemma 2 is easy to prove. To prove Lemma 3 we follow [23]: we f i r s t change b o u n d a r y conditions from the a r b i t r a r y s in the Lemma to
(O)-boundary-conditions.
Direct estimates on p a r t i t i o n
functions yield: PA(.I s) ~ exp(4BlaAI)PA(.IO) . Now from
Lemma 1 and simple combinatorics
(we connect together the
disconnected parts of r A t h r o u g h @A, which, for
I IrAIl_> claAI,
possibly
gives only a small
e r r o r ) we obtain PA( I I r A I l > c'laAI
I0) ~ e x p ( - c " B laAI) ,
from which Lemma 3 follows. Proof of Lemma 1.
We will use the following notation. We w r i t e Z(AI o) for a
p a r t i t i o n function in a subset AcZ 2, where we indicate a f t e r the vertical bar the ensemble defining the partition function and the boundary conditions. For example, PA(rls) = Z ( A I r , s ) / Z [ A I s ) where
in the
numerator we sum only o v e r the
(large-scale) contour. For a contour r its support is [r]
=
U B~F
B
,
(A.7)
, configurations for
which
r
is
a
46
and, f o r hcZ = its
boundary, ah, is t h e set of points of i t s complement which have
nearest neighbors in h. Now, we condition on the values of the spins in a [ r ] :
PA(rls) = ~-, P A ( r l s , s ' ) P A [ r , s ' l s ]
,
(A.8)
s ,
o
where t h e sum runs o v e r all StXR,a[1` ] ,
since,
b y d e f i n i t i o n of 1`, c o n f i g u r a t i o n s
in a[1`] belong to the r e s t r i c t e d ensemble X °
R,~[rl"
C o n d i t i o n i n g on the spins in a i r ] interaction has a range equal to one).
PA(1`ls,s ') = P[r](rls,s')
decouples t h e contour r from h\[1`]
(the
We can t h e r e f o r e w r i t e
= z([r]lr,s,s')/z([r]ls,s')
where s, s' define the b o u n d a r y conditions f o r
, [r]
(A.9)
(s outside A, s' inside h).
we shall p r o v e t h a t the RHS here is bounded b y e x p ( - c l 3 1 1 r l l ) ,
Now
with c i n d e p e n d e n t
of s and s'; b y ( A . 8 ) , this will o b v i o u s l y p r o v e the lemma. We can w r i t e
z([r]lr,s,s')
=
z
~' z ( [ r ] l r = , = , s , s ' )
F2 u
,
(A.10)
where t h e f i r s t summation is o v e r all possible families 1`= of t y p e - 2 boxes of £, and the second over families u of small scale contours in [1`] such t h a t f o r each box of £= t h e r e is a c o n t o u r of w w i t h s u p p o r t i n t e r s e c t i n g the box. Let [r]\[u]
= u Mi, where [u] = U [~'] , i ~ u
be the decomposition of [ r ] \ [ ~ ] small-scale c o n t o u r s ,
z([r]lr=,=,s,s ')
into connected components; w i t h a f i x e d family = of
z ( [ r ] 1 1 ` = , u , s , s ') is a sum o v e r c o n f i g u r a t i o n s belonging to a
r e s t r i c t e d ensemble in conditions s, s'.
(A.11)
each M i,
an ensemble determined b y ~ and
the b o u n d a r y
We may w r i t e : (A.12)
= e -BE(u) ]I Z R ( M i l s i) , i
where E(=)
= ]1
E(~)
,
and s i is the c o n f i g u r a t i o n on aMi d e t e r m i n e d b y s, d e f i n i t i o n of 1`2 ,
s' and u.
Moreover, b y t h e
47
[ r \ r =] c
U
~( i)#0
M.. l
[A.13)
We insert (A.12) into ( A . 1 0 ) ,
(]I Z R ( M i l s i ) ) / Z ( [ £ ] l s , s ' ) i
(A.10) into ( A . 9 ) and estimate the ratio
_< (]I Z R ( M i l s i ) ) / Z R ( [ t ] l s , s ' ) i
(A.14)
,
where in the denominator we used the obvious lower bound: z([r]ls,s')
-> ZR([r]ls,s')
.
We can now use the convergent Iow-fugacity expansion to estimate the ratio of the p a r t i t i o n functions of the restricted ensembles in ( A . 1 4 ) . Using a simple modification of the Proposition of Sec. II, ZR(AIs)
we can w r i t e Io9
for the boundary conditions s belong to the ~-restricted ensemble, as a
sum of a volume term and a boundary term: (A. 15)
log ZR(AIs) = - IAIBSR(BIc) * A~(A,B,s) , where ~R(BIs) is as at the beginning of this A p p e n d i x . The boundary term is AE(A,B,s) =
X
¢'(Ols)exp-4BlOl ,
(A.16)
where the sum is over all multiplicity functions ~ defined on the "particles" of the "gas" of elementary excitations of the restricted
ensemble, ¢
ak#~i means that the
s u p p o r t of one of the "particles" of ~ intersects aA, and Iol is the total number of " p a r t i c l e s " (counting multiplicities) in the support of ~. Finally, ¢'(Ols) is a sum of ~T(o)
(see Sect. 2) which depends o n l y on s t h r o u g h
its restriction to @~ - the
b o u n d a r y of the support of ~. The estimate (2.11) holds if we replace in it ~T(o) by ¢'(Ols). Thus we have: sup ~ I ¢ ' ( O I s ) l e - 4 B I l l ~ 0(e -4B) , X O:XssuppO
(A.17)
where the sum runs over all O whose s u p p o r t contains a f i x e d excitation X. We w r i t e the volume terms of (A.14) as: expB(; IMiI~R(BI~(i))-ItI~R(BI0]) s exp(-(e-4B*0(e-8B)) X i i:s(i)#0 exp(-½e-4BIrll) ~ exp[-ce-BIIPII) where r = =
r\r =
,
IMil)
(A.18)
, and where to obtain the f i r s t e q u a l i t y we used (A.2) and ( A . 3 ) ,
and the last one ( A . t l ) ;
c is a B-independent constant.
48 Now, in (A.14) we have also to estimate the b o u n d a r y terms. The family of the boundaries aMi consists of two subfamilies: one contained in [~] (coming from the small scale contours), same boundary (A.14).
and another one contained in a[£],
conditions,
s and
s',
in the
Since these boundary conditions
numerator
are the
on which we have the and the
denominator
same, the contributions
boundary term (A.16) from this part of the boundary,
of
to the
cancel each other in Xi log
ZR(MiIs i) and in log ZR([r]ls,s'). On the other hand, for the f i r s t part of the b o u n d a r y we obtain, using ( A . 1 7 ) , a term bounded in absolute value by c'e-4Bl~l
,
(A. 19)
where c' is independent of B, and
I=1=1[=]1.
Hence, we obtain f i n a l l y that (A.14)
is bounded by exp ( - c e B I I r Z l l
+ c'e-4~l=l)
.
By E(~)ZI~I, this inserted in (A.8) yields z([r] Ir=,u,s,s')/zR([r]
Is,s') <-- exp(-B'E(u)-ceBI Ir=l I) ,
(A.20)
where B' = B'(13) -= B - c'e -4B ,
(A.21)
with c' from ( A . 1 9 ) ; for B large, B' differs little from B. Now, inserting (A.20) into (A.10), and (A.10) into ( A . 9 ) , gives: Pk(rls,s') ~ exp(-ceBIIr=ll)XZexp-B'E(=) The
sum over
the families u of
(A.22)
. small-scale
contours
is
restricted
by
the
condition that, for each B~F = there exists at least one contour ~ u with [l~]t~B#~. Therefore, X2 exp (-B'E(u)) --u
][ B~r =
( X (I/m!) m_>l
XB ~...~ ~
<_. ]I ( ~ (1/m!)(I: B exp(-#'E(~')) m) , BsF = m_>l
exp(-B'X E(~'j)) j
(A.23)
where the superscript B indicates summation over contours ~" with [ ~ ' ] ~ B #~1.
49 Now, since E(~')>I~" I and, since E(~')>6 (see ( A . 5 ) ) , the usual Peierls' argument gives for B large XB exp (-B'E(~')) <-- c ' l B I exp(-6B') = c'e -½B , where we use IBl=expSB and (A.17);
(A.24)
c' does not depend on B.
(A.24)
inserted
into (A.23) gives for (A.22) the bound:
(exp(c'e-½B)-l) I Ir=ll <_ (c"e-½B) I Ir=ll
,
(A.25)
since e x p ( y ) - l = 0 ( y ) as y~0. Now, inserting this last estimate into (A.18) and going back to (A.8) concludes the proof.
REFERENCES 1. Aizenman, M. and Lieb E. H.: J. Stat. Phys. 24, 279 (1981) 2. Baxter, R. J. : J. Phys. C6, L 445 (1983) 3. Binder, K.,
Lebowitz, J.
L.,
Phani, M. K. and Kalos, M. H.:
Acta Metall.
29, 1655 (1981) 4. Blume, M.: Phys. Rev. 141, 517 (1966) 5. Bricmont, J . , El Mellouki, A. and Frohlich, J . : "Random Surfaces in Statistical Mechanics: Roughening, Rounding, Wetting . . . " . 6. Bricmont, J . ,
Preprint
Fontaine, J. R. and Lebowitz, J. L.:
J. Stat.
Phys. 29,
193
(1982) 7. Bricmont, J . ,
Kuroda, K. and Lebowitz, J. L.: J. Stat. Phys. 33, 59 (1983)
8. Bricmont, J . ,
Kuroda~ K. and Lebowitz, J. L.: "First Order Phase Transitions
'in Lattice and Continuous Systems: Extension of the Pirogov-Sinai Theory", to appear in Commun. Math. Phys.. 9. Bricmont, J . ,
Lebowitz, J. L. and Pfister. C. E.: Commun. Math. Phys.
78,
117 (1980) 10. Bricmont,
J.
and Slawny, J.:
"Phase Transitions in Systems with a Finite
Number of Dominant Ground States", in preparation. 11. Capel, H. W.: Physica 32, 966 (1966) 12. Coppersmith, S. N.: "The Low Temperature Phase of Stacked T r i a n g u l a r Ising Antiferromagnet", preprint, Brookhaven National Laboratory 13. Dinaburg, E. I. and Sinai, Ya. G.: Commun. Math. Phys. 98, 119 (1985) 14. Dinaburg, E. I. and Sinai, Ya. G.: work on the Potts Model, in preparation 15. Dobrushin, R. L.: Func. Anal. and Appl. 2, 31, 44 (1968) 16. Dobrushin, R. L.: Theory Prob. and Appl. 17, 582 (1972)
50 17. Dobrushin,
R. L.,
Kolafa, J. and Shlosman S. B.:
"Phase Diagram of the
Two-dimensional Ising Antiferromagnet (Computer-assisted Proof)", preprint 18. Dobrushin,
R.
L. and Zahradnik, M.:
"Phase Diagrams for the Continuous
Spin Models. Extension of Pirogov-Sinai Theory", preprint 19. Domb, C.: Adv. Phys. 9, 149 (1960) 20. Fisher, M. E. and Selke, W.: Phil. Trans. Roy. Soc. Lond. A302, 1 (1981) 21. Frohtich, J . ,
Israel, R.
B.,
Lieb, E. H. and Simon,
B.:
Commun. Math.
Phys. 62, 1 (1978); J. Stat. Phys. 22, 297 (1980) 22. Frohlich, J.,
Simon,
B.,
and Spencer,
T.:
Commun.
Math. Phys. 50, 77
(1976) 23. Gallavotti, G., Martin-Lof, A. and Miracle-Sole, S.: and Methematical Problems" (A. Lenard, ed.), pp.
In "Statistical Mechanics 162-204, Springer-Verlag,
Berlin, 1973 24. Gerzik, V. M. : Izv. Akad. Nauk SSSR Set. Mat. 40, 448 (1976) 25. Ginsparg, P., Goldschmidt, Y. Y. and Zuber, J . - B . :
Nucl. Phys.
B170, 409
(1980) 26. Glimm, J. and Jaffe,
A.:
"Quantum Physics",
Springer-Verlag,
New York,
1981 27. Glimm, J., Jaffe, A. and Spencer, T . : Commun. Math. Phys.
45, 203 (1975);
Ann. Phys. 101, 610 (1976); ibid., 631 28. Goldschmidt, Y. Y . : Phys. Rev. B24, 1374 (1981) 29. Gopfert, N. and Mack, G.: Commun. Math. Phys. 82, 545 (1982) 30. Griffiths, R. B.:
In "Statistical Mechanics and Field Theory" (C. De Witt and
R. Stora, eds.), Gordon and Breach, New York, 1971 31. Griffiths, R. B.: In "Phase Transitions and Critical Phenomena", Vol. 1, (C. Domb and M. S. Green, eds.), Academic Press, London and New York, 1972 32. Holsztynski, W. and Slawny, J . : Commun, Math. Phys. 66, 147 (1979) 33. Imbrie, J. Z.:
Commun. Math. Phys. 82, 261 (1981); Commun.
Math Phys.
83, 305 (1982) 34. Israel,
R. B.:
Apendix
B to "Convexity in the theory of Lattice Gases",
Princeton University Press, Princeton, New Jersey, 1979 35. Kotecky, R. and Shlosman, S. B.: Commun. Math. Phys. 83, 493 (1982) 36. Lanford,
O.
E.:
In
"Statistical Mechanics and Mathematical Problems" (A.
Lenard, ed.), pp. 1-113, Springer-Verlag, Berlin, 1973 37. Lebowitz, J. L. and Lieb, E. H.: Phys. Lett. 32A, 2 (1972) 38. Lebowitz, J. L. and Martin-Lof, A.: Commun. Math. Phys. 25, 278 (1972) 39. Lebowitz, J. L. and Penrose, O.: J. Math. Phys. 7, 98 (1966) 40. Minlos, R. A. and Sinai, Ya. G.: Tr. Mosk. Mat. Obshch. 17, 213 (1967); Mat. Sb. 73, 375 (1967); Tr. Mosk. Mat. Obshch. 19, 113 (1968) 41. Peierls, R.: Proc. Cambridge Philos. Soc. 32, 477 (1936)
51 42. Pirogov, S. A. and Sinai, Ya. G.: Teor. Mat. Fiz. 25, 358 (1975); Teor. Mat. Fiz. 26, 61 (1976) 43. Ruelle, D.:
"Statistical Mechanics:
Rigorous
Results",
Benjamin,
New York,
1969 44. Rueile, D.: Phys. Rev. Lett. 27, 1040 (1971) 45. Ruelle, D.: Commun. Math. Phys. 53, 195 (1977) 46. Seiler, Theory
E.:
"Gauge
and
Theories as
Statistical
a
Problem in
Mechanics",
Lecture
Constructive Quantum Notes
in
Physics,
Field
vol.
159,
Springer-Verlag, Berlin-Heidelberg-New Y o r k , 1982 47. Sinai, Ya.
G.:
"Theory of Phase Transitions:
Rigorous
Results",
Pergamon
Press, Oxford, 1982 48. Slawny, J . :
"Low Temperature Properties of Classical Lattice Systems:
Phase
Transitions and Phase Diagrams", to be published in Phase Transitions and C. Domb and J.
Critical Phenomena,
L.
Lebowitz, eds., vol.
10, Academic
Press, London, 1985. 49. Thompson,
C. J.:
"Mathematical Statistical Mechanics", Princeton
University
Press, Princeton, N. J . , 1979. 50. Villain, J. and Bak, P.: J. Physique 42, 657 (1981). 51. Villain, J . , Bidaux, R., Carton, J. P. and Conte, R.:
J. Physique 41, 1263
(1980). 52. Widom, B. and Rowlison, J. S.: J. Chem. Phys. 52, 1670 (1970) 53. Wu, F. J . : Rev. Mod. Phys. 54, 235 (1982) 54. Zahradnik, M.: Commun. Math. Phys. 93, 559 (1984) 55. Bortz, A. and Griffiths, R. B.: Commun. Math. Phys. 26, 102 (1972) 56. Gallavotti, G.: Commun. Math. Phys. 22, 103 (1972) 57. Basuev, A. G.: Theor. Math. Phys. 58, 171 (1984) 58. Cassandro, M. and Da Fano, A . : Commun. Math. Phys. 36, 277 (1974) 59. Galeb,
F.
F.:
Tr.
Mosk.
Mat.
Obshch.
44,
(1982) Engl.
Transl.:
Trans.
Moscow Math. Soc. 2, 111 (1983). 60. Kotecky, R. and Preiss, D.: in preparation 61. van der Waals volume: Physica 73 (1974) 62. Baxter,
R. J . ,
Enting,
I.
G. and Tsang,
S.
K.:
J.
Stat.
Phys.
22, 465
M. K. and Styer,
D.
F.:
J.
Stat.
Phys. 38, 413
(1980) 63. Lebowitz, J.
L.,
Phani,
(1985) 64. Dobrushin,
R.
L.
and
Shlosman,
S.
B.:
"The
Invariance of Gibbs States at Low Ternaperatures",
Problem
of
Translation
Soy. Sc. Rev. Ser.
C,
vol. 5 (1985) 65. Ruelle,
D. :
"Thermodynamic
Massachusetts, 1979
Formalism",
Addison-Wesley,
Reading,
Low T e m p e r a t u r e Continuous Spin Gibbs States on a L a t t i c e an d the I n t e r f a c e s between Them - a P i r o g o v S i n a i Type Approach
Milo~,
Zahradntk
F a c u l t y of Mathematics a n d P h y s i c s , Prague,
Charles University,
Sokolovsk~ 83
186 OO C z e c h o s l o v a k i a
§
0
Introduction.
The Main Result and the Basic Example.
I. C O E X I S T E N C E O F PHASES
§
1
Basic c o n c e p t s . A b s t r a c t P i r o g o v S i n ai S e t t i n g .
§
2
Contour Models.
§
3
I n v e s t i g a t i o n of the contour models.
II. I N T E R F A C E S
§
1
Geometrical s t r u c t u r e
of i n t e r f a c e s .
§
2
Wall Models and A g g r e g a t e Models.
54
§
0 .Introduction.
The M a i n R e s u l t a n d
The Pirogov Sinai theory, which so called Peierls
approach,
the Basic Example.
is one of the very general expressions
is often referred
should use only if all the "more common" In
this
namely
note, that
description
we
aim
this of
to contribute
theory
a
low
is really
temperature
to as
of the
theory
one
methods fail to work.
in favor a
a complicated
of the opposite
convenient
behaviour
and
of m a n y
point of view:
general
means
lattice spin
for the models
of
statistical physics. The
theory is not only a "method
of construction of the phase
diagram".
It gives, too, some rather precise information about the structure of coexisting phases. Even
in the uniqueness
by
another
the
region
methods of
a
region,
(e.g.
the theory gives some results hardly
about
coexistence
the behaviour
of
phases
of the "nonstable"
is the
more
obtained
phases)
characteristic
but
realm
of
applications of the theory. Here,
we
will
formulate
one
general
result,
illustrating
a possible
use
of the theory. The investigation was motivated by the example below.
MAIN
THEOREM
ASSUMPTIONS. 1) G e n e r a l
conditions.
We c o n s i d e r
a lattice
X
model with a configuration
space
:
2
and with a finite
range
hamiltonian
H(.~^ IX^')= Z
of t h e t y p e
~A(XA )
(0)
A/~ A # ~ , diam A £ 2bs ~-~A are translation invariant, bounded from below, continuous interac-
where tions on
X (A)
=
(~)~
,with values in ~ tl ~4 (~)}
,and
2f] is the "inte-
raction range". 2) E x i s t e n c e We a s s u m e noted by We a l s o states ~( 4"
of
of g r o u n d that
there
X+ I .Xassume ~'~ =
converging to
that
states. are there
,denoted ~+_
two t r a n s l a t i o n
invariant
ground
states
of
~-~
,de-
in the following. by
are
two " h o r i z o n t a l l y
3(+
for no"vertical
(~(*)% (resp.{X')~)
,
~( ~
s h i f t " of
if
translation
in the following,
invariant"
~(_+ ; b o t h {.~_+)~
tit -~ Oo
ground
such that
(reap. t ~
, (X~-)~
-~-
oo
).
55
Note. H o r i z o n t a l ( r e a p . v e r t i c a l ) s h i f t is a s h i f t $
=
( $4 ..... S~- )
and
3) Second differentiabilit~ of all ~ A
~
{ ~: ~
$4 =
S.~ = 0 (reap.
~ +$ ~
S~
where
.... ~%,.~=
0 ).
the positive definiteness of the Taylor
approximations. We
assume that the interactions
~A
~.X-)A, ('.X ~ ) A ' ( ~'+ ) A " We such that for any ~ ~ X such t h a t
I ~ ~ ~"
= ~--
are twice differentiable at any also assume
(~+)A,
that there is some
C > 0
J ~ / f ~ ~K
is sufficiently small, the inequality
H(x~+4~)-
H(X+)
>.. C I q / |
holds, and analogous inequalities hold also for ~ -
z
(1)
, )~_+ , ~_+ .
4) A Peierls (Gertzik, Pirosov, Sinai) type condition. We assume that for each d" ~
any ~
0 there is some
× satisfying the condition ~]~ _---0
of ~ ~
C
:
C(dD>o
such that for
for all except of a finite number
7_~,
(2) where
8(.t
:
and analogously for
~-
,
.~ _+
,
~_+
Note. We do not assume the smallness of ~ / must
be
supplemented
by
some
entropy
there. Actually,
bounds
the condition (2)
for the "energy
shells" of the
model, guaranteeing the existence of a Gibbs state at least for low temperatures -see
[ 1] Until now, the interactions
~A
were fixed. We assume further that there
is a two parameter family of hamiltonians of the type
=
$ ;~,;~'
o,o
A " diem A ~ We assume t h a t the f u n c t i o n s ~!
(having
analytical
d e p e n d i n g on
2( A
( ~, , ~ ' ,
) (see
XA
J'A ~I~# (.XA~
a r e r e a l a n a l y t i c a l both in
and not
) a n d a l s o " s u f f i c i e n t l y r e g u l a r " as f u n c t i o n s of the t r i p l e
[ I ]
).
5) D e s e n e r a c y removin~ c o n d i t i o n . We d e f i n e the q u a n t i t i e s
~c~,~,
~
e x t e n s i o n s to some n e i g h b o r h o o d of zero in ~ X ~
:
( f o r m a l l y - i t is e a s y to g i v e a p r e c i s e d e f i n i t i o n )
I 7-~-~I "~ ( H ~'~'(x+-~ - H ~ " : " ( ~ + - ) )
56
Note.
.SJ~k,~1 is the difference in the mean energies of
)(+ , ~("
whereas
~k¢~' is the difference in the mean interface energies of )(_+, ~_+. We assume that the mapping
{
}
( ;~ . ~' ) ~ ( 2,~,~,, . ~ , ~ , )
(~)
ha s an i n v e r t i b l e d i f f e r e n t i a l in (O,O).
RESULT. Then, under all these assumptions, the following is true: There is an analytical function
f,p . , ~ > ( x ( T ) , ~ , ' ( ~ ' ) ) ' } :
(o,%)
~
~2.
~s)
d e f i n e d for some ( s m a l l ) ' ~o > 0 a n d such t h a t :
I (for ~ >I 2) There are two translation invariant Gibbs states P ; , P ~ on X corresponding to the hamiltonian H ~CI')'~'(I") and the temperature • such that P ~ is a perturbation of the gaussian field ~ ' ~ arising if each ~ A is replaced by its second order Taylor expansion around (for
V" >/ 3)
There
are
two
on
corresponding
II X
T
horizontally
such that e.g.
if each
~ A
translation
to the h a m i l t o n i a n
P~
~(~ . (Analogously for P ~ invariant
Gibbs
states
). P~-~ , P ~
H ~ C ' P ) I ~ * C ' r ) , an d the temperature-I--
is a perturbation of the gaussian f i e l d d ~ ±
is replaced
by
its second
order
Taylor
arising
expansion
aroud
(.x_+~ A . Notes. 1) A g e n e r a l i z a t i o n to the case of p e r i o d i c a l g r o u n d states is p o s s i b l e , 2)
In
the
(~, ( ~ ) , 3)
case
I
there
P~
in
P~
even a one d i m e n s i o n a l m a n i f o l d of
all
possible
~' ( ' r ' ) ) .
Both the Gibbs
to
is
the
fields
"vertical"
P~
direction
in the d i r e c t i o n
,
PG
converge
("exponentially
r ~%r-~ oO and c o r r e s p o n d i n g l y ,
~V" ") - ~
fast")
to
they c o n v e r g e
"
References. The s t at em en t I is p r o v e n in the p a p e r [ 1 ] .)(+ The ~2 ]
in
,
statement
~"1"
II
. A short,
will
~k'
,
be
discrete
a
subject
(there ar e no
in of a
[1 3 paper
). which
is
now in p r e p a r a t i o n
spin v e r s i o n of the method used in I I is e x p l a i n e d
[3]. A standard reference on the Pirogov Sinai Theory is [ 4 3 " Some comple-
ments to this theory are given in [ 5 ] (for the discrete spin case). The line of s t u d y used in II was o r i g i n a t e d by the p a p e r [ 6 ~
.
The other relevant references are [ 7 ] Cwhich contain some results comparable to [ 1 ] ) an d a l s o the c o n t r i b u t i o n s of ]. Bricmont an d ]. Slawny in t h i s volume.
See also [i~ for further references.
57
Technically,
the proof
tecl%niques
from
for
of
Main
perturbed
Theorem
gaussian
is
fields.
based
on
some
Some of t h e s e
cluster
techniques
expansion we l e a r n e d
[ 8 "] B a s i c example
Z" Take
~
= (~)
. Consider+ a h a m i l t o n i a n
H (x)
=
+ nearest
neighbors
where ~(~($,~I) = I J ( $ and where ("double
~
~ ÷ I ~ + small p e r t u r b a t i o n
is given
as follows: ,'+
well potential")
Figure
1
,~o ~intuitively, verns
the
it is understood behaviour
a properly
"balanced"
- one centered If
we
At phases
has
The c h a n g e of i n t e r f a c e s
balanced
~4
change
(assuming
that of
of ~,
"one
- entropy
invariant
one e x p e c t s
height
the hill
of
the
in
two t r a n s l a t i o n
"hill
6"
may effect,
principle
this
go-
example.
invariant
For
phases
•
above
~ o
small effect on the coexistence
remains
"
(by
varying
of t h e p r e c e d i n g
"high").
however,
the phases
centered
may
expect
even
fighting"
phases
and the other one around
the
AI
the "energy
translation
a comparatively
between
9~a
that
the
value
around
further ) this
of
~+
very
around
two
substantially, ~ ÷ ,
equally
good
6-
the structure . For a "properly
possibilities:
either
"tunelling" directly from one well to the other or "going over the hill". Graphically, one expects two competing kinds of interfaces: ++++++++++++++++++++ ++++++++++++++++++++
or ++++++++++++++++++++ ++++++++++++++++++++
00000000000000000000 Hence t h e n o t i o n s of J ( _ + , ~ ) .
either
58
Note.
One
can
easily imagine
more
wells and
also more competing interfaces.
In fact, if one studies the coexistence of a maximal possible number of phases (and interfaces) then no additional difficulties arise in the general case. Some modifications are needed, however, interface diagram. a
combination
of
a
This was technique
if one aims to construct the full phase
not studied in detail yet but it seems that presented
there
with
the
technique
of [ 5 ]
will suffice to describe this more general situation. x
x
x
In the rest of this contribution we will sketch the main notions and ideas used in the proof of Main
Theorem
(mostly of its I part; the subject of inter-
faces being only touched). I.
COEXISTENCE
OF PHASES
§ I. Basic concepts of the Pirogov Sinai Theory. Abstract P.S. setting. We will present a generalized form of the theory, in the spirit of [
1 ] . See
also the contributions of ]. Bricmont and ]. Slawny in this volume. Notations. We consider the situation of Main Theorem if not specified otherwise. (Sometimes we will sketch a more general approach,
or illustrate our conside-
rations on the Example above). Traditionally, the theory is considered as a low temperature one. It is, nevertheless clearer to explain it in the case
~
= 1
where ~' denotes the tempera-
ture. By this we m e a n that the rescaled quantities =
T
(0)
=e
are used (another scaling of ,X is needed in the "nongaussian" case), The
models studied
by
the
theory must have a structure
the f o l l o w i n g two complementary concepts: f i r s t ,
described by
there is a notion of
I) almost ground confisuration (ground state in the original P.S. approach - for the discrete Spin systems). A second notion is that of 2) energy barrier (composed of contours), separating the regions with an almost ground configuration. The usual choice is as follows. I) Write the constant ground states
~+
and
X"
as
Take some s u f f i c i e n t l y l a r g e Ot~ (to be specified l a t e r ) neighborhood of
and denote by
~-+
the
~ -+ . Take
7_ ~
X+
--
X-
:
(U-)
(7)
59 for the sets of almost g r o u n d c o n f i g u r a t i o n s . See F i g u r e 2. ( I n the d i s c r e t e case, we take (Clearly,
~-+
=
~ 1~
).
O~ must not be choosen too l a r g e .
Actually,
O~
is small before the
scaling (6)).
I
II
I
Figure 2
I
1 I
I I
%C
Oi +
2) The d e f i n i t i o n of a contour s t a r t s with a notion of a correct p o i n t : is
said
to be a + correct p o i n t of X E X
if ~5~ ~[+ for a l l $£ ~.V'such t h a t The set of a l l n o n c o r r e c t p o i n t s of In the
discrete
s p i n case,
of .K
to some f i n i t e component of
(more p r e c i s e l y ,
(+, ~ ,
~ ~
~_""
6") correct
I S - ~ I • MJ. .)(
B? (
w i l l be denoted by B(-~) (=
) ).
one then defines a contour simply as a r e s t r i c t i o n B C~()
(which is c a l l e d the s u p p o r t of a
contour). In the c o n t i n u o u s s p i n case, level".
such a d e f i n i t i o n works only at the " c o n c e p t i o n a l
I n o r d e r to p r e v e n t some complications otherwise a r i s i n g in the t e c h n i -
cal p a r t s
of the theory it i s ,
notion of a contour.
however,
r e a s o n a b l e to i n t r o d u c e a more r e f i n e d
The f o l l o w i n g modification is used i n
[ 1~
: (one c a n
skip t h i s a n d r e a d only the Note below). "Wrapped" c o n t o u r s . i) Take s o m e / ~ > 2 ]
such t h a t "a b o u n d a r y c o n d i t i o n from
('1.1-) A¢ ) is only " s I i g h t l y felt" o n , t ,
dist
(t,
(q/T)
Ac
A¢
) >.. 2 ~
(analogously, }
i n the
gaussian approximation. it)
Replace
B ~ ( ~ ) by ~
f i x i n g .l( on ~[-
outside of
~(,I),
~.~) . This is s t i l l not the best choice b e c a u s e by
the c o n f i g u r a t i o n .)( may "tempt to go out" of ~ +
B.¢.~(.~) .
(This is
h a p p e n with a n additional" i n t e r a c t i o n ~ + S + ~ l t _ S ~ 2 X , [ ~ ,
Take, therefore, a suitable ( l a r g e ) ~ <
6" , denote by
I$1 = ~ ) . See F i g u r e 3. ~±
of a n d define g (-~) as the smallest set c o n t a i n i n g XIE~t for all ~EB{~(),dist(t,(g~X)) c )~ /~.
I @
~,~.--~.~
~*
resp.
not the case i n our Example b u t it may the ~ n e i g h b o r h o o d ~.,.~(~() 1;
mean valueof the ./conditioned field
/ouni ....., r y c o n d i t i o n (from ~
)
such t h a t
60
The contours
of
~ (~)
are then defined as restrictions
of ,~ to the connected components
.
Note. Most of t h e a r g u m e n t a t i o n
below w i l l u s e o n l y the f o l l o w i n g two p r o p e r t i e s
of c o n t o u r s : a) the s u p p o r t
of a c o n t o u r
b)
a
outside type,
of
contour,
+ or - ,
is a c o n n e c t e d any
almost
being prescribed
set
ground
configuration
can
live
(only
its
by t h e c o n t o u r ) .
An A b s t r a c t A p p r o a c h . It
is
sometimes u s e f u l
the n o t i o n s
of a l m o s t
to c o n s i d e r ground
an
"abstract
configuration
Pirogov Sinai
a n d of c o n t o u r s
situation"
where
are given as primi-
t i v e n o t i o n s at t h e v e r y d e f i n i t i o n of t h e m o d e l . A configuration
is t h e n a c o l l e c t i o n of t h r e e o b j e c t s :
i)
a f a m i l y of c o n t o u r s { P~ } ;
of
~..'WN
the
k~. s u p p P ~
prescription~"
1~_+ { [1~ }
a compatible one i.e.
there
is
to
live
whether
a nonconflicting in
X +
or
prescription
X-
).
t h e u n i o n of a l l t h e s e + c o m p o n e n t s .
× - C^') (=
of i t s
We d e n o t e
It r e m a i n s
ii) some ~ + ~ X +(]~+) where, usually, X+(A÷)=(q~+)A~rfor some ~ + ,
iii) some
s u c h t h a t for a n y c o m p o n e n t sign
by
(i.e.
A !
=
to s p e c i f y
and
^- )
Such a general approach will be really necessary when studying interfaces. Given such a triple (~ P~ } , ~ t ~ - ) we assume that its hamiltonian has the form X where
H
is a " g r o u n d
In t h e d i s c r e t e
energy"
spin case,
and
~
an especially
H (a)
is
a
"contour energy".
transparent
f o r m u l a c a n be o b t a i n e d :
= .+IA÷I+.61^-I
* 7_ X
(8')
w h e r e . ~ +{'} is the m e a n e n e r g y of t h e g r o u n d , s t a t e a ( + ( - ) The b a s i c 1)
Quick
"restricted Theorem,
assumptions
(usually,
an
fields"
X +
on
H
and
exponential ,
X-
~
are:
one)
decay
of
correlations
endowed w i t h t h e h a m i l t o n i a n
t h i s is a c o n s e q u e n c e of t h e c o n d i t i o n
in ~
both
. (In
the Main
(1)).
2) A P e i e r l s t y p e c o n d i t i o n ~(P)
~ Z I supp P
( w h i c h f o l l o w s from (2) - a f t e r 3) will
We w i l l occur.
situation of
assume
our
We e m p h a s i s e
which
coexisting
that
appears
phases.
outside
h a s to m o d i f y t h e m e t h o d ) .
,
some
'~
(9)
large
(6) - in t h a t c a s e ) .
hamiltonian
again
(If
I
that
(8)
is
such
that
i t i s n o t otlr t a s k
of t h e
point
"nonstable"
a
phase
there
of the
maximal
phases
are
transition
to d e s c r i b e possible
to be
the
number
expected,
one
61
§2. Contour models This
is the essential
approach
(than
construction
construction
that of
is very
[4
3
natural
of the theory.
~ )there.
(See
in the symmetric
We
present
[1 3 case
a more
general
for more details). (if there
The
is a symmetry
between the restricted fields, as well as between the contours separating them). In fact, the notion of a contour model was
first introduced in E 9 3
(which
is
the thorough study of the Ising model). Nevertheless
the
nonsymmetric
case
is the
more
important
and
characteristic
for the applications of the theory. Our emphasis will be on it, too. Realizations of the contour ensemble In
addition
to the notion of a "compatible
also the notion of an "admissible Pi , supp
P~, )
type +(or-)
~
family
of contours"
we
introduce
family of contours" i.e. such that dist (supp
2 for any
~
~ i l. We
say that a contour /'I is of the
if the external component of (supp [' )¢ is signed + (or-). Admissib-
le famillies of contours of one type only will be considered. In the symmetric
case,
flipping some contours) This has
no
analogy
notion of an
an admissible family of contours corresponds
(by
to some compatible family (uniquely up to +/- symmetry). in
the
"admissible
nonsymmetric
family
case
of external
but,
still, in any
case
the
contours of the type +" coincides
with the notion of a "compatible family of external contours of some configuration of the type
+". (A configuration
the region "surrounded by" The
central
idea
behaviour
of the
system
model
studying
by
B (.~)
of the
is of the type + if .X46 ~ * ).
Pirogov
of external
it as
Sinai
contours
a behaviour
realization
of a + contour
ensemble
theory
is then
to describe
of configurations
of the system
realizations of a suitable contour ensemble, A
"outside" of the
of the "real"
of external
contours
of
defined as follows:
is a pair
of two objects (compare
the
notion of a configuration in (8)): i) an almost ground configuration
E
X +
(on the whole space)
7_v"
=
(%(+)
it) a family of admissible contours i ~ }
,
of the type
+.
Analogously i s defined the - contour ensemble. Graphically,
a
behaviour
of a real configuration
is one contour ~ of the type + and one contour
~
(see Figure
regime
•
~
|
.
~ J
~
r
4a; there
of the type - inside ~ )
e
(
4
a)
62
is
modelled
by
a
behaviour
of
a
realization
the same v a l u e s on the " e x t e r i o r " of
P
P
containing
and
having
also
: F i g u r e ~ (b)
[~
Note. to
r-.~ ,2/
The a p p e a r a n c e
do w i t h
the
of a n o t h e r
"real"
p~me
P
as b e f o r e
+. c o n t o u r s
behaviour
inside
P
P
, -
P
inside
there
is
P
has nothing
no " f l i p p i n g "
in the
nonsymmetric case. Assuming before,
it
contour
of an
lent"
is
now a
that
natural
the idea
admissible
"ground to t r y
hamiltonian" to a s s i g n
system { PX ~
(in the s e n s e a b o v e )
~-~
remains
a proper
such t h a t
contour
the
same
weight
a model " e x t e r n a l l y
as
to a n y equiva-
to the r e a l one w o u l d be o b t a i n e d .
This is d e v e l o p e d in more d e t a i l
in the n e x t s e c t i o n ,
where the basic
framework
of the t h e o r y is p r e s e n t e d . D e f i n i t i o n of v a r i o u s Equivalence
partition
functions.
The C o n t o u r W e i g h t .
of e n s e m b l e s .
We n e e d some g e o m e t r i c a l n o t i o n s d e s c r i b i n g
a structure
of c o n t o u r s :
Figure 5
Write ~ A = Given a contour and put int (This ~_~
{,~ ~ A denote by
V{~) = ~
rI
(= the
is =
~
(int
:dist ( ~: , A t ~ = supp P
L] i n t P
union
a somewhat
a ~'I : ~4" ~
A
P
of a l l
where connected
unusual,
but
IT' )¢"
'
U ~_ ~
•
) ~ ~ Write ~ c
=
ext
P
U int
(for a c o n t o u r of t h e t y p e +) i n t components
convenient
~¢"
w h i c h a r e s i g n e d - ).
n o t i o n of an e x t e r n a l i t y ) .
~ T ~-1 = ~ V ( / " )
. Finally we write
of
~t P
we h a v e
the
: ( P ) ~_+ ~
P
of
~(A '
" a finite
exp(" H(~^I'~aA ¢)) ~ A
the "diluted" partition function; the integral being taken over all that for any contour
Writing
decomposition
Consider the + contour ensemble, for the definiteness. Given and some boundary condition ,X~A= E ('~+)~A¢ denote by
e(Az O(~Ac) = f
P
P
(lo) J~
such
63
] , Ac
dist (Vfp
) >I 2 are defined by extending
(Contours of ~ ^
(14) ~(A "naturally" by some
.XA¢
We w i l l also need a more r e s t r i c t e d notion of
~(~,
X(]Ac) =
~,
exp{- H ( X ^
f
#..)(A¢ )J a . ~
(ll)
where the "forbidden zone" ~ imposes another restriction dist (V(P), m ) >/ 2 ! to (IO).
Given a +contour p
write
(lf)
= V([ "e) ~ ¢~at~ and denote
V ~ = V a (~)
by
Or')
7_
] exp
(-H(Xv, ID+r, )
int n
(12)
(the "crystallic" partition function) the i n t e g r a l b e i n g ta k e n o v e r a l l p o s s i b l e of
"~
int
such t h a t
, extended "naturally to the whole
L; X int F
is a contour
~'~".
The following notion is a crucial one: Define a contour weight F ([~) by the formula F (~)
=
In ~ _ ( V ~ ~n! a+P ) - In ~
(P )
.(13)
Note. This can be interpreted as the "work needed to install the contour". In ~ ( ~ ) . (This is no the symmetric case, this is almost the same quantity as -longer true in the nonsymmetric situation). Finally define a hamiltonian assigned to a realization ~/^ = ( /~/2, /% O'' { ~ ] ) by An analogy of (iO) in the contour ensemble is
- / / . . ( A I F I , X ~ c ) = ] exp{-- ~ { ~ a J ~ A c) ~:~ /~/A (15) (diluted partition function of the contour ensemble) where the integral is taken over all realizations ~ A = ( X/~ , { /~¢" } ) satisfying (iOt) for any Theorem. Under (13) we have, for each finite A
"~_..(A, both for . ~ A ¢
X~^c ~
F , ")(i~Ac )
, resp
the system f
proof of the
the " l e v e l " of
(16)
.
the co n t o u r en s em b l e . The
= "2z.. ('A, .X~AC )
~ (q/+)~A©and (correspondingly, in the - contour ensemble) for
(#/-) 0 ^ ~
Note. Then the distribution of "~A ~
x^
,the relation
A
}
k]V~[~]and of
the external contours (of
) Is r e a l l y the same both in the r e a l and
This is the e q u i v a l e n c e of ensembles
theorem
is
rather
straighforward,
using
(used
in [
13
).
some i n d u c t i o n o v e r
(which is the maximal p o s s i b l e c a r d i n a l i t y of a " c o n c e n t r i c " ,
64
admissible
A
system of contours in
and using the very definition
each stage of the induction. See [ i ] ,
(13) at
theorem 4.2.6.
A few words about what picture is to be expected, looking on the external contours ~A¢
of the
system.
6 ( ~ + ) ~A¢.
Take
some
finite
]k
and
some
boundary
condition
There are two characteristic kinds of a behaviour of a typi-
cal configuration in
A
(see Fig. 6): either all the contours are "small" and
"rare" or the systems jumps - as quickly as possible - to the "stable" regime, thus forming a large contour near the boundary of a given set /~ . The latter behaviour volumes +
is characteristic for a nonstable phase
A . +
+
+
+
(of the type + ), for large
•
Figure 6 + +
+
0
4-
+
_ 4-
We
in
..~
+
are interested in the first behaviour
of the contour fact
+
and we will study it in the language
model. We will see later that the stability (of both phases)
equivalent
to the validity of the following condition
(for both
is
types
of a contour): Peierls type condition for the contour weight. For any contour
[~ ,
(17)
§ 3. I n v e s t i g a t i o n
of the contour models.
~(XA): H(~ A I~)
Using the notations
we
can
write
(13) as
C l e a r l y , what is needed is a precise estimate of the d i l u t e d p a r t i t i o n s functions
('A~M, ~aA ~ ?(= ~.(A~/~F,XaA,). Approximating model. In
the
arising
situation
of Main
Theorem,
if the second order
resp
X;
this is the corresponding
Taylor expansion of the potentials
) is taken in tead
gaussian ~ --
model
(around
A
(In the discrete spin case, the approximating model is a trivial one, the "constant model"
{ .)(# }
resp. ~ ~ - }
. Another types of approximating models m a y
be needed in other situations - see the contributions of ]. Bricmont and ].Slawny). Define the approximating partition function approx
approx all X A
(19)
65 (with
H
][
approx
rel
just described). Put
.
-I )
('AI M IF , xaAc ) _ ~(~lNiFlw(9/k¢) ( --~ approx
Estimates of
.(20).
-~ rel
The basic technical problem of the theory is an
estimate of the boundary terms
of the partition functions (20). Usually, such an estimate is obtained by some expansion techniques (see [ 1 ]
, § 2 where a selfcontained exposition of the tech-
niques can be found). We summarize the main result: Note.
Actually,
a mere simply
this
problem
fact of a phase by
using
basis of a
is not so central in the symmetric
transition can
the usual
"symmetric"
be
proven
without
Peierls argument.
"nonsymmetric Peierls argument"
case,
where
referring to i t In fact, (17) is the
- which is formulated in the con-
text of contour models. (To obtain a more precise information about the structure of phases, the forthcoming estimates are useful even in the symmetric case!). Theorem. Assume that (17) holds. Take some finite A E
(~+)~AC
and some
,X a A c
(see Figure 3!). Then
In ~rel(^,M,F,~a^,)= 2~rel I^I
+ A ( A , ~ , F A a A C ) (211
where
IA I and where Proof.
16 Ac
u M I
,
(2.20
sma,,
&
/~/rel is the "relative free energy".
(See [ i ]
"polymer we
."
). In the discrete case, this is a consequence of some standard
model"
obtain,
techniques
rather,
an expression of
some -~ rel
of cluster expansion.
"polymer
more precisely
of the following type:
fA, } i where .~+(-)is
In the continuous spin case,
perturbation of a gaussian",
i
the (positive mass)
approximating
gaussian
field and where
the
v
functions f ~
where
satisfy a condition of the type
(~ ( A } is the l e n g h t of a s h o r t e s t t r e e on
Partition
functions
of the
type
(23)
are
studied
A in
Theorem 2.3.8
of [ 1
]
Some k e y s t e p s a n d k e y w o r d s of the proof: 1) e x p a n s i o n of
# A into p r o d u c t s of Wick p o l y n o m i a l s
2) w r i t i n g the e x p e c t a t i o n
of ~ ' # A . £ as a sum of p r o d u c t s of s e m i i n v a r i a n t s A2
3)
(= t r u n c a t e d e x p e c t a t i o n s ) of s u b f a m i l l i e s of { # A.E } e s t i m a t e of s e m i i n v a r i a n t s of Wick p o l y n o m i a l s , u s i n g expansions;
e s t a b l i s h i n g the " t r e e d e c a y " (in the s p i r i t of
some [8 ]
diagrammatic ).
66
4)
expression of
-~rel
as a polymer partition function of the type
7_ tel ,_
7--
]7
(25)
{Tz} where
/~ ~
, defined and
estimated
of the type
I A ,r,I
£
by means of I)-3) satisfy a condition
i'rl 4('~)
o(
, ~. small,
5) a p p l y i n g the methods of c l u s t e r e x p a n s i o n s on In fact,
a
more precise
e s s e n t i a l l y proven in [ 1 ]
result
< ~
~
(26)
(25).
is o b t a i n e d (which is not formulated b u t
, § 2.3.8
):
Theorem. ~rel
In where
and
~,
= ~p
~
= I (J(t~t)A¢ , F )
~'~e
= /f~rellAI ~"I
(analogously,
~-~, )
"~,p
(27)
are translation invariant
satisfy a condition of the type (26). (This is not needed there but it is
substantial in the study of interfaces). Corollary. Under the condition (17), the relation (18) can be written
(if we
substitute (16) in its right hand side and use (21)) as F(P)
=
~(P)
÷ ~(r),acr,l
~),(~*(F)-,[-(F))[
where ~,~'~ ~ ) i s the free energy of the + tity of the type (21)) satisfies the estimate
[ A (~, ~) I -~ ~. liE' t and H
,
contour
~.
intl'[
model, ~ ( p l
~)
(28) (a quan-
small
~ is some "counterpart" to ~ , depending only on the hamiltonian approx (This term disappears in the discrete case).
Note. This relation seems to be more complicated than the "explicit" relation (18). Moreover ~+(F:) must be equal to ~%-~F) if the equivalence of ensembles holds. In fact, it is better to solve, instead of (28), the equation
F (P)
=
without a s s u m i n g ask whether
that
~
(F)
If this is t r u e ,
(~(r ~) .P ~(r) ~"
defines
= ,4,"
an
-i. A (r
~)
(29)
e q u i v a l e n t contour ensemble a n d
C/=)
then
(30)
the phase t r a n s i t i o n occurs and both phases are d e s c r i b e d b y
the c o r r e s p o n d i n g contour model. If not, a n d we have to modify our s t r a t e g y .
then the solution of (29) is f o r m a l ( ! )
Solution of (29). The method of fixed point theorem can be used
([ 4 ] ,
[ 1 ]
). Only the
solutions s a t i s f y i n g (17) are o b t a i n e d (but they are the only r e a s o n a b l e o n e s ) .
B7 It is clear t h a t ,
first,
one must check (17) for the contour weight ~ ( ~ ]
This is true in Main Theorem b u t the proof is a g a i n omitted there.
• ~(P).
It r e q u i r e s
some i n v e s t i g a t i o n of the s t r u c t u r e of contours ( p r e f e r a b l y , the "wrapped" o n e s ! ) .
seerl],
§ /4.3.
In the discrete
spin ~case,
-~
d i s a p p e a r s and (17) follows, for ~ ( r )
, easily
from the P e i e r l s condition of the type (1). F i n a l l y , the d e g e n e r a c y removing condition of the type (4) (for ,gJ
only)assures
that
hamiltonian
(30)
H
can
be
satisfied
for
a
"suitably
balanced"
original
. Thus, the point of a phase t r a n s i t i o n is found.
Concluding note
(to
§ 2 and
§ 3).
The method p r e s e n t e d i n this c o n t r i b u t i o n
is, in some sense, an " i n t e r s e c t i o n " of the methods of D J to the c o n t i n u o u s ses,
spin case
(= [ 1 ]
theory
but adapted
). I n the i n v e s t i g a t i o n of n o n s t a b l e p h a -
it seems to be e a s i e r to a d a p t the method of [ 5 ]
t a t i o n of the
' ~ 5]
. Most of the a r g u m e n -
(in p a r t i c u l a r the t e c h n i c a l estimates
(21),
(27))
remain
u n c h a n g e d except t h a t some c o n s t r u c t i o n s and estimates c h a r a c t e r i s t i c to
[5]
must be added. II. I N T E R F A C E S This part of the exposition is a very informal one. Essentially,
one has to combine the discrete version of the method
[ 3]
with
the techniques outlined in the first part of this exposition. Note.
There
will be
some
new
geometrical notions like walls etc. We
mentioned
yet earlier the need for replacing the "naive" definition of a contour by a more intricate
one
("wrapped
contours").
This
is (technically)
even
more
important
there. Nevertheless we will not do it in this introduction because, without going into details of the proof of the technical results of § 2 below no need for the introduction of "wrapped walls" would be seen on this introductory level. (There would be no change on the "conceptional level"). Notations. We again take .~+
,
.X-
X
, X _+ , ~ +
= (~
~
) ~¢"
( V"
,~>
3
)
Let
be as in the Main Theorem. We take the rescaling
(6) everywhere. Let
.X E
X
be such that e.g.
nite component of by
the symbol
~ (.X)
_~_
. The
X
=
.~ _+
almost everywhere. The infi-
will be called the support of an interface and denoted interface
-~
is then
the restriction of ~
on
ml Contours ration
.X
(the "naive" ones) are defined as before.
We will say that a configu-
(of the type above) is a "canonical" one if it has no contours.
(It is convenient configuration
~
to .)
study
a given
interface as
an
interface of a canonical
68
§ i. Geometrical The
geometrical
structure of an interface.
notions of this § are still some adaptations
of the notions
of [6] Take a c a n o n i c a l
configuration
A "perpendicular" restriction
of
p
.X
.X
= { {%
...
,'/-.~-_,, ( . ) ) }
on the c y l i n d e r
h a s the f o l l o w i n g p r o p e r t y : ~X~-
Cp
after a suitable
~(_+~ I ~ 0~
is c a l l e d ~.o~orrect one if the
{[Sl,...,S ~)
=
vertical
; ~
I$.,:-'t~2 / ~J¢~ }
s h i f t of
for all
4
~
for all
e E (]p
.X
, either
C~
or
(30)
I×~-~t~l (Accordingly, Note.
We
(30), and
p
such
say
that
is needed
Definition
as in part
~._+ and
a
any
such
I. We
~+_
Remove
O~
be "clearly
p
~'*_
).
is large but not
distinguishable
in
determined.
is a standard
one if no shift
definition.
from . J ~
remainder
component
~(_+ or
recall that
would
perpendicular
in the preceding
Split the
is of the type
shift would be uniquely
correct
of a wall.
perpendicular. on
~
that
a possible vertical
We of
we say that a perpendicular
use the same
"too large"
-~ o~
all the points belonging
into connected
W
is called
components.
a wall,
denoted
to some correct A
by
restriction
of
W
is
.~
called the support of kA/ . The union of all correct perpendiculars components, of the
which
ceiling
are
called
cylinder with ~
"flat" (as in the discrete be
clearly
ceiling
assigned
. The
spin case),
to them,
as
a
can be also splitted into connected
cylinders. ceilings
but,
A
ceiling
nevertheless,
height
is
an
intersection
are not necessarily
of some
absolutely
a unique
"height" can
perpendicular
intersecting
~hem.We do not specify any configuration on a ceiling . External plane
walls.
Taking. the
~_~'-4 C
projection
7fw" one can
("shadow")
of ~
define the notion of an
into
external
the
wall
horizontal analogously
as we did it for contours in part I. The
only
difference
included
into ext
"exterior have
is that F'
of a
wall
the same
type
while
all the
(for a contour W
[~
+ components
.X_+ or
~±
prescription
for ext
) and
(supp
of the type +), we
"only those ceilings cylinders
(
of
include
"touching
also the same
f'~
height
~
) were into the " which
as the "truly
external" one. This whether
one wall
is "inside"
W
and
another
int W
one.
will be used w h e n
In particular
specifying
this will be used w h e n
defining the notion of an external wall. Standard
walls.
We
external
ceiling
cylinder.
The
Figure
8 shows
define a height of a wall Standard
two standard
ceiling but does not change
walls
walls
are W
its type, while
as the height of the surrounding those ones , W
W
having
. The wall
changes
W
the height 0. elevates the
the type of the ceiling.
69
r e type ~
y ~
~ / j
N
Reconstruction s t a t e m e n t . Call the a
~/
u n i o n of a l l w a l l s of -/'2.
"compatible"
collection
of w a l l s
W
as the " t r u n c a t e d i n t e r f a c e " ../~w . _.O_~ where
c o m p a t i b i l i t y means
is
a nonconflicting
p r e s c r i p t i o n on i)
the types of the s u r r o u n d i n g c e i l i n g s ( " h o r i z o n t a l " c o m p a t i b i l i t y )
ii) the h e i g h t s of them . We claim t h a t there is a .... one to one correspondence between t r u n c a t e d i n terfaces a n d h o r i z o n t a l l y compatible collections of s t a n d a r d w a l l s .
§. 2. Outline of the further strategy. Wall and aggregate models. Fix some finite A and some truncated interface.~=J'~(~A).Assume that dist ( ~ , A ¢" )~ 2 for all walls W of .f~_~ and, analogously, for all contours of XA . Assume e.g. that the partition function (~! _~w) = f
~A
exp
is "externally of the type
.~_+ " and take
& H ( . X A I(.X~)~A¢ ) ~ ~ A
(31)
the integral being taken over ~II D(A described above, with a fixed _ ~ . (Notice that we integrate "through" the ceilings, too). Very roughly this partition function can be expressed as In ~ _ ( ~ i
_~_w) =
const
-
( ~ ICI t ~ I ~ l
+ ~
~(W,g))(32)
where const depends only on A , H ; ~ resp. ~ are the "mean interface energies of X_+ resp. ~± and where I C I (analogously, I ~I ) is the cardinality of the projection, to ~ " ~ - of all the ceiling points of the type J(_* . The quantities ~ (V~) are some "wall energies" assigned to each wall W of _~_~ . (They are defined similarly as the contour energies ~ + ~)(P) in part I, § 2). We know (from the reconstruction statement above) that . ~ W (and ~ W~g~ ) will be uniquely determined if we prescribe the "standard vertical shifts" of W,~ only. Thus, the behaviour of _ ~ is roughly-modelled by some "standard walls model" with a hamiltonian (assigned to any collection of standard walls { W 4 ~ ) of the following type:
This is
apparently
a Pirogov S i n a i type
contour" being obviously needed
model
(7)
(the " a b s t r a c t
notion of a
there - even if one "factorizes" the notion of
70
a wall by taking its shadow to ~_q'-'~ ). The model is a V'-"I dimensional one and we can, therefore, expect phase transition in it for ~" ~,- ~ .(This will be the phase transition in the type ( J(+ or ~_* ) of the external ceiling). The basic question now is about an exact substitute for (32). Note. We will formulate the result only. The methods of its proof are essentially those of
[I]
needed).
notion of an in part
§2..3.8
,
Perhaps
(as in the proof of (2~), with some generalizations
the only really new
approximating
I. Namely
gaussian
to avoid
to have
an
"above"
"below"
the interface
arises there is that the be defined
so simply as
problems in the expansions
approximating
is the horizontal translation
which
measure cannot
technical
is desirable and
problem
measure which is translation
(at least far from
invariancy
like (27-) it
-/~L~
invariant
). Also desirable
around the ceilings where the approxi-
mating measure changes "vertically from OA~*
to j
". Clearly,
re cannot be realized by a finite range quadratic hamiltonian.
such a measuIt can be, ho-
wever, realized by an infinite range hamiltonian of the type (we write ~ = A " _ ~ )
H appr°x
t x.o)
=
.
N,f: Nn M = { ~ } where ~UN (analogously, ~ N ) decay as
Io.,NI d
~
(N) being
~
<
,I
(35)
the lenght of a shortest tree on
in detail in Using now
or, , ~ ( ~ ) , [ 2 ]
, developing
N,~:,s:Nnt1={s,t~ orit}
further
~/
(This will be explained
some simple estimates of §2.2j[I]).
some adaptation of the techniques of § 2.3.8 [I ]
of (27); this will be explained in detail in
E 2 3
(as in the proof
), we obtain the following
result. Theorem. In
-
"~z_{'/~1-0-~ ) = const(A)
÷ ~J~,l -I"
~lgl *
+ T_
+ F_ where to
~
~ (~V)are
the wall energies,
~ (
. )
/~t ~/
)
I X = A ¢ and
(36)
are some "counterparts"
which are linear quadratic functions of Wn(~-t)~ where W = ( ~
~t'Ll~l~: and-
) + I
WZ)~)
~i[, are some quantities defined by means of the
+
contour models. They satisfy the following estimates:
i)
I ~('E,~/)l
ii)
I ,~- T
The quantities is true for
~'[W)
I
, ~(~,
/~rr, (if also,,~~'O ~
Note. The quantities
~
~ Ol~'¢('(~t~E) ~
g ~
) and
(37)
~
(38)
are translation invariant and the same are shifted).
disappear in the discrete spin case.
71
Another expression of ~ ( A, -O-~).
Fix _0.~ and ~
= AlL ~
A
and consider the limit
A ~' ( A
~, _.~.= )
i
(see Figure 9 ). Writing
A we obtain,
in the limit,
the following consequence of (36):
z
w,^
T~mf~¢
where only those c e i l i n g p o i n t s are counted which are "above/A~ "; a n d where
some modified
~
must be t a k e n
for those
~"
~
C
/J~
which i n t e r s e c t
AI,~. This c a n be w r i t t e n , with a n o t h e r
~7'
(see Figure 9)
~1~/
.~
Figure 9
' as follows:
~,
~
"l~"
a , ' $ ~ ( T % , Tx,) >~ 9
A
Y~
e+
¢
A
GIueing of
W
and
T
. Aggregates.
We
will ignore the quantities
as
,~t).
= exp ( - ~ Having
)
~
, for the simplicity of notations.
- i, the quantities
~
(lqriting
can be handled much the same way
the idea of a "wall model" in mind,
the relation (£0) can be rewritten
as follows: Write T cv ~V if the "shadows" of 1. Split the graph ~ ted components.
The
T
and ~
on
(on the set of all f W x ]
components
are called aggregates
~v"
have a distance
and i T X }
into connec-
and we can write (40),
u s i n g the notion of an a g g r e g a t e weight
as
4~m} the sum b e i n g t a k e n over a l l a g g r e g a t e s walls which are contained in .J'l~ .
whose u n i o n c o n t a i n s e x a c t l y those
72
Aggregate model. Notice that all the external walls of the system of all walls of some aggregate have
the same
height.
exterior of an
This is called the height of the aggregate
aggregate
secting the aggregate,
~
. The
is the union of all correct perpendiculars not inter-
having
the same
type and a same height as the "true
exterior". It is easily seen that the aggregate model with a hamiltonian
= is
oC l Cl
+ 7_.
a model of the type o u t l i n e d in
for the i d e a of a w a l l can be u n i q u e l y
)
.,C
§ 1, p a r t
( 31
1 . This is the exact s u b s t i t u t e
model. Namely any " c o m p a t i b l e " c o l l e c t i o n of aggregates
represented as a " h o r i z o n t a l l y c o m p a t i b l e " c o l l e c t i o n of s t a n d a r d
aggregates. We will not go into the details of this model, and conclude our exposition by few remarks on this model only: I) If we show that an aggregate model behaves like an "infinite ceiling of the type
~_+
"islands" by
(resp.
~
surrounded
the P.S.
theory)
) with
by
external
small
and
aggregates
rare
(but uniformly
distributed)
(a characteristic picture obtained
then a similar picture is obtained when
looking
on the
behaviour of the system of all walls. (External walls can be reconstructed from external aggregates). 2) We
have
to check
a Peierls type condition for
~(~)
. Such a condition
follows from (38) and the following inequality: Peierls type condition for ~ W ) .
~(W)
1> "c I W l
,
~
large
(/~f+)
(This is an znequality of the same type as (9). It follows from (2)). The condition for
where
~
~(~)
is then as follows:
I exp(-
~C~))I_~ ~
= ( ~. k~V~')L' ((~.'~,
Notice that while walls It turns out. however, replace
the
~(/~)
I ~ ~ (~)
assumption _> T
I~ I
~
, IE small ,
= ~ VV~'} ~
,
%<4
(/~5)
{ ~" ~
are connected this is no longer true for aggregates. that formulating the "abstract "contours ,
are
~" large,
connected holds
for
P. S. setting" one can
and
the
them"
by
Peierls condition a
more
general
assumption "contours are some objects, whose energy satisfies a condition I exp ( - ~ ( / " ) ) 1 ~
~., I/PI G[/'~('/'P{
~. small ,
~
< 4
,
"
(The notion of an e x t e r n a l contour must be g e n e r a l i z e d a c c o r d i n g l y ) . 3 ) The q u a n t i t y
exp ( - ~ [ ~ ) )
may be n e g a t i v e , because ~
This is u n u s u a l but cause no d i f f i c u l t i e s . (39) with n e w , positive ~ ,
and some new
may be n e g a t i v e .
(One can also take a v a r i a n t of (36), ~
, ~
).
73
4)
The
degeneracy
~, , 9, I
removing
the coexistence
condition
(4) assures
that for suitable values
of both the aggregate
(The proof of an existence of such
~ I kt
of
models really takes place.
is some elaboration of the simple
observation that a coexistence of both types of an interface is mainly controlled by a change in the mean ceiling energies Concluding
We tried to explain that it would general
version
useful in m a n y also
other
~ ,
remark. of the
Pirogov
applications.
interesting
be really desirable to have Sinai
theory,
Besides the applications
fields which
should
be
some reasonably
as this approach
proved
to be
sketched there, there are
investigated
from
this point of
view : i) Infinite range models (with rapidly decaying interactions) 2) Wetting phenomena 3) Impure models (e.g. with an external random field). While I) seems to be accesible without much change the problem 3) is apparently very
involved
as
some of its partial solutions
(Fr~Shlich, Fisher,
Spencer
and
Imbrie - see EIO] show. See the contributions of J. Bricmont
and J. Slawny
for some further information
about the various aspects of the theory [in particular, about 2), and also about some
discrete
spin models with infinitely m a n y
tion that m a n y
authors (Lebowitz, Bricmont, Pfister, Kuroda,
buted to the study of interfaces. type"
ground states). We want to men-
line of study.
Finally
we
and others) contri-
See CII J for the references along this "Peierls note that there
are m a n y
other
advances
of
the theory e.g. [12 ~ ) which are not reflected in our text. References. 1
Dobrushin R.L., Zahradnlk M., Phase diagrams for continuous spin systems; to appear in "Math. Problems of Star. Phys. and Dyn., ed. by R.L. Dobrushin, D. Reidel (1985)
2
Holick~ P., Koteck~ R., Zahradnlk systems; to appear
3
Zahradnlk M., Rigid interfaces; Seminar in Crit. Phenomena, R a n d o m Systems, Gauge Theories, Les Houches Summer School 8~, ed. by K. Osterwalder and R. Stora, North Holland, to appear (1986)
4
Sinai Ya.G.,
5
Zahradnlk M., An Alternate Version of P.S. Theory; Comm. 93,559-581 (1984)
6
Dobrushin R.L., Gibbsian state which describes coexistence of phases for a three dimensional Ising model; Theor. Prob. and AppI. 17,612-639 (1972) in Russian
7
Imbrie J.Z., Phase Diagrams and Cluster expansions for Low Temperature ~(~)2Models; Comm. Math. Phys. 82,261-304 and 305-393 (1981)
8
M.,
Rigid interfaces for continuous
Theory of Phase Transitions; New York, Pergamen
spin
Press 1982
Math. Phys.
Malyshev V.A., Cluster exPansions in lattice models of statistical physics and the quantum theory of fields; Uspechi Mat, Nauk 35,212 (1980) (in Russian. English translation in Russ.Math.Surv. 35, 1-62 (1980)
74
9
Minlos R.A., Sinai Ya. G. "Phase separation" phenomenon at low temperatures, in some lattice models of a gas, I, Matem. Sbornik, 72, No 3, 1967, 1967, 375-~48; II Trudy Mosk. Matem. Obsch. 19,1968, 113-178 in Russian
iO
FrDhlich ]., Disordered Systems; Lectures in Crit. Phenomena, Random Systems, Gauge Theories, Les Houches Summer School 84, ed. by K. Osterwalder and R. Stora, North Holland, to appear (1986)
ii
Pfister C.E., in Birkhauser 1983
12
Dinaburg E.I. Sinai Ya. G., An analysis of ANNI model by Peierls contour method;Comm. Math. Phys. 98,119-144 (1985) See [L i ]J ,
Sc~iling and Selfsimilarity in Physics,
[L 4 ]J , [ 1L1 ]
J
ed. by ~. FrDhlich,
for a more complete list of references
SPIN GLASSES,
E F F E C T I V E D E C R E A S E OF L O N G - R A N G E
INTERACTIONS
A e r n o u t C.D. van Enter SFB 123 Im N e u e n h e i m e r Feld 294 6900 Heidelberg,
Abstract:
FRG
L o n g - r a n g e s p i n - g l a s s m o d e l s w i t h random H a m i l t o n i a n s
H = -.Z J(i,j) li- j I-US.S. ~3 i 3 where the J(i,j)
are independent,
random variables with mean conditions,
identically distributed
E J(i,j)
= 0, s a t i s f y i n g some m o m e n t
in some respects b e h a v e like n o n r a n d o m models w i t h
Hamiltonians
H = - Z li- jl-2~SiSj.
This result is o b t a i n e d through a study of the t h e r m o d y n a m i c limit in any dimension,
and the absence of long range order and
the a s y m p t o t i c c o r r e l a t i o n decay in d i m e n s i o n one and two. Hence the e f f e c t i v e d e c r e a s e of the i n t e r a c t i o n tends to be twice as fast as the o r i g i n a l one.
§i
Introduction
Spin glasses are m a g n e t i c m a t e r i a l s as impurities
(like iron) w h i c h are b r o u g h t
into a n o n - m a g n e t i c e n v i r o n m e n t
(like gold). The
m a g n e t i c p r o p e r t i e s of the iron atoms in their m e t a l l i c host are s u p p o s e d l y d e s c r i b e d by the dilute R K K Y - H a m i l t o n i a n
76
H = -
E i,j
~.e. cos 2 K F I i - J l i 3
S.S. i 3
(1)
li -Jl 3 where
the
e. are 1
(site)
random variables
describing
the dilution.
The s y s t e m
randomness
(due to the site dilution)
long-range
interaction.
w i t h values
0 and
is c h a r a c t e r i z e d
i,
both by
and by an o s c i l l a t i n g
Experimentally
there occurs
a transition
in the s u s c e p t i b i l i t y . According glasses [1,2].
to Edwards
by p u t t i n g
r a n d o m bond
The H a m i l t o n i a n
H =-
with
and A n d e r s o n
(EA),
one can d e s c r i b e
interactions
on a regular
spin lattice
then b e c o m e s
E J(i,j)S.S. i,j 1 3
the J(i,j)
a distribution
(2)
as r a n d o m v a r i a b l e s which
only depends
w h i c h has zero average.
This
gives
which
are independent,
on the d i s t a n c e
with
li- Jl and
a r a n d o m bond m o d e l
for a
spin glass. These
are the m o s t g e n e r a l l y
physicists
there
and the results to p r e s e n t p robl e m s
is still which
sufficiently
Anderson
long-range there
has gone down
A spin-glass
rigorously
but
first
Among
how they b e h a v e are modest.
I will mention
there exists
(non-mean-field)
in a s u f f i c i e n t l y
at w h i c h
bour model
is to show that
for some
lattice
dimension
one can prove
models.
I plan some
are still open.
The first p r o b l e m
regular
spin-glass
a lot of c o n t r o v e r s y
some of these results,
which
transition
studied
transition
order parameter
r a n d o m bond m o d e l
high d i m e n s i o n
interaction.
last year,
a
lowest
for the n e a r e s t
from four to three
is c h a r a c t e r i z e d
on a
or w i t h
The c o n j e c t u r e d
is a t r a n s i t i o n
[1,5]
a spin-glass
by a p o s i t i v e
neigh-
[3,4]. Edwards-
77
qEA
= lim sup ~ {mi(Si)} 2 A÷~ ~ iEA
(3)
>
IAt Here the ~ denote the set of Gibbs states ditions).
N o n a n a l y t i c behaviour,
(or b o u n d a r y con-
due to G r i f f i t h s s i n g u l a r i t i e s
is e x p e c t e d to occur in a m u c h larger domain A second p r o b l e m is the following. (equivalent neighbour)
The S h e r r i n g t o n - K i r k p a t r i c k
m o d e l has p r e s u m a b l y a transition. W h a t
happens at low t e m p e r a t u r e s ?
Is it p o s s i b l e to give m a t h e m a t i c a l
m e a n i n g to the Parisi h y p o t h e s i s c o m p l i c a t e d low T structure? yes
[6-9].
[10-12]
and the c o r r e s p o n d i n g
In some respects the answer is
[13], though in a rather a r t i f i c i a l model.
Has this a n y t h i n g
to do w i t h the b e h a v i o u r in the physical d i m e n s i o n d = 3? The latest guess
[14,39]
is: P r o b a b l y not that much.
W h a t one can prove at p r e s e n t is that some of these q u e s t i o n s are w e l l - d e f i n e d dynamic limit
(for example,
there is a w e l l - d e f i n e d thermo-
[15,16] and a w e l l - d e f i n e d EA order p a r a m e t e r
[1,5].) and that there is no
spin-glass
phase for some m o d e l s
w h e r e the c o r r e s p o n d i n g ferromagnets do have a t r a n s i t i o n [7-9,17-24].
In the next section I w i l l p r e s e n t some of these
results. As a side remark,
I w a n t to m e n t i o n that there are some spin-
glass models w h i c h one can treat in a m a t h e m a t i c a l l y m o r e d e t a i l e d way. T h e y are o b t a i n e d by w e a k e n i n g the EA assumption. E i t h e r one can c o n s i d e r r a n d o m site models b o n d m o d e l s on n o n - r e g u l a r
(hierarchical)
[25-29] or r a n d o m lattices
[30,31].
78
§2
Lon@-range
interactions,
effective
decrease:
results
The models we will treat have the following H a m i l t o n i a n H = -
Z J(i,j) li- jI-~ds.s. i,j 1 ]
where the J(i,j)
are independent,
variables with mean
~
(4)
identically
distributed
random
J(i,j)
= 0 and a cumulant expansion with
finite radius of convergence.
The S i denote either classical or
quantum spins. The results which we have obtained are compared both with the w e l l - k n o w n
in the next table,
n o n - r a n d o m case and with previous
which have been obtained with the help of Bernstein's
results
inequality
for large deviations. SG [16c,20-23,38]
A) Thermodynamics
~>~
1
Non-random [32-35]
SG
(Bernstein)
[7, 16a, 17-19]
~>i
=>~"1
~>2
~>~
exist B) Unique Gibbs
e>l
3
(+C~property)
state d = 1 C) NO broken rotation symmetry
in n-vec-
>3
~>i
tor models d = 2 D) M c B r y a n - S p e n c e r bounds
~>i
3
a>2
in n-vector
models ..l<S0Sn>I < (d = i)
(d = 2)
Cst n- (e-½)
"exp{- (in n)l-e},,
Cst n -(~-I
n
-T
Cst n
-cst
exp{ oIE}
79
Except for the result in quotation marks which holds all the spin-glass
results are true almost surely with respect t o
the random variables optimal
§3
in L2-sense,
{J(i,j)}~
These results are supposedly
[24,36].
Proofs
A) Existence
of thermodynamics
To prove the existence of the thermodynamic energy density one uses subadditive
ergodic
For notational
[16c]
(for free boundary
theorem of A k c o g l u
simplicity we consider
a Gaussian distribution
limit of the free
of the J(i,j)
conditions)
and Krengel
the
[37].
Ising spins in d = 1 with with unit variance.
We write
~(i,j)
= J(i,j)li
To apply the theorem, subadditive a uniform
j I -(~
(5)
"
we have to check that the free energy is
and that the averaged
lower bound
Subadditivity Griffiths
--
free energy density
satisfies
(stability).
of the free energy follows
from an argument of
[40] w h i c h applies to all even spin models.
A stability bound for the averaged
free energy is derived as
follows
-~F A =
~
~
in Z(HA)
tr exp(-HA)
=
~
in tr exp(-HA) < (by Jensen's
= in tr
inequality)
~ • exp~(i,j)SiS j = Z li-j1-2~ i,j£A i,j.EA
(which is of order 0({A[)) In the last equality we used the fact that the J(i,j)
(6) have a
80
Gaussian
distribution
there
occur
alter
the
some
extra
argument conditions.
Take
A the
~i
= +i Then - "
the S i are of h i g h e r
This
Now we
interval the
that
terms
[16c].
boundary for
and
proves
add
energy
order,
spins. but
the r e s u l t
an a r b i t r a r y
[-IAI + 1,0]
surface
Ising
and
In g e n e r a l
they
for
do not
free
boundary
condition.
take ~ = {oi }
between
A and
with i6 IW the r i g h t h a l f - l i n e
is
~A =
Z ~(i, j) Si~ j : E W~.. i6 A i6A z j 61~
We estimate
(7)
its n o r m by
½
: i6A
9 61~
--
J
E i6A
E j 61~
] i - j [ -2~
½ :
3
= 0(I^I By the
law of l a r g e
it f o l l o w s
numbers
for n o n - i d e n t i c a l
)
random
(8)
variables
that
lim
= 0
J-almost
surely
(9)
A÷~ T ~ T
whenever
s > ½ and hence
f = lim
(zo)
FA
A÷- T ~ T
exists
and
the p r o o f
is i n d e p e n d e n t of A).
of b o u n d a r y
conditions.
This
finishes
81
Remark:
If {~l(i,j)} and {~2(i,j)}
interactions,
are two i n d e p e n d e n t sets of
an e s t i m a t e of the form
(6), w i t h tr r e p l a c e d by ~j. 1 the Gibbs state for H I = - Z ~l(i,j)SiSj, shows that the energy density is c o n t i n u o u s as a function of the interactions
I f ( H I + H 2) - f(Hl) I ~< Z ~ J
(ii)
~2(i,j)
B) U n i q u e n e s s of the Gibbs state in one d i m e n s i o n W h e n we prove the absence of transitions, f l u c t u a t i o n s suppress e n e r g e t i c a l l y If we c o n s i d e r the
W =
formula
(formal)
it turns out that thermal
"bad" configurations.
quantity
Z ~(i,j)SiS j i~0 j>0
(12)
(8) shows that IIWII = -
c o n f i g u r a t i o n appropriate. This looks p l a u s i b l e from
for
3 ~ ~ ~ by c h o o s i n g the spin
It turns out that IIwII < ~
for
(8) and was p r o v e n by K h a n i n
e > ~3 .
[17]. This
fact is basic for the d e r i v a t i o n of the results in the last (Bernstein)
column of §2 and restricts "its d o m a i n of validity.
However,
if we c o n s i d e r the c o r r e s p o n d i n g free energy expression,
w h i c h is
(6) w i t h H A r e p l a c e d by W, we see that this is finite
w h e n e v e r ~ > i. The a p p r o p r i a t e free e n e r g y - l i k e q u a n t i t y to c o n s i d e r is a r e l a t i v e e n t r o p y phase transition"
[33b,34b]. A w e a k v e r s i o n of "no
is the absence of s p i n - f l i p s y m m e t r y breaking.
One proves this by showing that the free energy cost of f l i p p i n g an infinite h a l f - l i n e almost surely
(which is a r e l a t i v e entropy)
is finite, J-
[20]. Let ~ be an extremal Gibbs state and ~' the
state w h i c h one obtains by flipping all the spins on a half-line. Then,
for the r e l a t i v e e n t r o p y it holds
0 ~ S(~ I ~' ) ~ S d~ ind~/d~ ' = -2~(W)
(13)
82
We write ~ = ~HI + H 2 + W right half-line
' with H I and H 2 living on the left and
respectively.
Then the relative entropy can be
rewritten:
S(~I~' ) = -~HI + H 2
(2W expW)
(14)
Z (H 1 + H 2) Z (H I + H 2 + W)
Both factors
in this product can be shown to be finite J-almost
surely by an estimate of the form the fact that
IKeKI ~ exp(2~)
we use the fact that
Z(H 1 +H2)
~
+ exp(-2K)
(variational
exp
(6). For the first term we use and for the second term
principle)
(15)
-~HI + H 2 ( W ) .
Z (H 1 + H 2 + W)
The important point to note is that one may interchange
the
average over the J's which occur in W with the m o d i f i e d
thermal
~H1 + H2 w h i c h d o e s n o t c o n t a i n
expectation
By translating
these
J's
anymore.
the origin one shows that ~ and the spinflipped
have a finite relative entropy with respect to each other, hence are absolutely
continuous w i t h respect
to each other and thus
identical because they are extremal Gibbs. notion of absolutely equivalent Remark:
continuous measures
For q u a n t u m spins the
gets replaced by quasi-
states.
To prove uniqueness
of the Gibbs state,
one applies a
similar estimate on a large
(finite)
shows that the interaction
energy between A and the far outside
is uniformly
small
region surrounding
A, and
[38]. The fact that in an operational
the Gibbs state is unique,
because all J - i n d e p e n d e n t
c o n d i t i o n s give rise to m u t u a l l y absolutely states was commented upon in [21].
sense
boundary
continuous
Gibbs
83
C) Unbroken rotation symmetry in two dimensions To prove a result of Mermin-Wagner type, one applies to a state a constant rotation in an inner region AI, keeps all the spins outside a large region A 2 m A 1 fixed and rotates the spins in A 2 - AI, over angles 0 i which linearly interpolate. We write ~Ai,A2for this transformed state (this idea has been applied by Fr~hlich and Pfister [34b] in the nonrandom case). The relative entropy S(aI~AI,A 2) is now bounded by a sum of terms of the form
{I -cos(0 i - 0j)}~(SiS j ~(i,j)) =
{i - cos(0 i - 0j)}~i,j(SiS j ~(i,j) expSiS j ~(i,j))
=
~i,j (expSiSj ~(i,j))
{ I - cos(0 i - 0j)}~i,j(SiS j ~(i,j) exp SiS j ~(i,j))
(16)
1 + ~i,j{exp(SiSj ~(i,j)) - i}
Short range contributions can be treated separately, along the lines of [34b], hence without loss of generality we can assume that the ~(i,j) are small. In fact, here we need a distribution for the J(i,j) with bounded support. However, more general distributions like Gaussians can also be treated. Thus we can make a convergent Taylor expansion of (16) in ~(i,j). The modified thermal average ~i,j' in which the term SiS j ~(i,j)
is subtracted
from the original Hamiltonian does not contain the random variable J(i,j) and hence can be interchanged with the expectation over J(i,j). The result is
m(16)
= { I - cos(0 i - 8j)}0( m ~(i,j)2).
(17)
84
In this way we derive Fr~hlich
and Pfister
(by reducing the estimates
to those of
[34b])
S(~I~AI,A 2) < ~
uniformly
(18)
in AI, if A 2 (dependent on A I) is chosen appropriately.
This proves the rotation D) M c B r y a n - S p e n c e r
invariance
of ~ [21].
bounds
To prove M c B r y a n - S p e n c e r for n-vector models,
I<SoSn> I ~ Z(H')
upper bounds on correlation
one uses the following
functions
inequality
[35a,b]
exp - (a n - a 0)
(19)
Z(H) where
H' = - Z ~'(i,j)SiS j
(20a)
~' (i,j) = cosh(a i - aj)~(i,j)
(20b)
and
By the variational
Z(H[!
principle
< e x p - ~ H , ( H ' - H).
(21)
Z(H)
The expression
~H,(H' - H) is estimated
relative entropy Following
Picco
in case C)
aj = a'~11Jl
for d = 1 a n d
[22].
[].9] and Messager,
we take the following
in a similar way as the
choices
Miracle-Sole
and Ruiz
[35c]
for the aj:
k and aj - aj_ 1 = ~ , j = i, 2 . . . .
(22)
85
- 1
alj I - alj
=
k
,
lJl
=
1 ....
N
(23)
j(in+j)l -
for d = 2. Again,
the r e w r i t i n g of
(21) in terms of m o d i f i e d thermal expecta-
tions allows us to reduce the e s t i m a t e s to the n o n - r a n d o m case of [35c]. Acknowledgements:
Most of this w o r k was done t o g e t h e r w i t h J.L. van
H e m m e n and J. Fr~hlich, w h o m I thank for p l e a s a n t collaborations. w a n t to thank the o r g a n i z e r s
I
for their i n v i t a t i o n to present it in
a p l e a s a n t conference.
References
[1] S.F. Edwards and P.W. Anderson:
J. Phys. F 5, 965
(1975).
[2] Reviews on d i f f e r e n t aspects of spin glasses can be found in the p r o c e e d i n g s of the H e i d e l b e r g C o l l o q u i u m on Spin Glasses. Ed. J.L. van Hemmen and J. M o r g e n s t e r n , in Physics
192
Springer Lecture Notes
(1983).
See also D. C h o w d h u r y and A. Mookerjee: [3] J. M o r g e n s t e r n and A. Ogielski:
Phys. Rep.
114,
Phys. Rev. Lett. 54,
928
1 (1984). (1985)
(transition d > 3). [4] a) R. F i s c h and A.B.
Harris: Phys. Rev. Lett. 38,
b) J. M o r g e n s t e r n and K. Binder: (1979). [5] A.C.D.
785
Phys. Rev. Lett. 43,
(1977). 1615
(transition d > 4).
van Enter and R.B. Griffiths:
Comm, Math.
Phys. 90,
319
(1983) . [6] R.B. Griffiths: [7] M. Cassandro, 229
(1982).
Phys. Rev. Lett. 23,
17
(1969).
E. O l i v i e r i and B. Tirozzi:
Comm. Math.
Phys. 87,
86
[8] A. Berretti: [9] J. FrShlich
J. Stat.
Phys.
and J. Imbrie:
[10] D. Sherrington
38, 483
(1985).
Comm. Math.
and S. Kirkpatrick:
Phys.
Phys.
9-8, 148
Rev. Lett.
(1985). 3_55, 1792
(1975). [ii] G. Parisi:
J. Phys. A i_~3, ii01,
[12] G. Parisi:
Phys.
[13] M. Cassandro,
Rev. Lett.
E. Olivieri
1887
(1980).
50, 1946
(1983).
and P. Picco:
[14] D. Fisher and H. Sompolinski: [15] The thermodynamic
Phys.
Rome preprint.
Rev. Lett.
limit for short-range
5_~4, 1063
(1985).
random interactions
has
been treated in a) R.B. Griffths
and L.J. Lebowitz: Phys.
J. Math.
b) F. Ledrappier:
Comm. Math.
c) P. Vuillermot:
J. Phys. A 1-8, 1319
d) J.L. van Hemmen and R.G. Palmer: [16] The thermodynamic
56, 297
Phys. 9,
1284
(1968).
(1977).
(1977).
J. Phys. A 15, 3881
limit for long-range
(1982).
random interactions
has
been treated in a) K.M. Khanin and Ya.G.
Sinai:
b) S. Goulart Rosa: J. Stat. c) A.C.D.
J. Stat.
Phys.
Phys. A 15, L 51
van Enter and J.L. van Hemmen:
2-0, 573
(1979).
(1982).
J. Stat.
Phys.
32, 141
(1983). [17] K.M. Khanin:
Theor. Math.
Phys.
4_~3, 445
[18] P. Picco:
J. Stat.
Phys.
32, 627
(1983).
[19] P. Picco:
J. Stat.
Phys.
3-8, 489
(1984).
[20] A.C.D.
van Enter and J.L. van Hemmen:
[21] A.C.D.
van Enter and J. Fr~hlich:
(1980).
J. Stat.
Comm. Math.
Phys. Phys.
39, 1 (1985). 9-8, 425
(1985). [22] A.C.D.
van Enter:
[23] L. Slegers, 267
J. Stat.
A. V a n s e v e n a n t
Phys.
4_~I, 315
(1985).
and A. Verbeure:
Phys.
Lett.
108 A,
(1985).
[24] The existence of transitions
in the corresponding
ferromagnetic
87
models
is proven
in
a) F.J. Dyson:
Comm. Math.
Phys.
b) J. FrShlich
and T. Spencer:
c) H. Kunz and C.E. Pfister: [25] D.C. Mattis:
Phys.
van Hemmen:
b) J.L. van Hemmen, 5_~0, 311
b) F. Benamira,
(1976).
3_~7, 778
(1976).
4_99, 409
(1982). Z. Phys.
B
contribution
J.P.
to 2a).
Phys.
Rev. Lett.
Provost and G. Vall~e:
5_O0, 598
(1983).
J. de Phys.
46,
(1985).
[29] D. Amit,
H. G u t f r e u n d
and H. Sompolinski:
[30] P. Collet and J.P. Eckmann: [31] P. Collet, 3-6, 89
(1982).
4_~6, 245
van Enter and J. Canisius:
Provost and G. Vall~e:
1269
Phys.
8_~4, 87
(1976).
Rev. Lett.
A.C.D.
Phys.
(1983).
c) J.L. van Hemmen, [28] a) J.P.
Comm. Math.
Rev. Lett. Phys.
(1969).
Comm. Math.
Phys. Lett. 5_66A, 421
[26] J. Luttinger: [27] a) J.L
12, 212
J.P. Eckmann,
Comm. Math.
J e r u s a l e m preprint.
Phys.
9-2, 379
V. Glaser and A. Martin:
(1984).
J. Stat.
Phys.
(1984).
[32] A treatment
of the thermodynamic
limit can for example be found
in a) D. Ruelle: b) R.B.
Statistical Mechanics,
Israel:
Convexity
P r i n c e t o n University
(among others)
Press,
Comm. Math.
c) J. Bricmont, 2_!i, 573
Princeton N.J.
(1979).
for one-dimensional
models
is
in
a) D. Ruelle: Comm. Math. b) H. Araki:
(1969).
in the theory of lattice gases,
[33] A b s e n c e of phase transitions proven
Benjamin New York
Phys. 9, 367 Phys.
J.L. Lebowitz
(1968).
4_44, 1 (1975). and C.E.
Pfister:
J. Stat.
Phys.
(1979).
d) M. C a s s a n d r o [34] T w o - d i m e n s i o n a l
and E. Olivieri:
Comm. Math.
Phys.
models where the rotation symmetry
8-6, 255
(1981).
is not broken
88
are treated in a) N.D. M e r m i n and H. Wagner: b) J. Fr~hlich
Phys.
and C.E. Pfister:
Rev. Lett. 17,
Comm. Math.
1133
Phys. 81,
(1966). 277
(1981). and references
mentioned
[35] M c B r y a n - S p e n c e r a) O. M c B r y a n
there.
estimates
are given in
and T. Spencer:
b) J. Glimm and A. Jaffe: New York, Heidelberg, c) A. Messager, Poincar~ 40, d) K.R.
Comm. Math.
Quantum Physics Berlin
S. M i r a c l e - S o l e 85
[36] G. Kotliar,
299
(1977).
§16.3, Springer-Verlag
(1981). and J. Ruiz: Ann.
Inst. Henri
(1984).
Ito: J. Stat. Phys. 29,
e) S.B. Shlosman:
Phys. 53,
Theor. Math.
P.W. A n d e r s o n
747
(1982).
Phys.
and D.L.
37, 1118
Stein:
(1978).
Phys. Rev. B 27,
602
(1983). [37] a) M.A. A k c o g l u and U. Krengel:
J. Reine Angew. Math.
323, 53
(1981). b) U. Krengel: York
Ergodic
theorems
§6.2, de Gruyter,
Berlin,
New
(1985).
[38] M. Campanino, [39] J. Fr~hlich:
A.C.D.
van Enter and E. Olivieri:
private communication.
[40] R.B. Griffiths:
Phys.
Rev.
176, 655
(1968).
in preparation.
ANALYTICITY IN SOME MODELS OF QUANTUM STATISTICAL MECHANICS
Huzihiro ARAKI Research Institute for Mathematical Sciences Kyoto University, Kyoto 606, JAPAN
Abstract
Application of C*-algebra approach to proof of analyticity
of correlation functions for some models of quantum statistical mechanics
is reviewed.
Analyticity of correlation functions of ground
states with respect to the asymmetry parameter magnetic field strength
~
¥
and the external
for the XY-model on one dimensional lattice
and that of equilibrium states with respect to the inverse temperature and the coupling constants
Jl
and
J2
for the Ising model on two
dimensional lattice are obtained for all real values of parameters except the singularity at the critical lines. The same method also gives an exact decay rate for the exponential clustering of the correlation functions of the latter model as a function of parameters.
§i.
Introduction
The C*-algebra approach is by now a well-established mathematical framework for quantum statistical mechanics of spin lattice systems. Recent study shows that somewhat abstract theory of C*-algebras
can
actually be used in finding physically interesting quantitative properties of some (exactly soluble) models of spin lattice systems such as the one-dimensional XY-model and the two-dimensional model.
Ising
In the present article, we review the results in [9] and [5]
about analytic properties of the correlation functions. For the one-dimensional
lattice with its sites lebelled by
integers
j E g, we consider the C*-algebra
algebras
~j
different
of
2×2
matrices,
jE g
j's), where elements of
~.
identity matrix and Pauli spin matrices J is the dynamical
system
( ~ ,a t )
~
generated by the
(mutually commuting for are linear combinations of the a ' a=x,y,z. ~(J)
defined by
The
X-Y model
go
~(A)=lim eitH(N)Ae -itH(N),
A~
(i.l)
N-~oo
(i.2) where
H(N) denotes H with summation restricted to ~ j , -N<j<__N. All ground states of the dynamical system (~,~t) are obtained in [8]. For the (finite-dimensional) subalgebra ~(N) generated by ~j, -N<~<_N, there is a unique eigenstate ~ N of H(N)E~(N) with the lowest eigenvalue. The limit ~l,,'(A)=limN÷ ~ ~N(A),
AGU~(N)N
(1.3)
exists and determines a state ~X,T of ~ . If either ]X]~I or y:0, ~k,y is the unique ground state of (~,at). If [k]
A £6@~,
t £& ~.
(1.4)
Theorem 1.1. The correlation functions (1.4). are real analytic functions of the parameters (k,y) for ~=~k,y except for the critical lines X=±l and Y = 0 , and for ~=~k,y ± in the re~ion Ik]
(1.5)
~,j=~ being the spin variable at the lattice site (i,j) and J's being nonzero real parameters, it is known [1,10] that there is a unique equilibrium state ~B for IB]<~ while there are exactly two extremal (and ergodic) equilibrium states @S,+ and @S,- for [B]>~, where ~ is the unique solution of (K~)(1/2)log{coth]~l[}=~IJ2](~K2). The correlation functions are
(1.6)
91
n ¢((i,j)~l
~i,j
(1.7)
) t
for a finite subset Theorem 1.2. functions of
I
of lattice sites.
The correlation functions (1.7) are real analytic
~' Jl
and
J2
for
~=~B' IBl<~c
and for
~=~B,±'
IBI>c. §2.
The CAR algebra and the spin algebra The Jordan-Wigner transformation cj=Tj(g~J)-i~ (Jy))/2, c~=Tj(o~J)+ig (jy))/2, (jEg),
bridges the spin algebra
~
(2.1)
and the CAR algebra grgCAR generated by
c's and c*'s satisfying the canonical anticommutation relations (CAR's), where T. is supposed to serve the role of the product of ) J , j ran~ing from the left end of the ene-dimensional lattice up to the site j-1. The difference with the case of a finite lattice is that T's are no longer an element of ~. Following the method introduced in [4], we extend the algebra ~ to
a~J
= ~+T~
(2.2)
by adding T=T I satisfying T=T*=T -I and TAT=@_(A) Here @_ is the automorphisms of ~ satisfying
O_(cY(j)) =
for
a=x,y,z
c j)
for
j>0
O(c (j) )
for
j<0
I
and
@ ( ~ J ))=~J ),
@
for all
AE~.
(2.3)
is another automorphism of @(c~J ))=-c~J ),
@ ( ~ J ) )=-~J )
~
satisfying (2.4)
for all j EL. The C*-algebra ~ is the crossed product of ~ by @_-action of the group Z 2 of two elements. The operator Tj in (2.1) is T multiplied by ~(j)...~(j-l) for j>l and by Z Z ~0)...~(j) for j
92
The automorphism 9 can be extended to ga such that 9(T)=T. It maps ~CAR onto itself and counts the even-oddness of c's and c*'s: G(cj)=-cj, 9(c~)=-c~.j The algebra ~ and ~CAR are split into a sum of
e-even
and
O-odd
parts (denoted by suffices ±)
with
the following relations. ~+= ~AR~
~
=T~CAR
(2 5)
In particular, this gives a specific isomorphism of the @-even part ~+ of the spin algebra ~ onto the even part ~_CAR of the CAR + algebra ~CAR. --CAR . Therefore the limit Since H(N) is 9-even, H(N)G ~+= ~+ (i.I) for
A~
CAR
makes sense within
~CAR.
exists and can be described as follows. {gj;j~g} For each
be in ~2(g), and h, we define
h=(f)
Let
be in
In fact the limit f~{fj; j ~ L }
and
B(h) = )~ (fjc~+gjcj). j~Z It satisfies
(2.6)
B(hl)*B(h2)+B(h2)B(hl)*=(hl,h2 )
r(f)=[~], and generates
~CAR.
r ~[Z)induces an automorphism
~(U)B(h)=B(Uh), The dynamics
st
of
~(U)
Any unitary of
~CAR
U
and on
B(h)*=B(Fh), X
satisfying
(h 6~'~). ~CAR
(2.7)
is given by
satisfying
Kr=-FK.
Combining exists for dynamics
(2.8)
K=K(~,y)
K*=K,
st
(2.9) for
~
A=T=(Cl+C~)a(1)x st
where
commuting with
~t=~(e iKt) for some
g--
~ = ~ 2 ( Z ) $ g 2 ( Z ) = ~ 2 ( Z ) ~ 2.
on ~.
and
~CAR,
we see that the limit (i.i)
and hence for any
AG~
defining a
93
§3.
Ground states of the XY model
A method of obtaining all ground states of (~,~t) from the knowledge of ground states of ( ~ C A R , ~ t) via ground states of (~+, ~t ) has been developed in some generality [8,7]. ~n the present model, the operator K above does not have an eigenvalue 0. This implies that ( ~ C A R ,st) has the unique (Fock) ground state
~E
characterized by
~E(B(hl)*B(h2))=(hl,Eh2
where
E=E(~,Y)
(3.1)
)•
is the spectral projection of
K=K(~,Y)
for
(0,+~).
If (~,y)#(0,+l), K has a continuous spectrum. This implies that the restriction of ~E to aL+-CAR_-~+ is the unique ground state of (~+, ~t )
and hence
~k, ~(A)=~E((A+®(A) )/2)
(3.2)
is the unique 0-invariant ground state of (~,~t), necessarily coinciding with (1.3). The equality of (1.3) with (3.2) persists for (~,y)
=(0, ±l). By the associated
equivalence
criterion
with
Fock
Z2-index
)
it was
shown
in
state
of
(~,~t)
(and
hence
is pure)
there
exist
disjoint
pure
the
states
for
(in terms [8]
cyclic
representations
of the
Hilbert-Schmidt
that
~
for
A,y I~I>__I
ground
states
'
mutually
is the and
for
~l,y,±
I~l
§4.
Strategy for the proof of analyticity
~+
norm
unique
~,y(A)=(~A,y,+(A)+~,y,_(A))/2
for
of
and
ground
y=0,
while
satisfying
(3.3)
~0.
For any two projections Ej satisfying E~=E.=E~j ~ J and Ej+FEjF=I (j=l,2), there exists a unitary UI2 commuting with F and satisfying UI2E2U~2=EI and hence satisfying
(4.1)
~E 2 (A)--~E1 ( ~(U12 )A ) for all
A E ~ cAR.
Since
o=~(-I)
always commutes with
~(U), ~(UI2)
94
is an automorphism of
w+--CAR = ~ + .
We use the following criteion for
the extendability of
a(U)
~+
Theorem 4.1. from
~+
unitary
from
(Evans and Lewis)
to an automorphism u
in
~+
~
to an automorphism The automorphism
o__ff ~
~
(U)
of ~ . extends
if and only if there exists a
satisfying
[~(U)@_~(U*)@_](A)=uAu* for all
AE~CAR'
Then
u
(4.2) can be chosen to satisfy
u@ (u)=l
and
^
is given by ~(A++A_T)=~(U)(A+)+~(U)(A_)uT A + ~~ +~
for
CAR
(4.3)
(and hence for any
A=A++A_TE~).
The strategy for the proof of analyticity is as follows. given
(~,~)
(~',y')
not on the critical lines
l~I=l
and
in a sufficiently small neighbourhood of
Theorem 3.1 to show that, for EI=E(~,y) and has an extension from ~ + to an automorphism
For
y=0, and for (~,¥), we use
E2=E(~',y') , ~(UI2) a of ~ . The formulas
(3.2) and (4.1) imply
(4.4)
~k, y,(A):~k,y(~(A)) • We show that of An .
A
~(A)
including
is real analytic in A=~tl(Al)'''~tn (An)
(~',¥')
for a certain class
with strictly localized
Then we obtain the real analyticity of (1.4) for all
(X,y)
AI,..., not
on critical lines. The real analyticity of Step i.
E~,y
analytic in (~,y)
~(A)
is proved in 3 steps:
is explicitly given (see Section 4) and is real (except on critical lines).
is a canonical construction of
UI2
For
IIEI-E2D < i, there
given in [9] (based on the angle
operator between 2 projections, see Appendix of [3]), such that is real analytic in (k,y) and (~',y') for the present case. Step 2. j~0.
Let
Let
(e_f)j=~jfj
I°:
(8_h)= e
g
for
with
:
h=().
Ej=I Then
for
j>0
o_=~(e_)
and
Ej=-I
UI2 for
and the condi-
tion (4.2) amounts to the condition that a(V) for v=u12e_u~2 e_ is inner and its implementer is in ~ +CAR . The criterion for this given in Section 8 of [8] is that
~-l
is in the trace class and
det V=l.
95
The trace class condition will be discussed determinant
condition
the property
rVF=V
multiplicity IIEz-E211
is automatically which implies
of the eigenvalue
is sufficiently
When
the trace class, trace class.
H
Then
det V=(-I) v
-I
small
I!V-III< 2, V-l=e iH
for
(<~/2) for
V.
if
in
with
(B,HB)
v
(~,Y)
The
due to
being the
is satisfied
if
EIUI2-111 < I.
J]Hll<w and
if
V-I
is in
with values in the
in Theorem 4.1 is given by
with the bilinear Hamiltonian
IIV-III < 2
ILV-III< 2
so that
H=H*,
is real analytic u
in the next section.
satisfied
exp i(B,HB)/2
([3]) and is real analytic in
(~,Y). Step 3. B(Uh) nomial ~(A)
If
U
is real analytic
is real analytic A
of
in
the size of localization with sufficiently
with
Bn
supported
sufficiently at(A)
in
(l,y)
region,
for any poly-
exponential [-n,n]
A.
Hence A.
of local algebra with
this real analyticity
in the interval p.
~(U)(B(h)) =
for strictly local
of the dimension
fast decreasing
large positive
then
~(U)(A)
all strictly localized
given by (4.3) is real analytic increase
in (~,y),
and so is
B's, which includes
Due to the exponential A
(X,Y)
tail,
and
extends to all i.e.
A=[B n
IiBnll ~ Cpexp-pn
(See Lemma 3.3 of [5].)
is shown to be such an operator with any choice of
for
In [2], p.
(cf.
Theorem 4.2 of [2].)
§5.
Trace class estimate We introduce
the Fourier series expansion
for
fGZ2(Z)
and
h G Z2(Z)SZ2(Z)
~(e)= [
einOfn,
(5.1)
h(8)=[~((881]
nEL
The operator
and
K
in (2.8) is then the multiplication
Icos8-1
-iysin@]
K(O)=2 [iysine
-cosS+lJ
E=E(~,y)
is the multiplication
of
(5.2) of
E(e)=(2~(e))-l(K(e)+~(e))
(5.3)
p(8)=2[(cose-l)2+y2sin2e] 1/2
(5.4)
96
where
D(e)#o
for all real
0
if
(l,y)
is not on critical lines.
The crucial property is that E(e) is holomorphic in ~, y, and z=e ie in the neighbourhood of real CI,y) not on critical lines and Izl=l.
This property persists for UI2 which is again a multiplica~ eie ). tion of UI2(e)=UI2( By a shift of integration contour yr={Z;IzI=r} for Ul2(z)zJ-k-ldz , ^
(Ul2)j,k=(2~i)-l~
(5.5)
YI we obtain the exponential decrease of where j>0
(Ul2h)j=~(Ul2)j,kh k. and for
Since
(UI2)j, k
q±z(l±8_)/2
for large lJ-kl, are projections for
j ~0, the identity
Ul2e_u~2e_-l=U12(q+-q_)U~2(q+-q_)-Ul2(q++q_)U~2(q++q_)
(5.6)
=-2(Ul2q+u~2q_+Ul2q_u~2q+) and the exponential decrease of
(UI2)j, k
(and hence that of
(U~2)j, k) for large lj-kl imply that q+U~2q , q U~2q + (5.6) are in the trace class due to the following Lemma.
and hence
Lemmla 5. I. lIXiltr:inf [ [(e£,Xe~) i [ [(ez,Xem) Z,m £,m
(5.7)
where the infimum is over all orthonormal bases
{e£}
and
{el}.
The inequality follows from the following estimate for the polar decomposition X=ulXI: HXlltr=trIXl=II(e~, m
due to
u*Xel)[=[l[(el,u*ez)(e£,Xel)l~ m i
[ I(ez,Xel) l
£~m
l(e~,u*eg)lal.
The trace class estimate above also much simplifies the HilbertSchmidt class estimate of E(l,y)-e_E(l,y)e_=2(q+E(l,y)q
+q E(l,y)q+)
in [5] necessary for judging equivalence of
~E
and ~e Ee "
97
§6.
The Isin$ model By the transfer matrix method, the correlation function
(1.7) can
be expressed as Schwinger functions of a ground state of a C*-dynamical (~,~) with the same spin algebra g~ as before and the system dynamics
~
given in [63:
¢(~(kl,ll)...6(kn,ln))= where
~(k,l)= H 6k,j'
~(aBik I (~x(ll)
°x(1)= H °(j)
jEI
-..~Bik n (~x(In)))
(6.1)
and the formula holds for
jEI x
¢=@B and ¢B,± with ~ = ~ B and ~B,+. Here ~B is the unique @-invariant ground state of (~,~), which is pure for IBI>Bc and is an average of mutually disjoint pure ground states ~B,± for IBI>Bc
([6]). The identification [6]
and for
~B,±
and
(6.i) has been attained for ~,±
~B
and
B
in
by using the preceding~ result along
with the cluster property in [5]. The operators ~k(~x(1)) and their products are of the type discussed in Step 3 of Section 4 (with any p) Thus the machinery explained for the ground states of the XY model works also for the present case and Theorem 1.2 follows. Details are refered to [5]. By using the exact information on the spectrum of the time translation generator (in the representation associated with 's), an exact decay rate for the exponential clustering of the correlation functions can be obtained. A typical result is as follows (Proposition 6.5 (2) and (3) of [5]): Theorem 6.1. lim where
(i)
For any polynomial
FI
~(~,0)
is a lattice translation,
~=4(IK2[-K{)
~=gB i_ff IBI
~÷+~ e£(6+e)I~(F1,T(£,0)(F2))-~(F1)9(F2)I=~
lim
6
F2
o_~f ~'s,
eI~I61¢(F1T(~,0)(F2))-¢(F1)¢(F2)I=0
i__ff IBI>B c and 6=2(K{,IK21) and (2) There exists polynomials given s>0 such that
where
and
and
~
are the same as (I).
(6.2) and
¢=gB,±
for any
(6.3)
98
References [I] [2] [3] [4] [5]
[6] [7]
Aizenmann, M., Co~amun. Math. Phys. 73, $3(1980). Araki, H., Commun. Math. Phys. 14, ~ 0 ( 1 9 6 9 ) . Araki, H., Publ. RIMS, Kyoto Univ. 6, 384(1970). Araki, H., Publ. RIMS, Kyoto Univ. ~0, 277(1984). Araki, H., Analyticity of correlation fuctions of the ising model in two dimensions, Prepreint RIMS-525, submitted to Commun. Math.
Phys. Araki, H., and Evans, D.E., Commun. Math. Phys. 91, 489(1983). Araki, H., and Matsui, T . , P r o g r . i n Phys. 1~0, 17-~1985).
(Statistical Physics and Dynamical Systems, eds. Fritz, J., Jaffe, A., and Sz~sz, D., Birkh~user). [8] Araki, H., and Matsui, T., Commun. Math. Phys. i01, 213(1985). [9] Araki, H., and Matsui, T., Analyticity of ground states of the XY-model, Preprint RIMS-522. Lett. Math. Phys. in press. [i0] Higuchi, Y., p.517 in Random Fields I, ed. Fritz, J., Lebowitz, J.L., and Sz~sz, D., Colloquia Societatis Janos Bolyai, voi.27 (North-Holland, Amsterdam-Oxford-N.Y., 1981). [ii] Matsui, T., Explicit formulas for correlation functions of ground states of the i dimensional XY model, Ann. Inst. H. Poincar~ Sect. A, in press.
K-TXEORY OF C*-ALGEBRAS IN SOLID STATEPHYSICS
Jean BELLISSARD Univeriit6 de Provence et Centre de Physique Th6ot'ique
MARSEILLE(FRANCE)
I)- C'algebras : an appealing but Incomolete tool ? The subject I would like to present here today pretends to use C"-algebras in concrete problems of modern physics, a mathematical tool which have been liable to controversies among the mathematical physicists for the last twenty years. When D. Testard, R. Lima and I started our program about solving some mathematical problems in the study of disordered systems, five years ago, we naively believed that using C"-algebras was just like using any other too] of mathematical physics like functional analysis or probability theory. However when we began to explain to the experts what we were able to do, not very much I confess, we realized that before trying to convince them, it was necessary to get really hard results. Since we knew perfectly well the limitations of these technics, we understood quite soon that what could be solved with it, was not as spectacular as was implicitly demanded by both the believers and the sceptics in the community. We spontaneously adopted what can be qualified as a schizophrenic attitude : on the one hand we presented an external work as reliable as possible, and on the other hand we kept unpublished for a very long time our results always present in the back of our mind as a guide in performing our program. What I will explain today have been presented almost in essence at an informal meeting in Marseille during the spring of 1981, apart from the chapter on the quantum Hall effect. Actually some traces of this work can be found in the recent literature. Several results were partially quoted in seminars, conferences [14], reviews [94], or even in some papers [17,56]. We finally published the first part very recently [19]. Reading this last paper will explain the way it has been received by the scientific community. At first sight, the scheme seems very simple and so efficient that it looks miraculous. And when miracles appear the world is immediately divided into two irreducible components : those who are true believers and those who just reject the claims. Fortunately, or perhaps unfortunately depending upon the side one prefers to choose, there is no miracle, at least in physics. In our problem, even though the scheme looks very simple, when one starts the proofs, one realizes immediately how heavy they are : all the machinery of C~-algebras and algebraic geometry is needed, and one looses quite rapidly the original physical intuition! Moreover many technical details remain unsolved or requlre very complicated and winding proofs. Clearly C*-algebras
100 are very appealing mathematical tools but they must be used together with other technics to become efficient. One can say that C"-algebras started their career in mathematical physics in the early sixties with the Haag-Kastler axioms [48] of what is known today as the Structural Field Theory. The hope was that the miraculous aspect of these objects would solve the mysteMous difficulties encountered in Relativistic Quantum Field Theory. Twenty years later it is quite clear that the flu went the other way : more have been learned from the physics to understand the mathematical tool than the converse. Moreover, very few came from the original motivation namely the understanding of the particle physics. On the contrary, like for many advances in physics during the l a s t thirty years, the concrete understanding came from Melds connected with Condensed Matter Physics. For example, the KMS condition, one of the major step in the Tomita-Takesaki theory [1"129]and in the classification of type III factors by A. Connes[25], was introduced in the field by R. Haag, H. Hugenholtz & ~ Winnink [47]. This improvement came after several mathematical physicists who started there work with the axiomatic Quantum Field Theory, had changed their mind and decided to study the Statistical Mechanics [1"124]. The attempt of J. Glimm and A., Jaffe [45] in using C~-algebras in constructive quantum field theory was soon abandoned and replaced by the probabilistic approach[M10,1,127]. However eight years ago a breakthrough was performed by A. Connes [26,27,30] who associated to any foliated manifold a canonical C~-algebra, playing the role of the basic object needed to apply the methods of algebraic geometry and to get topological invariants as well as a cohomotogy theory. He contributed in giving a concrete content to the K-theory, and created the cyclic cohomology, to take into account this new situation [28-34]. He also emphasized that non commutativity of the algebras was closely related to the ergodicity of the foliation, or equivalently to the fact that the set of leaves was not endowed with a non trivial topology. At about the same time, algebraic geometry entered in quantum field theory with the contribution of mathematicians to the classification of instantons in gauge field theories [M15]. Later on, they also interpreted the anomalies (as Adler's one) in term of topological obstructions. More recently the supersymmetries offered the mathematicians a formal tool for calculating simply the formulae appearing in the Index theory. Nevertheless the previous works are all performed on the ground of a semi classical quantization by mean of hypothetical Feynman path integralS: the algebraic geometry concerns usual fibre bundles, on usual manifolds. Nothing in this approach is essentially non commutative. The reason is that nobody knows how to built a theory for describing gauge fields in four dimensions, and one has the hope that something of the topological obstructions will remain in the final theory if any. However literally speaking there is an apparent contradiction between the fact that topological obstructions require smooth
I01
configurations of gauge fields, whereas it is well known in constructive field theory that almost surely no configuration is smooth. Topological invaMants actually reflect the properties of the observable algebra, more than those of the configurations. The Quantum Hall Effect (QHE) may be a good laboratory to test such an idea. What will be presented here concerns only the ordinary QHE, namely a one particle theory. The quantum mechanics of such a system is well understood. Moreover its has been demonstrated that, when the Fermi energy belongs to a gap, and at zero temperature, the Hall conductivity can be interpreted (in e21h units) as the Chern character of some fibre bundle above the Brillouin zone, at least when the magnetic flux through a unit cell, correctly normalized, is rational [99,9]. When it is irrational, the C"-algebra of observables becomes non commutative in a non trivial way : it is type II as was remarked a long time ago by A. Grossmann [46]. It is no longer possible to speak about Brillouin zone, and the previous interpretation of the Hall conductivity becomes meaningless. Nevertheless, thanks to the A. Connes proposal, the C"-algebra of observable can be interpreted as the algebra of continuous functions on an hypothetical Brillouin zone which does not exist as a geometrical object but has sufficientl~/many continuous "non-commutative" functions to be studied by the technics of the algebraic geometry. In particular, the Chern character still exists and varies continuously with the magnetic field, in such a way that the semiclassical approach gives the correct answer. It is widely admitted that the fractional QHE comes from many bodies interactions. A correct mathematical theory of it should require a Fock space, and a second quantized algebra of observable. It is therefore a (non relativistic) quantum field theory. Nevertheless there is a structural question here related to the calculation of the Hall conductivity : somewhere there must be some topological invariant. Is this related to the differential algebra introduced by Arveson [4] and used in the definition of the cyclic cohomology [32,33]? Is the vacuum a closed current in the sense of Connes ? Is the Hall conductivity given by a Chern character in the many body problem ? Besides the internal interest of the physical problem, which goes far beyond the analysis I will give here, it seems that investigating easier situations which are better understood in the physical set up than the intricate questions in particle physics, is an efficient way of improving our understanding oI~ the mathematical tool, and gives the hope.that in the next future, paradoxes will be solved. For this reason I feel that mathematical physicists should take more seriously the mathematical study of simple , realistic models of condensed matter physics where topological invariants exist in order to understand much better what to do in more ambitious questions like the structure of gauge field theories.
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Acknowled.gments : Thi. work benefited from many discussions both encouroelng or discourqing and from sevoral helps to visit many Institutes.givingme the opportunity to meet many exportl. The Slit of contributors to these exchanges would be too long to be reproduced hero, and they have been quoted in previous works, Let me thank them collectivelyhere. Let me bowevor express m y gratitude to A. Connes whese constant enthustum, efficiency and patience in explslning the properties and the importance of C*algebras, convinced me to start this program and to go beyond the formalism It offered it the beginning.
2)- Why are C*-alyebras needed to describe disordered systems ? A real sample used in experimental solid state physics is made of a finite assembly of atoms bound together by electrostatic interactions. However, even though finite and relatively smal] (few millimeters in size, sometime smaller in crystallography), any sample contains so many atoms that it is better described via the use of the infinite volume limit. On the other hand, it appears homogeneous at large scale while at atomic scale the disorder breaks any translation invariance. When studying electronic properties of such a crystal, it is common]y admitted that a one electron approximation is usually quite good. Co]]ective properties of the electrons gas are described through the model of a perfect Fermi gas. The disorder appears simply as external forces coming from the random positions of the atoms or impurities. In this set up, the SchrOdinger operator for an electron, acting on the space L2(i~) (D being the dimension of the crystal), is given by an effective hamiltonian of the form : (I)
H = =(h2/2m)& + :E i vi(x-x i)
where vI is the potential created by the i u~atom or impurity and xI denotes its position. The disorder may have several sources. One is given by the randomness of the position of the atoms. This randomness may come from the occurrence of many defects due to the way the sample has been prepared. It may as well come from structural reasons, namely the thermodynamical equilibrium favors the occurrence of some disorder : this is the case for amorphous materials (glasses) or quasi-crystals. Another source of disorder is also given by the impurities which modify the atomlc potential at random positions. In any case, to describe such a system in full generality, one usually introduces a probability space (t~, ~., I1), the points of which labelling the hamiltonian and describing in an implicit way the configuration of the material. In other words H becomes a function of co E t3. For obvious
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mathematical reasons, one demands that H be a measurable function of ~o, at least in the strong resolvent sense (we recall that Borel sets are the same for the norm and the strong topology in the algebra of bounded operator in a separable Hilbert space). To describe the macroscopic homogeneity of the material we just remark that translating the electron in the sample is equivalent to translating the atoms backward and since the sample looks almost the same at any place, this is just changing the configuration co in ~l. Therefore, there must be an action o~->Tx~ of the translation group Ro on ~l such that, if U(x) is the unitary operator representing the translation by x in the Hilbert space L2(l~) then: (2)
U(x). H~. U(x)" = HTxe
(covariance)
This action will be at least measurable and will satisfy the group property TxTy = T(x+y) . As we shall see in the end of this section,we may assume ~ to be actually a compact space and the I~° action to be given by a group of homeomorphisms. The probability measure will serve later on in dealing with "self averaging" quantities. The framework can be applied to the study of the phonon spectrum as well. For indeed, phonons are elementary mechanical excitations of the atoms around their equilibrium position. In the approximation where there is no phonon-phonon interaction, they are described quantum mechanically as a free Bose field, and only the one phonon hamiltonian must be considered : we must solve the corresponding classical problem. The classical eigenmodes are just eigenvalues of a discrete Schr6dinger-like operator with random coefficients. Again the randomness comes from the disorder inside the sample. In any case this operator will obey a covariance condition given by (2). Even though in practice a sample is given from the beginning and will not change at low temperature during the experiment, the physical energy operator is not the operator H= alone corresponding to the given configuration of the disorder. Otherwise it would imply the cholce of an origin of coordinates inside the sample, preventing the use of its homogeneity : there is no physical reason "a prlorl" to prefer one orlgln instead of one another ! Therefore the observable algebra must contain not only He but also the whole family of its translated. Since in general two elements of this family do not commute the observable algebra w i l l be non commutative in a fundamental way. When performing calculations in quantum mechanics, we need to compute several operators starting from the basic observables. In particular, we will need operators defined through series expansion and the question of convergence will be addressed. Therefore we need to define on the set of observables an algebraic structure which will allow us to make computations, and also a topology which will be essential in proving convergence of infinite
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series, Clearly the algebra obtained will be a W-algebra since we need to distinguish real numbers from the complex ones. However there is a wide choice of possible topologies depending upon the technical point of view we will choose, Nevertheless there is a canonical choice namely a topology which is of purely algebraic origin. There are two kinds of such topological "-algebras : the C"-algebras and the WW-algebras. For in the former case the norm of an element A is nothing but the spectral radius (a purely algebraic object) of A"/~ On the other hand a V/W-algebra is a C"-algebra with a predual (which is unique) [M25]. Equivalently it is a weakly closed "-subalgebra of B(I{,). The von Neumann theorem [M6], shows that it coincides with its bicommutant (again an algebraic property even though it may depend upon the representation). As we shall see the ,W-algebra built out of the translated of the original hamiltonian is too large to contain any relevant informations. For this reason we shall prefer the C"-algebra generated by the hamiltonian and its translated. In a sense this algebra is the smallest object which contains all the relevant physical informations. The next step in describing the formal framework consists in answering the question whether it is possible to recover the nature of the probability space ~ from the physical datas, Since the translated of the hamiltonian must belong to the algebra of observables, it is natural to built (} out of the family of self adjoint operators we obtain in this way. For this reason we introduce the following mathematical criterion to describe what we call "homogeneity". Definition : A bounded operator A on L2(IR D) is called homogeneous if the family of its translated { U(x)AU(x)* ; x~ ~ ] has a compact strong closure~ 0
l
Strong compactness means that for any finite family of vectors (pi,...,(pN ,and any ¢>0, there ls a finite set x I
.....
xN l n R D,such that each vector
U(x) A U(x)Wq)i (XE RD ) stays within the distance c of one of the vectors U(xj) A U(xj)"(I)i (l~j~ I"1)in L2(IRD). In other words, after translation, the operator A reproduces itself everywhere up to c. This is why we called it homogeneous. Actually this notion of homogeneity Is quite weak. Indeed the operator of multiplication by a continuous functlon vanishing at Infinity ls homogeneous according to our definition (It corresponds to the potential created by finitely many Impurities). Another example is given by the following : ProDosltion I : If V E L=(I~ D) then the resoivent of the Schr6dlnger operator I H = - ~ + V Is homogeneous,
o
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To prove this result, we use a Neuman series expansion and the expression of the Green function of the free laplacian to show that the mapping V -> (z-H) -I is strongly continuous if we endow L'(IR D) with the weak topology. Since any ball in L'(I~ D) is weakly compact and since translating the resolvent is equivalent to translating the potential alone, the set of translated of the resolvent lies in a strongly compact set. The main interest of this definition lies in the following construction. The hull ~ --Hull(A) of a homogeneous operator A is the strong closure of the set of its translated. By hypothesis it is a compact space. Moreover it is translation invariant. Thus I~° acts on ~ through a group of homeomorphisms which will be denoted by {T x ; xEIR D }: any 03 E ~] is a bounded operator as the strong
limit
of
some
sequence
A~= U(xk)AU(xk)"
and
we
have
TxCO --U(x)coU(x) ". Therefore, the map (x,co)-> Txco is continuous. In this way, associated to any homogeneous operator, we get a dynamical system which will also be called the hull of A. From the proof of proposition I it follows that if V belongs to L'(I~ °) the hull of the resolvent of H = -A ÷ V is homeomorphic to the weak closure of the set of translated of V in L"(IRD). For instance if V is continuous and vanishes at infinity, the hull is homeomorphic to the one point compactification of 