Statistical Mechanics and Field Theory: Mathematical Aspects

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 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[ _ 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 _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 [(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 ~(~') 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 gA > < 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) 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.

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 ~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. discontinuous.

At

this

transition

temperature

T(q)=l~(q)

the

m e a n energy

(a,b nearest neighbors) and the order parameter 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' (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 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) 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(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 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') 6 (see ( A . 5 ) ) , the usual Peierls' argument gives for B large XB exp (-B'E(~')) .. 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 ) ~


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

~


~ 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~'} ~

,

%~

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 = 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<jc. §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

jl and by Z Z ~0)...~(j) for j0

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

(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 IBIB 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.

102

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

103

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

104

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

105

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 I~°, namely a D-sphere, and, {~] is the set of non wandering points of the flow generated by the translations. More interesting is the case for which V is almost periodic in the sense of Bohr [M4]. Generalizing this case leads to the following definition : Definition A bounded operator A on L2(I~ D) is called almost periodic if the family of its translated {U(x)AU(x)";xE I~D] has a compact norm closure. o By mimicking the construction of the hull of an almost periodic function on I~, one can show that the hull of an almost periodic operator is an abelian compact group, and IRD acts as co-> co+ y(x)where y is a continuous homomorphism from IRo into £~ with a dense image. Following Mackey's notion of virtual group [71,72], such a dynamical system will be called an "abelian virtual group". Let us note that the previous construction is universal because any kind of dynamical system can be obtained in this way. For indeed if ~ is a compact space endowed with an IRD action through a continuous group T of homeomorphisms, and if v is any continuous function on £~, then for each coo in ~, the function x->v(T_.co o) = V(x) is uniformly continuous and the hull of the Schr6dinger operator it defines is homeomorphic the dynamical system (£~,T).

106

In our purpose, for most interesting examples, taking the WWalgebra generated by the family of translated of a homogeneous operator A in L2(IRD) will give rise to the full set of bounded operators. Therefore no specific property of the physical system we want to investigate will be seen out of it. For this reason we shall prefer to take the smallest possible Walgebra which contains A and all its translated, and which is closed in the norm topology : let C~(A) be this CW-algebra. Using a "3c argument", one can show, that if A is homogeneous, so does any element of CW(A), for the product is strongly continuous on bounded sets. Moreover, if co E Q, and i f (Xk)k>0 iS a sequence in IR° converging to co in

(we consider Ro as a subset of co here), then for any B E C*(A) the sequence U(x k) BU(Xk)* converges strongly to some operator B(o which depends only upon co and B. The map 11~: B~C*(A)->B(~ is a *representation of C*(A) into the space of bounded operators on L2(RD). Moreover, the map ~: co E Q-> ~o~ is pointwise strongly continuous and it satisfies the following covaMance condition: (2)

U(x) ~=(B) U(xP = ~T (o(B)

COE ~, X E RD, B e C*(A)

X

As a matter of fact, It cannot be more than strongly continuous In general for we have :

: the representation map n Is polntwlse norm continuous If and only If A Is almost periodic, or equivalently if its Hull (t3,T) is an abel lan vlrtual group. O

The proof Is just an elementary application of the definition. We now address the question whether one can compute expIIcltly the algebra C*(A). To each dynamlcal system (D,T) Is associated a natural C~algebra, namely the cross-product of C(Q), the algebra of continuous functions on ~, by IRD through T [M23]. What Is the connection between C*(A) and C*(Q,T) ? Let us recall the construction of C*(~,T). The vector space Cc(QxR D) of continuous functions on QxR o with compact support is endowed with a structure of *-algebra as follows :

(3)

a. b(co,x) = ~d°y a(co,y) b(T_,/co,x-y)

a*(co, x) - a(T_xco,-x)*

We consider now on @(IR D) the family of 'erepresentatlon ~ defined as : (4)

Tl=(a) ~,(x) - ~dOy a(T xco,y-x) vCy)

11~(a) is a bounded operator. A C'norm is then given by :

107

(5)

II a II : sup~

II %(a)II

Then C'(Q,T) is nothing but the completion of Cc(9Xl~D) under this norm. If a E C"(~,T), ~(~(a) is a strongly continuous function of co and satisfies the covarlance condition (2). Therefore, since 9 is compact, it is a homogeneous operator. We will say that A ls affiliated to C"(9,T) if there ls coo In ~, and a ~ C"(9,T) such that A : ~(~n(a). Not every homogeneous operator A Is affiliated to the C"aigebra of its hull. Two conditions are required for this: first of all A must be "diagonal dominant" namely it must be the norm llmit of a sequence An of operator for which there is Rn>0, such that <TI/~ q)>-0 whenever the distance between the supports of ~ and ~ is greater than R,. Secondly A must be regular, namely It ls the norm llmlt of operators with a continuous kernel. For Instance the operator of multiplication by V is never affiliated with C"(HulI(V)) even if It Is smooth. However we get : ProDosltlon 3: If V E L ' ( R D) then H : -A+ V Is affiliated to C"(£2,T) where I (Q,T)ls the Hull of the resolvent of H. O

To prove this result it is sufficient to compute the kernel associated to the operator A(f):f(-A)Vf(-A) where f Is some smooth function with an LI positive Fourier transform. This operator is strongly continuous with respect to V Jr L'(IR °) Is endowed with the weak topology. 51nce the weak closure of the set of translated of V is homeomorphic to the Hull of the resolvent of H this shows that A Is actually affiliated to C~(HulI(H)). Now, It is clear also that If we now take for f the Fourier transform of (1+p2)-1/2, A ls a norm llmlt of operators of the form A(g) wlth g rapidly decreasing. Then It is still in the C~algebra. Taking V=I we get the resolvent of -A which is therefore affiliated to C"(Hull(H)), and using a Neuman expansion of the resolvent of H we get the result.

108 3)- The tiaht bindina renresentation :

In this section we want to take into account that the samples studied previously look like discrete lattices, even if there is some disorder in them. As we shall see, there is a mathematical counterpart, namely the Poincar~ section [M26], which transforms a continuous Mow into a discrete one. Correspondingly there will be a C"algebra for the discrete flow. The converse operation is called the suspension of a discrete flow (or of the C"algebra). Let us consider the model given in S2-eq(1). Let us assume for simplicity that there ls only one species of atoms. This is not actually a restriction if all species have an homogeneous spatial distribution. We call R the smallest distance between any pair of atoms in the sample. We can think of H as depending on R by setting : (I)

V(x) = T. i v(x-Rxi))

where v is the atomic potential. If the disorder is not too big, we can label the atom by iEZ D : it is not too faraway from the ideal position it would occupy in a perfect crystal. Now, v is attractive, and therefore the atomic hamiltonian H a = -A + v must have bound states in order to bind the electrons. Let E be the energy of such a bound state and let ~ be the corresponding wave function. We will also call 6E the energy distance between E and the energy level closest to E. If we choose the energy to be zero at infinity, where the atomic potential vanishes, the wave function ~ decreases exponentially fast at infinity, and the rate of decreasing is IEI [MI]. Therefore if R is big enough, the functions ~i(x) = ~u(x-Rxi) are approximate eigenfunctions of H = -A + V, and E is the corresponding eigenvalue. Actually this is not quite correct unless we take the limit R->~. To take into account the corrections let us consider the matrix L on Z D defined by: (2)

Lij = = 6ij + O(e-EIR )

as R->,~

This matrix is non negative, and there is Ro such that if R > Ro, ILL-! II < 1 and therefore L-1/2 is well defined. Now we set : (3)

~i--~'j (L-ir~)ji~j

-->

--6ij

Thus we get an orthonormal basis of the subspace ~ of L2(I~ D) spanned by the ~i's. Moreover, II 9i - ~i II : O(e-EE ) as R->~. Let now P be the projection onto tl~. Then for R large enough the

109 spectrum of PHP is contained in the open interval J of width 6E/2 centered at E, which does not meet the spectrum of (I-P)H(I-P). To study the spectrum of H which is contained in this interval, there is a well known method, the first step of which goes back to Schur [101]. It was developed by Feschback and is nowadays called the projection method [101]. Let us define the following energy dependent operator (where Q=I-P): (4)

H(z) = PHP + PHQ {z-QHQ}-~HP

As a function of z it is holomorphic on the complement of the spectrum of QHQ, and therefore it is holomorphjc in a neighbourhood of the interval J. The main result about it is the following Proposition 4: (i) An energy c belongs to the part of the spectrum of H contained in J if and only if 6 belongs to the spectrum of H(c). (ii) ¢ is an eigenvalue of H with eigenvector q) if and only if c is an eigenvalue of H(c) with eigenvector Pp. Then q) can be recovered from the formula :

(5)

Q~ = {c-QHQ}-~3HP~

From the definition of ~i it is simple to show that for any @ in L2(IRD): (6)

-- E + ~'j,t = E <elf;i> + O(ll@lle-ER )

In particular it follows from (2),(3) & (6) that QHP is a bounded operator and that the correction PHQ {z-QHQ}-~HP is bounded by O( e-2EI~ ) as R->~ uniformly for zeJ. Now, the matrix of H(z) in the basis (~i)i is diagonal dominant namely. (7)


= O( e4~EIk i-xj I)

as R->~

Let us remark that Lij is actually a function of xi-xj, and therefore, there is a function ~ in L2(RD) such that ~i = U(xi)q~What we have done here was to replace the unbounded operator H on the continuum IR° by an infinite matrix acting on the sequences indexed by the lattice of atoms. As far as we are concerned with the local properties of the spectrum, the two problems are equivalent. The next step is to introduce a knew C'algebra which will contain the previous matrix. Following A. Connes [27], we need the notion of transversal. Indeed let us consider in ~ the set :

110

= = closure of {T,.coo ; i ~ Z D}

where coo ~ ~ represents V.

I

= is a "transversal" in the following sense (which differs slightly from the Connes definition, to take into account the topology of 9)" Definition

A closed subset = of ~ is called a transversal if for any continuous function f on I~D with compact support, the map (8)

(Co,f)->v~(f)= 7. x:T e~_= f(x) -X

is continuous, for the topology of uniform convergence on compact sets. 0

It is not hard to check that if _= is a transversal, for any o3 ~ ~ the set L(co) = {x~ I~° ; T_,co ~ _} is actually discrete. In addition, given a bounded set A in I~D , if N(co,A) denotes the number of points (or atoms) of L(o3) in ^, then by subadditivity, the limit : (9)

p = lira^ ted IAI-Isup6) N(co,A)

exists. It is a measure of the density of the atoms in the sample. Now let g be a bounded continuous function decreasing at infinity in such a way that Ilgll f(x-i.oo)=F(x) is continuous and bounded on G, and the quantity It(f): M(F) is a positive linear map such that It(1): I. By Riesz's theorem, It is a probability measure on ~3. (ii) is just a consequence of Birkhoff's ergodic theorem [M13]. (iii) expresses the fact that any continuous positive function f on {1 such that It(f): O, vanishes li-almost everywhere, and, by continuity, everywhere. The proof of (v) is a classical result in the study of dynamical systems (see [M5,M11,52]). o The main interest of the trace, is to give a simple mathematical tool which describes what is called by physicists the s e l f averaging observables namely, those physical quantities which are independent of the disorder. They are obtained through averaging a local observable over the sample. It has also been used, without notifying the formal definition of a trace, by R. Johnson & J. Moser [55], in dealing with the rotation number of a one dimensional Schr6dinger equation (see section 6 below). The first non trivial result using this formalism was provided by L. Coburn, Moyer and 1. Singer [24] who proposed to use C'algebras for extending the index theorem to pseudodifferential operators with almost periodic coefficients. This paper plaid a very lmportan[ role in the original work of A. Connes when he generalized the index theorem for pseudodlfferential operators on a follated manifold, differentiating along the leaves. Following this proposal Shubin [93] in the mid seventies, studied the C'algebra of zeroth order pseudodifferential operators with almost periodic coefficients. He also studied the properties of the trace and realized that there is a connection between the integrated density of states and the trace, a result described below. Before giving this formula, we recall that if "[ is a trace on a C'algebra at, then through the GNS construction [M19], there ls a representation l~ on the Hllbert space L2(at,%) (the completion of at under the Hilbert norm Ilall2 = [~(a*a)} 1/2 ) such that if ll is the canonical map from at into L2(at,%), we have : (6)

'[(a*bc) = H e is stongly continuous, Y. is the union of spectra of { ~ ; co in the support of It]. (iv) If co-> He is stongly continuous, and if Q admits periodic I Points for T in the support of p, then the set ~ cannot be nowhere dense. 0

The proof of (i) and (ii) is classical [see 66]. For (ill)and (iv) it suffices to remark that if co belongs to the support of ~t, it may be approximated by a sequence of points co' for which the spectrum of He. is precisely Y.. Since the spectrum does not increase by strong limit [M18], Y. contains the spectrum of He . If co is a T-periodic point, He admits a band spectrum [M7,M22,100]: it cannot be nowhere dense. The previous result applies In particular to the $chrOdinger operators on RD or Z D with a random potential to show that the spectrum ls actually connected [see 66]. In contrast, It has been shown that several examples of SchrOdinger operators with almost periodic potential, have a nowhere dense spectrum. According to the prevlous result, this ls Impossible for a flow havlng a perlodlc orbit in the support of an lnvarlant ergodic probability measure. However, thls property ls actually generlc In one dimension for limit periodic potentials, as was shown firstly by J. Moser [75], and by J, Avron & B. Simon [11]. This is also true for a generlc set or palrs (a,l~) for the almost Mathleu operator (seeS3 eq. 16) acting on 12(Z) as [18,see 2:3,53]: (1)

~(n+l) + ~(n-1) + 2~cos21l(~-na)~(n) = H~Cn)

n~Z

It is also believed to be generic for Schr~}dlnger operator with an almost periodic potential. Cantor spectra have been found also for different examples of Jacobi matrices in one dimension. For instance if in (1) the cosine is

120

replaced by a characteristic function of length a, numerical works indicate that this happens [63,79,80]. There is also a class of models [12,13,15,77], having the Julla set of a polynomial as spectrum. Nobody gave any example of non almost periodic Schr6dinger operator in one dimension having a Cantor spectrum, even though there is no reason "a priori" that Cantor spectrum is impossible for non almost periodic potentials. One interesting question in this respect is the following : Questton 3 : Let C be a Cantor set contained in a bounded interval of R. Let I~ be a probability measure on C the support of which being C itself. Let H be the operator of multiplication by x in L2(C,dl~) and let (pn),~o be the orthonormal basis generated by the Schmidt procedure from the set of monomials. Then the matrix of H in this basis i s tridiagonal [M9]. When is this matrix almost periodic ? Let us note however that the previous property is only generic. Non generic examples of one dimensional Schr6dinger operators on I~ with finitely many bands in their spectrum have been constructed, using the inverse scattering method, in connection with the KdV equation, by Dobruvin, Matveev and Novikov [38]. The situation in more than one dimension is still unclear. There is only one known class of examples of Schr6dinger operators, namely the discrete Laplace operators on "Serpinsky gaskets", which have a Cantor set as set of limit points of their spectrum. This was shown by R. Rammal [85, see I]. However, for the periodic case, Skriganov [95] shown that in 2 dimensions, at high energy the spectrum is connected. We believe that this property survives even for a large class of homogeneous potential, because of the "band overlapping". However it may be perfectly possible that at low energy, there is some Cantor spectrum. Nevertheless, the question remains in these cases to label the gaps, when the spectrum is a Cantor set. The solution to this problem is a question of taste : it depends whether one is a physicist or a mathematician. If one is a physicist, one will observe that the integrated density of states is locally constant outside the spectrum : it is constant on each gap (see fig 1 below). Since it is a strictly increasing function of the energy on the spectrum, two gaps separated by some non empty piece of the spectrum correspond to different values of the density of state. Thus a given gap J can be labelled by the value 1N(E)for E ~ J. A mathematician will not be very happy wlth thls method, for it seems rather arbitrary. He would like some more canonical way. Thinking a little bit about the problem, one realizes that there is a mathematical object which can be affected to a gap J of the spectrum, namely the eigenprojection P(J) =)~(H ~E) for any E ~ J. Since E does not belong to the spectrum, the function x->x(x ~E) is continuous on the spectrum and this eigenprojection

121

. . . . . . . . .

/

o

Fi 9. 1 : ' f i e integ~t~I density of s b ~ for a me dirn~iorel ~ire~r operator on a latUce or for any self ~ljoint Ix~nded operator affiliated to the Irrational rotation algebra.

still belongs to the C,algebra of the Hull ! This not a trivial property, for in a separable C" algebra projections are rather exceptional : for instance the projections of the C'algebra K of compact operators on a separable Hilbert space are all the finite dimensional ones. However giving the eigenprojection of the gap is certainly too much an information. For the spectrum does not change under unitary transformations. Thus only the equivalence class of such projections under unitary transformations is needed to label the gap. More precisely [MI 9,M25] : Definition Let #Ibe a C* algebra with a unit, and P, Q be two projections in A. Then P and Q are equivalent, and we will denote this relation by P = Q, if there are two elements U and V in A such that : (2)

P = UV

Q = VU

If P and Q are self adjoint then one can take V = U*. One could ,ask the question why we have chosen such an equivalence relation. For indeed as far as the spectrum as a set is concerned, any automorphism of the C'algebra would leave it invariant. Identifying two projections obtained one from the other through such an automorphism would diminish the set of equivalence classes in general. Our choice is motivated by the remark that unitary transformations are those automorphisms which are approximated by local automorphisms, when H is approximated by its restrictions on boundedsets. The next problem is to compute the set P of equivalence classes. Actually nobody knows how to do that in general ! However as remarked by

122

Grothendieck [M3], there is some s~ructure in this set provided one accepts to weaken a little bit our notion of equivalence. For indeed, if [P] denotes the equivalence class of the projection P, and if P,Q are orthogonal to each other (namely PQ = QP = O) then P+Q = P~Q is also a projection the equivalence class of which depending only upon [P] and [Q]. Therefore we get an addition in 113defined by : (3)

[P] + [Q] = [P'eQ °] for any P'~-Pand Q'~-Qwith P'Q = Q'P = 0

Again this addition is not defined for any pair of projections, for it may happen that there ls not enough room In A to rotate this pair In such a way that the two projections become orthogonal. However, if we enlarge ,,lt by adding the union over n > 0, of the nxn matrices with entries in ,AI.,this will work. indeed, we will identify P with the matrix P (4)

0

0

0

P=

in A ® M 2 0

0

0

P

Hence in ,~®M2 it is always possible to transform P and Q in ~1. into a pair of equivalent orthogonal projections. However this will not be true for any pair in ~I.®M2 . Thus we will replace AeM 2 by AeM2®M2 and if we keep going we eventually end up with the algebra ,ReK. Now this is enough for K,®IC is isomorphic to ~ and therefore A®t3 is isomorphic to ~ [ ® ~ K . For this reason •~®K, ls called the stabilized algebra. To conclude, the set P of equivalence classes of projections in the stabilized algebra ls endowed with an addition which is commutative. Then by "abstract nonsense", Grothendleck told us that there is a canonical group which can be constructed out of P exactly like we construct Z out of N : this group called Ko(~l), is the set of classes of pairs ([P],[Q]) where the equivalence is given by:

C5) ] [R]e P

[P] + [Q'] + [R] = [P'] + [Q] + [R]



([P],[Q])= ([P'],[Q'])

Then [P]-[Q] ls identified with the class of the palr ([P],[Q]), and the addltlon extends to a group law. Moreover let us note the followlng result : Proaorition 8 [Mlg] :

If A is a separable C'algebra, the group Ko(~D is

I abelian and countable

0

Now we remark that any trace • on ,,lt satisfies the two following properties :

123

(6)

(i) (ii)

if P~-Q if PQ = QP = 0

then ~(P) - ~(Q) then ~:(PeQ) = ~:(P) + ~(Q)

This shows that ~ defines amap ~ . on Ko(,,l.) which is actually a real valued homomorphism through the formula: (7)

%,([P]) = ~(P)

Moreover if • is faithful (namely a ~Oand a=O implies %(a)>0) then "~. is one-to-one. Coming back to our original problem we see that if J is a gap the number q:(P(J)) which, by 5hubin's formula, is identical to the value of the Integrated density of states on the gap, Is equal to the number ~c.([P]) and therefore' GaD labelling theorem ] • Let H be a self adjoint operator affiliated to C*(~,G), and such that its density of states exists with respect to a G-invariant ergodic probability measure I~ on (~. Then the values of the density of states on a gap of the spectrum belongs to the positive part of the countable subgroup of IR given by •[,.(Ko(C*(~,G))). o

I

Corollarv l(homotoDy invariance) • Let H be as before, let J be a gap in the I spectrum of H, and let ~(J) be the value of the integrated density of states on J. Then %(J) is invariant under small perturbations of H in the norm resolvent convergence. Even though this construction may appear rather complicate, it is actually very natural. For instance, the stabilized algebra of a dynamical system is isomorphic to the algebra of its suspension [29,30,8711 Thus stabilizing is nothing more than the inverse operation described in the section 3, namely the tight binding representation . The obvious question now is how to compute the K-group I Precisely this was one of the breakthrough in the early eighties, to give a calculation of these groups. The first important result was given in 1979 by Pimsner and Voiculescu [81,82] and it concerns the irrational rotation algebra: Theorem 1 : If a is an irrational number, let ~[a be the algebra generated by two unitaries U and V such that UV = e~ a VU, (i.e. A a = C*(T,Z), where Z acts on 11" through the rotation Ra ). Then : (1)on A a there Is a unlque trace "[ which ls falthful (prop.6). (ii) the group Ko(A ~) is isomorphic to Z 2. Its generators are given

124

by the equivalence classes of the identity I, and of the "Rieffel projection" PR[86]. (iii) The subgroup ~,(Ko(~lIa)) is equal to Z÷aZ. (iv) If h is any self adjoint element of •a, and J is a gap in its spectrum, there is a unique n e Z such that the integrated density of states takes on the value na-[na] on J.

o

The previous argument leading to the gap labelling theorem1, was presented in a meeting at Marseille in April 1981, and announced in the IAMS conference in Berlin in August 1981114]. On the other hand it can also been found in the review by B. Simon [94] on almost periodic SchrOdinger operators. This result was also proven in a direct way by Delyon and Souillard [35] in the very special case of the almost Mathieu operator (with an arbitrary continuous function V instead of the cosine, as a potential in (1)). They actually used a suspension technics together with the work of Johnson and Moser [55] who gave a version of the gap labelling theorem for the one dimensional SchrGdinger operator on R (see below). Let us also mention that the result of Pimsner and Voiculescu is more general in that they actually computed the K-group for a wider algebra, which is generated by the translation operator U in 12(Z) together with the operator X of multiplication by X(o.a](x-na). Since this new algebra contains the irrational rotation algebra, what they proved was that the K-group of the latter was not blgger than Z2. To prove that it was equal, they used the first work of M. Rleffel [86] who produced an example of projection PR, in the form of a tridiagonal matrix, the trace of which was equal to a. Nevertheless this extension of theorem 1 is important to notice for the SchrGdinger operator (1) with li X as a potential became popular recently because it represents the operator generating the phonon spectrum of a one dimensional quasi crystal. However, even though important, the previous theorem was just a first step toward a general theory. The next breakthrough was performed by A. Connes few weeks after he received the work by Pimsner and Voiculescu, and generalized it in giving a geometrical interpretation together with explicit formulae to compute the .image of the K-group by the trace homomorphism. Actually, as soon as 1977, A. Connes had proved a generalization of the index theorem for pseudodifferential operators on a foliated manifold [26,27,31], and this earlier result contained in essence what we are going to explain now. Definition

Let G be a Lle group of dimension D, acting freely and smoothly on a smooth compact manifold £~. Let XI,...,Xo, be vector fields generating the G-action on ~. Let also li be a G-lnvarlant ergodlc probability measure on Q (lf G ls not amenable li may or may not exist). The Ruelle-Sulllvan current [88] ls defined as the linear

125

form on the space of D-differential forms on 9 as follows"

(8)

< ] I ~l> = f~ dH

Then :] is closed and is positive (i.e. < 11 ~ > ~ o if TI is positive on each G-orbit) and this is a characterization of the Ruelle-Sullivan current [30]. To say that J is closed means that it is zero on exact forms TI =d~ (~ a (D-1)-differential form) and this is equivalent to say that I~ is invarlant by' G. Therefore :} defines a linear map denoted by [:1], from the D-cohomology group HD(fl,P) into I~. Now the main result by A. Connes [30] in our context is the following.

GaD labelline theorem 2 • Let Q,G as before. Then the countable subgroup of IR I given by the image Under the trace homomorphism of the Ko-group of C*(9,G) is equal to the image of the integer D-cohomology group HD(g,Z) by the homology class [J] of the Ruelle-Sulllvan current, o

In practice this means that we must compute a set of D independent D-cycles in 9, generating the homology group H°(Q,IR). Then, by duality, we must exhibit a set of D linearly independent closed and non cohomologous differential forms of degree D, the integrals of which on the previous cycles being integers. Then we evaluate I on this last set of form using (8). The subgroup of IR generated by the numbers that we obtain in this way gives the answer. Let us consider the special case where ~ is a torus of dimension v >D, and R o acts on it via constant vector fields al,...,o o . Let o{ be the v x D matrix the columns of which being given by the coordinates of the o~'s.To get a free action, this matrix must be "irrationar, namely there is no non zero vector m in I~I) with integer coordinates, such that a m be a vector with integer coordinates. Then the dynamical system is an abelian virtual group, and it corresponds to the hull of some 5chrGdinger operator on lqI) with a quasi periodic potential. We know explicitly a set of generating D-forms on T v, namely dcoiIA...AdrUOil ) where the dooi's are the differential of the coordinate functions. There Is also a unique Invarlant measure In thls case, namely the Haar measure dCoiA...Adr3v on the v-torus. The evaluatlon of (8) becomes elementary and this gives :

126

19] : In the quasi periodic previous case, the integrated density of states of a self adjoint operator H affiliated to C"(TV,R D) takes on positive values on the gap of H belonging to (9)

L = ~ (i) Z o~i)

where the a ~i) 's are the minors of maximal rank of the matrix a. 0

A special case is given by 0=1. Then a ts a column matrix, namely a vector in rl v, and the minors of maximal rank are just the coordinates of this vector. A typical potential admitting this dynamical system as a Hull, has the form :

(lo)

V ( x ) = V(CO-GX) = ~.. meZ v

e2i~-xa)m v(m)

x E R

where the right hand side represents the Fourier expansion of v E C(T v). Thus the coordinates of a represents the independent frequencies of V. The "frequency module" is defined as the group generated by the elementary frequencies, and we gel as a corollary [see 55]: ~

:

(i) Let H = -d2/dx 2 + V be a 5chr6dinger operator on R with a quasi periodic potential V. Then the integrated density of states on a gap takes on values in the frequency module of V. (ii) If H is any self adjoint pseudodifferential operator on IR with quasi-periodic coefficients, the same result holds, with the frequency module of the coefficients replacing the one of the potential. 0

The result (i) was first obtained in 1981 by Johnson and Moser [55, see 54], who used all the properties of a second order differential operator to prove it. The other result (1i) is a consequence of 5hubin's analysis, and of the gap labelling theorem 2. The virtue of our approach is that it extends as well to the tight binding approximation : Prooositio~ 8 : (i) Let (Q,G) be a dynamical system, where G is connected, and let =_ be a transversal. Then the stabilized algebras of C"(Q,6) and of C"(-=-) are isomorphic (one says that they are Morita equivalent). Consequently their K-group are isomorphic. (11) If 6=Z o, there ls a compact space ~' together with a I~-action such that (~,Z °) ls isomorphic to the dynamical system of a transversal of ~'. The system (t~',R °) is called the suspension of (k'2,zD ). 0

127

The proofs of this result are scattered in the literature. The concept of Horita equivalence is quite old [see 87], and the connection with the C'algebra of a transversal is the result of works by I't Rieffel [87], and A. Connes [30]. The suspension construction was well known from the experts in dynamical systems for a long time but we can find an exposition of it in [72(remark),96]. As a consequencewe get [see also 42]: Corollary 3 [19] : (i) Let H be a self adjoint operator on Z D affiliated to the CNalgebra of (TV,Z D) where m e ZD acts on T v via the translation by am where a is a v x D matrix the minors of which being rationally independent. Then the integrated density of states of H on a gap takes on values in the set of linear combinations with integer coefficients of the minors of a of any order (with the convention that 1 is a minor of zeroth order). (ii) Let H be the discrete laplacian on Z D with a quasi periodic potential V, the hull of which being the previous dynamical system. Then (i) apply to it. In particular if D=I, the integrated density of states of H on a gap takes on values in the set Z + L where L is the frequency module of V ¢

To finish with this section let us give few more results which were noticed in [14,17]. As we noticed in proposition 7, to get a nowhere dense spectrum we need flows without periodic orbits. This is actually not sufficient. It is also necessary that the image of the K-group by the trace be dense in R. Let us mention one example, noticed by A. Connes, of flow for which this does not happen. A. Connes [29] gave a series of sufficient condition for a discrete dynamical system (~3,Z) where ~ is a manifold, to insure that C"(£LZ) is simple without projection. In this case, obviously, the spectrum of any self adjoint element of C"(9,Z) has no gap. : let F be a discrete subgroup of SL(2,Z) with a compact fundamental domain, and let ~ by the compact space SL(2,1q)/i'. Let q~ be a minimal diffeomorphism of Q (namely such that any orbit is dense). The horocycle flow provides such an example. Then the Ko-group for this discrete flow is equal to Z. In particular since the flow is minimal, its C"algebra is simple [29,43], and any trace "[ is faithful. Therefore there is no non trivial projection otherwise there would be a non zero projection P with ~(P) ~Z. Since Z is discrete and ~ compact, ~ is normalized, and we must have 0 (~(P)~1. Thus ~(P)= land ~(I-P)=0 which implies P=I. Let now v be a continuous function on £~, and H be the operator on 12(Z) given by : (I l)

H~p(n)= ~p(n+l) + ~p(n-1) + v(q)-nco) ~p(n)

128

Then H has no gap in its spectrum. The next example [17] goes just in the opposite direction: ~ : l e t (i2)

us consider the 5chr6dinger operator on 12(Z) • H~(n)= ~(n+l) + ~(n-l) + hX(o.~i(x-na) ~(n)

where 1,a,13 are rationally independent. It was shown in [17] that the gap labelling theorem was different from what was obtained from the Pimsner Voiculescu theorem. The reason is that in the corresponding C~algebra of the Hull, there is a new projection, namely X(o,~] the trace of which being 13. For this reason the K-group is strictly wider and its image by the trace contains Z + Za + Zl3. The same phenomena would appear if instead of taking the irrational rotation by a we chose a Denjoy diffeomorphism of the circle [M14,04]. Let us note the last problem : : Thank to the gap labelling theorem, a necessary condition for a self adjoint operator H, affiliated to C*(~,G) to have a Cantor spectrum is that the image of the K-group by the trace homomorphism is dense in the real line. Another necessary condition is that Q be free of periodic orbits. Let us suppose that these two conditions hold. Has a generic self adjoint element of C*(Q,G) a Cantor spectrum ?

129

6)- The case of flaws : Joh.n,son's aporoach. When investigating the properties of one dimensional SchrSdinger operators with random potential, R. Johnson [56] gave another version of the gap labelling theorem, using the notion of rotation number already described by R. Johnson & J. Moser [55], and by M. Herman [52]. He actually found an explicit formula which turned out to be nothing but the Connes formula given in S5 eq.8. However, being more precise, this approach allows us to understand the origin of the "Ruelle-Sulllvan" current. Let us consider a compact metric space ~, together with a continuous flow tER->Ttco, and let I1 be an ergodic invariant measure on it. Let now co->M(co) be a continuous map on ~ with values in the set of 2x2 matrices with real entries. We now consider the ordinary differential equation:

(I)

dX/dt = M(T_tco)X(t)

X(t) E ~2

Without loss of generality we may assume that M(co) is traceless for all co. The solution of (I) can always be written as • (2)

x(t) = @(co,t) X(O)

where @(co,t) is a 2x2 real matrix with determinant one satisfying the cocycle properties (3)

¢(co,0) = 1

¢(co,t+s) = ¢(T_sco,t) ¢(co,s)

We will say that (1) admits an exponential dichotomy [89,90] if the trivial fiber bundle ~xl~ 2 splits into a direct sum of non zero subbund]es E+eE- such that : (i) if (co,X) ~ E-+, then (T_tco, ¢(co,t)X) e E-+ for all t~lR (invariance of E+-) (ii) there are positive constant K and r independent of (co,t) such that if (co,X)~ Et then II m(co,t)x II ~ Ke~('rt) An example of such flow is provided by a one dimensional SchrOdinger equation (where V is a continuous function on 9):

-d%/dt 2 + V(T_tco)~(t)= E ~(t)

(4) where we set '

Xl(t) = ~(t) (S)

X(t) -

I0

#E

M(CO) I_×2(t) = ~v'(t)IV'E

~E+V(co)/4E 0

130

It admits an exponential dichotomy provided E does not belong to the spectrum of the self adjoint operator described formally by (4) on L2(IR) [56,90]. More precisely, for each OOE~ there ls a solution u+(oo,t) (resp. u-Coo,t)) of (4) unique up to a multlplicative constant, decreasing exponentially at +~ (resp. -~) together wlth its flrst and Its second derivative. Let X+(oo,t)be solutions of (I) In E +, written in the form : X+-(oo,t) = r+(oo,t) e(e+(co,t))

(6)

where e(e) is the unit vector of components cose and sine. We remark that the line passing through X-+ is uniquely determined by e-+ (modulo ~). Thanks to the uniqueness of X-+, it is easy to see that the angle 8z(oo,t) is actually a function of T_tooonly, which we will denote by e± again. The rotation number of this solution is defined as the amount of angle per unit length namely : p+ = limt_,±" {@t(T4co) - e-+(oo)}/t

(7)

Let us assume that ~ is a manifold and that the flow T on It is defined through the vector field ~. By using the Birkhoff ergodic theorem, and provided e + is differentiable in oo, we get, for l~-almost every ooo:

(8) p-+ = llmt_>±. (l/t)~o t ds ae-+(T_sooo)/as = J'dp(oo)<de-+l(>(oo)= < ~ I de-+ >

where ~l is precisely the Ruelle-Sullivan current for the dynamical system. Let us remark that de -+ is closed but not exact in general for e t is defined modulo #, and in particular, its integral on any loop (or 1-cycle) in ~ is an integer multiple of r[. Therefore the cohomology class [de+/~] belongs to H~(~,Z). Thus we get: T.beorem 2 (R. Johnson) : (i) The rotation numbers of the flow (I) satisfies p+=p-, and p-+lll belongs to the image of the integer cohomology group Hi(•,Z) by the homology class of the Ruelle-Sullivan current. Moreover it is a homotopy invariant. (ii) If the flow (I) comes from the Schr6dinger equation (4), where E belongs to a gap of the spectrum, then the rotation number p-+Ml is equal to the integrated density of states and satisfies: (9)

~i(E) = p-+/~ = #E/~ fdp(co) {1 + cos2(e-+(oo)) V(co)tE)

131

Sketch of the proof : The first part is a consequence of the previous reasoning. The second is an application of the Sturm theorem : the number of zeroes of a solution of (4) in a finite interval [-L,L] with some boundary conditions and the number of the corresponding eigenvalues smaller than or equal to E, differ at most by 2. On the other hand the number of zeroes of ~ in [-L,L] differs from {8+-(T_LcO)-B+-(TLCO)}/~by at most 1. Dividing by 2L and lettlng L go to infinity gives the identity between the rotation number divided by ~ and the integrated density of states. From (5) we get :

(iO)

dO-+/dt = V'E {1 + cos2(O±) V(T4o~)/E } - (T4co)

which gives the last formula.

0

The previous analysis for flows extends to the case of discrete maps, namely: ( 11)

X(n+ 1) = M(T-nco)X(n)

n~N

The hypothesis will be now the following: M is a 2x2 real matrix valued continuous function with determinant one which is homotopic to the identity matrix. We then replace the discrete flow (~,T) by i t s suspension (S~,ST) where S~ is the quotient of ~xlq by the equivalence relation (o3,t)=(To3,t-1), and ST is the quotient of the flow (co,s)->(o3,t+s). In much the same way, we replace the discrete flow (co,X)->(T-lo3,M(o~)X) on the trivial bundle ~xl~ 2 by its suspension. Since M is homotopic ti the identity, S(Qxl~2) is homeomorphic to S~xiq 2. If (~,T) admits an exponential dichotomy, the suspension admits also an exponential dichotomy. Thus the previous analysis goes through. However we can now define the rotation number directly from (11) as was proposed by M. Herman [52], namely : any matrix in SL(2,1~) defines a diffeomorphism on the projective space pS(R), the set of lines in R2 which can be identified with the unit circle or with the torus T=Iq/Z. Thus we get a mapping which is continuous and such that f~ :e E T-> f(o3,e) ~ T is a diffeomorphism of T. Let us denote by f again the lifting or f on ~xl~. It satisfies : f(co,x+l)= f(co,x)+1 (xEl~). The rotation number ls then deflned as : (12)

p = 1~llmn_>~ (fT-n+l~o...of(o(X) - x}/n

That this limit exists formic-almost every oo and uniformly in x was one of the results of [52]. It is then easy to see that this rotation number is the same as the one defined through the suspension. Again also, this approach can be used to investigate a one dimensional discrete Schr6dinger operator of the form :

132

(13)

H~(n) = t(T-n-ico)~2(n+l) + t(TInco)~(n-1) + v(T-nco) ~(n) = E~u(n)

where v,t are continuous real functions on ~ and t is positive (actually it is sufficient that t be complex and non vanishing). Then the equation H~ = E~uis equivalent to (11) with: (14)

M(co)

=

IIE"X~)I t(T-'~)

t(~Y~T-'~)IO

In this case also the density of states coincides with p/~. The results in examples are therefore identical to those given in the previous chapter. Let us finish this section by describing an application of this framework to some problem in classical mechanics. Let F be a monotone twist mapping of the annulus A=Tx[O1], namely an homeomorphism which preserves the ends of A, Tx{0} and Tx{l}, which is area preserving and such that if y < y' then F(x,y) < F(x,y'). The restrictions of F to the ends define two diffeomorphism of the circle the rotation numbers of which will be called p(°) and p(1). It follows that p(O)< p(1).One example is given by the "standard map": (15)

F(x,y) = (x +y+ksin2~x, y+ksin2~x)

The Aubry-Mather theorem [7,8,57,58,75] shows that for any irrational p in the interval (p(o),p(1)), there is a closed invariant subset M(p) such that the restriction of F on it is conjugate through a homeomorphism, to the restriction of a diffeomorphism f of the circle of rotation number p to its (unique) minimal invariant set. For indeed [see I"114], any diffeomorphism f of [ has a unique minimal invariant set which is the full circle if it is smooth enough (depending on p), whereas it is a Cantor subset of T otherwise, as was firstly shown by Denjoy [37]. Thus M(p) is homeomorphic either to a circle or to a Denjoy set. If p is rational the situation is more involved [see 73]. Let F denote again the lifting of F on B=Rx[0,1]. Since F is area preserving, using the monotone twist property it can be shown [73] that there is a Ci periodic function h(x,x') on R2, called the generating function of F, such that y'=ah/ax'(x,x') and y=-ah/ax(x,x'). If we set (Xn,Yn) = Fn(x,y) the sequence {xn} satisfies the following non linear equation: (16)

ah/ax'(xn_t,x n) + ah/aX(Xn,Xn+1) = 0

which gives for the standard map : (17)

xn+1 + xn_1- 2X n +ksin2~x n = 0

133

If now (x,y) ~ M(p) the Aubry-Mather theorem means that there are f,g the lifting of two homeomorphisms of T, and ~cT., the minimal set of f, such that Xn = g(fn(~)) and that the rotation number of f is io. The linear stability of M(p) is described through the linear equation governing the evolution of an infinitesimal change 6xn = ~u(n) in Xn, namely, if h is in C2, through the equation (13) with ~ = ~., T = f, E=O and: (18)

t(~) =

a2hlaxax'(gof-l(~),g(~))

v(~) =

a2hlax'2(gof-l(~),g(~)) + a2hlax2(g(~),gof(~))

The rotation number of this discrete system is called here the amount of rotations. It has been studied by J. Mather [74] in the special case where p is a rational number and {x n} is periodic, where it is connected to the Morse index. He found a result which can be interpreted in term of K-theory. Let us give the result in the case where now p is irrational. If E=O does not belong to the spectrum of H, a property which is likely to be generic if one believes that the spectrum of H is a Cantor set, then the amount of rotation is nothing but the integrated density of states (if one normalizes the angle to 1 instead of tf). Therefore the gap labelling theorem I shows that it belongs to the image by the (unique) trace on C*(T.,f) of the Ko group of this algebra. The uniqueness of the trace comes from the uniqueness of an f-invariant measure on [ [MI4]. The K-group of this algebra has been computed by I. Putnam, K. 5chmidt & C. Skau [84]. I f H is the unique f-lnvariant probability measure on T, I1 is actually supported by ~. If 7. is a Cantor set then T\Z ls the union of at most countably many intervals which are the "gaps" of ~.. Then Q(f) is the set of values of g([,) = So~ d~t(ll) when [j varies through the gaps of ~.. Clearly Q(f) is countable, and it is known that if R(p) denotes the rotation by p, then gof = R(p)og, showing that Q(f) is also R(p) invariant in T identified with [0,I). Therefore there is a family of real numbers {),(i); I< i0. In particular one immediately sees that the two dimensional resistivity is also measured in Ohm whereas the three dimensional one is measured in Ohm x m. Therefore a two dimensional measurement of the Hall resistivity will give a measurement of the Ohm standard without any reference to any unit of length, an advantage only if the accuracy of the Hall measurement is high enough. This was accomplished a century after Hall, with the experiment of Von Klitzing, Pepper and Dorda [62] leading to what is called nowadays the Quantum Hall effect (see below). The next step is the L.D. Landau theory [68] developed in 1930. In a pure metal, like the copper or the gold, the charge carriers can be considered as

136

free independent particles. They constitute thermodynamically a perfect Fermi gas, and therefore at relatively low temperature, their energy lies below the Fermi level. The motion of each particle is governed by the Schr6dinger equation: (3)

H~u(x) = {-ihcWo'x - qA/c}2/2m ~u= E~u(x)

where A are the three components of the vector potential created by the magnetic field,q = -+e the carrier charge, and m their effective mass [M20]. In the two dimensional approximation, the motion in the direction perpendicular to the strip is frozen out. On the other hand if the sample is large enough, one can consider it as infinitelyextended, and it is therefore identified with R 2. The previous operator can be written as : (4)

H -- [Ki2+ K 2 }/2m

Ki = {-iha/~X i - q A i / c }

i=1,2,

We remark that the Y's satisfy the following canonical commutation relations : (5) [ K i , K2 ] = ihqB/c I

Therefore H appears as the hamiltonlan for a harmonic oscillator, and the energy spectrum is now : (6)

En : hcoc (n+ 112)

n = 0, t,2 ....

~c = IqlB/mc

coc is called the cyclotronic frequency and corresponds to the rotation frequency of the classical motion of a particle in the magnetic field B. When B->0 the spectrum becomes continuous. The current is now given by the vector valued operator : (7)

j=qv=(iq/h)[H,x]

=qK/m

Let us assume now that at time zero, an electric field E is turned on. Since we consider the infinite volume limit, the two directions in the plane of the sample are equivalent. Let us choose the 1-axis parallel to E, and let c be the modulus of E. The evolution of j after t=O, is now governed by the modified hamiltonian H(~) -- H + qc x I namely : (8)

J(t) = e iH(c~J~ J e 4~`~/h

Using the Ferml-Dlrac statistics, its thermodynamlcal average at inverse temperature 13=(kT)-i is then given by: (9)

<J(t)>~ = timA_>;~2 IA1-1 T r ( x ^ { ] +el~(H-EF}}-lJ(t))

137

where X ^ is the indicator function of the finite box A. We remark that at zero temperature, each Landau level corresponds to a density of states equal to the quantity B/@, where @ = hc/e is a quantum of magnetic flux. This can be seen by replacing j by 1 and the Fermi distribution by the eigenprojection on the given level in the right hand side of (9). If (n6) is the charge carriers density per unit area, we see that the number of level which can be filled is v = (n6)@/B, a quantity called the band f i l l i n g . It turns out that the right hand side of (9) can be computed exactly: it is made of a time periodic function of t at the cyclotronic frequency. The constant part is parallel to the 2-axis, and its amplitude is • (10)

Ija v I = lim~_>= I t -1J'o~ dt <j(t) >t = (e2/h)c v(t~) V(6) : ~.. n~O {! +e p(~ c (n+t/2)-E F)] -1

The resistivity is now a 2x2 matrix such that E : pj and p-t=o is the conductivity. From the previous calculation, it follows that both matrices are antidiagonal, which means that -

(11)

p =

p

0

0"=

aX× =O

OxY=OI

0 =0 xy

%--o I

0 = e2/h.v(t~)

I n particular, this free system is at the same time a perfect conductor and a perfect insulator in the electric field direction I In absence of a magnetic field the situation would be just the opposite. The diagonal coefficients of p which in this case vanish, are called m a g n e t e r e s i s t i v i t y . Actually a real sample is never strictly two dimensional. The motion in the magnetic field direction is governed by the kinetic part p32/2m with a zero boundary condition on the upper and the lower sldes of the strlp. Since thls part commutes with H, it simply adds a dlscrete set of elgenvalues to each Landau level. They exhlblt therefore a slight broadening but do not change the conclusion. On the formulae (10) & (1 1) one sees that the factor v(6) admits two ]lmlting behaviors. On the one hand at reasonably high temperature (i.e. for 6 small), the discrete sum In (10) can be replaced by an integral If 6hcoc Is small. However 13EF is generally quite big. In the limit where I3ho~c -> 0 and 6EF -> = we get the classical result a = e2/hv = (n6)ec/B, whereas in the limit of low temperature (l.e. ~-> =), v(6) converges to a step functlon taking on Integer values (see flg.3 below). There are two kind of experlmenta] devlces [M2] which are used in the measurement of the Quantum Hall effect. The first one Is the metaloxyde-semlconductor fleld effect transistor (MOSFET) the other one is a

138

ET

Nall

5e2~ h 4~ 2 h

o,T,oo,

h I

2

5

4

5

Fiq. 3 : The c | ~ l ~11 ~ r ~ t i v i t y (hi~ Leml~r~t~'e) end tJ~ qu~Lum one (eL zero Leml~ture) for a two dimensional free Fermi gas as a funcUon of l~e ~ filling v=(n~)hc/eB - (nS)~/B. EXl~rlmentally v can be varied eltt~r by changing the rnagneLic field B or by d ' ~ n g ~ c ~ carriers densiLy n.

heteroJunction (e.g.AlxGai_zAs-AsGa or InP-lnxGai.~As). In these devices one creates a thin layer, called an inversion layer, at the interface between two components, where electrons are confined and have a two dimensional motion. In a heterojunction, the principle is a little bit different [M2,97] but leads also to the existence of a thin layer of electrons with a low density. However they are practically better to use nowadays for the interface is free of impurities, and themobility of the electrons in the inversion layer is usually much higher.Thus quantum effects are likely to appear at higher temperature. The previous theory of the free Fermi gas must be improved in practice to take into account the influence of the crystalline structure of the substrate on the electrons. In particular, the effective potential will give rise to a broadening of the Landau levels. There is no reason that the previous result, namely the quantization of the Hall conductivity at low temperature may survive. On the other hand the direct conductivity ( ~ does not vanish usually, and exhibits some oscillations when varying the magnetic field or the charge density carrier : this is the de Haas-Shubnikov effect [M20]. However numerical calculations, based upon approximate theories, taking into account the disorder, and performed during the seventies [M2,2] predicted that such a quantization may survive. The question was to know with what accuracy. In 1975, the japanese group of S. Kawaji [1"12] had already observed some deviation from the classical law (11) at low temperature. The samples used at this time improved rapidly afterward, and in 1980 two groups S. Kawaji and K. Wakabayashi [59,60] in Japan and K. von Klitzing, G. Dorda and M. Pepper [62] from the Max-Planck Institut at Grenoble observed that the Hall resistance admits some plateaux when varying the band filling. The von Klitzing group also observed that these plateaux corresponded to integer

139

mu]tlp}es of e2/h for the Hail conductivity, wlth an accuracy better than 10-s (see flg. 5 below)! Since this time the experiment has been performed by several groups, and the accuracy Is better than 10-71 Since e2/h ls a physical

Upp ImV

nl~

~[

[I

UHIm¥ 25

]~

II

p-SUBSTRATE HALLPROBE [IL~ [~ RDRAIN

)]

, F ~ ! SURF~Ec~.Ne. I

2O

15

)

i

/

Upp

SIO'5

'

ol

l/l/ 5 n=O

10

n=l

. 20 n=Z

1'5

V 25 J

. vg~v

Fiq. 4 :Recording of the Hall voltage, UH , and the volb~je 6"op bet.we~ the pot~Ual probes, U~, as a funcUm of the ~te voltage at T-1.5 K. The rre~Uc field h 1ST.

The oscillaUon up to i ~ L~dau level n=2 is shown. The Hall v~It~je and U~ proporUmel to pxyend Pxx respecUvely (p=~-lsee eq. ll).The Inset shows a Lop view of the device with a length of L=4OOIJrn, a width of W=50~am, and a distance between the potential probes of" Lpp=13Ol~m, (Taken from rer, [62])

constant related to the fine structure constant, the Hall effect provides an independent way of checking the validity of the quantum electrodynamics. As explained already, it also provides a measurement of a universal unit of resistance (RH; e2/h = 6453,2 ohms ) and a new Ohm standard. The main surprise was not the Hall effect itself but the extremely high accuracy of the result. The idea was that it may have a very deep and universal origin. The first explanation was given by Laughlin [6g] who related this quantization phenomena to the gauge invariance of the hamiltonian describing the electron. More precisely, he considered a sample having the form of a ring (see fig. 5 below) of perimeter L and width 6. He assumed that a magnetic field of constant modulus was crossing the loop perpendicularly to it. The Hall current is then given by the adiabatic derivative of the total electronic energy U of the system with respect to the magnetic flux through

140

T

Fla.5 : Left :diagram of metallic loop. Ri~t :density of states without (top) 9El with (bottom) disorder. Region of delocalized states are shaded. The dashed line indicates the Fermi level (Taken from ref. [69])

loop. This may be obtained through a "Gedanken" experiment, by adding a flux q~ in the middle of the loop, and increasing it slowly in such a way that the system be constantly in equilibrium. If the additional magnetic field 6B created by the flux q~, vanishes on the loop, we can describe it with a uniform vector potential 6A = ~/L pointing around the loop. Let x be the coordinate around the loop and y-the coordinate transverse to it. We also assume that a transverse electric field £ exists due to the Hall effect. The one-electron hamiltonian is given in the Landau gauge by :

(12)

H(6A) = {(px-q6A/c+qBy/c) 2 + p2 }/2m" + W(x,y) + e£y

with periodic boundary condition with respect to x and Dirichlet boundary conditions at y=O and y=& We can change this operator through a gauge transformation multiplying the wave functions by the phase factor ei 2 ~ where (~=~/4) is the ratio between the additional flux and the quantum of flux. If a is not an integer, the new wave functions have a discontinuity at x=O. This is a unitary transformation which suppresses the 6A dependence in the first term of (12) but modify the domain of H(6A). The advantage of this representation is that in this frame, H(6A) becomes a period one function of the parameter (z. If we now consider the situation where the potential W can be neglected, the one-electron wave function has the form : (13)

~u(x,y)= e I

~

h(y-a) e-((y-a)/r)2

where h is some Hermite polynomial properly normalized. The corresponding energy is linear in a. Changlng (x from zero to one, namely changing the flux through the loop by one quantum unit, has the effect of changing a into a-A6A/B. Since this transformation maps the system back to itself, the energy increase due to it results from a net transfer of n electrons from one • edge to the other [69]. The energy increase is then given by neV where V is the

141

Hall voltage between the edges. On the other hand the current is given by I m cAU/A~ m ne2Vlh. Hence the Hall conductance (in two dimension it equals the conductivity) is an integer multiple of e21h. If the,system is now dirty we must assume that the Fermi level lies in a gap of extended states. In this case the Hamiltonian is still periodic in a and the same argument holds even though now we have no excitation of quasiparticle across the gap, and the charge is transferred only through the extended states with energy close to the Landau ]eve] as in (13). The number of electrons transferred in this way may be different but it is still an integer. Clearly the argument must be supplemented by some more rigorous study. More recently Avron and Seiler [I0] produced an argument, using [9,99], which made the Laughlin argument rigorous under the condition that the Fermi level lies in a true gap. Again in this latter work, the topology of the sample seems essential. We should be able to avoid such a constraint which does not fit with the experiment. Several works were useful to investigate each part of the argument. In his early work, using weak-scattering calculation, Ando & Aoki [3] showed that the presence of an isolated impurity does not affect the Hall current. A similar result was obtained by Prange [83] for a 6-functions impurity, to the leading order in the drift velocity c:F_,IB.Later on Thouless [98], showed indeed that such an Invarlance holds at least for weak disorder, when a true gap occurs between Landau levels. An important step was performed by Den Nijs, Kohmoto, Nightingale and Thouless [99] (N2KT) who recognized that the Hall conductance for a perfect crystal is given by a homotopy invariant quantity over the two dimensional torus given by the Brillouin zone when the Fermi level lies in a gap. Avron, Seiler and Simon [9] showed that it is actually the Chern character of some fibre bundle over this torus. This torus sits in the momentum space rather than in the real space, and is independent of the topology of the sample. The very high stability of the result under perturbation is explained by the homotopy invariance of the Chern character. However, they required the magnetic field to satisfy a rationality condition, namely that the flux through one unit cell be a rational multiple of the quantum of flux. Up to now, no rigorous proof is available yet, when the Fermi level lies in a gap of extended states, and the topology of the sample is arbitrary.

142 O)-

The ouontum Hall effect and Connes's cohomoloo_g :

In thls section, we intend to describe a mathematical framework using C*-algebras In order to give a local proof of the quantum Hall effect, Independent of the topology of the sample. We follow the ldea of the N2KT paper to express the Hall conductivity in term of a non commutative Chern character. The main approximations are the following : - we consider only the one electron theory. - we work with a 2D disordered sample of inflnite size. In this approach however, we shall miss one point, essential for a physicist, namely, that the result will be valid only if the Fermi level belongs to a gap of the spectrum of the one body hamiltonian. We shall indicate what should happen if the Fermi level belongs only to a gap of extended states. The Schr6dinger operator for a charge carrier in the sample submitted to a uniform magnetic field perpendicular to it is given by" (!)

HB = l/2m{(Pi-eAl/c) 2+ (P2-eA,2/c)2)} + V~(Xl,X 2) = HO+ Ve

where m ls the effective mass of the charge carrier, e its electric charge, and co represents the effect of the Impurities on the potential. As previously, co will be taken in a compact metr|sable space (l w l t h (2)

V~(XI,X2) = v(TICO)

v e C(Q)

X = (X1,X2) E I~2

Let us introduce the vector valued operator K = p-eA/c and for ~ E R2 and f a function in the Schwartz space S(1~2),we set' (3)

W(~) = exp(il~K/h)

w(f) =

f(E)

These "Weyl operators" fulfill the following commutation relations • (4)

W(l~)W(~') = W(E+{') ei~aE'E' a = B/e = eB/hc

I;*~'=(~16'2- ~'l~ 2)

It ts a well known fact that the C"-algebra generated by the W(f)'s is isomorphic to the algebra of compact operators iI(,on a separable Hilbert space. Moreover the resolvent of the free part of the hamiltonian belongs to this algebra. We shall denote by A the C~-algebra generated by operators of the form V~W(f) with V as in (2) and f t n the Schwartz space. This algebra can be abstractly reconstructed as follows [see M23] : we consider the space of continuous functions with compact support on ~xR 2 endowed with the following structure of "algebra :

143 (5)

ab(co,l~)= JR2d2l~' a(o~,l~')b(T41.co,~-[,') einal~'l~" a*(o~,l~) = a(T ~o),-I~) *

We built ,,1 as in the section 2 (equ.5) with the representations ;I~ acting on L2(R2)

•

(6)

ne(a)ku(x) =/1~2d2~ a(T.=oJ,l~-x) ejTkd~'x ~(~)

One can check that the resolvent Re(z)--(z-Hm)-i of the hamiitonian Hm has the form rle(r(z)) for some r(z) In ,,1. If now I1 is a T-invariant ergodic probability measure on ~, one can define a trace "[~ on a as before by the formula (3) of the section 4. In addition we have a differential structure [28,32,33] through the data of the following two derivations of ~1. (7)

6ja(o),l~) -- 2111F~ja(o),{~)

J= 1,2

or ~e(6i a) = 2i11 [ I%(a), Qj] where the Oj's are the operators of multiplication by xj in L2(IR2). The main remark of Thouless et a]. in [99] consists in writing the Hall conductivity by mean of the Kubo formula, in a way which identifies it with a Chern character. In our notation this gives the following • PrODOSi t i 0 pt 1 : The expression of the Hall conductivity according to the Kubo formula at zero temperature is given by"

(8)

oH = e2/h (1/21~1) T~{PF[6iPF,62PF]}

where ~e(PF) is the eigenprojection of He on energies smaller than or equal to the Fermi energy EF. This formula is valid provided theFermi energy lies in a gap of H~ (p-almost surely),

o

Before using this result, let us comment about the way this formula is derived. l)-The proof of it is just a matter of calculation once the Kubo formula is accepted. To compute the Kubo formula however, one should use the following steps : (i) as In the section 7 (eq. 7-10) one must compute the current at time t after turnlng on an electric field e In the x I direction ; (ii)

144 at zero temperature, the Fermi Dlrac function reduces to the projection PF ; (iii) one computes the time average of the statistical average of the current, and (iv) we compute the first order term as the electric field ¢ goes to zero. Actually, in practice one usually exchanges the time averaging and the limit ~->0. The control of this exchange has been performed in this particular case by R. Seiler [92], for a finite size sample. Since there is no thermal dissipation in this problem, the difficulties in doing this seems purely technical. One does not know how to extend the Seiler proof in our framework case. 2)-In the course of the calculation, one uses the fact that the Green function of H(~, decreases exponentially fast at infinity when the Fermi

energy lies in a gap. Following the arguments given by Prange [83], Thouless [98], and Halperln [50], It is likely that the calculation extends to the case for which the Fermi level lles In the pure point spectrum, for In thls case, since the states of energy close to EF are localized, the Green function still decreases fast enough at infinity to insure the convergence of the integrals. However, as shown by FrGhlich and Spencer [44], one must be careful with "almost sure" properties. 3)-It has been proved rigorously recently [20] that in 2D, in the framework of a tight-binding approximation, all states are localized at high disorder, and the spectrum is pure point. At lower disorder, only the states corresponding to energies in the band edges are localized. Once we are ready to admit the previous formula, one remarks with A. Connes that the expression : Ch(P) = (1/2 ill) %~{P[GIP,62P]}

(9)

where P is a projection in A, satisfies the following properties: (10)

(i) (ii)

if P=Q then if PQ = QP = 0 then

Ch(P) = Ch(Q) Ch(PeQ) = Ch(P) + Ch(Q)

This shows that Ch actually defines a mapping on the K-group of A, and that it is a homomorphism. It is therefore sufficient to know a set of generators of the K-group in order to know what are the possible values of Ch(P). Thank to the proposition 11, the Quantum Hall effect will be established in this set up once we are able to answer yes to the following questlon : Question 5 : Let (~,T,~2,p) be a dynamical system as before, and let A(QXT~2,a) be the corresponding algebra constructed according to eqs.(5-6). Then is the Chern character Ch(P) of any projection in this algebra an integer ?

145

The answer to this question is yes in a certain number of cases. First of all, if (~ ls reduced to a point the algebra is nothing but the algebra of compact operators and the answer is trivially yes. More involved is the case for which V is periodic. In this case, Q is a two dimensional torus ]-2 and the algebra ls Isomorphic to the stabilized the rotation algebra A . The answer In

this case is still yes thanks to the Pimsner & Voiculescu [82,86], Rieffel [] who computed the two generators of the K-group, and to an explicit calculation of A. Connes [30], who computed their Chern character. Note that In this previous case, the spectrum is likely to be a Cantor set [18,23,53], and the gaps are not necessarily associated with Landau levels. At last, the answer is still yes at small disorder, for in this case, H~ is close to an operator (in the norm sense) for which the Chern character can be explicitly computed. Using the homotopy invariance of Ch(P), the value of Ch(PF) is still an integer as far as EF stay in a gap while Lurning on the disorder. Let us note at last the two homotopy invariance results : Propositiol

12 : (i) Let us assume that the potential V in (I) depends continuously (in the norm sense) upon a parameter ~. Then the Chern character Ch(PF(~,)) is independent of ~k as far as the Fermi level stay in a gap. (ii) The Chern character Ch(PF) is continuous in a (i.e. or the magnetic field) as far as the Fermi level stay in a gap.

o

The first result is just the usual homotopy invariance of the class of K-theory of a projection. The second one can be shown by considering the universal algebra obtained as the union over a of the previous ones [41]. Then the trace "c~ can be shown to be a pointwise continuous function of a.

146

RprEN,O,III $hubin's formuln for n discrete notion Let G be a discrete, countable amenable group. Let ~ be a compact space on which O acts by a family of homeomorphisms such that the map (~,x)->x-loo be continuous. Let I.t be a G-lnvariant ergodlc probability measure on ~ and as in the $4, let T~ be the corresponding trace on C*(Q,G). Let H be a bounded self adjoint operator on 12(G) and o~0 in ~, h in the C'algebra of the dynamical system (Q,G) such that : (I)

H = Tie(h)

For A a finite subset of G let X^ be the projection onto 12(A) in 12(G),and let H A be the restrictionof H on A namely

(2)

HeA = X^ TI (h) X^

Since A is finite, H A

is finite dimensional, and its dimension is IAI, the

cardinality of A. The density of states was defined in the $4 eq.2, and gives rise to a probabilIty measure def Ined b y

(3)

I d~l'~(E) f(E) = I im~_~ I A nl-iTr (f( HeA .))

where A n is a FBlner sequence in G. In the sequel we will drop the index n. The first result we need ls • Lemma 1 • for !~ almost every oo, and all f E C(~), the limit (3) exists and is I equal t o (41

J' cI]NI"(El f(E) = '[. (f( h )) 0

Proof : For any finite A the right hand side of (3) defines a probability measure (Rlesz's theorem) on the convex hull of the spectrum of h, for if we denote it by MA(f), we get (i)

~IA(f)is linear In f

(ii) ~IA(1) =I Consequently we have

and NA(f)~ 0 if f~ 0

147 (ill)

IN^(f)I ~ II f II

for all A

On the other hand from the definition of the trace given in 54 eq.5 we have (5)

"cp(f(h)) =limlal._~, I A l - l T r ( f ( H ( ~ ) X A ) f o r p a l m o s t a l l c o

In much the same way, for each A, the right hand side of (5) defines another probability measure on the convex hull of the spectrum of h which we denote by ~^(f). We know from the Birkhoff theorem [M13], that the limit exists p almost surely. The result will be achieved if we prove that the difference NA(f) - ~^(f) converges to zero as IAI goes to infinity. Using the Stone-We;erstrass theorem it is sufficient to prove the result when f is a monomial. Let k be a positive integer. Then dropping the index co (for k-1 we get zero) : (6)

I A I {NA(X k) - %^(xk)} = Tr{x^ ( Hk - (XAHX^)k)} =

: T, l(j(k-1 Tr{x^ Hi (1 -XA)(Hx^ )k-j} ~ (k-l)II H IIk-2 Tr{x^ H(I-xA)Hx^} Since H = TI(h) with h E C*(~,G), for any ¢ > O, there is h(c) in Cc(~XG) such that II H - H(¢) II < c, if H(c) = ~(h(c)). Replacing H by H(¢) in the right hand side produces an error 6 = (k-l) II H II k c IAI. On the other hand we get : (7)

Tr[x^H(c)(1-x^)H(c)X^} : T..xe^ T.yt ^ Ih(c;x-l.co,x-~/)l2

Now since G is discrete and h(c) has a compact support, there is F finite in G such that x-~/¢ F implies h(c;x-l.oa,x-k/) : 0 for all o~£2. Let C be the maximum of h(c; co,x) over QxG, then : (8)

Tr{XA H(c)(1 -XA )H(c) X^ ) ~ C I Aa(Aa) I

Therefore, for any c > O, there is C(c) > O, such that : (9)

{N^(Xk) - ~^(Xk)} ~ (k-l) II H IIk c + C(c)(K-I)II H Ilk-21 AA(AF)I/IAI

Since the group G is amenable, F is finite and A ls a member of a F~lner sequence, the last term of the right hand slde converges to zero as I^1->=. Since ¢ is arbitrary, we achieve the result. o

148

Lemma 2 ' The function E-> %,(x(h(E)) is increasing from 0 to I, and is

locally constant outside the spectrum of h. If E is a point of continuity of thls function, then" (10)

f'l~(E) - ~p(x(h~E))

for I~ almost all co 0

Proof : The first assertion is obvious. The left hand side of (10) is defined through the weak the limit of a sequence of probability measures on the convex hull of the spectrum of h, as was proved in lemmal. The result is a consequence of a well known result on weak limits of probability measures on a a complete metric space, o

Remark: In general the function IN(E) is not continuous. For if h is a projection in C*(~,G), then clearly, M(E) is a step function with a discontinuity at E--I. Conversely, if it is discontinuous at E, then E is an eigenvalue of H of infinite multiplicity and the eigenprojection is actually in the C* algebra. The question is whether such projections exist. The answer depends upon the C'algebra. For an abelian virtual group (T V, ZD) the answer is yes. This was proved firstly by M. Rieffel [86] on the irrational rotation algebra, and his argument extends in the general case. He even exhibited a projection given by a continuous function with compact support. However, there are examples of dynamical systems (Q,Z)such that the corresponding CWalgebra has no projection and is simple [29] (see also S5)

149

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156

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QUANTUM FIELD THEORY WITHOUT CUTOFFS: RENORMALIZABLE AND NONRENORMALIZABLE A. Kupiainen Research Institute for Theoretical Physics, Helsinki University Helsinki, Finland

i. INTRODUCTION Constructive quantum field theory set as its goal to construct models of quantum fields in the continuum with nontrivial interactions.

In the 70's, this goal was fullfilled in the context of

superrenormalizable

models

[see [i] for references],

whereas the

renormalizable case remained an enigma. In fact, some doubts whether renormalizability

is a sufficient condition for the

existence of nontrivial theories were raised by the work of Aizenmann and FrShlich

[2], [3], who presented "almost proofs" of

4 the conjecture that the lattice regulated ~4 theory with positive coupling has only a trivial

(gaussian) continuum limit.

In these notes I would like to present some work done jointly with K, Gawedzki on the question of necessity/suffieciency renormalizability

of

for the existence of these theories. What will

be presented is a rigorous construction of nontrivial renormalizable theories,

the Gross-Neveu model in two d i m e n s i o n s

4 and the 44 model with negative coupling constant as well as a nonrenormalizable

version of the Gross-Neveu model. We should

mention that by different methods, Feldman, Magnen, Rivasseu and Seneor have succeeded constructing the Gross-Neveu model [4], 't Hooft and Rivasseu model and Felder

[5],

4 [6] the planar negative coupling 44

[7] a nonrenormalizable

version of it.

158

2. EFFECTIVE ACTIONS

The problem of continuum limit in (euclidean) QFT is to try to establish the limit of the Green functions

l

-sA

SA = J D~e

I

~(xi)/

-SA D~e

(I)

as the ultraviolet cutoff A is taken to infinity° S A in (i) is some action involving the cutoff A both explicitly

(in the form of

lattice spacing or momentum cutoff) and implicitly in the various couplings gi(A) occurring in it (see below). It is advantageous to look at this problem from a slightly different point of view, that is, to study the continuum limit of

the effective actions. Vaguely, the effective action for momenta ~, S~ FF describes the theory

A

(its Green functions)

for momenta

~ and is produced from S A by performing a partial functional integral:

we integrate out fluctuations ~ of the fields having

momenta ~ ~ IPl ~A,

schematically,

Concrete ways of realizing

(2) will be sketched below. Continuum

limit means now that we are to find the bare couplings gi(A) s.t. the limit

s FFA = FF. lira

A

(3)

SA

exists in some strong enough sense. The Green functions of the theory are determined by the family Is I ~FF ~ = 0 '

provided we gain

sufficient control of it (as will be the case below).

159

3. THE OLD PERTURBATIVE

APPROACH

To contrast with the rigorous analysis the standard perturbative amounts to computing expansion

sketched below,

let us recall

approach to the continuum limit. This

(2) by means of a formal perturbation

in the couplings gi(A). One obtains a formal sum

EFFA

[ N (~) = Z ) dXlo.odX N F~(Xloo.XN,A)w

~ (xi)

(4)

N

where the vertex functions

F{ are given in terms of a formal power

series in the coupling(s)

F~(xi,A)

= ~ g(A) n F~n(Xi,A) . n

As is well known, models)

as A ÷~.

the coefficients

(5)

F{n in (5) diverge

This should come as no surprise;

is a relation between the data scale ~ and the data

(F~) describing

(in most

after all

(5)

the theory at the

(g(A)) at scale A. Such a relation

in most

interesting cases should be singular.

The old renormalization to save

(5) by expressing

of renormalized scale

theory can now be formulated as an attempt gi(A) as functions

couplings giR describing

(the renormalization

the theory at some fixed

point ~)

g(A) = ~ g~~ Cn(A,~) . n (6) and

(formal power series)

(6)

(5) combine to a series

F (xi'A)

n -N = Z gR F~n(Xi'A'~) n

(7)

160

and the theory is called renormalizable if the coefficients ~ have a limit as A ÷ ~ .

In such a case, all the effective actions ~A FF

have a computable expansion in terms of a finite amount of data giR at scale ~. The theory is nonrenormalizable in case the above procedure doesn't work: typically in such a case an infinite number of couplings R

gi

(gi (A)) are needed, invalidating the whole approach.

Finally, the work of Aizenman and FrShlich teaches us that a

%

theory may be perturbatively renormalizable, i.e~ lim (A) exist, A~ 4 but still be trivial for a nonperturbative reason: for lattice 44 with positive coupling the only value of gR that may be achieved is zero.

4. THE WILSON APPROACH

In the late 60's and early 70's a new approach to the continuum limit was developed by K. Wilson

(see [8] for references). The idea

is, that rather than trying to establish a direct relation between S A and S ~ FF

(as in (5)) we reduce the cutoff little by little,

establishing a flow of effective actions EFF EFF S A + S A / L ÷ SA/L2 ÷ ... ÷

~FF ÷

...

(8)

in some "space of actions". Here L is some number of order unity. To bridge a connection to dynamical systems, it is advantageous to make a trivial change of variables

(to dimensionless quantities)

so that the cutoffs A/L n are all the same. We define a "Hamiltonian"

161

s A (~) = HA (A-P{ (~))

and s i m i l a r l y field,

for

~

(9) EFF

. D in

(9) i s t h e s c a l e d i m e n s i o n o f t h e

w h i c h we have t o d e t e r m i n e i n each model s e p a r a t e l y

(see

below), (9) amounts to the formulation of the continuum limit.as a scaling

limit; we are to find a sequence of Hamiltonians H A and to look at larger and larger distances as A ÷~: N = AN~< ~ SA ~(Axi)>HA.

(i0)

Thus we study the flow of Hamiltonians

R HEFF HA + A/L ÷ " ' °

(11)

each of cutoff 1 and related to each other by the Renor~nal~zation

Group Transformation (RGT) R e-RH(~)

= I D ~ e-H(L-~ ~ (~) +4)

(12)

which is a nonlinear map in some "space of Hamiltonians" terms of

R

H. In

the continuum limit is thus expressed as

H~FF = lim R N H

N+~ The canonical

.

(13)

LN~

example of (13) is obtained if

point H*. Then,

R has some fixed

if H A is made to approach the stable manifold of

H* in a suitable way, the H~ FF will lie on the unstable manifold of H , which will then parametrize

the continuum

in our case the space H is complicated of

R

limit. Obviously

and a very good knowledge

is needed to establish such a picture°

concrete examples where such an analysis

We now turn to

in fact may be carried out.

162

5. THE RENORMALIZABLE CASE: FERMIONS

We take S A the Gross-Neveu model of two dimensional Dirac fermions with a four-fermion coupling:

SA =

f d2x [ ~ i ~ A ~ - g ( A ) ( ~ ) 2 ] .

Here ~ = (~)

are Grassmann variables

(14)

(they belong to an infinite

dimensional Grassmann algebra; see below and [9] for a rigorous formulation) where e = 1,2 are the Lorentz indices and i = I,...,N, N > I are vector internal symmetry indices, suppressed below. The cutoff A is put to the propagator in momentum space: (i ~A) -I (p) = ~ 2

e-p2/A2 "

(15)

P The theory (15) is renormalizable but not superrenormalizable. Thus g(A) is dimensionless, which is reflected in the corresponding Hamiltonian

HA = I[~i~1* - g(A)(~,)21 .

(161

1

We took D = ~ so as to keep the coefficient of the gaussian part unity. The RGT is now explicitly given as

exp [-RHA[ ~)] = e

.

I

D X exp [ -(X,F-IX) - g(A)V(L-½~ (L) ÷X) ] (17)

(V = (~)2) where the fluctuation covariance is

163 2 F(p) = ~

(e-p

2 2 - e -L p ) o

(18)

P Note how F has mass of O(i) and UV cutoff too. The point of studying a fermionic theory is, that a rigorous analysis of

R

is especially simple. In fact it turns out, we may

use standard perturbation theory to evaluate and

(5) expand

(17) in powers of g(A)

(17)! Just as in (4)

(~# = ~ or ~):

H EFF = E I dxl .. "dXN FA/L(Xl,A) N ~ ~#(xi) A/L N

(19)

with again N

FA/L = Zg(A)n n However,

FN A/Ln "

(20)

this time both the sums over N and n converge. To

understand this, note that FNA/Ln is given in terms of all connected graphs with of

n

4-point vertices and N legs with lines carrying F

(18). There are ~ n! such graphs which in the case of bosonic

theories leads to the divergence of

(20) in n. However,

in the

case of fermions minus signs occur saving us. More explicitly,

the

free expectation N

r

(21)

in the covariance F is given by determinant det F(x i - yi ) of a N × N matrix. Since F has both UV and infrared

(IR) cutoffs, (21) is

bounded by (const) N. This observation leads to the convergence of (20) in quite a straightforward way. As for (19), recall that the graphs entering

(20) are local,

since F is. Thus convergence of the integrals in (19) follows,

164

for ~ # = O (i), as well as the sum over N since the graphs carry explicit g

O(N)

factors.

In fact we may now set up the space

where R acts as a nice Banach space. We take H 6 ~ of the form

H = / [ ~i~l ~ + 6 Z ~ i ~

- g(~)2

(22)

] + Z I F N ( x i ) w ~ ( x i) N

- (6 Z, g, F N)

Here ~ = (~, ~ ~), i.e. we allow also derivatives

of ~. These appear,

since the local parts 6Z and g are extracted from F 2 and F 4 leaving extra derivatives

on the fields. The structure of the local

parts follows from syn~etries.

Introduce next the norm

(for

some

small go )

IIFN II = go-N ]C dx2"''dxNFN

(Xl'''XN) eZ(xl---x N)

with 2 the shortest connected graph on {xl...XN},

I1~11 = I ~ 1

(23)

and for H:

+ Igl + : IIr~II

(24)

N

In this way ~ gets a Banach space strucure. RH perturbatively,

the boundedness

leading to the cenvergence

We may again evaluate

of fermionic expectations

of the expansion.

The coordinates

(6Z', g', F 'N) of RH are analytic functions of those of H and we get

g' =g- ~292- ~3g3- ~(g,~Z,rN) (25) F 'N = L~NF N (L') + YN (g' ~Z, F N)

where some leading terms were separated.

The coefficient

82 is

165

negative;

thus the g direction is marginally unstable. D N is the

scale dimension of F N. Since we separated the marginal parts in (22), the F N directions are unstable. Indeed, in terms of the norm (23)

II L

~N FN

(~')ll

< ~

-dN

IlrN[I,d N

> 0

(26)

It is now a straightforward matter [9] to establish the continuum EFFA limit of H ~ in the norm

(24). Only a fine tuning of g(A) is

needed, it turns out, that only the three first perturbative terms in (25) are needed, we take g(A)-I

-i

= gR

83

- 82 logA/~ + ~2 log(l - gR82 logA/~)

(27)

Here 8i are related to 8i ; they are the first two coefficients of the perturbative 8-function. H A

With the choice

(27)

= O(g(~)).

How about the Green functions? For them we need H E~FF for all ~. A However, as ~ decreases,

g ~ increases,

eventually leaving the

region where perturbation converges. Thus we cannot study this way the infrared behaviour of our model. Nevertheless,

adding an

EFF explicit mass term to (14) the full set of H ~ may be constructed as well as the Green function which satisfy Euclidean axioms guaranteeing the existence of Wightma~

QFT. We expect to control

the "massless" theory for N, the number of components,

large

(but

finite) when the model is expected to show dynamic mass generation and symmetry breaking

[i0].

166

6. THE NONRENORMALIZABLE CASE

By a slight modification of the free propagator, we may make the Gross-Neveu model in d=2 nonrenormalizable.

Namely, consider again

(14), this time with 0~

I

(i~A)-i (p) = ~

2 -el2 d x e -~ p e

(28)

A-2 Note that for A = ~ (28) becomes e ~ / p 2 - e series nonrenormalizable.

HA =

I (~i~l~

and the perturbation

The Hamiltonian now is

- g(A) A 2 e ( ~ ) 2 ) .

(29)

Note, how the coupling has a non-zero "dimension" just as formally in d=2 +,e. All the analysis of the previous section goes through; the new F is again massive. We get the recursion for

I~

= ~2e

g~

I' = L - 2 e l -

as

82 12 + "'"

(30)

This time I is stable at zero. However a non-gaussian fixed point is seen to emerge for I = 0(£). It is a standard matter of nonlinear analysis to construct [Ii] a H • = (O(e),O(e),O(e2)) turns out to have one unstable direction, to I. The sequence of bare theories

which

roughly corresponding

(29) is shown to intersect

the stable manifold of H • at a point

(0, I

c

,0), Then the

continuum limit exists, provided we choose

=

=

+

(l -

where 9~ is the largest exponent at H ~.

I c)

(31)

167

The massless theory exists now as well, provided I ~ I . The IR c is then governed by the gaussian fixed point. For I > I c again a dynamic mass generation is expected and the present method only yields the massive theory with explicit mass term. Needless to say, the resulting theory is not reflection positive since even the free theory us not. However we feel, that this example shows that nonrenormalizable theories indeed should make sense.

7. B O S O N I C T H E O R I E S

4 The approach sketched above is applicable to the ~4 model with some modifications [12], [13]

HA =

Z

. Now we have

½(V~(x)) 2 + ½~(A) ~ (x) 2 + ~I(A) ~(x) 4

(32)

x6Z4 (Here it is more advantageous to work with lattice cutoff, the RGT is then given by the block spin transformation). The main modification to the fermionic case comes from the unboundedness of ~: The perturbation expansion for RH diverges as nl. However, the main point of R is, that the fluctuation integral is massive. The role of the perturbation expansion will now be played by a convergent high-temperature expansion. The analysis will be more complicated since one has to keep separately track of the small EFF and large field properties of H ~ (for a pedagogical exposition, see [14] ). In the region of field space where ~ is not too large a representation (19) again emerges with a convergent N-sum. The F ~ h a v e

168

perturbative and small non-perturbative contributions, again leading to a recursion of the type

(25) where B and

XN are under

good control. Summarizing, for ~(A) > 0 in (25) the I recursion is 2 ~' = h - B 2~

+ ...

(32)

with 82 > 0 this time. The gaussian fixed point has only a gaussian unstable manifold for I(A) small: all such continuum limits are trivial. This proves the conjecture of Aizenmann and Fr6hlich in the case of small bare soupling. It should be noted that this is in no contradiction with the renormalizability of the model, i.e. the existence of the limit A ÷ ~ of ~ (A) A n

in ( 7 ) .

Indeed

renormalizability manifests itself in the fact that if we keep the cutoff very high

(O(e-C/gR)), then for ~ ~ RA/~

172

exists for each

~

and is a non-degenerate function of

g~en . This notion

should be distinguished form the perturbative renormalizability where the existence of the limit in (2) is required only order by order in the formal expansion in powers of

g~en . A plausible scenario for renormalizability of a family of

interactions has been put forward by Wilson : In the space of Hamiltonians we have a fixed point

H,

of the RG transformations with finite dimensional

expanding and infinite dimensional contracting manifolds, see Fig.l. The family H(g i)

crosses transversally the contracting manifold. It should be then possible

to choose that

H(g~are(A))

converging to the contracting manifold in such a way

R A/~ H r bare-~A)) -- converge to a Hamiltonian on the expanding manifold of ~gi

The continuum limit

(A ÷ ~) theory would be then described by effective Hamiltonians

Heff(~)

lying on the finite dimensional manifold independent of the detailed

form of

H(g i)

(universality in field theory).

(3)

lim Heff(~)

We would say that the fixed point

One would also have

= H, .

H,

governs the continuum limit and the ultra-

violet behavior of the field theory.

i rubY

H.

H, .

173

A possible stategy for the demonstration of renormalizability of an interaction may consist of establishing the above scenario.

This can be approached by

approximate analysis (perturbative RG, numerical analysis of the approximate RG recursions, Monte-Carlo RG) or done with full rigour. Only the latter case will be discussed here.

Exact analysis of Wilson's RG is a formidable task. The simplest situation occurs when the continuum limit is governed by the gaussian fixed point i.e. by H,

quadratic in fields.

Such field theories are called asymptotically free.

One may achieve then that the whole relevant flow of the RG takes place in the vicinity of

H,

where the perturbative expansion around the gaussian theory

provides a reliable tool for the control of the flow. In fact all the models of quantum fields constructed until recently like (sine-Gordon) 2 , (abelian Higgs)2, 3

P(~)2 ' (~4)3 ' Y2 '

(see e.g. [2]) were asymptotically free

(or could be solved exactly by mapping into free fields as the Thirring and Schwinger models). Indeed, the methods developed by the so-called constructive QFT seemed suitable for any asymptotically free super-renormalizable theory i.e. one with a finite number of divergent Feynman graphs.

It was generally expected,

however, that treatment of the renormalizable case would be much more difficult. As the first constructions [3] of the asymptotically free renormalizable models obtained recently have shown this was an excessively pessimistic point of view. In particular, for the purely fermionlc Gross-Neveu model in 2 dimensions (the ~(i~+m)~-g(~) 2

theory), the RG transformations are given by convergent pertur-

bation expansions, which largely simplifies the analysis, see A. Kupianen's contribution to this volume. However, we are also able to show the existence of the continuum limit of the bosonic negative coupling

4

theory in four

euclidean dimensions [4]. This theory is renormalizable and asymptotically free, although it describes a metastable system and lacks as a result the so-called Osterwalder-Schrader positivity. On the other hand, as many rigorous and numerical

174

studies suggest [5]

the positive coupling '

~4

theory, lacking asymptotic

4

freedom, seems to have no continuum limit. Although the main problem in the field, the construction of non-abelian four-dimensional asymptotically free and renormalizable gauge theories has not been solved yet, it seems now within reach, see [6] and references therein.

The case of

~44

raises the question whether the asymptotic freedom is a

necessary condition for the non-perturbative renormalizability. There are strong rigorous arguments to the contrary. If in the Gross-Neveu model we replace the free fermionic propagator

_~

by

~

we render the theory perturbatively

non-renormalizable. Nevertheless, for small

s > O , it is still possible to

renormalize the theory non-perturbatively by constructing the continuum limit governed by a non-gaussian fixed point [7], see also A. Kupianinen's lecture in this volume. For small

g

the new fixed point is close to the gaussian one,

so that again the whole flow of the RG can be studied by the convergent perturbation expansion. The Green functions of the model are not infinitely differentiable in the weak physical coupling what explains the difficulties of the perturbative renormalization. Similar analysis works for the planar

4 ~4+s theory

[8]. Also for the Gross-Neveu model in 3 dimensions in the limit when the number of fermionic species (flavors) goes to infinity one can establish the Wilson's scenario with the similar RG flow as in the

~

case. Although genuine

non-renormalizable field theories remain to be constructed, it seems that they may be consistent in the presence of non-gaussian fixed points of the RG as argued in [9]. This opens the possibility of occurrence of a realistic nonrenormalizable sector at ultra high energies which should perhaps be given more thought in model building and future Monte-Carlo RG simulations.

175

References. [i]

K.G. Wilson, J. Kogut, Phys. Rep. 12C (1974), 75.

[2]

J. Glim, A. Jaffe, Quantum Physics, A functional integral point of view, Springer, New York 1981.

[3]

J. Feldman, J. Magnen, V. Rivasseau, R. S~ngor, Phys. Rev. Lett. 54 (1985), 1479 and Ecole Polytechnique preprint. K. Gawgdzki, A. Kupiainen, Phys. Rev. Lett. 54 (1985), 2191 and IHES preprint.

[4]

K. Gawgdzki , A. Kupiainen, Co~un. Math. Phys. 99 (1985), 197 and Nucl. Phys. B2 (1985).

[5]

J. Fr~hlich, Nucl. Phys. B2OO

FS4

(1982), 281,

M. Aizenman, R. Graham, Nucl. Phys. B225

FS9

D. Callaway, R. Petronzio, Nucl. Phys. B240

(1983), 261, FSI2

(1984), 577.

[6]

T. Balaban, Co, nun. Math. Phys. 99 (1985), 389 and references therein.

[7]

K. Gaw~dzki, A. Kupiainen, Phys. Rev. Lett. 55 (1985), 363 and IHES preprint.

[8]

G. Felder, ETH preprint.

[9]

K. Symanzik, Conmlun. Math. Phys. 45 (1975), 79, G. Parisi, Nucl. Phys. BIOO (1975), 368.

R e n o r m a l i s a t | o n g r o u p m e t h o d s for circle m a p p i n g s Oscar E. Lanford HI HIES 91440 Bure~sur-Yvette France I review here a number of recent results all having as central theme the application of renormalization group methods to the study of the iteration of circle mappings. I will concentrate on mappings which are smooth but which have critical points and hence are not diifeomorphisms. This review will be divided into two quite distinct parts. The first part is devoted to the theory of circle mappings with a special rotation number, the golden ratio. The theory in this case is by now relatively complete. None of the results described in this section are my own; they are due independently to Feigenbaum, Kadanoff, and Shenker[3] and to Ostlund, Rand, Sethna, and Siggia[7]. A second section discusses how to extend this analysis to general rotation numbers. In contrast to the first section, the analysis in this part focuses on the variation of rotation number with parameter rather than on the detailed dynamics of individual circle mappings. This part of the subject is much less developed than is the study of special rotation numbers. The main new idea which will be presented in this section is an extension of the standard renormalization group analysis--which studies the consequences of the existence of a hyperbolic fixed point for a renormalization operator--to a situation where a renormalization operator leaves invariant a hyperbolic set more complicated than a fixed point or periodic cycle. This extended renormalization group analysis appears to apply to the theory of circle mappings with general rotation numbers and makes strikingly strong predictions about the dependence of rotation number on parameter in one-parameter families. Golden ratio rotat|on n u m b e r . We will use the term circle m~ppin¢ to denote a continuous (usually in fact analytic) strictly increasing mapping f of R to itself satisfying the circle mapping identity

f ( x + 1) = f(x) + 1. The reason for calling such an f a circle mapping is that it induces, by passage to quotients, a continuous one-one mapping of the circle R / Z to itself, and, conversely, any continuous one-one mapping of the circle to itself which is orientation-preserving ("increasing") is induced by an f satisfying these conditions and unique up to an additive integer. The circle mapping identity can equivalently be expressed as the statement that f(z) - z is periodic with period one, or that f commutes with z ~ z + 1. A critical circle mapping will mean one satisfying f ( 0 ) = 0; i.e., one having a critical point which, for convenience, we situate at the origin. Since circle mappings are by definition increasing, this implies .f"(0) = 0 also. The rotntion nsmber of a circle mapping f, denoted by p(f), is defined by p(f) -- lim f ' ( z ) - z n -...I.O0

n

177

The definition makes sense because, by a simple argument due to Poincax~, the limit on the right exists and is independent of z. The golden retio will mean ( V ~ - 1)/2. In this section, we reserve the symbol # to denote this number. What we axe going to b e studying is =nallltic critical circle mappings with rotation number ~r. The Fibonacci sequence Qn is defined by the recursion relation Q . + , = Q. + Q . - ,

with initial conditions Q1 = Q2 = 1. The Fibonacci sequence is closely related to the golden ratio; one connection, of which we will make heavy use, is the identity ,,Q,, _ Q._,

=

which can easily be proved by solving explicitly the recursion relation defining the Fibonacci sequence. From this identity, if f is a circle mapping with rotation number ~, then = I

I.

(x) - Q . _ ,

(again a circle mapping) has rotation number (-1)'~+1~ '~, which goes rapidly to zero as n goes to infinity. This suggests that the f~ should converge (in some sense) to the identity mapping. We will look at the limiting behavior, in the immediate vicinity of the critical point 0. For that purpose, we define A(")=fn(O)

and

q,~(z)-~1---~fnf)~("-1)z ~

Numerical experiments strongly indicate that the A(n) converge to zero (consistent with the guess that the f , converge on a coarse scale to the identity) and that the ~, (which describe the fine-scale behavior of the f , ) converge to a limit ~* which doesn't depend on which f we start from provided always that f is an analytic critical circle mapping with rotation number ~r. A completely elementary argument shows that, if the I/, constructed from some f converge to a limit r/*, then q* must be quite a remarkable function: It satisfies a

functional equation

(where A* denotes #*(0)). Furthermore, writing ~* for ~* rescaied by A*:

we find that r/* and ~* commute, i.e., r/* has the unusual property of commuting with a non-trivial rescallng of itself. On the other hand, neither r/* nor ~* has any reason to satisfy the circle mapping identity, and, in fact, the limits found in numerical experiments do not satisfy it. The proof of the functional equation for V* goes as follows: We begin by observing that the recursion relation for the Fibonacci sequence, the circle mapping identity, and the definition of the ft, imply /~+l = f~ o / ~ _ , .

178

Rescallng this equation by A('~) and reorganizing in a straightforward way gives: 1

1

where

A(,~) A,~ -~ A(n_i) = r)~(0).

Since, by assumption, t/,~ converges to a limit F/*, ),,~ converges to a limit A*, and we can thus take the limit of the above formula for t/n+, to get the functional equation. We can now easily put the above into a renormalization group setting: We introduce, along with D/,,, the function (,, defined by &(z) =

1 f._,(~("-')z). A("-')

i.e., by rescaling fa-1 by the same factor as f,~ was rescaled by to produce Tb~. It should be noted for later use that $,~ and $,L-, commute (since they are both essentially iterates of the same mapping f), and thus (a and ~h (for any given n) commute. The calculation sketched in the preceding paragraph shows that (z) =

and it is immediate that

(.+I (x)

=

1

This leads us to introduce a "renormalization operator" T acting on appropriate pairs ((, ~) and producing new pairs (~, O) by q(z) ---- ~

o ~(Az)),

~(z) ---- ~q(Az)

where A =- 17(0).

By what we have just shown

(ff the sequence of q's and ~'s is generated starting from a circle mapping f), although T "doesn't depend on $'. In our formalism, the ,1~ and (,~ are defined only starting at n -- 2, but we can clean up the notation slightly by observing that if we introduce f , ( f ) = (~1, q,) by

then we get ((n, ~,~) = T ' ~ - l f l ( f )

for all n ~ 1.

Various choices are possible for the space on which the renormalization operator is to act. We will describe one of the possibilities which seems to work reasonably well. The description will be semi-formal in that we will suppose we are dealing with functions defined and well-behaved on the whole real axis, although for serious applications of renormalization group ideas it is necessary to select bounded domains in the complex plane in such a way that the operator T, acting on functions defined on those domains, gives functions with strictly larger domains.

179

Our space of mappings will consist of pairs (~, #) of real valued functions which are defined, analytic, and strictlyincreasing on R, with ~/'(0)- 0, normalized by ~(0) = I, and satisfying ~(z)z; rlo~(z)>z for allz. These latter conditions can be interpreted as follows: The first says that ~ moves points to the left; the second (which actually follows from the first and third) says that ~ moves points to the right; and the third says that the action of ~ dominates that of ~. The pairs ( ~ , ~,~) constructed from circle mappings have a further property which is extremely important for the analysis--they commute. Formally, we would like to restrict ourselves to working in the space of comrastin¢ pairs. For technical purposes, this may not be convenient--the set of commuting pairs may not form a space with good properties--and so, instead of exact commutation, it may be better to require only that o r/and ~/o ~ agree up to some finite order at the origin. Either exact commutativity or approximate commutativity to any given order is preserved by the renormalization operator. In our semi-formal discussion, we may as well consider the space of exactly commuting pairs. We will use Ai to denote the space of all pairs satisfying the above conditions. It is easy to check that the formally defined operator T, acting on a pair (~, r/) in ~4, gives a new pair again in A{ if and only if the original pair satisfies 1t~(z) < z

for all z,

i.e., if and only if the action of ~ is more than hal/as strong as that of ~, (in addition to the conditions defining ~ . ) Thus, if we let D denote the subspace of Al satisfying this condition, T becomes a well-defined operator mapping/) into ~ . In terms of this machinery, we reconsider the construction of a sequence (~n, r/n) in At starting from a circle mapping f, dropping the requirement that the rotation number of f be exactly or. If f is a critical circle mapping, it is easy to check that the pair fl(f) defined above is in A{ if and only if the rotation number of f is (strictly) between 0 and 1, and is in/) if and only if it is between 1/2 and 1. More generally, it is not difficult to show that, for n --- 2, 3,...

p(f) i* (,trictlll ) bettse¢mQ,~/Qa+I and Q,,-I/Q,~ if and os,ly if fl(f) is in D(T'~-I). Hence, in particular, fl(f) is in P(T ~) for all n ifand only if f has rotation number ~r. I stress that this is not a difficultor deep result;its proof uses only some very elementary theory of continued fractions (and a standard trick about rotation numbers to make sure the borderline cases come out as asserted). The renormalization group picture now goes as usual: T has a fixed point (the pair (~*,#*) introduced above as the limit of rescaled .f,~'s;the functional equation satisfied by #* just expresses the fixed point property.) This fixed point is hyperbolic, with one-dimensional unstable manifold and, hence, stable manifold (criticalsurface) of codimension one. Since anything in the stable manifold is in the domain of all powers of T, we expect that the stable manifold will be, locallyat least,the set of pairs with rotation number ~r. Motion in Ai transverse to the stable manifold--in particular, motion along the unstable manifold--means changing the rotation number. The validity of this picture can be proved, using estimates proved by computer. This has been done by B. Mestel[6], whose results are described in detail in a preprint of

180

Rand [S]. Partial results (existence of the fixed point) have been obtained by de la Llave and the author• I have been told by K. M. Khanin that Sinai and his collaborators in Moscow also have some results along these lines, but I do not know any details. Rand's preprint extends Mestel's work considerably, using traditional analytic (noncomputer) methods (but starting from Mestel's computer-verified estimates). In particular, Rand shows that Mestel's fixed point really can be obtained as a limit of rescaled fn for an analytic circle mapping f, and that the stable manifold really is locally the pairs with rotation number ~r. As usual, the renormalization group picture leads to many precise quantitative statements about '~niversal rates". Here is one sample: Consider a one-parameter family of critical circle mappings f~ which crosses the stable manifold transversally; for notational simplicity, assume that the crossing occurs for p equal to zero• Then

p(&)

-

p(fo) ~

where the " ~ " means that the ratio of the two sides is bounded and bounded away from zero for small enough/~. The exponent, which is "universal", i.e., is the same for all families which cross the stable manifold transversally, is given by -21og(~)/log(l~D, where 6 -- -2.83361.-. is the expanding eigenvaine of the linearization of T at the fixed point. General rotation number.

We will now rework the above construction of a renormalization operator to make a more general operator which can be used to study critical circle mappings with arbitrary rotation numbers. This will involve a certain amount of repetition in a more general framework of things done in a concrete way in the preceding section. The construction we describe is due to Ostlund, Rand, Sethna, and Siggia[7]. We begin by recalling some elementary facts about continued fractions. First, as a typographical convenience, we will write Jr1, r 2 , . . . , rn] for 1

1 rl +

1

r2+

1 rn-1 +

rn

If p is a number in (0, 1), we put rz--int(I)

and

pz=frac(1)

sothat

We repeat this process as ,long as we can: If p,~ ~ O, we put

r~+1 = i n t ( 1 )

and

=

1 p = ~ rz + Pz

181

It is well-known that: • The process terminates in a finite number of steps if and only if p is rational. If the process terminates at the n-th step, then p = Jr1,..., rn], rn _> 2, and there is no other representation of p in the form [81,..., S~n] with the si strictly positive integers and sm _> 2. Without this last condition, there is a trivial ambiguity in the continued fraction representation of rational numbers, illustrated by 1 2=

1 1"

1-1-1

When we speak of the continued fraction representation of a rational number, we always mean the unique representation whose last entry is at least two. • If p is irrational, p = tim [rl, r ~ , . . . , r , ] , . -..*00

and no other sequence of strictly positive integers has this property In other words, an irrational number has a unique continued fraction representation, and this representation can be generated by the iterative procedure defined above. We write p = [rl, r , , . . . ] as an abbreviation for the above limiting relation. In the preceding section, we constructed the renormalization operator as acting on a space of commuting pairs of mappings of the line, rather than as acting on circle mappings themselves; circle mappings f were identified with commuting pairs via f ~ (z + 1 , - f ( - z ) ) . We will proceed in the same way here, but we formalize these considerations with a little more generality. When we speak of a commuting pair in what follows, we mean a pair of continuous strictly increasing mappings (~, r/) of R to itself which commute with each other and which satisfy in addition ~(z) > z

for all z.

We will say that a commuting pair is normalized if ~(0) = I and critical if ~f(O) = O. A non-normalized commuting pair (~, r/) can be normalized by a rescaling, i.e., by replacing it by the essentially equivalent pair (~/-1 ~(~z), ~/-lr/(~fx))

where ~f = ~(0).

All the commuting pairs we encountered in the preceding section were normalized and critical. The notion of rotation number generalizes from circle mappings to commuting pairs: If f = (~, r/) is a commuting pair, we define p(f)=sup{P:p, qEZ, q>0,~o~q(z) 0. Loosely, the reversed pair (r/, ~) ought to have rotation number 1/p. With the conventions we have adopted, this doesn't make sense, since p > 0 implies r/(z) < z for all z, whereas the first elements of our pairs are defined to move points to the right. We can deal with this, and also produce a normalized pair, by making a negative resc~llng. Thus, we form the pair (l~()~z), I~(Az))

with ~ = I/(0)(< 0).

It is easy to see that this pair satisfies all the conditions, is normalized, and does indeed have rotation number liP. It also follows easily from the definitions that, if (~,q) is a normalized commuting pair with rotation number p, then (~, ~ o f/) is also a normalized commuting pair but has rotation number p - 1. With all this in mind, we are ready to give the formal definition of the renormalization operator. For r = 1, 2 , . . . , we let/)r denote the set of normalized critical commuting pairs ((, I/) with rotation number strictly between (r + 1) -~ and r -1. We then define Tr on/)r by

and it follows easily from the above that what is produced is a normalized critical commuting pair with rotation number 1

r e (o, 1),

i.e., is a member of ~ . We have thus constructed a renormalization operator Tr for each strictly positive integer r. The operator TI is what we called simply T in the preceding section. The domains of the operators for different r's are disjoint, so we can, as a notational convenience, think of all these operators taken together as a single operator oO T, i.e., we define an operator T with domain P ~- ur=lDr by T f = Trf

when f ~ Dr.

183 As already asserted, the domain D is exactly the set of all pairs in ~ with rotation number which is not the reciprocal of an integer. If, for r -- 2, 3,..., we let ~r denote the set of f in ~ with rotation number r -1, we get a disjoint decomposition

J~

(U,°°=,D,) U (U~,1',).

The renorm~lization group picture developed in the first section can be expressed in the present language as the assertion that TI has a fixed point in the neighborhood of which it is expansive in one direction and contractive in an others. It is natural to ask whether the same is true for Tr for general r. Numerical studies for a few other values suggest that it is; if so, then there is a theory paralleling that developed in the first section for an rotation numbers of the form [r, re...]. It even seems that any finite product of Tr's has such a fixed point; if this is so, there is a theory for each qs~dr~ti¢ irrational rotation number. It appears, in fact, that something considerably stronger is true--that there is a large open domain in ~t on which T acts expansively in one direction and contractively in an the others. This expansive-contractive character of T is referred to loosely as hyperbolivitu. The kind of hyperbolicity property which seems to hold is analogous-although with some important technical differences--to that of the Smale horseshoe map. In the remainder of this article we will explore the consequences of T's having such a property. This exploration will be semi-heuristic; the objective is to develop a picture rather than to make mathematically precise conjectures about exactly what hyperbolicity properties T is likely to have. Indeed, as will be seen, there is an important point on which it isn't yet clear exactly what one should expect. The exploration will be guided by the well-known theory of the Smaie horseshoe (for an exposition in the spirit of this review, see [5]) and by some standard or almost-standard results from the general theory of hyperbolic sets which it should be straightforward to extend to the situation at hand once precise hyperbolicity properties have been established. I expect that (if careful numerical studies confirm the validity of this picture), it should eventually be possible to derive it f~om a precise set of statements about the action of 7" which could, in principle and perhaps also in practice, be proved rigorously by computer. I stress, however, that this is still far in the future. At this time, the principal justification for the picture is ~physical~--it provides an economical explanation for some non-trivial phenomena which aren't understood otherwise. There is also a certain amount of indirect evidence for it (see, for example, Farmer and $atija[2] and Cvitanovi~, Jensen, Kadanoff, and Procaccia[1]), and, so far as I know, no evidence against it. For simplicity of exposition, we will assume the domain on which T acts hyperbolically is all of ~ . The sets Di, ~i defined above organize .~t into "layers", as illustrated in Fig 1. In this figures rotation number increases upwards; DI is on top, ~ immediately below it, D~ below that, and so on. Each Tr maps the corresponding/~r (a short wide rectangle) into a long thin rectangle running the full height of ~(. The assertion that the image of Dr runs the full height of ~ is nothing mysterious; it is simply a geometricai translation of the statement that the image of the interval ((r + 1) -1, r -1) under

p

fr

(1/p) is an of (0,1)).

Consider a sequence roe rl, r~,.., and ask what the sets P(Tr, Tr,_~ ... %0) look like. Take first n = 1. Then

184

r

r

[///I

Figure I. As shown in Fig. 2, Dr, n (Tro Pro) is a short rectangle running across Tro/~ro, and since we are supposing that Tro is expansive vertically and contractive horizontally, its pre-image will be a shorter rectangle running all the way across ~(. Repeating the argument for larger values of n, we see that the domains P(Tr,,T,.,=_I "'" Tro) are a sequence of shorter and shorter rectangles running all the way across J~. It is a routine application of standard techniques from the theory of hyperbolic sets to show that what they converge to is a smooth codimension-one hypersurface, which we will denote by WS(r0, r t , . . . ) . Note, however, that from the definition of the Tr's it follows immediately that

E P(TrnTr,_l ""Tro)

¢=~

p(f) lies between [ro, rl,...,rr,]

and [ro, r],...,r,= +1],

and hence that

rl,...] In other words, WS(ro, rl,...) is exactly the set of ~'s with rotation number [ro, r t , . . . ] . We now need express a technical reservation. The operators Tr become more and more singular as r ~ c¢. It is not evident, heuristically, that we should expect the hyperbolicity of the action of T to hold uniformly in r (i.e., all the way down to the bottom of the figures we have been drawing). It may turn out that T is not uniformly hyperbolic on all of Ur~-lPr but that it is uniformly hyperbolic on U~=lDr for each R, without any uniformity in R. Under this weaker assumption, the above analysis

185

!

Figure 2. of WS(ro, r l , . . . ) works only for bounded sequences. The behavior of Tr for large r is interesting and apparently non-triviaL At this time it is only partially understood, so it is not possible to say with any confidence whether hyperbolicity uniform in r is likely to be true. For the remainder of this exposition, we will describe results which follow from the assumption of uniform hyperbolicity; there are corresponding (slightly weaker, and more complicated) result which follow from the weakened hyperbolicity assumption. Having described the structure of the domains of products of ~ ' s , we next describe the images of these products. We have already noted that Trl Prl is a thin rectangle running the full height of ~ . Since ~o contracts horizontally, it must map Dro n ~ Dr~ into a still thinner rectangle, again running the full height of ~t and contained, of course, in ~oDro. (See Fig. 3.) Continuing in this way we see that, given any sequence r0, rt,..., the images of the products ~oTr~-.. Tr~ form a decreasing sequence of thinner and thinner rectangles of full height collapsing down to a smooth curve which we will denote

byW~(ro,rt,...).

For any doubly infinite sequence r = (..., r - t , to, r~,...), the curve W~'(r_t, r_~,...) and the hypersuffaee WS(ro, rl,...) intersect in a single point which we will denote by ~b(r). The mapping ~ is continuous and =

where r is the left shift mapping. The image of ~b, which we denote by ~, is mapped to itself by T; it is the analogue, in the present analysis, of a fixed point for the renormalization operator in standard renormalization group analyses. Numerical studies suggest

186

t,,

---------_..__

L I

Figure 3. that ~ is one-to-one (There is, nevertheless, no convincing general reason why it must be, and the analysis we are going to present does not require it.) If it is, then X is an imbMded oo.shi~, with a Cantor-set-like structure. In any event • is something only slightly more general than a hyperbolic set in the classical sense of Anosov-Smale with one-dimensional unstable manifolds and codimension-one stable manifolds. It may be worth noting explicitly that the set • is most definitely ~ t a strange attractor. It might reasonably be described, however, as a s t r 6 ~ e saddle. What we have been doing so far is just elaborating on what we really mean when we say that T acts expansively in one direction and contractively in all the others, on an adequately large region of the space of commuting pairs. We now want to extract some consequences from this picture. These consequences concern one-parameter families in ~ , and it will be convenient to restrict the class of one-parameter families we consider and to impose some normalizations on how they are parametrized. From now on, when we speak of a one-parameter family f~ in M, we always mean: • The rotation number is a non-decreasing function of/~. • The parameter/J runs from 0 t o 1. • The rotation number is 0 for p -- 0, 1 for/~ = 1, and strictly between 0 and 1 for/J strictly between 0 and 1. Given such a family, and a rational number p/q between zero and one, the set of p's where p(f~) -- p/q is an interval of non-zero length; we will refer to this interval as the ph=se.loel~'ng interval with rotation number p/q.

187

Consider, now, such a normalized one-parameter family ~'~. For any r = I, 2 , . . . , the

set of/Ys where ~ ~ D, is an interval (/~;-, p~). As/~ runs over this interval, p(Ef~,) runs from 1 to 0. By making an affine (decreasing) change of parametrization, we replace the interval (#7,/~,+) by (0,1) and recover the normalization. This gives a new one-parameter family, which we will denote by f[d. Explicitly:

Repeating this process, we define recursively

(The order here is chosen so that f[r,,r2...,r,l is obtained by applying Tr, Tr2"" Tr. in that order to f , for p in the appropriate subinterval and reparametrizing.) With this notation, we can formulate two principal technical results on which our analysis of the length of phase locking intervals rests. The following two propositions, while n o t quite standard, are not difficult to prove, using standard methods from the theory of hyperbolic sets, given appropriate hypotheses on the global hyperbolicity of T. The first result concerns ways of parametrizing the curves W ~. For each r0, r~, r , , . . . , "f,o maps part of W~(rl,r2,...) onto all of W*(ro, rl, r2,...). What is claimed, in essence, is that there is a unique way to parametrize all the curves W u in such a way as to make this action of T affine in the respective parametrizations. P r o p o s i t i o n 1. There is a unique continuous way of writing a//the curves W~(rl, as normal/zeal one parameter families W~(rl, r,,...) so that, for al/to, rl, r , , . . . , (W~, (r~, rx, . ..))[,o1 = W ~ ( r o , r l , r 2 ,

r$,...)

. ..).

The second proposition says that applying T repeatedly to an "arbitrary" arc gives a sequence of arcs converging to one of the W a. P r o p o s i t i o n 2. There is an open set l / o f normalized one-parameter fami//es (containing in particular all the W~(rl, r2,...) such that, for any f~ E // and any rl, r z , . . . , the sequence of one-parameter families ...,

converges to W~(rl,r,,...). expoaent/a//y fast in n.

..... ,.1

...

F o r / i x e d f,, the convergence is uniform in rx, r~,.., and

We will refer to the set U of the above proposition as the ~niversalitll class. As noted, U is non-empty, since it contains the parametrized unstable manifolds. There are other ways of constructing many elements of//. For example, if fu is any one-parameter family crossing the stable manifold WS(r0, r l , . . . ) transversally, it is easy to see that q[r...... rol is in U for sufficiently large n. Roughly, U is the set of all one-parameter families which cross all the stable m_~nifolds transversally and for which this trausversality is sufficiently uniform. We will write ~(r01rl, r2,...) and "r(rolrl, r 2 , . . . ) for the lengths of the parameter intervals where W~(rl,r~ . . . . ) is in 16o and D,o respectively. The functions ~r and

188

are a convenient way of representing the metric structure of the hyperbolic invariant set Z. We are now going to argue that this metric structure determines a large number of ~universal properties" of one-parameter families of mappings. Let f~, be in the universality class, and for any r o , . . . , r,~ let S ( r o , . . . , rn) denote the length of the parameter interval where f~ is phase locked with rotation number [ro,..., r.]. G ( r 0 , . . . , rn) denote the length of the parameter interval where f~ E/)(Tr, --- "fro). S(ro,..., r.

)

s(r, lr,-l,...,ro) denote G(ro,...,r,~_~) g ( r . l r . - l , . . . , ro) denote

G(ro,...,r.) G(ro,...,r,-d

It follows readily from the definitions that we can also identify s(r~[r,~_l,..., ro) and g(r~,[r,~_l,..., ro) with the lengths of the parameter intervals where f[r.-1 ..... re] is in Vr. and Dr. respectively. Thus, from Proposition 2, for large n,

d r , l r . - l , . . . , ro) ~ ~r(r, lr,-l,..., to, r-l,...)

g(r, lr,,-l,... ,ro)

~

~(r, lr,,-,,..., to, r_l,...)

Here, r_l, r - s , . . , can be chosen arbitrarily. These approximations hold in the precise sense that the ratio of the two sides differs from one by a quantity which is exponentially small in n, uniformly in r,~, r,~-z, .... N o t e that the right-hand sides do not depend on the particular one-parameter family f~, from which we started. With these observation in mind, we write

S(ro, rl,...,r,) G(ro,rl,...,r,_z)

G(rx,ro)G(ro)

S(ro, r , , . . . , r,) = G(ro, rl,..., r,,-l) G(ro, r l , . . . , r,-2) "'" G(ro) = ,(r, l r , - t , . . . , ro)g(r,-llr,-2, ..., to)-." g(rllro)¢(ro) This differs from

~ ( ~ 1 ~ - 1 , r,-2, . . . ) ~ ( ~ - 1 1 ~ - 2 , . . . ) " "

~(rllro, r-l...)~(rolr-1...)

by a factor which is bounded, and bounded away from zero, uniformly in (n, r~, m - i , ...). Since this latter expression does not depend on which one-parameter family f~ we started from, we see in particular that

i I ~ ~) , . ~ ~ ' ) ,re ,,~y two m ~ n ~ , , o! th~ . , i ~ , , ~ i t y ~ , , , , , d qSO)(ro...., r.) , , d rn) are tke corresponding lenotks of phase-locking intervals, tkcn tlte r~tios

S(2)(ro,...,

s O ) ( r o , . . . , r,,)

sC2)(ro, ..., r.) ~re bo~n&d, and bosnde~l etoaU from zero, sni.tormlti in (n, r , . . . . , ro) More loosely formulated: Up to bounded corrections, the lengths of all phase locking intervals are universal, i.e., the same for all members of the universality class.

189

This is only one of a large set of consequences which follow from this approach to the study of the lengths of phase-locking intervals. For example, it is a very small step to extend the above analysis to show that the Hausdortf dimension of the set of #'s where p(f~) is irrational is the same for all f~'s in the universality class. This universality was discovered numerically by Jensen, Bak, and Bohr[4]. Moreover, using methods coming from classical statistical mechanics, one can derive a prescription for determining this universal Hausdortf dimension from the quantities ~ (to Irl ...).

References. 1. P. Cvitanovi~, M. H. Jensen, L. P. Kadanoif, and I. Procaccia, Renormalization, unstable manifolds, and the fractal structure of mode locking, Phys. Rev. Lett. 55 2. J. D. Farmer and I. I. Satija, Renormalization of the quasiperiodic transition to chaos for arbitrary winding numbers, Phys. Rev. A. 5 (1985) 3520-3522. 3. M. J. Feigenbaum, L. P. Kadanotf, and S. J. Shenker, Quasi-periodicity in dissipative systems: a renormalization group analysis, Physica 5D (1982) 370-386. 4. M. H. Jensen, P. Bak, and T. Bohr, Complete devil's staircase, fractal dimension, and universality of mode-locking structure in the circle map, Phys. Rev. Left. 50 (1983) 1637-1639. 5. O. E. Lanford, Introduction to the mathematical theory of dynamical systems, in Chaotic Beh~viour of Deterministic Systems, Les Houches Session X X X V I (1981), G. Iooss, R. H. G. Helleman, and R. Stora eds. (North Holland, Amsterdam, 1983) 5-51. 6. B. Mestel, P h . D . Dissertation, Department of Mathematics, Warwick University

(1985). 7. S. Ostlund, D. Rand, J. Sethna, and E. Siggia, Universal properties of the transition from quasi-periodicity to chaos in dissipative systems, Physica 8D (1983) 303-342. 8. D. Rand, Universality for critical golden circle maps and the breakdown of dissipative golden invariant tori, Cornell University Laboratory of Atomic and Solid State Physics preprint, September 1984.

CORRELATIONS

AND FLUCTUATIONS

IN CHARGED FLUIDS

Ph. A. Martin Institut de Physique Th@orique Eeole Polytechnique F@d@rale de Lausanne PHB-Ecublens CH-1015 Lausanne, Switzerland

I.

INTRODUCTION

We describe

in this lecture specific properties

of Gibbs states of char-

ged particles which are not present in equilibrium interacting with short range forces. sum rules, complete shielding,

These new properties,

reduced fluctuations...,

tal or partial screening of the Coulomb thods

(Sine-Gordon

be constructed ge symmetric a review.

formalism),

field theoretical

[i]) or cluster expansions

and cluster

and general view point

: we explore the assuming

conditions.

that equi-

We treat general multicomponent

as well as the jellium model of point charges moving on a uni-

neous fluid phases in the crystal

of density

R ~

pI

in dimension

(the one dimensional

Coulomb

[5, 6 ] ). Inhomogeneous

(OCP). We consider Only homoge~ = 2,3, and havei~noresults

some recent results

in their simplest

details

and completeness.

MULTIPOLAR

the

it by hard walls are of considera-

can be found in [7,8,9,10,ii].

In this text, we state the relevant propositions guments

for

system is solvable and well un-

fluid phases obtained by submitting

system to external fields or confining ble interest,

II.

from this

extended state exists and that it obeys appropriate

formly charged background

derstood

(char-

[2,3] , see also [4] for

follow naturally

imposed by the long range Coulomb potential,

librium equations systems,

inequalities

analysis.

Here we adopt a non contructive

the infinitely

the to-

limit of the state can

in some cases by means of correlation

systems

multipolar

express

force. Using field theoretic me-

the thermodynamic

Then, most of the above properties

constraints

states of particles

and sketch the main ar-

form. We refer to the quoted references

for

SUM RULES

The screening properties

of the Coulomb

in terms of the excess particle ticles are fixed at

~...~

force

are conveniently

density at ~ , ~C~lql... 9a)

expressed , when par-

191

(q, "'" q,'-)

P = (~j×) ge

e~

~'-~1

denotes the position × of a particle of species ot and char,

~qq~

=

~

~(x-

×~)

and

~(q1.-.~)

are the cor-

relation functions of the infinitely extended state. A phase with good screening properties

(plasma or Conducting phase) is

characterized by the fact that the excess charge density carries no multipole of any order I

A 4~ ~ : ~ ~- e~ ?c~l ~...~)

where traints

~(~)

for

is

=

(2)

o

an h a r m o n i c p o l y n o m i a l

~ = 0,i,2..-, ~ =

on

~

. This set

of cons-

1,2,... ( ~ , ~ s u m rules), which do not exist

for short range forces, express that typical configurations in a plasma phase shield the multipoles induced by specifying any arrangement of the system's charges. The case

~= O

(resp. ~ = I ) are called the charge

(resp. dipole) sum rules. The general ( ~ )

sum rules can be established

by two different methods. Proposition i

~12]

The states constructed by ~rydges the Debye screening regime

les for all

~o

0, 1 .

and Federbush [2], and Imbrie [3] in

~e%/{D

, ~=

~


and g~p

=

i s the quantum mechanical charge-charge correlation

(~W~ ~ 1 7 ~ ~'i~.

is the plasmon frequency.

(47) can be derived from the energy-entropy

13 ~'A* ['H,A'J>

~ ,~

[AI

(51)

Inserting this in (49) leads immediately to the result (47). It should be stressed again that the relation (47) is specific to the OCP and does not extends to multicomponent systems. In a general quantum charged system, the proper generalization of the Stillinger-Lovett (9)-(10)

condition

involves the two point Duhamel function

g'(x) = where ~ )

-.'

~ ~ C'=(x~ ~ ( o ) >

(52)

is the imaginary time displaced charge operator formally gi-

yen by ~ ( ~ p ( - H _ ~ l ~ ) ~ p ( ~

)

In the linear response theory,

the shielding of an infinitesimal test charge leads to the same constraint~ (9) or (i0) with

~

(x)

in place of ~(x) , i.e. in three

dimensions ~7]

I + ~"~

~'~x IxJ~' ~C~) = o

(53)

There is also a non perturbative proof of (53) (the quantum mechanical analog of Prop. 3) relying on the assumption that the imaginary time displaced charge correlations have suitable spatial decay properties and satisfy the ~ = ~

sum rules (for details, see ['28]).

207

REFERENCES i

J. Fr~hlich,

Y.M. Park

: Comm. Math.

2

D. Brydges,

3

J. Imbrie

4

See the articles by J. Fr~hlich and T. Spencer, D.C. Brydges and

P. Federbusch

Phys. 57, 235 (1978)

: Comm. Math.

Phys. 73, 197 (1980)

: Comm. Math. Phys. ~_77, 515 (1983)

P. Federbusch,

M. Aizenman,

J.L. Lebowitz

in the Proceedings

the Erice Summer School, Ed. Velo and Wightman, 5

S.F. Edwards, J. Math.

6

M. Aizenman,

Ph.A. Martin

(1981)

: J. Math.

:

: Ann. Phys. 85, 303 (1974)

: Comm.Math.

Ph.A. Martin

P. Federbush

Plenum Press

: J. Math. Phys. ~, 778 (1962), A. Lenard

Phys. ~, 533 (1963), H. Kunz

Ch. Lugrin, 7

A. Lenard

of

Phys. 78, 99 (1980)

Phys. 23, 2418 (1982)

: Surface effects in Debye screening,

preprint Michi-

gan University 8

Ch. Gruber, J.L. Lebowitz,

9

B. Jancovici

Ph.A. Martin

: J.Chem. Phys. 75, 944 (1981)

: J. Stat. Phys. 28, 43 (1982), 29, 263 (1982), 3~, 8o3,

(1982)

io

J.L. Lebowitz,

Ph.A. Martin

: Phys. Rev. Lett.

11

B. Jancovici,

J.L. Lebowitz,

Ph.A. Martin

tions in an Inhomogeneous

5~4, 1506 (1985)

: Time-Dependent

Correla-

Plasma, to appear in

J. Stat. Phys. 12

J.R. Fontaine,

13

Ch. Gruber, Ch. Lugrin~ Ph.A. Martin

14

L. Blum, Ch. Gruber, J.L. Lebowitz, 48, 1769

Ph.A. Martin

: J. Stat. Phys. 3~, 163 (1984) : J. Stat.Phys. Ph.A. Martin

2~, 193 (1980)

: Phys. Rev. Lett.

(1982)

15

J. Fr~hlich,

T. Spencer

: Comm. Math.

16

F. Stillinger,

R. Lovett

17

Ph.A. Martin,

Oh. Gruber

18

Ph.A. Martin, T. Yalgin

19

J.L.

Lebowi~z

20

J.L.

Lebowitz,

21

A. Alastuey,

B. Jancovici

: #. Stat.

22

J. Benfatto,

Oh. Gruber,

Ph.A. Martin

Phys. 8~4, 55 (1982)

: J. Chem. Phys. 49, 1991 (1968) : J. Stat. Phys.

31, 691 (1983)

: J. Stat. Phys. 22, 435 (1980)

: Phys. Rev. A 27, 1491 (1983) Ph.A. Martin

: J. Stat. Phys.

34, 287 (1984)

Phys. 34, 557 (1984) : Helv. Phys.Acta 57, 63 (1984)

208

23

A. Alastuey,

Ph.A.

Martin

24

Ph.A. Martin,

Ch. Gruber

: Phys.

25

Ph.A. Martin,

Ch. Oguey

: J. of PhySics

26

M. Fannes,

27

G. Mahan

28

Ch. Oguey

A. Verbeure

: Many Particle : Ph.D.

: J. Stat.Phys.

: Comm. Physics

Dissertation,

3~,

405 (1985)

Rev. A 3~, 512

Math.

(1984)

A 188, (1985)

Phys.

55,

125 (1977)

- Plenum New-York EPF-Lausanne

(1981)

(1985)