Lecture Notes in Physics Edited by H. Araki, Kyoto, J. Ehlers, MQnchen,K. Hepp, Z0rich R. Kippenhahn,MLinchen,H. A. Weidenmeller, Heidelberg and J. Zittartz, Kbln Managing Editor. W. BeiglbSck
257 Statistical Mechanics and Field Theory: Mathematical Aspects Proceedings of the International Conference on the Mathematical Aspects of Statistical Mechanics and Field Theory Held in Groningen, The Netherlands, August 26-30, 1985
Edited by T.C. Dorlas, N.M. Hugenholtz and M. Winnink
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo
Editors
T.C. Dorlas N.M. Hugenholtz M. Winnink Institute for Theoretical P~ysics, University of Groningen P.O. Box 800, Groningen, The Netherlands
ISBN 3-540-16777-3 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-16777-3 Springer-Verlag NewYork Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2153/3140-543210
PREFACE
This volume of LECTURENOTES IN PHYSICS contains the text of the lectures presented during the International Conference on the Mathematical Aspects of S t a t i s t i c a l Mechanics and Field Theory, held in Groningen, the Netherlands, 26-30 August 1985. Some of the participants have provided us with an abstract of the poster(s) they presented during the poster sessions. We have incorporated these abstracts as we]l in this volume. As is unavoidable, in those abstracts claims are presented without proof. Interested readers are encouraged to contact the authors. The organizers of this conference thought i t a good idea to have a conference on mathematical physics in the Netherlands where this subject is receiving more attention nowadays. We hope, among other things, to have given an impetus to mathematical physics in the Netherlands. We want to express our sincere gratitude to the s c i e n t i f i c advisory board, the lecturers, the participants, the Congress Bureau of Groningen University, the secretary of the I n s t i t u t e for Theoretical Physics Ms. M. Boering, and last but not least to the sponsors of this conference, Philips Research Laboratories, Shell Nederland BV, IAMP, Groninger Universiteitsfonds, Koninklijke Nederlandse Akademie van Wetenschappen, College van Bestuur RUG, Fakulteit der Wiskunde en Natuurwetenschappen RUG and Subfakulteit Natuurkunde.
T.C. Dorlas N.M. Hugenholtz M. Winnink
Scientifc Advisory Committee: H. Araki, J. FrOhlich, R. Haag, N.M. Hugenholtz, J.L. Lebowitz
CONTENTS
A MODEL FOR CRYSTALLIZATION: A VARIATION ON THE HUBBARD MODEL T. Kennedy, E.H. Lieb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FIRST ORDER PHASE TRANSITIONS AND PERTURBATION THEORY J. Bricmont, J. Slawny . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
LOW TEMPERATURE CONTINUOUS SPIN GIBBS STATES ON A LATTICE AND THE INTERFACES BETWEEN THEM
-
A PIROGOV-SINAI TYPE APPROACH
M. Zahradnik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
SPIN GLASSES, EFFECTIVE DECREASE OF LONG-RANGE INTERACTIONS A.C.D. van Enter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
ANALYTIClTY IN SOME MODELS OF QUANTUM STATISTICAL MECHANICS H. A r a k i
......
~. ..........................................................
89
K-THEORY OF C*-ALGEBRAS IN SOLID STATE PHYSICS .J. B e l l i s s a r d
..............................................................
99
QUANTUM FIELD THEORY WITHOUT CUTOFFS: RENORMALIZABLE AND NONRENORMALIZABLE A. Kupiainen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
157
RENORMALIZATION GROUP METHODS IN RIGOROUS QUANTUM FIELD THEORY K. Gaw~dzki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17]
RENORMALIZATION GROUP METHODS FOR CIRCLE MAPPINGS O.E. Lanford I I I
..........................................................
]76
CORRELATIONS AND FLUCTUATIONS IN CHARGED FLUIDS Ph.A. M a r t i n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
190
MODELS OF STATISTICAL MECHANICS IN ONE DIMENSION ORIGINATING FROM QUANTUM GROUND STATES H. Spohn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
209
WHY DO BOSONS CONDENSE? J.T.
Lewis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
234
Vl
BLACK HOLES AND QUANTUM MECHANICS G. ' t Hooft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
257
QUANTUM FIELD THEORY AND GRAVITATION R. Haag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
258
A REMARK ON ANTIPARTICLES H.J. Borchers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
269
NOTES ON THE CANONICAL ANTICOMMUTATION RELATIONS R.V. Kadison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
281
A DIFFERENTIAL-GEOMETRIC SETTING FOR BRS TRANSFORMATIONS AND ANOMALIES D. K a s t l e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
295
POSTER ABSTRACTS RENORMALIZATION OVA HIERARCHICAL FERMION MODEL IN TWO DIMENSIONS T.C. Dorlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
313
GRIBOV COPIES AND ABSENCE OF SPONTANEOUS SYMMETRY BREAKING IN COMPACT U(1) LATTICE HIGGS MODELS Ch. Borgs, F. N i l l
........................................................
319
RENORMALIZATION GROUP APPROACH FOR THE ISING MODEL AS AN APPROXIMATE SOLUTION OF THE DIAPHANTIAN SYSTEM OF EQUATIONS S. Rabinovich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
320
TRANSFORMATION OF THE TRANSFER MATRIX FOR THE ISING MODEL UNDER DECIMATION S. Rabinovich . . . . . . . . . . . . . .
...............................................
321
SOME RESULTS CONCERNING THE LOCALIZATION PROBLEM IN ONE-DIMENSIONAL QUASlPERIODIC SYSTEMS D. P e t r i t i s
...............................................................
322
RANDOM WALKS ON RANDOM LATTICES W.Th.F. den H o l l a n d e r , P.W. K a s t e l e y n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
323
THE ROLE OF TRANSVERSE FLUCTUATIONS IN MULTIDIMENSIONAL TUNNELING A. Auerbach, P. van Baal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
323
Vll
SELF-SIMILAR TEMPORAL BEHAVIOR OF P4~NDOMWALKS IN RANDOM MEDIA J. Bernasconi, W.R. Schneider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
324
LOOP EXPANSIONS IN THE PRESENCE OF GRIBOV COPIES F. N i I l
....................................................................
325
NON-EXISTENCE OF LONG-P~ANGE-ORDER FOR A CERTAIN ID MODEL AND THE SOLITON PICTURE H. Grosse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
325
QUANTUM FIELDS OUT OF THERMAL EQUILIBRIUM W. Schoenmaker, R. Horsley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
326
STATISTICAL MECHANICAL APPROACH TO THE CONSTRUCTION OF FOUR DIMENSIONAL BOSON FIELD THEORIES G.A. Baker, J.R. Johnson, J.D. Johnson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
326
CORRELATION LENGTHS AT ZERO TEMPERATURE IN ONE DIMENSION J.-J.
Loeffel
..............................................................
327
HORIZONTAL WILSON LOOPS IN FINITE TEMPEPJ~TURE LATTICE GAUGE THEORIES C. Borgs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
327
THE RANDOM BOND POTTS AND ASHKIN-TELLER MODELS P.L. C h r i s t i a n o , S. Goulart Rosa J r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
328
SPECTRAL LINE SHAPES IN QUANTUM MARKOV SCATTERING H. Maassen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
328
EXACT BOGOLIUBOV LIMITS FOR THE BASSICHIS-FOLDY MODEL AND CONTINUED FRACTIONS D. Masson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
328
A M o d e l for C r y s t a l l i z a t i o n : A Variation on t h e H u b b a r d M o d e l
Tom Kennedy and Elliott H. Lieb Departments of Mathematics and Physics Princeton University Jadwin Hall, P.O. Box 708 Princeton, NJ 08544
Much attention has been paid to the question of proving the existence of long range order in model statistical mechanical systems in which the basic atomic constituents interact with short range forces. An important example is a lattice spin system in which the spin at each site represents the localized spin of an atom located at that site and where the short range, pairwise interaction (Ising or Heisenberg) reputedly comes from an interatomic exchange energy. Another problem - so far unsolved - is the existence of periodic crystals which are supposed to come from short range (e.g. Lennard- Jones) interatomic potentials. In the real world, however, these interactions are not given a-priori; it is ultimately itinerant electrons and their correlations that give rise to the long range ordering. In other words, a deep unsolved problem is to derive magnetism or crystallization from the SchrSdinger equation - or some caricature of it. The construction of a simple model based on itinerant electrons, and the rigorous derivation of ordering from it, is a challenge for mathematical physicists. A lattice model of itinerant electrons that is believed to display ferro and antiferromagnetism - if it could be solved - is the Gutzwiller-Hubbard-Kanamori model [1-3]. We are also unable to solve it, but we have succeeded in proving that a simplified version of it does display crystallization. It is a toy model but it is, to our knowledge, the first example of this genre. Roughly, it has the same relation to the Hubbard model as the Ising model has to the quantum Heisenberg model. Here we shall give a brief report of our results, the full details of which will appear elsewhere [4]. The Hubbard model, which is the motivation for our model, is defined by the second-quantized Hamiltonian
_
x c:oc o + 2 u a
x,y•h
xTn ,
(1)
x~.A
with the following notation: a --- +1 denotes the 2 spin states of the electrons; A is a finite lattice; cxa is a fermion annihilation operator for a spin a electron at x E A; n~a = c ~ c ~ a is the number operator for spin a at x. Electrons interact
only at the same site with an energy 2U, and tx~ = ty~ is the hopping amplitude from x to y. The crucial assumption will be made that A is the union of two sublattices A U B such that tx~ = 0 unless x E A,y E B or x E B , y E A. The number of sites in A,A, and B are denoted by IAI, IAI and IS]. The two sublattices need not be isomorphic. Thus, for example, a face-centered cubic lattice is allowed with A=face centers and B = c u b e corners. A is said to be connected if every x, y E A can be joined by a "path" through nonzero t's. In our model we assume that one kind of electron (say a = - 1 ) does not hop. One can say that these electrons are infinitely massive. The Hamiltonian is then x,yEh
xEA
with nx = c~cx (the subscript a is omitted since the dynamic electrons have a = +1) and with W ( x ) = +1 if a fixed electron (a = - i ) is at x and W(x) = 0 otherwise. It will be recognized immediately that (2) is just a fancy way to say that the movable electrons are independent, with a single particle Hamiltonian
h=T+V,
(3)
with T being the IAt-square matrix t~y and Vxu = 2UW(x)4~. It is convenient to write h = h + U with h--T+US (4) with S ~ = sx~xy, and sx = 1 (resp. -1) if x is occupied (resp. unoccupied). The {sx) are like Ising spins. We shall henceforth call the movable particles "electrons" and the fixed particles "nuclei". This terminology is most appropriate if U < 0, for then H does represent a lattice system of electrons and nuclei in which all Coulomb interactions except the on-site electron-nucleus attraction and the on-site infinite nuclear repulsion are regarded as "screened out". This conforms with the spirit of the original Hubbard model. The electron number, the nuclear number and the total particle number, all of which commute with H, are, respectively Ne=Enx, xEA
1
Nn=~E[sx+ll=EW(x), xEh
JC=N~+Nn.
(5)
xEh
It is to be emphasized that W does not represent a disordered potential. We take the "annealed", not the "quenched" system. The ground state for fixed N~ and N,t is defined by taking the ground state of H (with respect to the electrons) for each W and then minimizing the result with respect to the location of the nuclei. The ground state energy will be denoted by E(Ne,N,~). Likewise, for
positive temperature, we take Tre-~H with respect to both the electron variables and the nuclear locations. Since txv = 0 for x, y on the same sublattice, the spectra of H and H H are invariant under t~y --* - t x v (all x, y). There is also a hole-particle symmetry. If c~ ---* c~,cx ---* c?=,n~ --* 1 - nx, t=y --* - t x y , then H(U) --~ H ( - U ) + 2UNn. If W(x) ---* 1 - W ( x ) , then H(U) --~ H ( - U ) ÷ 2UNe. A similar s y m m e t r y holds for H H. Thus, the U > 0 and U < 0 cases are similar - - from the mathematical point of view. Our results are of two kinds. The first concerns the ground state which we prove always has perfect crystalline ordering and an energy gap (defined later). T h e second concerns the positive temperature ( 1 / k T = / 3 < oo) grand canonical state. For large/3 and dimension d > 2, the long range order persists. For small /3 it disappears and there is exponential clustering of the nuclear correlation functions.
The Ground State T h e o r e m 1: (a) Let U < O. Under the condition )4 = Ne + N,~ < 2JAJ, the ground state (i.e. we minimize E(N~,N,~) over the set N~ + Nn <_ 2JAJ) occurs for Ne = Nn = JA I and a minimizing nuclear configuration is W(x) = WA(X) =-- l(x e A),O(x • A). Under the condition J4 < 2IBJ, the ground state is Ne = Nn = ]SJ and the B sublattice is occupied (W = WB). If A is connected, these are the only groundstates, i.e. if ]AJ > JBJ the ground state is unique; if ]AJ = JB I it is doubly degenerate. No assumption is made about the sign or magnitude or periodicity of the txv other than try = 0 for x, y E A or x, y E B. (b) Let U > O. Under the condition )4 >_ JAJ + JB], there are two ground states: Ne = IAI,N,~ = JBJ,W = W s and Ne = JBJ,Nn = JAJ,W = WA. If A is connected, these are the only ground states. The condition N = JAJ is called the half-filled band. If JAJ = JBJ = JAm/2, the crystal occurs at the half-filled band. If A is a cubic lattice, for example, this means that the ground state is a cubic lattice of period v/2 oriented at 45 ° with respect to A. Theorem 1 relies heavily on the fact t h a t the electrons are fermions. The ground state would be completely different if they were bosons. For bosons and for A a cubic lattice, the nuclei would all be clumped together in the ground state instead of being spread out into a crystal. By using rearrangement inequalities it is possible to describe this clumping quantitatively. Next, we define the energy gap. Actually two different definitions are of interest. First, let
E()¢) -
min{E(N~,N,~)lNe + Nn = M}.
(6)
T h e chemical potential is defined by ~()4) _= E()¢ + 1 ) - E(~/).
(7)
We say there is a gap of the first kind at A{ if ~ ( , ~ / ) - ~ ( J ~ / - 1) __~ E1 > 0
(s)
with el being independent of the size of the system. We say there is a gap of the second kind at Ne, Nn if E ( N e + 1, Nn) + E(Ne - 1, Nn) - 2E(Ne, Nn) >_ 62 > O.
0)
In other words, the nuclear number is fixed in the second definition. A gap is one indication t h a t the system is an insulator, for it implies that it costs more energy to put a particle into the system than is gained by removing one. The first kind of gap is relevant if one views our model as an approximation to the Hubbard model; the second is relevant from the "electrons and nuclei" point of view. T h e o r e m ~. Assume that h is not only connected but that every x, y E lk can be connected by a chain with Itabl >_ 6 for every a,b on the chain. Also, assume that IITI[ <_ r. The following energy gaps exist with e2 >_ ea > 0 and depending only on 6, r and U: U
O: First kind at )4 -- IAI + ISl. Second kind at Ne = IAI,Nn = IBI and at Ne = ]B],Nn = [A I. In order to give the flavor of our methods, the proof of Theorem 1 will be given here. T h e proof of Theorem 2 is more complicated. Proof of Theorem 1: Let Ax _< A2 -< .-. be the eigenvalues of h in (4). They depend on the nuclei. For Ne electrons the ground state energy, E, of H satisfies
1%
(a0) j-----1
,~" < 0
But T r h = UY]~sx = 2UNn - UIAI and Ihl = ( T 2 + U 2 + u g ) 1/2 with Jxy = t~y[sx + su]. Since the function 0 < x --* z 1/2 is concave, f ( y ) = T r { T 2 + U2 + y U J ) 1/2 is concave in y E [-1,1]. But f ( - 1 ) = f(1) (since spec(h) is invariant under T -* - T ) , so f(1) _< f(0), with equality if and only if g - 0. Thus E _> UXt - ~1 U]A] - ~ T r ( T 2 + U 2) a/~
(11)
If A is connected, the only ~ays to have J = 0 are either W = WA or WB. Consider U < 0 and N <_ 21A ], whence, from (11), 1 E >_U(21A ]- ~lhl) - ~ Vr(T 24-
U2) 1/2"
(12)
If W = WA then, as is easily seen, h has precisely IAI negative and ]B I positive eigenvalues. Thus, if W = WA and Ne -- IAI, then (12) is an equality. The other cases are similar. []
Grand Canonical Ensemble
First, we define the partition function E. A nuclear configuration is denoted by S = {sx},s~ = +1, and the As- are the eigenvalues of h in (4). If #,~,pe are the nuclear and electronic chemical potentials, I^1 E= Z
1 exp[~3pn( Z sz + IAI)] r I {14- exp[-~(A s 4- u - ~,)]).
S
x
(13)
j=l
The product in (13) is just the well known Fermi-Dirac grand canonical partition function for the electrons, We want to choose #e,#n so that (N~) = ½1AI and 1 (N~) = ~IAI, or ( ~ sz) = 0. From the fact that if T -~ - T , spec(h) --~ spec(h), one has that when S -+ - S , spec(h) -~ -spec(h). It is then easy to see that the desired chemical potentials are #e = #n = U. Since ~ ,~j = U ~ sz, (13) becomes in this case (after dropping an irrelevant factor 2iAle3vhAI/2) E= ~
exp[-flF(S)]
(14)
S
with
- 3F(S)=
I^1 ~ gncosh(~3)~s')= Tr£ncosh[~3h] jml
1 2 q- U 2 4- U J)1~2]. = Trgncosh[~3(T
(15)
Thus, (14) is like an Ising model partition function but with a complicated, temperature dependent "spin-spin" interaction, F(S), given by (15) in terms of the eigenvalues of h. With respect to this "spin measure" we can talk about the presence or absence of long range nuclear order in the thermodynamic limit.
6 •
In order to discuss this limit we henceforth restrict ourselves to a translation invariant nearest neighbor hopping on a cubic lattice in d dimensions. What we are able to prove is summarized in the schematic figure below and in T h e o r e m 3: For all U and su~eiently large fl there is long range order
for d >_ 2 (the same kind as in the ground state). For all U and sul~ciently small/~ there is none; indeed there is exponential decay of all nuclear correlation functions.
13 Presumably there is no intermediate phase, but we cannot prove this. For large U,/~e is clearly linear in U. For small U, we have the bound on the lower /~c ~ U -1/(2+d) and the bound on the upper/~c ~ I~nUIU -2. Our guess is that the true state of affairs is/~c ~ U --2. For large/3 we use a Peierls argument; for small/~ we use Dobrushin's uniqueness theorem. A sketch of our proof omitting many important details - is the following. For simplicity, we here consider only large U > 0. Define, for x > 0, P(x) = ~ncoshx 1/2. (16) -
[We note in passing that P is concave, and we see from the last expression in (15), using the proof in Theorem 1, that F(S) has its minima at precisely the
same values of W (or S) as in Theorem 1.] P is a Pick (or Herglotz) function with the representation
1,2 ~ = + x]-' x - 1 / 2 t a n h x 1/2 = E [ C k + ~)
P'(x) =
(17)
k=O
1 2 (T 2 + U 2 ~i U J) with gz~ = tx~(sx + sy). We are interested in x = ~/~ L o n g R a n g e O r d e r (large/~): Choose S, and then define antiferromagnetic contours in the usual way. If y is a connected contour component and r + is the whole contour, we want to prove that A -- F(F + y) - f ( r ) > C[ l for a suitable constant C = C(U). Obviously, Jr+~ = J r + Jr. To remove % we change sx to - s x inside ~. For 0 < t < 1, define J(t) = Jr + tJ~. Then, assuming for simplicity that r lies entirely outside ~, we have, by differentiating (15) and using (17), that
-/~A=
/~2U k~0
dtTr(GkJ~),
(18)
o
with Gk = [(k+ ~) 1,2 1r2 + ~ /~2(T2 + U 2 + U Jr + tug~)] -1
(19)
Integrating by parts, this becomes --~/k = - ( ) ~ 4 U 2 / 1 6 )
dr(1 - t)TrGkJ~GkJ~ + . . .
(20)
[The ... terms in (20) come from t = 0 in the partial integration. They are small and easily bounded for large U, but they have to be treated more judiciously when U is small.] If A, B, D are matrices with A > B >__0, then T r A D ? A D >_ T r B D ? B D . Also, A _> B > 0 =~ 0 < A -1 <_ B -1. Moreover, T 2 < (2d) 2 and U J ~_ T 2 + U 2. Using this, we have the matrix inequality
ak > [(k + --
:
+
2((2d)2 + V2)]-,.
(21)
2 /
Summing on k, A > (const.) U2(C2d) 2 + U2)-3/2Tr(J~)2. Clearly Tr(J~) 2 = (const.)]~]. Thus, for U large, C(U) > (const.)U -1, and thus long range order exists in d >__2 if/~/U is large enough. A b s e n c e of Long R a n g e O r d e r (small j3): Dobrushin's uniqueness theorem [5-7], together with the modification in [8], gives the following criterion for
exponential clustering. We have to bound the change in F when we change the spins at x and y (taking the worst case with respect Vo the other spins). Call this f~u" T h e requirement is that for some m > 0 and all x,
Z fzvexp[mlx - Y]] < 1.
(22)
Y
By an argument similar to the preceding (large U)
A~ z (Z3u2/16)~ T~a~J~a~J~ k
(~aU2/16)
~ la~(~,y)l~
(23)
k
Here, Jz = ifzT + Tifz and Gk(x, y) is the x, y matrix element of (19). To implement (22) we now require an upper bound on Gk(x,y) that has exponential decay. For this purpose the Combes-Thomas argument is ideal. Let Q be the matrix with elements Q~u = 6~ve n~ and with ]n I = 1. Then
QGklQ -1 = G-~1 + Rk - Lk.
(24)
The "remainder" Rk can be bounded for large U: IIRkH < C~2U for some constant C. Similar to (21), for large U
a ; ~ _> [(~ + 1 ~ 2 , + 8 ~ U ~ ] = ~
(25)
Thus,
IIL~-'II < [ ~ k - c~2u] -1.
(26)
Since I(L;1)=~l _ IIi;'ll, we have from (24), I(QCkQ-~)~I < IlL;Ill, and thus I(ak)~l _< exp[~. (y - x)]{~k - cz2u} -1.
(27)
This holds for all n, so summing on k, fx~ < (const.) U-le -21x-yl. Hence, (22) holds with m = 1 if ~/U is small enough. The support of both authors by the U.S. National Science Foundation, grant PHY 8116101 A03, is gratefully acknowledged.
References
1. M. C. Gutzwiller, Phys. Rev. Letters 10, 159-162 (1963), and Phys. Rev. 134, A923-941 (1964), and 137, A1726-1735 (1965). 2. J. Hubbard, Proc. Roy. Soc. (London), Ser. A276, 238-257 (1963), and 277, 237-259 (1964). 3. J. Kanamori, Prog. Theor. Phys. 30, 275-289 (1963). 4. T. Kennedy and E. H. Lieb, An itinerant electron model with crystalline or magnetic long range order (in preparation). 5. R. L. Dobrushin, Theory Probab: and Its. Appl. 13, 197-224 (1968). 6. L. Gross, Commun. Math. Phys. 68, 9-27 (1979). 7. H. FSllmer, J. Funct. Anal. 46, 387-395 (1982). 8. B. Simon, Commun. Math. Phys. 68, 183-185 (1979). 9. J. M. Combes and L. Thomas, Commun. Math. Phys. 34, 251-270 (1973).
FIRST ORDER PHASE TRANSITIONS AND PERTURBATION THEORY
J. Bricmont and J. Slawny I n s t i t u t de Physique T h e o r i q u e , 2, ch. du Cyclotron Universite Catholique de Louvain B - 1348 L o u v a i n - l a - N e u v e , Belgium Center for T r a n s p o r t T h e o r y and Mathematical Physics V i r g i n i a Polytechnic I n s t i t u t e and State U n i v e r s i t y Blacksbur9, V i r g i n i a 24061, U . S . A .
I. INTRODUCTION While in most situations the state of
a system changes
smoothly when e x t e r n a l
parameters such as pressure, magnetic field, or temperature are varied, t h e r e are also sometimes sudden jumps in the d e n s i t y , occur
at
Statistical
some values of these external Mechanics
is to
understand
the magnetization or the e n e r g y t h a t
parameters. these
One of the basic goals of
phase
transitions
starting
from
a
microscopic description. Since the invention of the Peierls' argument [41, 3 0 ] , have been shown to e x h i b i t f i r s t - o r d e r
phase transitions
some basic questions remain unanswered.
For example:
a great v a r i e t y of models [21, 26,
22].
However
- We do not have a proof of the occurrence of f i r s t - o r d e r phase transitions f o r a reasonable model of a simple f l u i d .
For a simple l i q u i d - v a p o u r phase t r a n s i t i o n ,
we cannot go much beyond van der Waals t h e o r y [61, 39]. - We do not have a complete u n d e r s t a n d i n g of the Gibbs phase rule, even for classical lattice systems (for partial results see [34, 4 5 ] ) . that, t y p i c a l l y ,
One would like to show
given a set of parameters, n phases will coexist on a manifold of
codimension n-1 in that space. -
From
a more
practical
point
of
view,
one
would
like
to
have
reliable
techniques to compute the phase diagram of alloys with interactions of interest. tn these lectures,
we shall discuss how the Pirogov-Sinai t h e o r y [42, 47, 48]
and some of its extensions [33, 8,
10, 13, 18, 14, 60]
p r o v i d e at least partial
answers to these questions. The main emphasis will be on the use and justification of p e r t u r b a t i o n theory.
11
To
develop
a perturbation
theory,
we
need
a
reference
system
which
is
"completely" known. At low temperatures such a reference system may consist of zero temperature states, or ground states, y i e l d i n g low temperature expansions. The f i r s t proof of convergence of a low temperature expansion was given by Minlos and Sinai [40] for the Ising model. Minlos and Sinai reformulated this model in terms ground
of a gas of contours separating states
occur,
t h e r e f o r e one could
At
regions of the lattice where d i f f e r e n t
low temperatures,
activity
of
use methods developed to control
this
gas
is
small
and
low a c t i v i t y expansion to
prove convergence here. We review this in Sect. II. However, it is often useful, and sometimes imperative, to have a p e r t u r b a t i o n t h e o r y around more general objects than ground states,
which we call restricted
ensembles. Such an ensemble consist of a subset of the phase space, t o g e t h e r with a measure (usually a Gibbs state) on it, which gives a better approximation to the true
phase than
a ground
state.
A
restricted
ensemble often
consists
fluctuations around a ground state as in the case of continuous
of
small
spin models or
quantum field t h e o r y [26, 33, 18]. However, this notion is useful in other contexts as well, as we shall t r y to demonstrate on the following examples: -
Lattice spin
related
by
Pirogov and
systems
symmetries
Sinai
of
with the
a finite
number
This
Hamiltonian.
(when the
of
number of ground
ground
states
is the
situation
state
is f i n i t e ) .
which
are
not
considered by We give an
introduction to t h e i r t h e o r y by discussing a simple example in Sect. 111. Lattice
spin
systems
with
an
infinite
number
of
ground
states.
In
this
situation, the t h e o r y is far from complete. The notion of a restricted ensemble is very
useful
in many
cases where an
infinite
number
of
ground
states
occurs
because it allows us to reduce this i n f i n i t e set to the consideration of a f i n i t e set of 9round states called dominant ground states (Sect. IV). Fluids.
T h e r e are v e r y few results on phase transitions for one component
fluids (see however [34]),
but several results are available f o r phase separation in
mixtures [44, 37, 8]. Restricted ensembles are essential in this case since t h e r e is no notion of a ground state for continuum fluids. The reference system here is an ideal or almost ideal gas (Sect. V ) . We discuss an extension of the Pirogov-Sinai t h e o r y to
- L a r g e e n t r o p y models.
the
q-state
Ports model for
q - l a r g e and for
the temperature where q÷l
phases
coexist, [8, 14, 60]. q of these phases are small perturbations of the corresponding ground
states
but
there
is
an additional
high-temperature
phase which
is
in
equilibrium with the other phases only because of its large e n t r o p y . This phase is a small p e r t u r b a t i o n of the restricted ensemble which is maximally disordered (Sect.
Vi).
12
II. THE ISING MODEL: LOW TEMPERATURE EXPANSION One of the basic open problems in the t h e o r y of f i r s t - o r d e r phase t r a n s i t i o n s is to understand
the phase diagram of
simple f l u i d s .
T h e r e are deep open
problems
concerning c r y s t a l l i s a t i o n but even t h e l i q u i d - g a s t r a n s i t i o n is understood the f o l l o w i n g lattice-gas approximation (see, intermolecular
potential
usually
long-range attractive part. Zd
(d
is the
potential,
us
of
instance,
a
[49,
short-range
only in
Sect. 5 - 2 ] ) :
repulsion
The
and
of
a
Let us c o v e r t h e continuum w i t h the cells of the lattice
dimension of
let
consists
for
require
space)
that
each
and,
in
cell
place of
the
repulsive
be occupied
by
at most
part
of
the
one p a r t i c l e .
F u r t h e r m o r e , one assumes t h a t the a t t r a c t i v e potential is insensitive to the position of the p a r t i c l e in the cell.
Then one obtains a lattice gas model, with a v a r i a b l e
p(a) at each lattice point a, which is equal 1 if the cell is occupied, and 0 if it is empty.
A
change
f e r r o m a g n e t i s m ; the
of
variables,
p(a)
=
a t t r a c t i v e potential
½(l+o(a)),
of the lattice
leads
us
to
gas yields
a
mode]
of
a ferromagnetic
interaction of the magnetic system. The simplest version of
such a f e r r o m a g n e t i c model is the
nearest n e i g h b o r
Ising model: H = -
Y. oCa)~Cb) - h~: oCa) , a
oCa) = ±1 ,
(2.1)
w h e r e means t h a t a and b are nearest n e i g h b o r s , nearest model,
neighbor
(n.n.)
pairs,
called
especially at low t e m p e r a t u r e s ,
bonds. but
Many
before
and the sum is o v e r all
results
are
discussing
known
them,
for
this
we want
to
stress t h e simplifications introduced by t h i s lattice a p p r o x i m a t i o n . Since the
lattice is given
a priori,
we do not have the problems (massless
e x c i t a t i o n s ) associated with the formation of c r y s t a l s . -
We have also i n d i r e c t l y i n t r o d u c e d a symmetry into the problem: H in (2.1) is
i n v a r i a n t under the change of h into -h and a(a) into - o ( a ) f o r all lattice sites a. This
symmetry
transition,
yields
if any,
a very
drastic
simplification:
is expected to occur at h=0, i.e.
because
of
it,
the
phase
at a temperature independent
value of h. This is c e r t a i n l y not the case when t h e r e is no symmetry between the phases
e.g.
in
a
liquid-gas
transition,
where
the
pressure
for
which
phase
coexistence takes place depends in a n o n - t r i v i a l way on the t e m p e r a t u r e . - Finally, t h e r e is a w e l l - d e f i n e d reference system about which one can t r y
to
c o n s t r u c t a low t e m p e r a t u r e expansion, namely the g r o u n d states. For h > 0 (resp h < 0)
t h e r e is a unique g r o u n d state w i t h all spins equal to +1 ( r e s p .
h=0 one has both of these g r o u n d states.
-1).
For
13
We shall recall now a d e r i v a t i o n of the low temperature expansion for the free energy of the Ising model, [19], to converge.
an expansion which,
in this case, can be proved
It will be convenient to define the free energy with the g r o u n d - s t a t e
energy substracted,
like
in ~(BI +) below: the
+ after the vertical
bar indicates
which g r o u n d state is considered. Thus ~(BI ÷) = - B-11im IA1-1 log ZCAI ÷) , A where
IAI
is the number of
sequence of
finite
subsets
points in A. The limit is taken over an expanding of
the
lattice
with
not
too
large
boundaries,
like
NxN-squares with N*~, or, more generally over Van Hove sequences, [43, Chapter 2]. Furhermore, Z(EI +) =
~: exp -13HA(oI+)
,
(2.2)
(5
and HACoI÷) = Z ( a C a ) a ( b ) - l ) In
(2.2)
the
configurations
s~Jm is
(2.3)
.
over all
are extended
complement of A,
configurations
to all
the
and the summation
Hamiltonian H^(-I ÷) is as in
(2.1),
o=(o(a))a~ A in A;
lattice by
is o v e r all the but with
more
involved
than
necessary for
when we discuss the
in
a(a)=l
bonds
(2.3)
for
these
a in
the
intersecting A.
The
h=O and the energy of the ground
state subtracted: o ( a ) o ( b ) replaced by o ( a ) o ( b ) - l . a manner
setting
The expansion will be derived in
this
model.
however,
be useful
simplicity
and in o r d e r to make the calculations e x p l i c i t ,
This
Pirogov-Sinai t h e o r y
derivation in
Sect.
3.
will, For
we shall work in two
dimensions. Let us start by computing the f i r s t o r d e r term in the expansion of ~(151*). 13 large, flips
the leading term in the p a r t i t i o n function (2.2)
i.e.
occupied
from
configurations
by plus-spins.
where all the
Restricting the
sites
For
comes from isolated spin
adjacent to a minus-spin
sum in the partition
are
function to these
configurations one obtains approximately, by adding the contributions of O, 1, 2 . . . . spin flips: Z ( A I + ) = I + I A l e - 4 B + ½ 1 A I ( I A I - 1 ) ( e - 4 B ) 2 + . . . = ( I + e -4~ ) I^1,- e x p ( I A l e-4B) .
(2.4)
This yields 13~(61+) = - exp-41~, to leading o r d e r . One could obtain in this fashion higher o r d e r terms but convergence of this expansion is best proved after an introduction of contours - the basic notion of
14
Peierls' argument: Consider a c o n f i g u r a t i o n o=(o(a))a~Z2 equal +1 e v e r y w h e r e , w i t h a possible exception of a finite set. Define the b o u n d o r y B(o) of o by B(o) = {: o(a) # o ( b ) } . Now decompose B(o) into connected components in the usual way: associate to each bond the bond of the dual lattice which is p e r p e n d i c u l a r to it, and decompose the r e s u l t i n g set of lines into connected pieces, called contours.
For each c o n t o u r ~"
set [~'] = {a = z d : t h e r e exists b such t h a t ~ ~'} . We mention t h a t this definition of contours which is convenient f o r the Ising model needs to be modified f o r more general models. In the special case of the Ising model this general notion would be defined as follows.
Cover t h e lattice w i t h squares of
side R (where R is chosen to be l a r g e r than the
range of the
here is 1) and define a square C to be r e g u l a r if a l c
i n t e r a c t i o n , which
(:the r e s t r i c t i o n of o to C) is
equal to the r e s t r i c t i o n to C of one of the g r o u n d states ( i . e .
o(a)=+l f o r all a¢C
or o(a)=-I
a connected set of
f o r all asC).
Then
a contour
is a pair made of
i r r e g u l a r squares t o g e t h e r with the r e s t r i c t i o n of o to these squares. One i m p o r t a n t p r o p e r t y of this general definition is t h a t a c o n t o u r is no l o n g e r a p u r e geometrical object b u t
r a t h e r it is a subset of the lattice t o g e t h e r With a c o n f i g u r a t i o n on it
from which one can read off the b o r d e r i n g g r o u n d states. With
any
of
the
two
above
definitions,
contours
enjoy
the
following
two
i m p o r t a n t geometric p r o p e r t i e s :
P r o p e r t y 1.
They are closed in the sense t h a t f o r any c o n t o u r ~', z d \ [ ~ "] ( o r
the complement of the s u p p o r t of ~' in the general case) can be decomposed into several connected components o n l y one of which is i n f i n i t e .
The l a t t e r is called t h e
e x t e r i o r of ~' while the f i n i t e components form the i n t e r i o r of ~;, Int~'. there
is a unique g r o u n d
state
(+ o r
-)
along
the
b o u n d a r y of
Moreover,
each of
these
components. This p r o p e r t y is obvious in the case of the Ising model; it also holds w h e n e v e r the number of g r o u n d states is f i n i t e b u t does not,
in general, when t h e
number of g r o u n d states is i n f i n i t e (see Sec. 4).
Property
2.
The
number of
contours
of
length
n
(having
n bonds)
and
containing a f i x e d bond is bounded b y c n w h e r e c is a constant ( d e p e n d i n g only on the lattice and the number R e n t e r i n g the general definition of c o n t o u r s ) . Using
the f i r s t
property
subfamily of outer contours, the c o n t o u r s .
we can
define,
for
any
family
of contours,
the
namely those t h a t do no lie in the i n t e r i o r of any of
15
Let us consider the p a r t i t i o n f u n c t i o n Z(~') of all the c o n f i g u r a t i o n s h a v i n g o n l y one o u t e r c o n t o u r ~" and consider the Hamiltonian ( 2 . 3 ) . Then
Z(~') = e x p ( - 2 B l ~ l ) ]I Z ( I n t i E I E ( i ) ) i
where
,
(2.5)
I EI is the number of bonds in ~', t h e p r o d u c t runs o v e r all the connected
components Int.~" of the i n t e r i o r of ~', and
~(i) = ÷ o r - d e p e n d i n g on t h e g r o u n d
state w h i c h , b y P r o p e r t y 1, b o r d e r s I n t . L
Finally, Z(AI ÷) is defined b y (2.2) and
1
I
Z(A I - ) is defined s i m i l a r l y , b u t w i t h o ( b ) = - I f o r b in the complement of A. Now f o r any A, we can w r i t e Z(AI +) = where
Z H Z(~), w ~sw
the
sum
(2.6)
is over oli compatible
families u of outer contours
product over contours of the family; compatiblity means
in A and the
here that the supports of
the contours are disjoint and that no contour lies in the interior of another contour of the family. The
{important) factorization property implicit in (2.5) derives here from the
fact that w e have a fixed ground state in the exterior of all outer contours. Now
to get our final expansion w e
use the symmetry of the Hamiltonian under
the global spin-flip relating the two ground states, which implies that for any A,
Z(AI ÷) = Z ( A I - )
•
(2.7)
Using (2.7) and ( 2 . 6 ) , one t r a n s f o r m s (2.5) i n t o : Z ( k l +) =
~
][
w ~sw
exp(-2Bl~l)]I Z(Intk~l+) k
,
(2.8)
with the s u m extended over the same set of w as in (2.6). This expansion is in a form which is suitable for iteration; iterating it one gets ZCAI*) =
~. ]T z(~') ,
zC~')=exp-2f31~'J
,
(2.9)
w h e r e the sum now runs o v e r all families u of contours t h a t are n o n - i n t e r s e c t i n g , w i t h o u t the c o n s t r a i n t t h a t the
contours are o u t e r c o n t o u r s .
The
identity
(2.9)
shows t h a t Z(A1 +) is equal to the p a r t i t i o n f u n c t i o n of a gas of an i n f i n i t e n u m b e r of "species of hard-core
partic|es"
exclusion
(all c o n t o u r s ' - m o d u l o t r a n s l a t i o n s )
(non-intersecting
contours),
interacting through
and h a v i n g
activity
z(~).
a
This
implies immediately t h a t f o r any c o n t o u r ~ t h e p r o b a b i l i t y t h a t i t is a c o n t o u r of a c o n f i g u r a t i o n is bounded b y exp-2t~l~rl ( P e i e H s ~ b o u n d ) . The basic estimate t h a t allows us to c o n t r o l this system is:
16 Z
z(~') + 0 (exponentially fast) as IB ~ - ,
(in the case considered here the sum is bounded b y
where b is any f i x e d bond
exp-4B,
for
(2.9a)
B large enough).
This
follows
easily
from
the fact
that
z(;O
is
decreasing v e r y fast when IZl is increased, here z(~)=exp-2t~'l, and that there are not too many contours with given IZl ( P r o p e r t y 2). The convergence of the low temperature expansion follows now from the results on low a c t i v i t y expansion for the gas of contours:
Define a m u l t i p l i c i t y f u n c t i o n as
a map from the set of contours into non-negative integers, with a f i n i t e support. A complex-valued function ¢ defined on the multiplicity functions can be identified with the formal power series
X ~(O)z O ,
(2.10)
where the sum is over all possible m u l t i p l i c i t y functions, and
O = ]I Z~,~(~)
Z
Z~, being a variable associated to ~. Define now ¢ as follows:
¢(~)=1
if ~(~')<1
for all ~" and the contours in the
support of 0 are pairwise disjoint, and ¢(~)=0 otherwise.
Then,
(2.0) is the value
of the formal power series (2.10) for z~,=exp(-2Bl~l) if ~ck, and z~,=0 otherwise. Let cT be the function corresponding to the logarithm, in the sense of formal power series, of <~.
Proposition ([23, 40, 4 6 ] ) . X
I~T(o)z 0'1 ~ 0
I f Iz~l < exp(-2Bl~'l) Call 7;) then f o r any c o n t o u r ~'
as
B~ ~
(2.11)
~:~(~')#0 F u r t h e r m o r e , setting in
X ~T(~)zO z~,=O f o r series
(2.12) ~ not
from
contained in it, one obtains f o r each
which
log
Z(it[ ÷) is
obtained
by
A an absolutely
substitution
expansion implies in t u r n that
log ZCAI +) = -
I A I B ~ ( B I ÷) ÷ ACA,6)
,
with B~(81 +) being an analytic f u n c t i o n of z=exp-2B, while
convergent
z~,=exp-281~q.
This
17
la(A,~)l
s 8(S)IaAI
(2.13)
,
with 6 (i~)=0(exp(-213)) • Given the expansion ( 2 . 1 2 ) , the p r o o f of the rest of the Proposition is easy. Using this phases
expansion,
[40,
invariant
23]:
Gibbs
one obtains a detailed d e s c r i p t i o n
There
are e x a c t l y t w o
states)
related
spontaneous magnetization. of
(+)-spins,
by
pure
the
phases
spin-flip
of the
(i.e.
low t e m p e r a t u r e
extremal
symmetry,
translation
with
opposite
In each phase, t y p i c a l c o n f i g u r a t i o n s consist of a "sea"
f o r example, w i t h
small islands of the opposite phase.
f u n c t i o n s are e x p o n e n t i a l l y c l u s t e r i n g and analytic in e x p ( - 2 ~ ) . a complete d e s c r i p t i o n of the l o w - t e m p e r a t u r e phases.
Correlation
Thus one has q u i t e
However, in o r d e r to obtain
these results one has to use many of the simplifications of the Ising model t h a t are not available in more realistic systems. Not much needs to be changed if we consider h i g h e r spins ( i . e . ,
c(a) assuming
the values - n , - n + l . . . . . n, w i t h the same Hamiltonian o r f e r r o m a g n e t i c system w i t h a f i n i t e number of g r o u n d states.
However, if we deal with a continuous d i s t r i b u t i o n
of a ( a ) , o r w i t h quantum f i e l d theories [ 2 6 ] , then complications arise. Let
us consider,
for
example,
an
Ising
model w i t h
continuous
spin:
the
Hamiltonian H is as before, (2.1) w i t h h=0, b u t a(a) is now u n i f o r m l y d i s t r i b u t e d in the interval
[-1,+1],
instead of t a k i n g t h e values +1 and -1 o n l y . As in the Ising
model, t h e r e are two g r o u n d states , t h e ( + ) - g r o u n d state which is +1 e v e r y w h e r e and t h e
(-)-ground
symmetry. the g r o u n d
state.
The g r o u n d
states
are again
related b y the s p i n - f l i p
However, one has now the f o l l o w i n g problem: even in the f i n i t e volume, state c o n f i g u r a t i o n s have p r o b a b i l i t y zero and t h e r e are excitations of
the g r o u n d states with ( r e l a t i v e ) e n e r g y which is a r b i t r a r i l y small. To deal with this problem one can i n t r o d u c e two r e s t r i c t e d ensembles: +
X R = {o: o(a) > 0, all a ~ Z d } , X R = {0: o(a) < 0, all a ~ Z d } . Then
one can
o(a)o(b)<0,
as in the
complement of ensembles.
define
contours Ising
a contour,
If one t r i e s
as connected
model. the
Thus,
families in
configuration
to expands t h e
of
bonds
each connected belongs
to
partition function
one
for
which
component of of
the
the
restricted
into a sum o v e r o u t e r
c o n t o u r s , as in the ( 2 . 6 ) , one does not have any more a f a c t o r i z a t i o n into p a r t i t i o n f u n c t i o n s of o u t e r contours. However, it t u r n s out t h a t the r e s t r i c t e d ensembles are v e r y d i l u t e at low t e m p e r a t u r e s . Namely, one has a c o n v e r g e n t expansion f o r these ensembles
one
derives
h i g h - t e m p e r a t u r e expansion,
from
which
o r from
all
the
properties
low f u g a c i t y
obtained
expansion
usually
of a d i l u t e
from
gas.
In
18
particular the factorization in (2.6)
holds in the sense that one can construct a
low-temperature expansion for
complete model which
the
takes
the
form
of
a
combined expansion, fi r s t into contours and then a "high-temperature expansion" in each restricted ensemble; we refer to [9] for detailes. Such a combined expansion has appeared f i rs t in other contexts, particularly in lattice and continuum models of Quantum Field Theory, see [26] and references therein. For other extensions of the Pirogov-Sinai theory to continuous-spin systems we refer to [18] and to Zahradnik's contribution to the present volume. One can avoid this combined expansion and obtain directly the Peierls' bound on the probability of a contour,
as done originally in [55].
many examples where chessboard estimates,
based on the
This is one of the Reflection
Positivity
property of the model, [26, 21], proved to be very useful. To
clarify
the
importance of
the
symmetry
in
the
preceding arguments
consider the Ising model, but with an external field. Now the weight e x p ( - 2 B l ~ l ) of a contour is modified slightly, which for a small field is not essential. However, (2.7) would no longer hold and the iteration procedure in (2.8, converge. Recall that (2.7)
comes from the symmetry.
2.3) would not
Since such symmetry is
expected for a realistic liquid-gas model, a fi r s t step may be to extend the analysis above to lattice models without symmetry.
This is quite non t r iv ia l and is the
content of the Pirogov-Sinai theory.
III. MODELS WITHOUT SYMMETRY This section is divided into four subsections. First we introduce a simple model on which the main features of the Pirogov-Sinai "theory can be most easily explained, and we discuss its phase diagram from the point of view of perturbation theory. Then we explain some of the ideas of the Pirogov-Sinai theory. In Sect. 3, we state the
general
result
of
Pirogov and
Sinai
for
this
model
perturbation theory for the phase diagram in more details.
and
we discuss
the
Finally (Sect. 4) we
generalize our model and explain some recent applications to the theory of random surfaces [5]. 1. Phase diagram of the Blume-Capel mode/. This model, [4, 11], is defined by the Hamiltonian H =
X
( s (a ) - s (b ) ) =
,
s(a) = 0,±1 .
(3.1)
19
It has three ground states, {+), (-) and (0), which are equal everywhere to +1, -1 and O, respectively. The model has still the s(~) -~ -s{a) symmetry but only the (+) and (-) ground states are related by this symmetry; the (0) ground state is in a class by itself. One could expect that at temperatures low enough the model has three pure phases which
are small perturbations of the three ground states.
We will
now give a
perturbation-theoretic argument showing that this is not true. More precisely, we will argue that for the original Hamiltonian (3.1) there is no low temperature phase which
is a small perturbation of the (+)-ground
perturbation
of
the
Hamiltonian
such
a
state,
phase may
and that for a suitable
appear;
similarly
for
the
( - ) - g r o u n d state. Suppose that in a large but finite region A the (+)-boundary conditions produce such a low temperature phase, i.e. that a typical configuration of such a system is +1 almost everywhere, with small excitations distributed rarely but homogeneously in A.
At very low temperatures the main contribution to the partition function Z(AI +)
+
+
+
0
+
+
0
+
0
+
0
+
0
+
+
0
0
+
+
+
0
0
0
+
0
+0 ÷0
+
0
+
0
+
000
0
+
+0
0 0 0
0 + 0
0
+
+0
0 - 0
0 0 0
0
+
+0
0 0 0
0
+
+0
0 + 0
0 0 0
0
+
+0
0 0 0
0 - 0
O+
+0
0 0 0
+0 + +
A' 0 +
0 +
0 +
0 +
0 +
0 +
0 +
0 +
+
0
0
0
0
+
+
+
+
0
+
0
+ +
Fig. 1. Typical configurations of the low-temperature "(+)-phase" of the model (3.1): tunneling to the {O)-phase.
of this system would come f i r s t from the ground state and then from the excitations of lowest energy, for which s(a)=O at a number of isolated points of A. Thus, see (2.4), Z(AI +) = 1 + IAle -4~ + ½1Al(IAl-1)(e-4~) 2 + . .. = e x p ( I A l e -4B) .
(3.2)
20
Similar a r g u m e n t yields f o r the p a r t i t i o n f u n c t i o n of the l o w - t e m p e r a t u r e ( 0 ) - p h a s e Z(A}0) = 1 * 21Ale -413 + ½(21al)(21Al-1)(e-4!~) = * . . . since here
l o w e s t - e n e r g y excitations
= exp(21AI e-4B) ,
can be obtained in
(3.3)
two ways:
s(a)
can
be
changed e i t h e r to *1 o r to -1. Now, consider the c o n t r i b u t i o n to Z(AI+) of c o n f i g u r a t i o n s of Fig. inside
A one has a s l i g h t l y
NxN-square
and
/,'
is
smaller
an
region A' with
(N-2) x ( N - 2 ) - s q u a r e ,
the
(0)-phase.
the
1.
If
contribution
Here
A is an of
such
c o n f i g u r a t i o n s to the p a r t i t i o n f u n c t i o n is e x p ( - 4 ( N - 2 ) 4 g ) • Z(A'I0) = e x p ( - 4 ( N - 2 ) 4 B + 2 ( N - 2 ) 2 e - 4 ~ ) which f o r large N is l a r g e r than Z ( A I * ) . is unstable
against insertions of
the
,
(3.4)
This shows t h a t the conjectured ( * ) - p h a s e
(O)-phase,
and
t h a t the assumption
of the
existence of a l o w - t e m p e r a t u r e ( * ) - p h a s e leads to a c o n t r a d i c t i o n . This p e r t u r b a t i o n - t h e o r e t i c a r g u m e n t can be cast into a form which hints at a more systematic development (see the end of Sect. 4). i n t e r p r e t e d as a p a r t i t i o n f u n c t i o n
The sum in
(3.2) can be
of a "gas" of lowest e n e r g y excitations of the
( + ) - g r o u n d state. The free e n e r g y of this
gas is denoted
~R(131+) (R
stands here
f o r " r e s t r i c t e d ensemble", as explained in t h e i n t r o d u c t i o n ) . Thus
13~R(131* ) = and
_
SR(131-)=$R(I~I*).
I~$R(1310) =
(3.5)
e-4B , o(e -41~) ,
I n t e r p r e t i n g (3.3) in a similiar manner:
_ 2e-413 + o(e -41~) .
(3.6)
We stress the p o i n t t h a t here, in c o n t r a d i s t i n c t i o n to the Ising model, the p a r t i t i o n function depends on the b o u n d a r y conditions in an essential way. Since the RHS of that
one can
potentials
infer
(3.4)
is essentially
exp(-B(laA'l*lA'l~R(~10))),
the l o w - t e m p e r a t u r e phases
~R(BI*), ~R(131-),
and
SR(~I0)
by
comparing the
it appears
thermodynamic
of the c o r r e s p o n d i n g r e s t r i c t e d ensembles.
Namely, we will say t h a t the ( 0 ) - g r o u n d state is dominant since it has more lowest e n e r g y excitations than the o t h e r g r o u n d
states, o r e q u i v a l e n t l y since the leading
term of the l o w - t e m p e r a t u r e expansion of t h e free e n e r g y of the r e s t r i c t e d ensemble of the gas of the low e n e r g y excitations is smaller f o r (0) than f o r (+) o r ( - ) . above
argument
suggests
that
only
dominant
ground
states
l o w - t e m p e r a t u r e phases. One can in fact p r o v e (see A p p e n d i x ) :
give
rise
The to
21
Theorem.
For B large,
model (3.1).
there is only one translation i n v a r i a n t Gibbs state f o r the
In typical
configurations
of this state the d e n s i t y of sites a where
s(a)=0 tends to 1 as B -~ ~. This theorem, and its proof, are included here f o r pedagogical reasons.
The
method of the proof that we use is applicable in a much broader context (see [10] and Sect.
IV).
The result
itself is a special case of the uniqueness
results of
Preiss (unpublished) and Zahradnik [54]. Remarks: (11 Using correlation inequalities one can show as in [38] that there is actually a unique Gibbs state for this model. inherent
in
our
method.
However,
in
this
(2) The restriction
particular
model,
to large B is
we do not expect
several phases at higher temperatures. The p e r t u r b a t i o n - t h e o r e t i c argument indicates how to obtain the thermodynamic phases corresponding to the g r o u n d states (+) and (-) of ( 3 . 1 / .
Namely, p e r t u r b
BH by replacing it with BHg = BH
-
gY. s(a) = a
(3.71
Now f o r any g > 0 we have only two g r o u n d states,
(+) and ( - ) . T h e y are related
by symmetry of the Hamiltonian and, f o r temperatures low enough (depending on g) a Peierls
argument,
similar
coexistence of two phases.
to the
one used
for
the
Ising
model,
proves
the
However, one expects that there will be a line g(B) on
which three phases, (+), (-) and (0) will coexist. The
perturbation-theoretic
argument which
excluded
the
(+)-phases
for
g=0
yields now that g(B) is given to o r d e r exp-4B by the condition that B~R(B,gl +) - g = B~R(B,910) , -g on the LHS being the energy of the ( + ) - g r o u n d state. (see (3.5),
The f r e e energies are
(3.6)):
B~R(B,gl0) = _ (eg+e-g)e -413 + o(e -4B) ,
B~R(B,gl ÷) = _ e-ge-4B + o(e -4B) .
Thus one expects the line of coexistence to be given by: g(B)=exp-4B+higher o r d e r terms. restricted
One
can
obtain
a better
approximations
for
g(B)
by
considering
the
ensembles which include h i g h e r - o r d e r excitations of the ground states,
excitations with energies not exceeding some cutoff e n e r g y E (two adjacent 0 spins in the + ground state or one - spin in this ground state, etc.1. However this would g i v e only the asymptotics of the line g(B) of coexistence; to prove the existence of g(B), and the fact that the obove computations yield a curve asymptotic to g(l~) we
22
need more sophisticated arguments, since t h e r e is no evidence t h a t the p r o c e d u r e o u t l i n e d above will converge when E is increased. The existence of 9(B)
is one of
the main results of the Pirogov-Sinai t h e o r y , specialized to the present model:
2.
The Pirogov-Sinai theory
We will sketch now the proof, a c c o r d i n g to Pirogov and Sinai [ 4 2 , 4 7 ] ,
that there
exists a line 9(B) on which the t h r e e l o w - t e m p e r a t u r e phases coexist. Define: Z(kl*)
= X exp-BHk(sl*)
,
(3.8)
S
with H =
X ( s ( a ) - s ( b ) ) = - 9X s(a) = • a
We let eo, e+ and e
(3.8a)
denote the e n e r g y p e r lattice site of the c o r r e s p o n d i n g g r o u n d
states:
eo(g)
e,(g) = - g .
= 0 ,
\
We i n t r o d u c e also the
partition
functions
with
the
same b o u n d a r y conditions
as
before, b u t with no s u b s t r u c t i o n of the g r o u n d state e n e r g y : ZS(A) = ( e x p - l A I B e ) Z ( A l ~ ) Define
contours,
,
almost
~ = O, ± . as
in
the
Ising
model,
as
subsets
of
the
lattice
composed of connected families of bonds f o r which sCa)~s(b), t o g e t h e r w i t h a c o n f i g u r a t i o n on it.
Unlike in t h e case of the Ising model, here tlle e n e r g y of a
contou r , E(~)
=
(sCa)-s(b)) = ,
E
w h e r e t h e summation is o v e r all t h e bonds of ~', depends not o n l y on its length b u t also on t h e n e i g h b o r i n g c o n f i g u r a t i o n . With these d e f i n i t i o n s one has the expansion: Z(AI*) = E
]I exp(-BE(~')) ]I ( e x p l l n t i ~ ' l B e *) z ~ ( i ) ( I n t i ~') , ~'E~
(3.9)
I
w h e r e t h e sum is o v e r all families of o u t e r contours in A, and the second p r o d u c t runs o v e r all the connected Components of t h e i n t e r i o r of the c o n t o u r ~', indexed b y i. E(i)=O, * o r - according to t h e value of t h e spins along t h e b o u n d a r y of Int.~. I
23
Now, we would like to iterate this expansion, as we did in the case of the tsing model, in o r d e r to obtain e v e n t u a l l y a sum over families of pairwise nonintersecting, but
not
necessarily
outer,
expansion for Z ( A I * ) ,
contours.
This
would
yield
a
convergent
polymer
p r o v i d e d the resulting activities of the contours are small
enough. Such an expansion would allow us to d e r i v e all the desired properties of the
(*)-phase,
like
Peierls'
properties of pure phases. (0)- and the (-)-phases,
bound on
probability
of
contours,
and
clustering
We would like also to have a similar expansion f o r the and the condition of coexistence of such a three phases
should determine 9(1~). To obtain this, we w r i t e (3.10)
z ~ ( i ) ( I n t i ~) = ( z ~ ( i ) ( I n t i ~ ' ) / Z + ( I n t i ~ ' ) ) . Z + ( I n t i ~) , as we did in the
Ising model, except that
the ratio was equal to 1 t h e r e ,
by
symmetry. Inserting (3.10) into (3.9) and i t e r a t i n g , we obtain the expansion Z(AI +) = ~
~
where
sum
the
z+(~)
is
,
(3.11)
over
all
families
¢0 of
non-overlapping
contours
in
A,
not
necessarily o u t e r ones, and the activities are z+(~') = exp(-BE(~')) ]t ( z ¢ ( i ) ( I n t i 2 r ) / Z + ( I n t i ~ ' ) ) i
.
(3.12)
Note that the families of contours o v e r which we sum in (3.11) are not, in general, families of
contours
of configurations
of our model
(unlike
in the
Isin9
case).
Indeed, because of o u r induction method, all contours have the ( * ) - g r o u n d state in their
exterior,
even those t h a t
may be in the
(-)-
or
(0)-interior
of another
contour. The convergence of the polymer expansion depends on an estimate like (2.3a) and would clearly
hold here
if,
in
(3.12),
we had only the factor
exp-13E(~),
because E(~)->cl~rl, where I~1 is the number of bonds in ~'. So we would like to show that the factors Z ° ( I n t i ~ ) / Z * ( I n t i ~') are not too large. From general arguments we know o n l y that
(3.13)
Ilog Z°(A) - log Z+(A)I s const, laAI Since the b o u n d a r y of Int~" is just
~" we have to f i n d out how does the const,
in
(3.13) depend on IS. From the i n t u i t i v e p i c t u r e described in Sect. 1, we expect that it should grow like l~ if we are not on the coexistence line. Indeed, if the (O)-phase is
the
dominant one,
for
example,
then,
in
a system
with
the
(*)-boundary
24
conditions, a contour " t u n n e l l i n g " to the (O)-ground state will develop itself and its length
will
be
approximately equal
to
[~AI.
Its
weight
will
be of
exp(-[5]aA]) which will give a constant approximately equal to [5. if the const,
the
order
And vice versa,
in (3.13) grows like [5 then the polymer expansions (2.2a) about the
ground states are out of control and there is no reason to expect that we are on the coexistence line. Now, what is the expected behaviour of the const, of (3.13) on t h e c o e x i s t e n c e line?
From the
results
obtained
for
the
Isin9
model
(the
convergent
polymer
expansion of the Proposition), we expect t h a t log ZS(A) = - I ^ I B ~ ( B ) + a~(A,B) , where in the f i r s t
(3.14)
~=*,0,-,
(volume) term $ is the bulk free energy
(independent of the
b o u n d a r y conditions); A~(^,[5) is the b o u n d a r y term which we expect to satisfy the bound (2.13): la~CA,[5)l
~ ~([5)I~AI
(3.15)
,
where 6([5)~0 e x p o n e n t i a l l y fast as [5~. Now, the common volume terms cancel out in (log Z°(A)-Iog Z+(A)),
and t h e r e f o r e the bound
(3.15) implies that the const,
in
(3.13) is of o r d e r 6([5). Thus, we have two extreme situations: either the const, in (3.13) grows like [5 f o r large enough ~, (away from the coexistence line), or it tends e x p o n e n t i a l l y to zero with [5 (if we are on the coexistence l i n e ) . T h e r e f o r e we can expect to be able to construct 9([5) by imposing the condition that
(3.15)
holds f o r ~=+, - and 0
simultaneously. How does one show that such a line 9([5) exists? If we had only to deal with the contours of lowest energy (isolated zeros in the + ground state) then, as our calculation above shows, one could set g([5)=exp-4[5*higher o r d e r terms, and one would get equality of the b u l k free energies for all the ground states, and also the bound (3.15)
to o r d e r exp-4B.
It is f a i r l y obvious that we could still find a
line 9([5) if we had only a f i n i t e number of contours
(up to lattice t r a n s l a t i o n s ) .
But we have an i n f i n i t e number of them. However, 9([5) can be constructed via an i n f i n i t e sequence of approximations for the following reasons: when the size I~l of the contour Condition
If increases,
of [24]
E(~') grows
and [42]).
p r o p o r t i o n a l l y to it:
So, r o u g h l y speaking,
E(~')_>c[~'l
(the P e i e r l s
if we h a v e a line gn([5) such
that we would have coexistence if we considered only contours of size less than or equal to n, then, to include contours of size n*l we shall only have to modify gn by
an
amount
like
exp(-[5(n+l)).
This
gives
approximations to the t r u e phase coexistence line g([5).
a
convergent
sequence
of
25
We stress the importance of the Peierls Condition in the above arguments. Once a notion of a contour can be introduced in a model in such a way t h a t the Peierls Condition holds the most essential elements of the Pirogov-Sinai t h e o r y can be taken o v e r
( f o r instance, see [57, 59] f o r an extension of the theory to some
models with an i n f i n i t e number of g r o u n d states, and the model of Sect. 4). When the
Peierls
Condition
does
not
hold,
one
can
sometimes
substitute
for
"averaged" version - the Peierls' bound, f o r a suitably defined contours,
it
its
as done
in [8] and the work described in Sect. IV.
3. Genera/ phase diagrams and perturbation theory We start with the general result of Pirogov-Sinai, specialized to this model. Let us introduce an additional p e r t u r b a t i o n of the Hamiltonian (3.7): ~ H ( g , h ) = I~H - g~: s(a) = - h~: s(a) . a a
(3.16)
Then we have the following zero-temperature phase diagram in the ( g , h ) - p l a n e . The ground state is unique except along t h r e e half-lines: there are two ground states, (*)
and ( - ) ,
h=-g;
on the half-line g>0 h=0; two ground states,
and (-)
(*)
and (0), f o r g<0
and (0) for g<0 h=g. Of course, for g=h=0 one has three ground
states (see Fig. 1 in [48]). A set of perturbations such as gZs(a)=*h~:s(a) above "removes completely the degeneracy of the ground states" since e v e r y possible combination of the original ground states appear on this z e r o - t e m p e r a t u r e phase diagram on manifolds of the correct dimension (n ground states coexist on manifolds of codimension n - l ) . Pirogov-Sinai t h e o r y then says that f o r small Igl
and
Ihl
(here
The
in fact f o r all
(g,h)),
and for low temperatures, the phase diagram is a small deformation of the
one at
T=0.
Thus
we
have one
point,
g(l~),
with
three
phases,
three
lines
emanating from that point with two phases, etc.. Moreover, this t h e o r y allows one to j u s t i f y the p e r t u r b a t i o n expansion, which was our
starting
point.
Let us explain
how does one obtain
the
perturbation
expansion of the phase diagram in a more general situation [48]: Define an excitation of a ground state as a configuration which coincides with this g r o u n d
state everywhere,
with a possible exception of a finite set of sites.
into connected components, called elementary excitations, in such a way that the ( r e l a t i v e ) energy of an excitation with respect Decompose
every
excitation
to the g r o u n d state is just the sum of the (relative) energies of the components. Note t h a t this decomposition may d i f f e r from that into contours. For example, in the Ising model, a large bubble of minus-spins in the ( * ) - g r o u n d
state with a plus-spin
in its middle is an elementary excitation g i v i n g rise to two contours: one separating
26 the bubble from the surrounding plus-spins, and another one separating the central spin from the bubble. To compute the asymptotics of the lines and surfaces of phase coexistence up to an
err o r of
order
exp-BE,
we consider for
ensemble consisting of its excitations with
each
ground
state the
restricted
elementary components of energy not
exceeding E. This we call the gas of excitations of the ground state G with energy not exceeding E. For example, in the Blume-Capel model considered above, the gas of f i r s t - o r d e r excitations of the ( * ) - g r o u n d state, i.e. excitations with energy not exceeding 4,
consists
surrounded by sites
of all
configurations
b where s(b)=+l.
with
isolated
sites
where
The free energy ~R(BIG) of
s(a)=0, this
gas
(denoted PE G in [48]) has a convergent (polymer) expansion, similar to the one in the Proposition, if B is large enough (depending on E). Indeed, convergence of this expansion rests on an estimate of the form (2.9a) which clearly holds here: we have a finite number of elementary excitations (modulo translations) and therefore in a suitable analogon of (2.9a) one has to sum over a finite set. The
definition of
~R'
and the
construction of
the
phase diagram which
follows, makes sense for a much larger class of systems, than those to which the Pirogov-Sinai theory applies, many of them with an infinite number of ground states (see Sect.
IV).
(regularity) support.
Namely, it is enough that the Hamiltonian satisfies the following
condition:
The
energy
of an
excitation
tends
to
infinity
with
its
Note that this condition is not satisfied by the one-dimensional Ising
model (see Sect. 4 for more interesting examples). To obtain the line of coexistence of all the phases, we set equal the free energies of the restricted ensembles corresponding to all the ground states, and solve for g,
h (in our example h will always be zero, by symmetry). This is what
we did in our f i r s t - o r d e r calculations.
To obtain the surface of the coexistence of
the (+)- and (0)-phases, for example, we again set the corresponding free energies equal, and impose the condition that the restricted (-)-ensemble has a higher free energy. This yields an asymptotic expansion for the line and for the surfaces of phase coexistence [48]. We do not expect this expansion to be convergent. However the coefficients of the terms in this expansion are easy to compute. Also,
it often
happens that knowing the f i r s t few terms in the expansion gives the topological structure of the phase diagram (i.e.
the incidence properties of the lines and
surfaces of the diagram). This structure is not affected by higher o r d e r which only give a better numerical approximation to the true phase diagram.
terms
27
4. A p p l i c a t i o n s to s u r f a c e models
Let us consider t h e same Hamiltonian s(a)
take
2n'1
values:
(3.1) as in the Blume-Capel model, b u t
s ( a ) = - n , - n * l . . . . . n.
Blume-Capel model implies t h a t t h e r e will
Then
the
same analysis
as
in
let the
be a unique phase at low t e m p e r a t u r e s
(see b e l o w ) . Also, we can add to ~H 2n p e r t u r b a t i o n s which l i f t the d e g e n e r a c y of the g r o u n d states, f o r example, ~: k: k#0, and
~ ( k ) ~: 6 ( s ( a ) - k )
,
I kl<-n
obtain
a
complete
phase
diagram
for
small
temperatures
and
small
1~.
C o n s i d e r i n g the lowest e n e r g y e x c i t a t i o n s , we deduce t h a t the coexistence between the
phase
corresponding
to
s(a)=n
]~(n)-iJ(n-1)=exp(-413)+o(exp-413),
and
while
the
one
for
coexistence
s(a)=n-1
between
will
the
(0)-
occur
for
and
the
(-*)-phases will o c c u r f o r ~(*1) = exp(-4n213) * h i g h e r o r d e r terms . Indeed the ( n - 1 ) - g r o u n d the ( n ) - g r o u n d
(3.17)
state has more excitations of e n e r g y equal to 4 than
state ( l i k e in the Blume-Capel model w h e r e n=l) w h i l e t h e 0 and -'1
g r o u n d states are identical in t h e i r low e n e r g y excitations up to e n e r g y 4n z which are g i v e n , in the 0 g r o u n d state, b y s(a)=-*n, s(b)=O f o r l a - b l = l . It t u r n s out t h a t this discussion has an i n t e r e s t i n g application to the t h e o r y of random
surfaces
Is(a)-s(b)l replaced
by
(see
[5]).
First,
observe
that
if
we
replace
(s(a)-s(b)) = by
in the Hamiltonian (3.1) t h e r e are no dramatic changes: n = in (3.17) is n but
particular there
the
phase diagram
is a unique
has the
same q u a l i t a t i v e
phase at low t e m p e r a t u r e s .
properties,
However,
this
in
model is
isomorphic to the SOS model of a surface which f l u c t u a t e s between two hard walls (s(a)
is the h e i g h t of the surface above site a and the walls are at height -n and
+n). Now, c o n s i d e r a d i f f e r e n t model, w h e r e the interface is constrained to be above one
fixed
wall,
say
at
height
0:
the
model
H=F.Is(a)-s(b) I and the c o n s t r a i n t s(a)>0 (all a).
is
defined
by
the
Hamiltonian
Let us tie the surface to t h e wall
(s(a)=0) on the b o u n d a r y of a box A and ask f o r the value of l i m < s(0) >A .
(3.18)
It is well known [16, 23, ,56] t h a t if t h e r e is no wall this limit is f i n i t e at low t e m p e r a t u r e s if the dimension of the lattice is l a r g e r than o r equal to 2 ( a c t u a l l y , f o r at t e m p e r a t u r e s if d_>3, [6, 2 9 ] ) . However, the presence of the wall induces an effective repulsion of the surface so t h a t
the
limit
(3.18)
is i n f i n i t e f o r
all
dimensions at low t e m p e r a t u r e s
(and,
28
presumably,
at
all
temperatures).
This
fact
is
an
easy
consequence
of
the
Pirogov-Sinai t h e o r y in the form discussed above. Indeed,
< s('0) >A > < s(0) >A,2n '
(3.19)
w h e r e on the RHS we have added the c o n s t r a i n t s(a)<_2n f o r all a. i n t u i t i v e and actually follows from FKG-inequalities.
This is f a i r l y
Now, change the variables on
t h e RHS of (3.19) to s ( a ) ' = s ( a ) + n . The s ( a ) ' variables are e x a c t l y like in t h e model discussed above
(]s(a)'l<_n).
Since t h e r e
is a unique
phase ( a c t u a l l y , a unique
Gibbs state) f o r this model at low t e m p e r a t u r e s which moreover is concentrated on small p e r t u r b a t i o n s of the ( 0 ) - g r o u n d state, < s(0)'> A ( ] s ( a ) ' l < n )
will go to zero,
as A-~®, i n d e p e n d e n t l y of the b o u n d a r y c o n d i t i o n s . I n s e r t i n g this into (3.19) we get
lim
< s(0)
>A ~ n .
But since n is a r b i t r a r y ,
our claim is p r o v e d . For more d e t a i l s , we r e f e r the reader
to [ 5 ] .
IV. INFINITE NUMBER OF GROUND STATES; OPEN PROBLEMS We will
now discuss on a simple example some of the e x i s t i n g extensions of the
Pirogov-Sinai
theory,
and
some of the
open
problems r e g a r d i n g
low-temperature
b e h a v i o r of classical lattice systems. The model is a version of the f a m i l i a r spin-½ a n t i f e r r o m a g n e t on a square lattice.
It is chosen here because it i l l u s t r a t e s most of
t h e points we want to make. A t t h e expense of complicating t h e notation we could have discussed models of alloys which are of some importance f o r applications (see Fig.
5 of [ 4 8 ] ) .
Let a(b)=-+l be a spin-½ v a r i a b l e at a point b of the simple square lattice Z 2 and let
H = Hnn - =Hnn n
hM
( * H v) ,
(4.1)
where H
nn
= E o(a)o(b) ,
H
nnn
= E o(a)a(b) ,
M = E o(a) ,
w i t h t h e sum e x t e n d e d over all pairs of nearest neighbors of the lattice in Hnn, o v e r all pairs of n e x t nearest n e i g h b o r s in Hnn n, and o v e r all points of t h e lattice
29
in M;
the term H v,
see
[4.3),
will be included
later,
when we will
consider a
t h r e e - d i m e n s i o n a l " s t a b i l i z a t i o n " of this model. The
antiferromagnetic
Hamiltonian
c h e s s b o a r d - l i k e g r o u n d states.
Hnn
is
well
known
to
have
two
To obtain t h e z e r o - t e m p e r a t u r e phase diagram of H
one can g r o u p its terms as follows: H = E [ ½ I; o ( a ) o ( b ) - ¢ ~ oCa)o(b) - ¼hi: oCa)] = i: i ° , [] nn nnn [] w h e r e t h e f i r s t sum is o v e r all elementary squares [] ( " p l a q u e t t e s " ) of the lattice, the first
sum in
the square
bracket over
(all f o u r )
p l a q u e t t e ; the second o v e r the t w o pairs of n . n . n . ' s
pairs of
n.n.'s
within
the
w i t h i n the p l a q u e t t e , and the
last o v e r the f o u r points of the p l a q u e t t e . Now, minimization of i o yields Fig. 2:
g=2
[7g=3 /
(+;) o
g=l
(::)-+,
,o
(':> N Q=4 s o\ Fig. 2.
\
C o n f i g u r a t i o n s of the s q u a r e minimizing i ° are obtained by
indicated on the f i g u r e ,
r o t a t i n g those
g is t h e number of ( p e r i o d i c ) g r o u n d states (obtained by
p a t c h i n g the indicated c o n f i g u r a t i o n s of s q u a r e s ) . e n t r o p y per lattice site of the set of all g r o u n d
If g is i n f i n i t e s indicates t h e states. The
Pirogov-Sinai t h e o r y
applies w h e n e v e r g is f i n i t e .
For
points
of t h e
lines of
this
phase diagram minimizing
plaquettes are those of each of t h e adjacent open regions. in t h r e e g r o u n d
states f o r the points of
the
configurations
of
the
For example, this results
h a l f - l i n e h=4,
~>0 and a n o n - z e r o
30 entropy
for
the
half-line
h*4¢=4,
~<0.
In
the
later case,
any
configuration
obtained from the ( * ) - g r o u n d states by flipping spins on a set of points no two elements of which are n.n.'s or n . n . n , s is a ground state again. In each of the
(three) open regions with
a finite g the ground states are
equiva/ent in the sense that they are related by translations and rotations which are symmetries of the Hamiltonian. For the (9=2) - and (g=4)-regions the Peierls' argument of Dobrushin [15] and Gerzik [24] yields two and four low-temperature pure phases, respectively. The Pirogov-Sinai theory applies both in these regions In the last region
and in the neighborhood of each point of the half-line h=4, ~>0.
the phase diagram is best computed using the perturbation expansion, which is justified
with the
help of
the
Pirogov-Sinai theory:
Let SR(B,~,h[G=nnn)
and
~R(B,¢,hlG°) be the free energies of the restricted ensembles of excitations of the corresponding ground states, with energies not exceeding E.
For h=4 the lowest
order excitations are obtained by flipping one of the spins in such a way that the only non ground states plaquettes are those with PZ-configurations. This can be done in n l ( G ° ) = l ways per lattice site in the case of the G o ground state, and in nl(G=nnn)=½ ways per lattice site for G=nnn. Hence G O dominates in order 1 and only
the
G°-phase
SR(B,¢,h(B) IG°) G°-phase
and
is
present
at
low
= SR(B,¢,h(B) lG=nnn) the
O=nnn-phases,
temperatures.
for the line
yields
Indeed,
h(B)
the
equation
of coexistence of the
h([B)=4-(1/2l~)exp-8¢B*o(exp-8¢B),
in
accordance with the argument based on domination. The rest of the plane: the interval DB and the strip with base BC, including the points of its boundary, is not covered by the Pirogov-Sinai theory.
In fact
there are only a few rigorous results about the system in this region, not even a formal
perturbation
Iow-temperatu re
theory
behavior
which (see,
would however,
three-dimensional stabilization of this model).
lead
to
below,
plausible the
conjectures
discussion
of
on a
For, for the values of the parameters
on the boundary of the strip the system admits local changes of the ground states without a change in the energy. Hence, the number of ground states in a square of edge-length L is of order c L2, and therefore the ground-state entropy is non-zero. Consider, for instance, for h+4¢=0 and ~<-½ (i.e.
on the half-line emanating from
the point B of Fig. 2) the ground state G=nn consisting of alternating rows of plus- and minus-spins. Flipping any number of minus-spins which are not nearest neighbors produces PZ-plaquettes only,
which are of the same energy as those
occurring in G=nn. Hence one will have an infinite number of kinds of excitations with
the same energy and the standard perturbation expansion does not make
sense. Following [63], points of the zero-temperature phase diagram for which the ground-state entropy is non-zero will be called super-degenerate. This perturbation expansion makes also no sense for (¢,h) on the interval DB or in the strip, though here local change of a ground state does not yield a ground
31
state any more:
now some of the excited
configurations can be changed locally
without changing their energy. We will discuss this in detail for the interval DB, after describing the ground states and low-energy excitations. All the periodic ground states of the system are among those which can be obtained from G=nn by flipping all spins on a number of lines parallel to one of the coordinate
axes.
For
instance
flipping
spins
on
every
second
line
yields
a
G2nnn-ground state. The plaquette P~ obtained from the ground state plaquettes P= by flipping one of the (-)-spins has the lowest energy
= {o(P 1) - Cn(p 2) . Hence the f i rs t excitation energy is 4s. An excitation with this energy is obtained when one (-~)-spin is fli.pped in any of the ground states. However flipping any interval of spins which begins and ends with a minus, like -*- or - * - + - * - ,
yields
again an excitation with four Pl-plaquettes and energy 4~. Thus if A is a square of side L, the coefficient of exp-134¢ of the expansion of (1/IAI)log Z(AI G2) is of order at least L, as L -~ ®, and hence has no thermodynamic limit. However if we ignore the fact that the free energy of the gas of low energy excitations does not exist, but just compare the coefficients of the low-temperature expansion of log Z(AIG=nn) and log Z(AIGSnnn) we would say that the ground state G=nnn is dominant since G= admits twice as many excitation with, say, (-+-)-intervals flipped (both nnn horizontal and vertical) than G=nn (only vertical or only horizontal); in a similar way G2nnn wins not only over G2nn but over any other periodic ground state. It is not clear at all whether these considerations lead to right conclusions about low-temperature phases
of the system (see,
however,
concerning a three-dimensional stabilization of the model).
below, for
results
For, most likely, the
low-temperature expansion of the free energy of the system starts not with exp-41~¢ but with exp-21~z, due to presence of nonlocalized excitations: flipping a half-line of alternating spins of a ground state Which starts
with
a minus-spin one gets a
configuration with only two Pl-plaquettes. Indeed such non-localized excitations do contribute to low-temperature expansion in one-dimensional models, or ferromagnetic models in any dimension, [32, 48]. Though some models with similiar features have been treated in a number of works
(see [50, 51], for example), the prob/em of
computation of the low-temperature expansion for general models of this type seems to be open. We return now to a discussion of the super-degenerate points (the boundary of
the
strip;
non-zero
low-temperature
expansion
zero-temperature
limits
of
ground-state does
not
equilibrium
entropy). make
states
sense zero
Although at order
the
these of
standard
points, a
the
perturbation
32 expansion, in a sense, are of interest here. A discussion of these limits clarifies the situation conceptually and yields n o n - t r i v i a l information on the low-temperature phase diagram [62, 64, 17, 63]. We explain f i r s t the situation on the example of the antiferromagnet in external field [62, 64, 17]. Consider the model (4.1) with ¢=0. We let h-~0 as T-K) (see Fig. 3).
More
precisely we approach the point h=0, T=0 along one of the inclines, i.e. h(T)=4*h'T,
we set
where h' is the inverse slope of the line; for h'=0 one has the usual
zero-temperature limit. Substituting this into ( 4 . 1 ) , with 13=T-1, we obtain (4.2)
I3H = 6(Hnn - h ( T ) M ) = ~H° - h'M ,
where Ho=Hnn-4M. Now one would expect that as 13-*® the term ISH° will suppress the contribution to equilibrium states of all but the g r o u n d state configurations of H° [62].
Thus in
the limit one obtains a state which is supported by the ground state configurations of H
O
with
interaction defined by -h'M.
This
is indeed the case (Slawny,
1980,
unpublished; [64, 17]). The g r o u n d states of H° are supported by configurations minimizing {n for each
plaquette
D.
Inspection
of
Fig.
2
shows
that
these
are
exactly
the
configurations in which no minus-spin is a nearest neighbor of another minus-spin.
T
i / // I
I I
/
I l
~ I I I I I
h I I I I
,Jl
I
Z Fig. 3.
Blowing-up the super-degenerate points:
approaching
the
point
(T=O,h=4)
from
different
the limiting states obtained by directions
are
identified
equilibrium states of a lattice gas with d i f f e r e n t values of the a c t i v i t y z.
with
33 In an isomorphic lattice gas language in which plus-spins are identified with empty
sites and minus-spins with occupied, one obtains a model in which no two particles are allowed to occupy the same or n e i g h b o r i n g sites and the a c t i v i t y is z=exp-2h'. This
picture suggests
that close to a super-degenerate point one should use a
p e r t u r b a t i o n expansion about the limiting state of the hard core lattice gas and to i n t e r p r e t the properties of the system u n d e r consideration in terms of the later model. We refer to [62, 63, 64, 17] for an implementation of these ideas. More generally,
for
any
H° with
non-zero g r o u n d - s t a t e
entropy
limits of
equilibrium states of 13Ho+H', as 6-~, are equilibrium states, in the sense of [65], of the Hamiltonian H' restricted to the set of ground states of Ho, and the limits of the pressure and entropy are equal to those of the limiting system (Slawny, 1980, unpublished).
The
reader may want
to work
out
the
limiting
systems
for
the
super-degenerate points of Fig. 2 d i f f e r e n t from C, and the two-parameter families of interactions obtained by approaching these points from d i f f e r e n t directions. We now pass to a three-dimensional "stabilizat'on' are
able
to
develop
an
extension
of
the
of the model f o r which we
Pirogov-Sinai
theory
describing
its
low-temperature behavior. The model is on a simple cubic lattice Z 3, with the Hamiltonian (4.1), but with the
term
Hv
included
("stacking"
of
[12]):
the two-dimensional
layers,
with
i n t r a l a y e r interaction H2=Hnn-~Hnnn-hM, are coupled by the term H
v
= -J~' o(a)o(a+v) , a
J > 0 ,
(4.3)
where v is the unit vector in the vertical direction. Since the coupling constant J is positive, the ground states of the f u l l Hamiltonian are obtained by repeating in each horizontal plane the same g r o u n d
state of
H =. Hence the zero-temperature
phase diagram remains as on Fig. 2, with the change that the g r o u n d - s t a t e e n t r o p y is now zero everywhere: the number of g r o u n d states in a cube of edge-length L is of o r d e r c L (as L -~ =) for the points of the strip and the interval L= order c on the lines where the e n t r o p y was non-zero p r e v i o u s l y .
DB, and of
Now, in the later case, there is still no perturbation t h e o r y since there is an i n f i n i t e number of kinds of excitations with given energy, as in the (g==., s=0)-case in two dimensions.
For,
the two-dimensional model admits a local change of the
g r o u n d state w i t h o u t a change in the energy. number
of
neighboring
planes
one obtaines
Repeating this change in a f i n i t e a configuration
in
which
only
the
vertical bonds going up from the uppermost plane and down from the lowest plane are " e x c i t e d " ,
independently of the
Thus,
"cL-case '' in two dimensions,
as the
number of planes affected the
"cL=-case ''
by the
in three
change.
dimensions
admits no standard perturbation expansion and the problem of the n a t u r e of the
34
low-temperature phase (phases?) is open (see [12] f o r a discussion of a simi[iar situation). sense
However for all o t h e r (~,h) the standard p e r t u r b a t i o n expansion makes
and
conclusions
derived
from
it
are
justified
by
an
extension
of
the
Pirogov-Sinai t h e o r y of [10]. We will now discuss in some detail the example of the i n t e r v a l DB. First of all we note that for the points of the i n t e r i o r of the interval the r e g u l a r i t y condition of Sect. 111.3 is satisfied. states are dominant there.
Furthermore, the two G=nnn-ground
For, as in the two-dimensional version, the lowest o r d e r
excitations are obtained by f l i p p i n g one minus-spin. and nl(G)=½
for any ground
obtained
flipping
by
a
state G.
This
has now e n e r g y 4~+2J,
If J is not too large,
(-+-)-interval
(energy
4E+6J),
with
the next o r d e r is n2(G=nnn)=t
and
n2(G2nn)=½, and intermediate values of n 2 for other periodic ground states. Hence, G=nnn
is
dominant
in
second
order,
and
therefore
one
would
expect
two
low-temperature phases. T h a t this conclusions are correct follows from proof ("Peierls' bound for large-scale contours")
[10].
The main points of the
are explained in the A p p e n d i x ,
especially in the proof of Lemma 1.
V. CONTINUUM FLUIDS The f l u i d models f o r which most results are known about phase transitions, and the Widom-Rowlison model [52] in p a r t i c u l a r ,
concern phase separation in mixtures.
In
the simplest version of this model, one considers two species of particles A and B, with
no interaction between particles of the same species and with a hard-core
exclusion between A- and B-particles.
Then Ruelle proved,
[44], that the A - r i c h
and the B-rich phases coexist if the activities z(A) and z(B) of the two species are equal and large enough. This
result holds also if t h e r e is a
suitable interaction
among the A particles and among the B-particles p r o v i d e d t h a t t h e y are identical, i.e. t h a t the A-B symmetry of the model still holds [44, 39, 58]. Comparing = this to the Isin9 model, we could say that we have kept the symmetry between the two phases b u t we do not have a lattice any more, perturb important.
around.
In
fact
this
is
an
nor do we have ground states to
example
where
A natural problem is an extension of this
situation (the interaction between the A-particles between
the
B-particles).
This
is
done
in
[8]
restricted result
ensembles
are
to a non-symmetric
is d i f f e r e n t from the interaction but
under
rather
restrictive
assumptions on these interactions (see below). The
restricted
ensembles
considered
by
Ruelle
(in
a
somewhat
different
terminology) are defined as follows: Cover the space with non-overlapping cubes so that if we have an A particle in some cube then the h a r d - c o r e forbids B particles
35
to be in any of t h e adjacent cubes.
Define one r e s t r i c t e d ensemble b y
t h a t t h e r e be at least one A - p a r t i c l e in each cell, B - p a r t i c l e in each cell.
requiring
and another w i t h at least one
Now, if we have a c o n f i g u r a t i o n o f one of the r e s t r i c t e d
ensembles in one region of space and of the o t h e r in another region, t h e h a r d - c o r e condition implies t h a t some of the cells must be empty. Define then a c o n t o u r ~ as a connected
set of
empty c e l l s and
a c t i v i t i e s empty cells are u n l i k e l y ,
I~1 as the number of cells in ~r. For large and we can a p p l y the Peierls a r g u m e n t .
t h e A - B s y m m e t r y , one shows t h a t f o r l a r g e z ( A ) = z ( B ) ~" is bounded b y e x p ( - 0 ( z ( A ) ) l ~ ' l ) . separation is s t a n d a r d .
Using
the p r o b a b i l i t y of a c o n t o u r
Going from this estimate to the p r o o f of phase
The new f e a t u r e of this
p r o o f is the i n t r o d u c t i o n
of t h e
r e s t r i c t e d ensembles in which outside of a c o n t o u r we have a gas and not j u s t one c o n f i g u r a t i o n . Note t h a t in the A - r e s t r i c t e d ensemble only A particles occur while in t h e actual A - r i c h phase t h e r e will be some (small) "islands" of the B p a r t i c l e s . Using an extension of the Pirogov-Sinai t h e o r y in which the g r o u n d states are replaced b y r e s t r i c t e d ensembles, A-A
and the
B-B interactions
[8] p r o v e an A - B phase separation, even if the
b r e a k the A - B
symmetry.
However,
one must then
r e q u i r e t h a t each of the r e s t r i c t e d ensembles be d i l u t e in the sense t h a t t h e y are both w i t h i n the region of convergence of t h e i r Mayer expansion. Such a limitation is not necessary in t h e A - B symmetric case. So, at p r e s e n t , we need e i t h e r s y m m e t r y or
diluteness
which
one-component f l u i d :
prevents
us from
one could i n v e n t
dealing
with
a liquid-gas
transition
in
a
r e s t r i c t e d ensembles f o r t h e l i q u i d and the
gas phases b y locally f i x i n g the f l u i d d e n s i t y at a p p r o p r i a t e values b u t t h e r e is no s y m m e t r y between these ensembles and i t is not clear what kind of diluteness would hold f o r the l i q u i d r e s t r i c t e d ensemble. Of course, one can obtain a one-component fluid
model
particles.
from
the
Widom-Rowlison
Then t h e A-B
A-B
phase separation
model translates
by
"integrating
out"
the
B
into a l i q u i d - g a s t r a n s i t i o n .
T h e e f f e c t i v e interactions between the A - p a r t i c l e s induced b y the B - p a r t i c l e s can b e computed e x p l i c i t l y when t h e r e is no i n t e r a c t i o n among the B - p a r t i c l e s since, f o r a fixed
configuration
of
the
A-particles,
the
B-particles
form
a free
gas
[52].
U n f o r t u n a t e l y these interactions are q u i t e u n r e a l i s t i c f o r a model of a one component fluid.
V I . THE POTTS MODEL A t each lattice site we have a v a r i a b l e i n t e g e r ; the Hamiltonian is H = -
Y. 6 ( s C a ) , s ( b ) ) ,
s(a) t a k i n 9 values 1 . . . . . . q, w h e r e q is an
36 where 6 is the Kronecker delta-function and the summation is over pairs of nearest neighbors of the lattice. We shall consider q large and follows [53]:
d>2.
The phase diagram is expected to be as
There are q ground states s(a)=m (all aCzd), m=l . . . . . q, and, for
temperatures low enough, there are q phases which are small perturbations of these q ground states.
As the temperature is raised, we do not reach a critical point, as
one does in the Ising model (which correspond s to q=2), but rather a f i r s t - o r d e r transition. <6(s(a),s(b))> discontinuous.
At
this
transition
temperature
T(q)=l~(q)
the
m e a n energy
(a,b nearest neighbors) and the order parameter <6(s(a),m)> are For temperatures above T ( q ) ,
there is only one phase; at T ( q ) ,
there are q * l phases, one high-temperature phase and q low-temperature phases. These latter phases are referred to as the "ordered phases" because, in each of them, the mean energy is close to 1 (for large q) i.e. typical configurations have most of their bonds "unbroken" (s(x)=s(y) if x and y are nearest neighbors). high-temperature phase is called "disordered" have most of their bonds "broken" (s(x)~s(y)
The
because its typical configurations if x and y are nearest neighbors),
and the average energy tends to zero as q-~®. There are many configurations with broken bonds for large q, and thus the disordered phase has large entropy which compensates for its smaller energy than that of the ordered phases, which explains why all these phases can coexist. This model has been analyzed using the reflection positivity property of the interaction by Kotecky and Shlosman [35].
Here we want to consider it from the
point of view of the Pirogov-Sinai theory because it illustrates nicely the use of restricted ensembles
[8,
14, 60].
Also, there is a vague analogy between this
model and a liquid-gas transition: the ordered phases have a lot of energy (like the liquid phase) while the disordered one is favored by entropy (like the gaseous phase). There is a more precise analogy between this model and the Blume-Capel model
(3.7) with small positive g: at low temperatures, there are two (ordered) phases and, at some higher temperature T(g) we reach a f i r s t - o r d e r transition point where three phases coexist. g=0 Hamiltonian,
The third phase corresponding to the ground state G o of the
has larger
entropy
(of
low energy
excitations) than
the two
ordered phase and becomes the unique phase for temperatures above T ( 9 ) . Returning to the Potts model, we define q *l (the ordered ones) are trivial; configuration: s(x)=m (all x),
they consist of the corresponding ground state
m=l . . . . . . q.
The last ensemble is made of all the
configurations for which all bonds are broken: XR D = (s: s(x) ~ s(y)
restricted ensembles: q of them
if x and y are n . n . ' s } ,
37
together
with the
Gibbs
state
induced by
H on
it:
the conditional p r o b a b i l i t y
d i s t r i b u t i o n s of this Gibbs state in finite volumes are just the normalized counting measures, since H=0 on X D R" The free energy of these ensembles is easy to compute.
For the ordered
ensembles, it is just the ground state e n e r g y , equal to -d (d bonds per lattice site in d dimensions).
For the
disordered ensemble, the free energy equals,
for q
large enough h, _B-l(iog q , f(q-1))
,
(6.1)
where f is an analytic function, and f(0)=0
(see [8, proof of Theorem 5 ] ) .
One expects t h e r e f o r e phase coexistence between the ordered phases to occur f o r B(q) "- Clog q ) / d
(to leading o r d e r ) .
Let us write B=B'log q.
Now we should
t h i n k of log q as playing the role of B in the Blume-Capel model while B' plays the role
of g.
restricted
For Io9 q-~ we expect the following: ensemble is
Blume-Capel model. G ÷, G-.
And f o r
similar
to the
For B'>l/d,
unique
for
if B'
state
G o of
the
the ordered ensembles replace the g r o u n d states
t~'=l/d we have coexistence of the ordered and the disordered
ensembles (just like at g=0 we had three g r o u n d states). more precise if one restricts
A c t u a l l y , the analogy is
oneself to quantities which are i n v a r i a n t u n d e r the
symmetry of the Hamiltonians, i.e.
under the global spin f l i p for the Blume-Capel
model or under global permutations of the spins in the Potts model.
Then one has
only one ground state for g=0 and one restricted ensemble for 13'#l/d. For the Blume-Capel model, the Pirogov-Sinai t h e o r y proves the existence of a line g(B)
on which
all three
p e r t u r b a t i v e l y (see Sect. 3). gives
similar results
phases coexist.
Moreover, one can compute g(B)
An extension of the Pirogov-Sinai t h e o r y [8, 14, 60]
f o r the
Potts model:
f o r any q large enough t h e r e
B'(q),
B ' ( q ) - ~ l / d as q-~,., for which q+l phases coexist. To f i n d
the
perturbation
expansion f o r
B'(q)
one has to compute the free
energies of the gases of excitations of the restricted ensembles (which replace the ground
states),
then
set
them
equal
and
solve for
B'(q).
The
lowest
order
excitations of the disordered ensemble are given by one unbroken bond < x , y > (i.e. s(x)=s(y)), site.
surrounded by broken bonds.
T h e r e are d such excitations per lattice
One looses a factor of q (to leading o r d e r )
in the sum over configurations
( e n t r o p y factor) but one gains a factor exp 13 on the energy side. Combining these two c o n t r i b u t i o n s ,
one obtains the free e n e r g y of this
new ensemble where all
bonds, adjacent to unbroken bonds are broken; this is similar to the gas of isolated spin flips in Sect. 2. To leading o r d e r , it is equal . B-l(iog q+f(q-1)) _ (d'expB)/Bq ,
(6.2)
38 where
If(q-1)l
~ 0(l/q)
,
(6.3)
since f is analytic and zero at the origin. For
the
ordered
ensemble,
the
lowest
order
excitations
corresponding ground state e v e r y w h e r e apart form one point.
are
equal
to
the
This yields (6.4)
- d - (q-1)/B.exp(-2dB)
f o r the leading o r d e r terms of the free e n e r g y of the restricted ensemble. Setting equal (6.2) and (6.4), term of o r d e r zero.
w r i t i n g B=B'Iog q and solving f o r B'one obtains 1/d as the
Inserting B'=l/d in the second term of (6.2) and using (6.3)
one obtains -d - 0 ( ( q log q ) - l )
= _ B,-1 _ d/(B'lo9 q ) q ( 1 - 1 / d )
, 0 ( ( q log q ) - l )
,
which, to leading o r d e r , yields B'(q) = 1/d * 1 / ( q ( l " l / d ) l o g
q) .
(6.5)
This coincides with the leading o r d e r terms of the exact answer [53] for d=2: B(q) = B'(q)log q = ½log q * log ( l * ( 1 / V q ) ) = I o g ( l * 4 q ) . However, (6.5)
holds for all dimensions, and higher o r d e r corrections can be
obtained by including higher o r d e r excitations (see also [25] and [28]).
Acknowledgements. J. B. thanks A. El Mellouki, J. Frohlich, K. Kuroda and J. L. Lebowitz for discussions and collaboration on some of the topics of the lectures, and Center
for
hospitality.
Transport J.
S.
Theory
thanks Joel
and
Mathematical
Lebowitz
and
Physics
Daniel Styer
at
Virginia
for
Tech
for
discussions,
and
Insitut de Physique Theorique de Universite Catholique de Louvain for hospitality; his w o r k on this paper has been supported in p a r t by NSF g r a n t MCS 8301709.
APPENDIX: PROOF OF THE THEOREM OF SECT. II1.1 We explain
first
the main
ideas of the
proof.
We want
to
show that,
at low
temperatures, there is a unique phase which is a small p e r t u r b a t i o n of the ground
39 state G o .
If G o was the unique g r o u n d
state, we would define t h e contours of a
c o n f i g u r a t i o n s as connected regions w h e r e s does not coincide w i t h G o . Then we would t r y
to p r o v e a Peierls'
(0)-boundary conditions.
bound f o r
From this
such
contours
in the Gibbs
estimate one would deduce,
state w i t h
using an idea of
Gallavotti and Miracle-Sole [23] which is recalled below (see Lemmas 2 and 3 ) , typical
configurations
of
any
translation
invariant
Gibbs
state
are
that small
p e r t u r b a t i o n s of t h e g r o u n d state G = and t h a t the l o w - t e m p e r a t u r e phase is unique. However, such a simple scheme of p r o o f has to be modified here, because t h e r e are t h r e e g r o u n d states, G , G , G o , not one. C l e a r l y , t h e r e is no damping f a c t o r m
the
Hamiltonian,
and
hence
no
Peierls'
bound,
for
the
regions
where
a
+
c o n f i g u r a t i o n coincides w i t h e i t h e r G
or G
and which would be r e g a r d e d as p a r t
of a c o n t o u r according to the scheme o u t l i n e d above. We know,
h o w e v e r , from the discussion of Sect.
III t h a t if we associate w i t h
each g r o u n d state the r e s t r i c t e d ensemble consisting of its lowest e n e r g y e x c i t a t i o n s then
the
(0)-restricted
r e s t r i c t e d ensemble. ensembles
replacing
ensemble has the
lowest free
energy
among these t h r e e
This suggests t h a t we use o u r f i r s t idea b u t w i t h r e s t r i c t e d ground
states:
the
contours
of
a configuration
s
are
now
connected subsets of the lattice on which t h e r e s t r i c t i o n of s does not belong to the (0)-restricted there will
ensemble. If we t r y
to estimate t h e p r o b a b i l i t y
be some damping f a c t o r
coming from
the
inclusion
of these c o n t o u r s , of the
low e n e r g y
e x c i t a t i o n s . The p r o b a b i l i t y t h a t a c o n f i g u r a t i o n s r e s t r i c t e d to a region A belongs to t h e ( * ) - r e s t r i c t e d
ensemble, is bounded b y the r a t i o of the p a r t i t i o n f u n c t i o n s ,
in /~, of t h e gas of low e n e r g y excitations of ( * ) to the p a r t i t i o n f u n c t i o n of the gas of l o w - e n e r g y excitations of (0). This ratio is a p p r o x i m a t e l y equal exp(-IAlexp-4~)
,
(A.1)
(see (3.5), (3.6)). However, at low t e m p e r a t u r e s this damping f a c t o r becomes v e r y small and it is not obvious
how to
control
the
sum o v e r
all the
contours.
To
deal w i t h
this
problem, we shall define contours on a large ( t e m p e r a t u r e - d e p e n d e n t ) scale. Cover the lattice with boxes B(i) Then,
from
(A.1)
we
r e s t r i c t e d a given box B ( i ) This
whose size is chosen so t h a t
can
deduce
that
the
belongs to the ( * ) - r e s t r i c t e d
c o a r s e - g r a i n e d d e s c r i p t i o n causes
e x c i t a t i o n s than the lowest o r d e r ones and,
I B ( i ) l e x p ( - 4 ~ ) ~- as l~®.
probability
an obvious
that
the
configuration
ensemble goes to zero as problem: t h e r e
are o t h e r
if we define o u r contours on such a
large scale, we loose some localization of these e x c i t a t i o n s . Namely an excitation w i l l have t o be in a box B ( i )
b u t i t may be a n y w h e r e in t h a t box. Moreover, we may
have many d i f f e r e n t e x c i t a t i o n s in a g i v e n b o x . So i t seems t h a t we are back to t h e
40 problem of controlling the sum over the contours. Observe however that the energy of the lowest excitations above the lowest ones (whose energy is equal to 4) is 6 ÷
(corresponding e.g. to two adjacent sites with s(a)=0 in the ground state G ). The p r o b a b i l i t y that such an excitation (or any higher energy excitation) is found in a region k can be bounded, by the usual Peierls' argument, by Iklexp(-6B) Therefore,
if we choose the size of the boxes in such away t h a t
with c between 4 and 6, say 5, then, for any box B ( i ) ,
]B(i)l=exp(cB)
the following two events
will be v e r y unlikely when B -~ =: the
configuration
restricted
to
B(i)
belongs
to
the
(+)-
or
(-)-restricted
ensemble. there an excitation (of any of the three ground states) of energy greater that 4 in B ( i ) . From this one deduces rather easily that a typical configuration belong to the ( 0 ) - r e s t r i c t e d ,ensemble in most of these
large
scale boxes,
and from this
the
uniqueness of the phase is deduced using the ideas of Gallavotti and Miracle-Sole mentioned at the beginning of this a p p e n d i x . Of course, the numbers 4 and 6 used here are accidental; what matters is the discreteness
of
the
set of
excitation
energies.
We may
summarize
the
above
discussion as follows: one associates with each energy E of elementary excitations a distance scale ¢ = exp(½BE) in the following way: consider a temperature dependent family of boxes k(B) of volume exp(BE').
If E' > E,
IA(I~)I
>> g2 and there will be
many excitations of energy E in k(B), f o r large B. "Many" means that we are almost in the thermodynamic limit, i.e. the free energy of the gas of excitations of energy <_ E is equal to its bulk contribution plus a ( r e l a t i v e l y ) small b o u n d a r y term. On the o t h e r hand, if E' < E, then, as 13 -~ ®, it becomes v e r y u n l i k e l y to find one
excitation of energy E in A(B).
even
It is the presence of these d i f f e r e n t distance
scales associated with the d i f f e r e n t excitation energies that makes the proof outlined above work. Now we define the restricted ensembles more precisely and discuss t h e i r main properties. Let X = (-1, 0, *1} z= be the configuration space of the model (we set d=2 for simplicity) and let X R = (sEX: s(a)=0,+l for all a, and if s(a)=0 then s ( b ) = * l if I b - a ] = l }
41 be the restricted ensemble of configurations of the gas of lowest energy excitations of the ( * ) - g r o u n d state. X R is defined similarly, and XRO = {s s X: if s(a)=-I or *1 then s(b)=0 if I b - a l = l } The
subscript
R at
X
("restricted
ensemble")
indicates
that
only
elementary
excitations up to some o r d e r (here up to f i r s t order) are considered. E A (¢=+,0,-) is the set of restrictions to A of the configurations For A c Z 2, XR, of X R. We define the p a r t i t i o n function of the restricted ensemble with boundary conditions s'¢XR,Ac as Z R ( ^ I s ' ) = ~: exp-BHA(s) , where the sum is over szX R which are equal to s' on Ac, and HA(s)
= ~; (sCa)-s(b)) = ,
where the sum is over pairs of n . n . ' s
with at least one point in A. These
p a r t i t i o n functions depend on the restriction
of s' to a~. only, and t h e r e f o r e the
same notation will be used f o r any s'¢XR, M with ~t~ c M. It is easy to realize that ZRCAI*) is just the partition function of a hard-square lattice gas: at each lattice site we can have a particle (corresponding to spin zero) and no two particles can be adjacent. The a c t i v i t y of a particle is equal to the Boltzmann weight of a zero spin,
z=exp-413. This
identification helps to realize that,
small, we have a convergent low a c t i v i t y
for
expansion f o r this
13 large,
The free e n e r g y is analytic in z and we have good clustering correlation functions.
i.e.
for z
restricted ensemble. properties of the
In p a r t i c u l a r , we may write, f o r all A
log ZR(AIs) = - IAI~$R(I~I*) * o(e-41~)l~AI , where B~R([~I+)
= - e -4~ * O(e -81~) ;
see (3.5).
Similar formulas hold f o r Z R ( A I - )
(A.2)
and for ZR(AIO),
except that
in the
latter case we have B~R(B]O) = - 2e -4~ * 0(e -8B) .
(A.3)
42
Now we introduce the contours.
T h e y will be defined on two d i f f e r e n t scales.
On the f i r s t scale, we just have the o r d i n a r y contours with e n e r g y large enough. The second, large, scale will depend on I~ and will be used to take advantage of the difference in the number of low e n e r g y excitations of the (0)- and the (+)-
and
( - ) - g r o u n d states. Let s be an excitation (Sect. Define its retouch, r e t ( s ) , that
Ib-al=l,
lowest
excitations
especially 111.3) of the ( * ) - g r o u n d
by setting r e t ( s ) ( a ) = * l
and r e t ( s ) ( a ) = s ( a )
energy
III,
of
s,
otherwise. but
not
state.
if s(a)=0 and s ( b ) = * l for b such
Thus,
in
ret(s)
the excitations
we remove all the
of
higher o r d e r .
For
÷
example, if ssX R then ret(s)
is the ( * ) - g r o u n d
state. The retouch of excitations
of the o t h e r ground states is defined similarly. A configuration s is retouched if s=ret(s).
Let s be such a configuration.
boundary is the set of pairs where s(a)#s(b). configuration s is a pair ~'=([~'],s[~])
(the support of ~'),
where [~']
Its
A small scale contour ~; of a is a maximal
connected subset of the boundary of ret(s) and s[~,] is the restriction of s to [~']; this definition of [~'] is d i f f e r e n t from t h a t of Sect.
II, but closely related to it.
The e n e r g y of ~ is (A.4)
E(~) = ~ ( s ( a ) - s ( b ) ) = , where the sum is over pairs of n . n . ' s contained in [~'].
Note that since we defined
contours by starting with a retouched configuration the lowest e n e r g y excitations (s(a)=0,
s(b)=*t,
for la-bl=l)
appear in the restricted ensembles but do not give
rise to contours, as they would with the usual definitions. In p a r t i c u l a r , we have min~, E(~') : 6 ,
(A.5)
not 4, and it follows immediately from ( A . t ) The
large
scale contours
that E(~)->I~'I = number of bonds in [~r].
are defined as follows:
Let
L(13)=exp(513/2)
(the
reason f o r this choice will become clear l a t e r ) . Cover Z 2 with large boxes B(i) = B(O) * ½Li ,
i ~- Z = ,
where B(O) is a square of side L(B) centered at the o r i g i n ;
a regular
box
of a configuration
s if
SlB(i)
t
O
XR, B ( i ) ,
IB(i)i=exp(5B). and
it
B(i) is
is irregulor
otherwise. Thus we may distinguish between two types of i r r e g u l a r boxes B(i) of a configuration s: ÷
t y p e 1: SlB(i ) s XR, B(i) U X R , B ( i ) t y p e 2: The support of a small scale contour of s intersects B ( i ) .
43
A l a r g e scale c o n t o u r r is a connected family of i r r e g u l a r boxes.
I trll
= number of boxes in r ,
Thus,
in o u r case,
Irl
We set
= number of sites in r .
Irl is bounded from above and from below by a constant times
exp(SB) I I r l l Now we state t h r e e
Lemmas the
last two of which
are f a i r l y
Lemma 1. The p r o o f of the theorem follows also in a b y these lemmas, [ 2 3 ] ,
standard,
given
now s t a n d a r d way from
so that we shall c o n c e n t r a t e on the p r o o f of the new i n g r e d i e n t
(Lemma 1) and sketch only the rest of the p r o o f s . kemma 1. Assume t h a t 6 is l a r g e e n o u g h . T h e r e e x i s t s a c o n s t a n t c such t h a t , o f o r all f i n i t e A c Z 2, all b o u n d a r y c o n d i t i o n s s z XR,AC and all c o n t o u r s r c A,
PA(rJs) <- exp(-cl311rl
I)
,
w h e r e PA(-Js) is t h e Gibbs measure i n A w i t h b o u n d a r y c o n d i t i o n s s.
If f is a function on X M, M c Z 2, we i d e n t i f y it w i t h the c o r r e s p o n d i n g f u n c t i o n on X w h i c h depends on the r e s t r i c t i o n of the c o n f i g u r a t i o n to M o n l y , and we w r i t e f C(M).
For A contained in M and s~X we define
A(S) =
T. f ( s ' ) P A ( s ' i s )
,
w h e r e the sum is o v e r c o n f i g u r a t i o n s s' in A. Lemma 2.
o
For a n y f i n i t e s u b s e t M of the l a t t i c e , a n y f z C(M) and a n y s ~- XR,
the thermodynamic limit
0 = l i m A(S) A e x i s t s and does n o t d e p e n d on s z X oR.
Moreover,
IA(s) - < f > ° l
<-I I l l l® e x p ( - c l 3 d ( M , Z Z \ A ) ) ,
where
positive
c is
large-scale
a
units:
constant
and
d(
d(A,B)=L(13)-linf{la-bl:
E u c l i d e a n d i s t a n c e of a and b, f o r example.
,
)
is
acA,
the
b=B}
distance
,
between
where
la-bl
sets
in
is
the
44
For any configuration s ~ X and f i n i t e subset A of the lattice consider the set of the large-scale contours of s which
intersect both A and its complement.
Let
rA(S) be the union of intersections of these contours with A - an analogon of the "open contours" of the IM of [23]. Lemma 3 [23].
There are
two positive constants, c and c', such that
PA( I I r A I l > cl~AI is) -< e x p ( - c ' ~ l a A I ) f o r ony f i n i t e A, s s X, and B Iorge enough.
Given the lemmas, we can prove the theorem: As in [23], we have to show that for any f i n i t e subset M of the lattice, a n y f¢C(M), and any s~X, l i m l A l - 1 ]~ <~(a)f>A(S ) = o , A a
(A.6)
where ~(a) is the translation by a (acting on functions) and the sum is over all a such t h a t M + a c A. Now, conditioning on rA:
<.>A(S) = ~: PA(rAIs)<.>(rA)
-
By Lemma 3 we may assume that r A covers a small p a r t of A; indeed, IrAI <- O(expS•)l I r A I I <- O(expSB)laAI , which i s much smaller than
IAI,
f o r IAI large.
Decompose the complement of r A in A into connected components: For most a of the LHS of ( A . 6 ) , definition of contours, restricted
ensemble
the configuration on the b o u n d a r y of A(i)
X oR.
A\rA=UiA(i).
$(a)f will be in C ( A ( i ) ) for some i.
Thus,
for
most
a,
<$(a)f>([' A)
But by our
belongs to the
will
be
equal
to
<$(a)f>A(i)(s) for some A(i) and some s~X~. Now we can use Lemma 2, which says that <$(a)f>A(i)(s) is almost equal to o provided a*A° is far from the b o u n d a r y of A(i).
But
again, since
IrAI
is much smaller than
covered by the A(i)'s with IA(i)I_>IAI l - s ,
IAI
the fraction
of A that is
z=O, goes to 1 as A-~ ®. So we can assume
A(i) to be large and to have a small b o u n d a r y (since the latter is contained in rA); thus,
f o r most a, we are far from the b o u n d a r y of A(i)
and this concludes the
p roof. Now, given Lemma 1, it is rather easy to prove Lemmas 2 and 3. we
observe
that
in
(O)-restricted-ensemble.
the
complement
However,
for
this
of
the
contours,
ensemble
we
have
For Lemma 2, we a
have
the
convergent
45
Iow-fugacity expansion. Now, combine this expansion with a (large-scale) contour expansion.
Convergence of the combined expansion then
follows from
Lemma 1,
which gives a suitable damping factor f o r the contours, and the following remarks. When we
sum over all contours
containing a given
contours having a given (large scale) size I l r l l = n
box,
the
number of
grows like c n where c does not
depend on L([}). - When we consider the combined expansion, we have objects defined on two d i f f e r e n t scales: the contours and the graphs of the low f u g a c i t y expansion in the restricted ensemble. However, one shows t h a t the sum over all graphs connected to a given box B in a contour is at most e x p ( O ( e x p - 4 ~ ) l a B I ) , where the boundary aB of B has a size L(ib)=exp(5B/2) which is much smaller than exp(4~) and, t h e r e f o r e , this c o n t r i b u t i o n is a small correction to the weight of the contour. Once
we have
a convergent
expansion for
the Gibbs
state with
boundary
conditions in the (O)-restricted-ensemble, Lemma 2 is easy to prove. To prove Lemma 3 we follow [23]: we f i r s t change b o u n d a r y conditions from the a r b i t r a r y s in the Lemma to
(O)-boundary-conditions.
Direct estimates on p a r t i t i o n
functions yield: PA(.I s) ~ exp(4BlaAI)PA(.IO) . Now from
Lemma 1 and simple combinatorics
(we connect together the
disconnected parts of r A t h r o u g h @A, which, for
I IrAIl_> claAI,
possibly
gives only a small
e r r o r ) we obtain PA( I I r A I l > c'laAI
I0) ~ e x p ( - c " B laAI) ,
from which Lemma 3 follows. Proof of Lemma 1.
We will use the following notation. We w r i t e Z(AI o) for a
p a r t i t i o n function in a subset AcZ 2, where we indicate a f t e r the vertical bar the ensemble defining the partition function and the boundary conditions. For example, PA(rls) = Z ( A I r , s ) / Z [ A I s ) where
in the
numerator we sum only o v e r the
(large-scale) contour. For a contour r its support is [r]
=
U B~F
B
,
(A.7)
, configurations for
which
r
is
a
46
and, f o r hcZ = its
boundary, ah, is t h e set of points of i t s complement which have
nearest neighbors in h. Now, we condition on the values of the spins in a [ r ] :
PA(rls) = ~-, P A ( r l s , s ' ) P A [ r , s ' l s ]
,
(A.8)
s ,
o
where t h e sum runs o v e r all StXR,a[1` ] ,
since,
b y d e f i n i t i o n of 1`, c o n f i g u r a t i o n s
in a[1`] belong to the r e s t r i c t e d ensemble X °
R,~[rl"
C o n d i t i o n i n g on the spins in a i r ] interaction has a range equal to one).
PA(1`ls,s ') = P[r](rls,s')
decouples t h e contour r from h\[1`]
(the
We can t h e r e f o r e w r i t e
= z([r]lr,s,s')/z([r]ls,s')
where s, s' define the b o u n d a r y conditions f o r
, [r]
(A.9)
(s outside A, s' inside h).
we shall p r o v e t h a t the RHS here is bounded b y e x p ( - c l 3 1 1 r l l ) ,
Now
with c i n d e p e n d e n t
of s and s'; b y ( A . 8 ) , this will o b v i o u s l y p r o v e the lemma. We can w r i t e
z([r]lr,s,s')
=
z
~' z ( [ r ] l r = , = , s , s ' )
F2 u
,
(A.10)
where t h e f i r s t summation is o v e r all possible families 1`= of t y p e - 2 boxes of £, and the second over families u of small scale contours in [1`] such t h a t f o r each box of £= t h e r e is a c o n t o u r of w w i t h s u p p o r t i n t e r s e c t i n g the box. Let [r]\[u]
= u Mi, where [u] = U [~'] , i ~ u
be the decomposition of [ r ] \ [ ~ ] small-scale c o n t o u r s ,
z([r]lr=,=,s,s ')
into connected components; w i t h a f i x e d family = of
z ( [ r ] 1 1 ` = , u , s , s ') is a sum o v e r c o n f i g u r a t i o n s belonging to a
r e s t r i c t e d ensemble in conditions s, s'.
(A.11)
each M i,
an ensemble determined b y ~ and
the b o u n d a r y
We may w r i t e : (A.12)
= e -BE(u) ]I Z R ( M i l s i) , i
where E(=)
= ]1
E(~)
,
and s i is the c o n f i g u r a t i o n on aMi d e t e r m i n e d b y s, d e f i n i t i o n of 1`2 ,
s' and u.
Moreover, b y t h e
47
[ r \ r =] c
U
~( i)#0
M.. l
[A.13)
We insert (A.12) into ( A . 1 0 ) ,
(]I Z R ( M i l s i ) ) / Z ( [ £ ] l s , s ' ) i
(A.10) into ( A . 9 ) and estimate the ratio
_< (]I Z R ( M i l s i ) ) / Z R ( [ t ] l s , s ' ) i
(A.14)
,
where in the denominator we used the obvious lower bound: z([r]ls,s')
-> ZR([r]ls,s')
.
We can now use the convergent Iow-fugacity expansion to estimate the ratio of the p a r t i t i o n functions of the restricted ensembles in ( A . 1 4 ) . Using a simple modification of the Proposition of Sec. II, ZR(AIs)
we can w r i t e Io9
for the boundary conditions s belong to the ~-restricted ensemble, as a
sum of a volume term and a boundary term: (A. 15)
log ZR(AIs) = - IAIBSR(BIc) * A~(A,B,s) , where ~R(BIs) is as at the beginning of this A p p e n d i x . The boundary term is AE(A,B,s) =
X
¢'(Ols)exp-4BlOl ,
(A.16)
where the sum is over all multiplicity functions ~ defined on the "particles" of the "gas" of elementary excitations of the restricted
ensemble, ¢
ak#~i means that the
s u p p o r t of one of the "particles" of ~ intersects aA, and Iol is the total number of " p a r t i c l e s " (counting multiplicities) in the support of ~. Finally, ¢'(Ols) is a sum of ~T(o)
(see Sect. 2) which depends o n l y on s t h r o u g h
its restriction to @~ - the
b o u n d a r y of the support of ~. The estimate (2.11) holds if we replace in it ~T(o) by ¢'(Ols). Thus we have: sup ~ I ¢ ' ( O I s ) l e - 4 B I l l ~ 0(e -4B) , X O:XssuppO
(A.17)
where the sum runs over all O whose s u p p o r t contains a f i x e d excitation X. We w r i t e the volume terms of (A.14) as: expB(; IMiI~R(BI~(i))-ItI~R(BI0]) s exp(-(e-4B*0(e-8B)) X i i:s(i)#0 exp(-½e-4BIrll) ~ exp[-ce-BIIPII) where r = =
r\r =
,
IMil)
(A.18)
, and where to obtain the f i r s t e q u a l i t y we used (A.2) and ( A . 3 ) ,
and the last one ( A . t l ) ;
c is a B-independent constant.
48 Now, in (A.14) we have also to estimate the b o u n d a r y terms. The family of the boundaries aMi consists of two subfamilies: one contained in [~] (coming from the small scale contours), same boundary (A.14).
and another one contained in a[£],
conditions,
s and
s',
in the
Since these boundary conditions
numerator
are the
on which we have the and the
denominator
same, the contributions
boundary term (A.16) from this part of the boundary,
of
to the
cancel each other in Xi log
ZR(MiIs i) and in log ZR([r]ls,s'). On the other hand, for the f i r s t part of the b o u n d a r y we obtain, using ( A . 1 7 ) , a term bounded in absolute value by c'e-4Bl~l
,
(A. 19)
where c' is independent of B, and
I=1=1[=]1.
Hence, we obtain f i n a l l y that (A.14)
is bounded by exp ( - c e B I I r Z l l
+ c'e-4~l=l)
.
By E(~)ZI~I, this inserted in (A.8) yields z([r] Ir=,u,s,s')/zR([r]
Is,s') <-- exp(-B'E(u)-ceBI Ir=l I) ,
(A.20)
where B' = B'(13) -= B - c'e -4B ,
(A.21)
with c' from ( A . 1 9 ) ; for B large, B' differs little from B. Now, inserting (A.20) into (A.10), and (A.10) into ( A . 9 ) , gives: Pk(rls,s') ~ exp(-ceBIIr=ll)XZexp-B'E(=) The
sum over
the families u of
(A.22)
. small-scale
contours
is
restricted
by
the
condition that, for each B~F = there exists at least one contour ~ u with [l~]t~B#~. Therefore, X2 exp (-B'E(u)) --u
][ B~r =
( X (I/m!) m_>l
XB ~...~ ~
<_. ]I ( ~ (1/m!)(I: B exp(-#'E(~')) m) , BsF = m_>l
exp(-B'X E(~'j)) j
(A.23)
where the superscript B indicates summation over contours ~" with [ ~ ' ] ~ B #~1.
49 Now, since E(~')>I~" I and, since E(~')>6 (see ( A . 5 ) ) , the usual Peierls' argument gives for B large XB exp (-B'E(~')) <-- c ' l B I exp(-6B') = c'e -½B , where we use IBl=expSB and (A.17);
(A.24)
c' does not depend on B.
(A.24)
inserted
into (A.23) gives for (A.22) the bound:
(exp(c'e-½B)-l) I Ir=ll <_ (c"e-½B) I Ir=ll
,
(A.25)
since e x p ( y ) - l = 0 ( y ) as y~0. Now, inserting this last estimate into (A.18) and going back to (A.8) concludes the proof.
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Low T e m p e r a t u r e Continuous Spin Gibbs States on a L a t t i c e an d the I n t e r f a c e s between Them - a P i r o g o v S i n a i Type Approach
Milo~,
Zahradntk
F a c u l t y of Mathematics a n d P h y s i c s , Prague,
Charles University,
Sokolovsk~ 83
186 OO C z e c h o s l o v a k i a
§
0
Introduction.
The Main Result and the Basic Example.
I. C O E X I S T E N C E O F PHASES
§
1
Basic c o n c e p t s . A b s t r a c t P i r o g o v S i n ai S e t t i n g .
§
2
Contour Models.
§
3
I n v e s t i g a t i o n of the contour models.
II. I N T E R F A C E S
§
1
Geometrical s t r u c t u r e
of i n t e r f a c e s .
§
2
Wall Models and A g g r e g a t e Models.
54
§
0 .Introduction.
The M a i n R e s u l t a n d
The Pirogov Sinai theory, which so called Peierls
approach,
the Basic Example.
is one of the very general expressions
is often referred
should use only if all the "more common" In
this
namely
note, that
description
we
aim
this of
to contribute
theory
a
low
is really
temperature
to as
of the
theory
one
methods fail to work.
in favor a
a complicated
of the opposite
convenient
behaviour
and
of m a n y
point of view:
general
means
lattice spin
for the models
of
statistical physics. The
theory is not only a "method
of construction of the phase
diagram".
It gives, too, some rather precise information about the structure of coexisting phases. Even
in the uniqueness
by
another
the
region
methods of
a
region,
(e.g.
the theory gives some results hardly
about
coexistence
the behaviour
of
phases
of the "nonstable"
is the
more
obtained
phases)
characteristic
but
realm
of
applications of the theory. Here,
we
will
formulate
one
general
result,
illustrating
a possible
use
of the theory. The investigation was motivated by the example below.
MAIN
THEOREM
ASSUMPTIONS. 1) G e n e r a l
conditions.
We c o n s i d e r
a lattice
X
model with a configuration
space
:
2
and with a finite
range
hamiltonian
H(.~^ IX^')= Z
of t h e t y p e
~A(XA )
(0)
A/~ A # ~ , diam A £ 2bs ~-~A are translation invariant, bounded from below, continuous interac-
where tions on
X (A)
=
(~)~
,with values in ~ tl ~4 (~)}
,and
2f] is the "inte-
raction range". 2) E x i s t e n c e We a s s u m e noted by We a l s o states ~( 4"
of
of g r o u n d that
there
X+ I .Xassume ~'~ =
converging to
that
states. are there
,denoted ~+_
two t r a n s l a t i o n
invariant
ground
states
of
~-~
,de-
in the following. by
are
two " h o r i z o n t a l l y
3(+
for no"vertical
(~(*)% (resp.{X')~)
,
~( ~
s h i f t " of
if
translation
in the following,
invariant"
~(_+ ; b o t h {.~_+)~
tit -~ Oo
ground
such that
(reap. t ~
, (X~-)~
-~-
oo
).
55
Note. H o r i z o n t a l ( r e a p . v e r t i c a l ) s h i f t is a s h i f t $
=
( $4 ..... S~- )
and
3) Second differentiabilit~ of all ~ A
~
{ ~: ~
$4 =
S.~ = 0 (reap.
~ +$ ~
S~
where
.... ~%,.~=
0 ).
the positive definiteness of the Taylor
approximations. We
assume that the interactions
~A
~.X-)A, ('.X ~ ) A ' ( ~'+ ) A " We such that for any ~ ~ X such t h a t
I ~ ~ ~"
= ~--
are twice differentiable at any also assume
(~+)A,
that there is some
C > 0
J ~ / f ~ ~K
is sufficiently small, the inequality
H(x~+4~)-
H(X+)
>.. C I q / |
holds, and analogous inequalities hold also for ~ -
z
(1)
, )~_+ , ~_+ .
4) A Peierls (Gertzik, Pirosov, Sinai) type condition. We assume that for each d" ~
any ~
0 there is some
× satisfying the condition ~]~ _---0
of ~ ~
C
:
C(dD>o
such that for
for all except of a finite number
7_~,
(2) where
8(.t
:
and analogously for
~-
,
.~ _+
,
~_+
Note. We do not assume the smallness of ~ / must
be
supplemented
by
some
entropy
there. Actually,
bounds
the condition (2)
for the "energy
shells" of the
model, guaranteeing the existence of a Gibbs state at least for low temperatures -see
[ 1] Until now, the interactions
~A
were fixed. We assume further that there
is a two parameter family of hamiltonians of the type
=
$ ;~,;~'
o,o
A " diem A ~ We assume t h a t the f u n c t i o n s ~!
(having
analytical
d e p e n d i n g on
2( A
( ~, , ~ ' ,
) (see
XA
J'A ~I~# (.XA~
a r e r e a l a n a l y t i c a l both in
and not
) a n d a l s o " s u f f i c i e n t l y r e g u l a r " as f u n c t i o n s of the t r i p l e
[ I ]
).
5) D e s e n e r a c y removin~ c o n d i t i o n . We d e f i n e the q u a n t i t i e s
~c~,~,
~
e x t e n s i o n s to some n e i g h b o r h o o d of zero in ~ X ~
:
( f o r m a l l y - i t is e a s y to g i v e a p r e c i s e d e f i n i t i o n )
I 7-~-~I "~ ( H ~'~'(x+-~ - H ~ " : " ( ~ + - ) )
56
Note.
.SJ~k,~1 is the difference in the mean energies of
)(+ , ~("
whereas
~k¢~' is the difference in the mean interface energies of )(_+, ~_+. We assume that the mapping
{
}
( ;~ . ~' ) ~ ( 2,~,~,, . ~ , ~ , )
(~)
ha s an i n v e r t i b l e d i f f e r e n t i a l in (O,O).
RESULT. Then, under all these assumptions, the following is true: There is an analytical function
f,p . , ~ > ( x ( T ) , ~ , ' ( ~ ' ) ) ' } :
(o,%)
~
~2.
~s)
d e f i n e d for some ( s m a l l ) ' ~o > 0 a n d such t h a t :
I (for ~ >I 2) There are two translation invariant Gibbs states P ; , P ~ on X corresponding to the hamiltonian H ~CI')'~'(I") and the temperature • such that P ~ is a perturbation of the gaussian field ~ ' ~ arising if each ~ A is replaced by its second order Taylor expansion around (for
V" >/ 3)
There
are
two
on
corresponding
II X
T
horizontally
such that e.g.
if each
~ A
translation
to the h a m i l t o n i a n
P~
~(~ . (Analogously for P ~ invariant
Gibbs
states
). P~-~ , P ~
H ~ C ' P ) I ~ * C ' r ) , an d the temperature-I--
is a perturbation of the gaussian f i e l d d ~ ±
is replaced
by
its second
order
Taylor
arising
expansion
aroud
(.x_+~ A . Notes. 1) A g e n e r a l i z a t i o n to the case of p e r i o d i c a l g r o u n d states is p o s s i b l e , 2)
In
the
(~, ( ~ ) , 3)
case
I
there
P~
in
P~
even a one d i m e n s i o n a l m a n i f o l d of
all
possible
~' ( ' r ' ) ) .
Both the Gibbs
to
is
the
fields
"vertical"
P~
direction
in the d i r e c t i o n
,
PG
converge
("exponentially
r ~%r-~ oO and c o r r e s p o n d i n g l y ,
~V" ") - ~
fast")
to
they c o n v e r g e
"
References. The s t at em en t I is p r o v e n in the p a p e r [ 1 ] .)(+ The ~2 ]
in
,
statement
~"1"
II
. A short,
will
~k'
,
be
discrete
a
subject
(there ar e no
in of a
[1 3 paper
). which
is
now in p r e p a r a t i o n
spin v e r s i o n of the method used in I I is e x p l a i n e d
[3]. A standard reference on the Pirogov Sinai Theory is [ 4 3 " Some comple-
ments to this theory are given in [ 5 ] (for the discrete spin case). The line of s t u d y used in II was o r i g i n a t e d by the p a p e r [ 6 ~
.
The other relevant references are [ 7 ] Cwhich contain some results comparable to [ 1 ] ) an d a l s o the c o n t r i b u t i o n s of ]. Bricmont an d ]. Slawny in t h i s volume.
See also [i~ for further references.
57
Technically,
the proof
tecl%niques
from
for
of
Main
perturbed
Theorem
gaussian
is
fields.
based
on
some
Some of t h e s e
cluster
techniques
expansion we l e a r n e d
[ 8 "] B a s i c example
Z" Take
~
= (~)
. Consider+ a h a m i l t o n i a n
H (x)
=
+ nearest
neighbors
where ~(~($,~I) = I J ( $ and where ("double
~
~ ÷ I ~ + small p e r t u r b a t i o n
is given
as follows: ,'+
well potential")
Figure
1
,~o ~intuitively, verns
the
it is understood behaviour
a properly
"balanced"
- one centered If
we
At phases
has
The c h a n g e of i n t e r f a c e s
balanced
~4
change
(assuming
that of
of ~,
"one
- entropy
invariant
one e x p e c t s
height
the hill
of
the
in
two t r a n s l a t i o n
"hill
6"
may effect,
principle
this
go-
example.
invariant
For
phases
•
above
~ o
small effect on the coexistence
remains
"
(by
varying
of t h e p r e c e d i n g
"high").
however,
the phases
centered
may
expect
even
fighting"
phases
and the other one around
the
AI
the "energy
translation
a comparatively
between
9~a
that
the
value
around
further ) this
of
~+
very
around
two
substantially, ~ ÷ ,
equally
good
6-
the structure . For a "properly
possibilities:
either
"tunelling" directly from one well to the other or "going over the hill". Graphically, one expects two competing kinds of interfaces: ++++++++++++++++++++ ++++++++++++++++++++
or ++++++++++++++++++++ ++++++++++++++++++++
00000000000000000000 Hence t h e n o t i o n s of J ( _ + , ~ ) .
either
58
Note.
One
can
easily imagine
more
wells and
also more competing interfaces.
In fact, if one studies the coexistence of a maximal possible number of phases (and interfaces) then no additional difficulties arise in the general case. Some modifications are needed, however, interface diagram. a
combination
of
a
This was technique
if one aims to construct the full phase
not studied in detail yet but it seems that presented
there
with
the
technique
of [ 5 ]
will suffice to describe this more general situation. x
x
x
In the rest of this contribution we will sketch the main notions and ideas used in the proof of Main
Theorem
(mostly of its I part; the subject of inter-
faces being only touched). I.
COEXISTENCE
OF PHASES
§ I. Basic concepts of the Pirogov Sinai Theory. Abstract P.S. setting. We will present a generalized form of the theory, in the spirit of [
1 ] . See
also the contributions of ]. Bricmont and ]. Slawny in this volume. Notations. We consider the situation of Main Theorem if not specified otherwise. (Sometimes we will sketch a more general approach,
or illustrate our conside-
rations on the Example above). Traditionally, the theory is considered as a low temperature one. It is, nevertheless clearer to explain it in the case
~
= 1
where ~' denotes the tempera-
ture. By this we m e a n that the rescaled quantities =
T
(0)
=e
are used (another scaling of ,X is needed in the "nongaussian" case), The
models studied
by
the
theory must have a structure
the f o l l o w i n g two complementary concepts: f i r s t ,
described by
there is a notion of
I) almost ground confisuration (ground state in the original P.S. approach - for the discrete Spin systems). A second notion is that of 2) energy barrier (composed of contours), separating the regions with an almost ground configuration. The usual choice is as follows. I) Write the constant ground states
~+
and
X"
as
Take some s u f f i c i e n t l y l a r g e Ot~ (to be specified l a t e r ) neighborhood of
and denote by
~-+
the
~ -+ . Take
7_ ~
X+
--
X-
:
(U-)
(7)
59 for the sets of almost g r o u n d c o n f i g u r a t i o n s . See F i g u r e 2. ( I n the d i s c r e t e case, we take (Clearly,
~-+
=
~ 1~
).
O~ must not be choosen too l a r g e .
Actually,
O~
is small before the
scaling (6)).
I
II
I
Figure 2
I
1 I
I I
%C
Oi +
2) The d e f i n i t i o n of a contour s t a r t s with a notion of a correct p o i n t : is
said
to be a + correct p o i n t of X E X
if ~5~ ~[+ for a l l $£ ~.V'such t h a t The set of a l l n o n c o r r e c t p o i n t s of In the
discrete
s p i n case,
of .K
to some f i n i t e component of
(more p r e c i s e l y ,
(+, ~ ,
~ ~
~_""
6") correct
I S - ~ I • MJ. .)(
B? (
w i l l be denoted by B(-~) (=
) ).
one then defines a contour simply as a r e s t r i c t i o n B C~()
(which is c a l l e d the s u p p o r t of a
contour). In the c o n t i n u o u s s p i n case, level".
such a d e f i n i t i o n works only at the " c o n c e p t i o n a l
I n o r d e r to p r e v e n t some complications otherwise a r i s i n g in the t e c h n i -
cal p a r t s
of the theory it i s ,
notion of a contour.
however,
r e a s o n a b l e to i n t r o d u c e a more r e f i n e d
The f o l l o w i n g modification is used i n
[ 1~
: (one c a n
skip t h i s a n d r e a d only the Note below). "Wrapped" c o n t o u r s . i) Take s o m e / ~ > 2 ]
such t h a t "a b o u n d a r y c o n d i t i o n from
('1.1-) A¢ ) is only " s I i g h t l y felt" o n , t ,
dist
(t,
(q/T)
Ac
A¢
) >.. 2 ~
(analogously, }
i n the
gaussian approximation. it)
Replace
B ~ ( ~ ) by ~
f i x i n g .l( on ~[-
outside of
~(,I),
~.~) . This is s t i l l not the best choice b e c a u s e by
the c o n f i g u r a t i o n .)( may "tempt to go out" of ~ +
B.¢.~(.~) .
(This is
h a p p e n with a n additional" i n t e r a c t i o n ~ + S + ~ l t _ S ~ 2 X , [ ~ ,
Take, therefore, a suitable ( l a r g e ) ~ <
6" , denote by
I$1 = ~ ) . See F i g u r e 3. ~±
of a n d define g (-~) as the smallest set c o n t a i n i n g XIE~t for all ~EB{~(),dist(t,(g~X)) c )~ /~.
I @
~,~.--~.~
~*
resp.
not the case i n our Example b u t it may the ~ n e i g h b o r h o o d ~.,.~(~() 1;
mean valueof the ./conditioned field
/ouni ....., r y c o n d i t i o n (from ~
)
such t h a t
60
The contours
of
~ (~)
are then defined as restrictions
of ,~ to the connected components
.
Note. Most of t h e a r g u m e n t a t i o n
below w i l l u s e o n l y the f o l l o w i n g two p r o p e r t i e s
of c o n t o u r s : a) the s u p p o r t
of a c o n t o u r
b)
a
outside type,
of
contour,
+ or - ,
is a c o n n e c t e d any
almost
being prescribed
set
ground
configuration
can
live
(only
its
by t h e c o n t o u r ) .
An A b s t r a c t A p p r o a c h . It
is
sometimes u s e f u l
the n o t i o n s
of a l m o s t
to c o n s i d e r ground
an
"abstract
configuration
Pirogov Sinai
a n d of c o n t o u r s
situation"
where
are given as primi-
t i v e n o t i o n s at t h e v e r y d e f i n i t i o n of t h e m o d e l . A configuration
is t h e n a c o l l e c t i o n of t h r e e o b j e c t s :
i)
a f a m i l y of c o n t o u r s { P~ } ;
of
~..'WN
the
k~. s u p p P ~
prescription~"
1~_+ { [1~ }
a compatible one i.e.
there
is
to
live
whether
a nonconflicting in
X +
or
prescription
X-
).
t h e u n i o n of a l l t h e s e + c o m p o n e n t s .
× - C^') (=
of i t s
We d e n o t e
It r e m a i n s
ii) some ~ + ~ X +(]~+) where, usually, X+(A÷)=(q~+)A~rfor some ~ + ,
iii) some
s u c h t h a t for a n y c o m p o n e n t sign
by
(i.e.
A !
=
to s p e c i f y
and
^- )
Such a general approach will be really necessary when studying interfaces. Given such a triple (~ P~ } , ~ t ~ - ) we assume that its hamiltonian has the form X where
H
is a " g r o u n d
In t h e d i s c r e t e
energy"
spin case,
and
~
an especially
H (a)
is
a
"contour energy".
transparent
f o r m u l a c a n be o b t a i n e d :
= .+IA÷I+.61^-I
* 7_ X
(8')
w h e r e . ~ +{'} is the m e a n e n e r g y of t h e g r o u n d , s t a t e a ( + ( - ) The b a s i c 1)
Quick
"restricted Theorem,
assumptions
(usually,
an
fields"
X +
on
H
and
exponential ,
X-
~
are:
one)
decay
of
correlations
endowed w i t h t h e h a m i l t o n i a n
t h i s is a c o n s e q u e n c e of t h e c o n d i t i o n
in ~
both
. (In
the Main
(1)).
2) A P e i e r l s t y p e c o n d i t i o n ~(P)
~ Z I supp P
( w h i c h f o l l o w s from (2) - a f t e r 3) will
We w i l l occur.
situation of
assume
our
We e m p h a s i s e
which
coexisting
that
appears
phases.
outside
h a s to m o d i f y t h e m e t h o d ) .
,
some
'~
(9)
large
(6) - in t h a t c a s e ) .
hamiltonian
again
(If
I
that
(8)
is
such
that
i t i s n o t otlr t a s k
of t h e
point
"nonstable"
a
phase
there
of the
maximal
phases
are
transition
to d e s c r i b e possible
to be
the
number
expected,
one
61
§2. Contour models This
is the essential
approach
(than
construction
construction
that of
is very
[4
3
natural
of the theory.
~ )there.
(See
in the symmetric
We
present
[1 3 case
a more
general
for more details). (if there
The
is a symmetry
between the restricted fields, as well as between the contours separating them). In fact, the notion of a contour model was
first introduced in E 9 3
(which
is
the thorough study of the Ising model). Nevertheless
the
nonsymmetric
case
is the
more
important
and
characteristic
for the applications of the theory. Our emphasis will be on it, too. Realizations of the contour ensemble In
addition
to the notion of a "compatible
also the notion of an "admissible Pi , supp
P~, )
type +(or-)
~
family
of contours"
we
introduce
family of contours" i.e. such that dist (supp
2 for any
~
~ i l. We
say that a contour /'I is of the
if the external component of (supp [' )¢ is signed + (or-). Admissib-
le famillies of contours of one type only will be considered. In the symmetric
case,
flipping some contours) This has
no
analogy
notion of an
an admissible family of contours corresponds
(by
to some compatible family (uniquely up to +/- symmetry). in
the
"admissible
nonsymmetric
family
case
of external
but,
still, in any
case
the
contours of the type +" coincides
with the notion of a "compatible family of external contours of some configuration of the type
+". (A configuration
the region "surrounded by" The
central
idea
behaviour
of the
system
model
studying
by
B (.~)
of the
is of the type + if .X46 ~ * ).
Pirogov
of external
it as
Sinai
contours
a behaviour
realization
of a + contour
ensemble
theory
is then
to describe
of configurations
of the system
realizations of a suitable contour ensemble, A
"outside" of the
of the "real"
of external
contours
of
defined as follows:
is a pair
of two objects (compare
the
notion of a configuration in (8)): i) an almost ground configuration
E
X +
(on the whole space)
7_v"
=
(%(+)
it) a family of admissible contours i ~ }
,
of the type
+.
Analogously i s defined the - contour ensemble. Graphically,
a
behaviour
of a real configuration
is one contour ~ of the type + and one contour
~
(see Figure
regime
•
~
|
.
~ J
~
r
4a; there
of the type - inside ~ )
e
(
4
a)
62
is
modelled
by
a
behaviour
of
a
realization
the same v a l u e s on the " e x t e r i o r " of
P
P
containing
and
having
also
: F i g u r e ~ (b)
[~
Note. to
r-.~ ,2/
The a p p e a r a n c e
do w i t h
the
of a n o t h e r
"real"
p~me
P
as b e f o r e
+. c o n t o u r s
behaviour
inside
P
P
, -
P
inside
there
is
P
has nothing
no " f l i p p i n g "
in the
nonsymmetric case. Assuming before,
it
contour
of an
lent"
is
now a
that
natural
the idea
admissible
"ground to t r y
hamiltonian" to a s s i g n
system { PX ~
(in the s e n s e a b o v e )
~-~
remains
a proper
such t h a t
contour
the
same
weight
a model " e x t e r n a l l y
as
to a n y equiva-
to the r e a l one w o u l d be o b t a i n e d .
This is d e v e l o p e d in more d e t a i l
in the n e x t s e c t i o n ,
where the basic
framework
of the t h e o r y is p r e s e n t e d . D e f i n i t i o n of v a r i o u s Equivalence
partition
functions.
The C o n t o u r W e i g h t .
of e n s e m b l e s .
We n e e d some g e o m e t r i c a l n o t i o n s d e s c r i b i n g
a structure
of c o n t o u r s :
Figure 5
Write ~ A = Given a contour and put int (This ~_~
{,~ ~ A denote by
V{~) = ~
rI
(= the
is =
~
(int
:dist ( ~: , A t ~ = supp P
L] i n t P
union
a somewhat
a ~'I : ~4" ~
A
P
of a l l
where connected
unusual,
but
IT' )¢"
'
U ~_ ~
•
) ~ ~ Write ~ c
=
ext
P
U int
(for a c o n t o u r of t h e t y p e +) i n t components
convenient
~¢"
w h i c h a r e s i g n e d - ).
n o t i o n of an e x t e r n a l i t y ) .
~ T ~-1 = ~ V ( / " )
. Finally we write
of
~t P
we h a v e
the
: ( P ) ~_+ ~
P
of
~(A '
" a finite
exp(" H(~^I'~aA ¢)) ~ A
the "diluted" partition function; the integral being taken over all that for any contour
Writing
decomposition
Consider the + contour ensemble, for the definiteness. Given and some boundary condition ,X~A= E ('~+)~A¢ denote by
e(Az O(~Ac) = f
P
P
(lo) J~
such
63
] , Ac
dist (Vfp
) >I 2 are defined by extending
(Contours of ~ ^
(14) ~(A "naturally" by some
.XA¢
We w i l l also need a more r e s t r i c t e d notion of
~(~,
X(]Ac) =
~,
exp{- H ( X ^
f
#..)(A¢ )J a . ~
(ll)
where the "forbidden zone" ~ imposes another restriction dist (V(P), m ) >/ 2 ! to (IO).
Given a +contour p
write
(lf)
= V([ "e) ~ ¢~at~ and denote
V ~ = V a (~)
by
Or')
7_
] exp
(-H(Xv, ID+r, )
int n
(12)
(the "crystallic" partition function) the i n t e g r a l b e i n g ta k e n o v e r a l l p o s s i b l e of
"~
int
such t h a t
, extended "naturally to the whole
L; X int F
is a contour
~'~".
The following notion is a crucial one: Define a contour weight F ([~) by the formula F (~)
=
In ~ _ ( V ~ ~n! a+P ) - In ~
(P )
.(13)
Note. This can be interpreted as the "work needed to install the contour". In ~ ( ~ ) . (This is no the symmetric case, this is almost the same quantity as -longer true in the nonsymmetric situation). Finally define a hamiltonian assigned to a realization ~/^ = ( /~/2, /% O'' { ~ ] ) by An analogy of (iO) in the contour ensemble is
- / / . . ( A I F I , X ~ c ) = ] exp{-- ~ { ~ a J ~ A c) ~:~ /~/A (15) (diluted partition function of the contour ensemble) where the integral is taken over all realizations ~ A = ( X/~ , { /~¢" } ) satisfying (iOt) for any Theorem. Under (13) we have, for each finite A
"~_..(A, both for . ~ A ¢
X~^c ~
F , ")(i~Ac )
, resp
the system f
proof of the
the " l e v e l " of
(16)
.
the co n t o u r en s em b l e . The
= "2z.. ('A, .X~AC )
~ (q/+)~A©and (correspondingly, in the - contour ensemble) for
(#/-) 0 ^ ~
Note. Then the distribution of "~A ~
x^
,the relation
A
}
k]V~[~]and of
the external contours (of
) Is r e a l l y the same both in the r e a l and
This is the e q u i v a l e n c e of ensembles
theorem
is
rather
straighforward,
using
(used
in [
13
).
some i n d u c t i o n o v e r
(which is the maximal p o s s i b l e c a r d i n a l i t y of a " c o n c e n t r i c " ,
64
admissible
A
system of contours in
and using the very definition
each stage of the induction. See [ i ] ,
(13) at
theorem 4.2.6.
A few words about what picture is to be expected, looking on the external contours ~A¢
of the
system.
6 ( ~ + ) ~A¢.
Take
some
finite
]k
and
some
boundary
condition
There are two characteristic kinds of a behaviour of a typi-
cal configuration in
A
(see Fig. 6): either all the contours are "small" and
"rare" or the systems jumps - as quickly as possible - to the "stable" regime, thus forming a large contour near the boundary of a given set /~ . The latter behaviour volumes +
is characteristic for a nonstable phase
A . +
+
+
+
(of the type + ), for large
•
Figure 6 + +
+
0
4-
+
_ 4-
We
in
..~
+
are interested in the first behaviour
of the contour fact
+
and we will study it in the language
model. We will see later that the stability (of both phases)
equivalent
to the validity of the following condition
(for both
is
types
of a contour): Peierls type condition for the contour weight. For any contour
[~ ,
(17)
§ 3. I n v e s t i g a t i o n
of the contour models.
~(XA): H(~ A I~)
Using the notations
we
can
write
(13) as
C l e a r l y , what is needed is a precise estimate of the d i l u t e d p a r t i t i o n s functions
('A~M, ~aA ~ ?(= ~.(A~/~F,XaA,). Approximating model. In
the
arising
situation
of Main
Theorem,
if the second order
resp
X;
this is the corresponding
Taylor expansion of the potentials
) is taken in tead
gaussian ~ --
model
(around
A
(In the discrete spin case, the approximating model is a trivial one, the "constant model"
{ .)(# }
resp. ~ ~ - }
. Another types of approximating models m a y
be needed in other situations - see the contributions of ]. Bricmont and ].Slawny). Define the approximating partition function approx
approx all X A
(19)
65 (with
H
][
approx
rel
just described). Put
.
-I )
('AI M IF , xaAc ) _ ~(~lNiFlw(9/k¢) ( --~ approx
Estimates of
.(20).
-~ rel
The basic technical problem of the theory is an
estimate of the boundary terms
of the partition functions (20). Usually, such an estimate is obtained by some expansion techniques (see [ 1 ]
, § 2 where a selfcontained exposition of the tech-
niques can be found). We summarize the main result: Note.
Actually,
a mere simply
this
problem
fact of a phase by
using
basis of a
is not so central in the symmetric
transition can
the usual
"symmetric"
be
proven
without
Peierls argument.
"nonsymmetric Peierls argument"
case,
where
referring to i t In fact, (17) is the
- which is formulated in the con-
text of contour models. (To obtain a more precise information about the structure of phases, the forthcoming estimates are useful even in the symmetric case!). Theorem. Assume that (17) holds. Take some finite A E
(~+)~AC
and some
,X a A c
(see Figure 3!). Then
In ~rel(^,M,F,~a^,)= 2~rel I^I
+ A ( A , ~ , F A a A C ) (211
where
IA I and where Proof.
16 Ac
u M I
,
(2.20
sma,,
&
/~/rel is the "relative free energy".
(See [ i ]
"polymer we
."
). In the discrete case, this is a consequence of some standard
model"
obtain,
techniques
rather,
an expression of
some -~ rel
of cluster expansion.
"polymer
more precisely
of the following type:
fA, } i where .~+(-)is
In the continuous spin case,
perturbation of a gaussian",
i
the (positive mass)
approximating
gaussian
field and where
the
v
functions f ~
where
satisfy a condition of the type
(~ ( A } is the l e n g h t of a s h o r t e s t t r e e on
Partition
functions
of the
type
(23)
are
studied
A in
Theorem 2.3.8
of [ 1
]
Some k e y s t e p s a n d k e y w o r d s of the proof: 1) e x p a n s i o n of
# A into p r o d u c t s of Wick p o l y n o m i a l s
2) w r i t i n g the e x p e c t a t i o n
of ~ ' # A . £ as a sum of p r o d u c t s of s e m i i n v a r i a n t s A2
3)
(= t r u n c a t e d e x p e c t a t i o n s ) of s u b f a m i l l i e s of { # A.E } e s t i m a t e of s e m i i n v a r i a n t s of Wick p o l y n o m i a l s , u s i n g expansions;
e s t a b l i s h i n g the " t r e e d e c a y " (in the s p i r i t of
some [8 ]
diagrammatic ).
66
4)
expression of
-~rel
as a polymer partition function of the type
7_ tel ,_
7--
]7
(25)
{Tz} where
/~ ~
, defined and
estimated
of the type
I A ,r,I
£
by means of I)-3) satisfy a condition
i'rl 4('~)
o(
, ~. small,
5) a p p l y i n g the methods of c l u s t e r e x p a n s i o n s on In fact,
a
more precise
e s s e n t i a l l y proven in [ 1 ]
result
< ~
~
(26)
(25).
is o b t a i n e d (which is not formulated b u t
, § 2.3.8
):
Theorem. ~rel
In where
and
~,
= ~p
~
= I (J(t~t)A¢ , F )
~'~e
= /f~rellAI ~"I
(analogously,
~-~, )
"~,p
(27)
are translation invariant
satisfy a condition of the type (26). (This is not needed there but it is
substantial in the study of interfaces). Corollary. Under the condition (17), the relation (18) can be written
(if we
substitute (16) in its right hand side and use (21)) as F(P)
=
~(P)
÷ ~(r),acr,l
~),(~*(F)-,[-(F))[
where ~,~'~ ~ ) i s the free energy of the + tity of the type (21)) satisfies the estimate
[ A (~, ~) I -~ ~. liE' t and H
,
contour
~.
intl'[
model, ~ ( p l
~)
(28) (a quan-
small
~ is some "counterpart" to ~ , depending only on the hamiltonian approx (This term disappears in the discrete case).
Note. This relation seems to be more complicated than the "explicit" relation (18). Moreover ~+(F:) must be equal to ~%-~F) if the equivalence of ensembles holds. In fact, it is better to solve, instead of (28), the equation
F (P)
=
without a s s u m i n g ask whether
that
~
(F)
If this is t r u e ,
(~(r ~) .P ~(r) ~"
defines
= ,4,"
an
-i. A (r
~)
(29)
e q u i v a l e n t contour ensemble a n d
C/=)
then
(30)
the phase t r a n s i t i o n occurs and both phases are d e s c r i b e d b y
the c o r r e s p o n d i n g contour model. If not, a n d we have to modify our s t r a t e g y .
then the solution of (29) is f o r m a l ( ! )
Solution of (29). The method of fixed point theorem can be used
([ 4 ] ,
[ 1 ]
). Only the
solutions s a t i s f y i n g (17) are o b t a i n e d (but they are the only r e a s o n a b l e o n e s ) .
B7 It is clear t h a t ,
first,
one must check (17) for the contour weight ~ ( ~ ]
This is true in Main Theorem b u t the proof is a g a i n omitted there.
• ~(P).
It r e q u i r e s
some i n v e s t i g a t i o n of the s t r u c t u r e of contours ( p r e f e r a b l y , the "wrapped" o n e s ! ) .
seerl],
§ /4.3.
In the discrete
spin ~case,
-~
d i s a p p e a r s and (17) follows, for ~ ( r )
, easily
from the P e i e r l s condition of the type (1). F i n a l l y , the d e g e n e r a c y removing condition of the type (4) (for ,gJ
only)assures
that
hamiltonian
(30)
H
can
be
satisfied
for
a
"suitably
balanced"
original
. Thus, the point of a phase t r a n s i t i o n is found.
Concluding note
(to
§ 2 and
§ 3).
The method p r e s e n t e d i n this c o n t r i b u t i o n
is, in some sense, an " i n t e r s e c t i o n " of the methods of D J to the c o n t i n u o u s ses,
spin case
(= [ 1 ]
theory
but adapted
). I n the i n v e s t i g a t i o n of n o n s t a b l e p h a -
it seems to be e a s i e r to a d a p t the method of [ 5 ]
t a t i o n of the
' ~ 5]
. Most of the a r g u m e n -
(in p a r t i c u l a r the t e c h n i c a l estimates
(21),
(27))
remain
u n c h a n g e d except t h a t some c o n s t r u c t i o n s and estimates c h a r a c t e r i s t i c to
[5]
must be added. II. I N T E R F A C E S This part of the exposition is a very informal one. Essentially,
one has to combine the discrete version of the method
[ 3]
with
the techniques outlined in the first part of this exposition. Note.
There
will be
some
new
geometrical notions like walls etc. We
mentioned
yet earlier the need for replacing the "naive" definition of a contour by a more intricate
one
("wrapped
contours").
This
is (technically)
even
more
important
there. Nevertheless we will not do it in this introduction because, without going into details of the proof of the technical results of § 2 below no need for the introduction of "wrapped walls" would be seen on this introductory level. (There would be no change on the "conceptional level"). Notations. We again take .~+
,
.X-
X
, X _+ , ~ +
= (~
~
) ~¢"
( V"
,~>
3
)
Let
be as in the Main Theorem. We take the rescaling
(6) everywhere. Let
.X E
X
be such that e.g.
nite component of by
the symbol
~ (.X)
_~_
. The
X
=
.~ _+
almost everywhere. The infi-
will be called the support of an interface and denoted interface
-~
is then
the restriction of ~
on
ml Contours ration
.X
(the "naive" ones) are defined as before.
We will say that a configu-
(of the type above) is a "canonical" one if it has no contours.
(It is convenient configuration
~
to .)
study
a given
interface as
an
interface of a canonical
68
§ i. Geometrical The
geometrical
structure of an interface.
notions of this § are still some adaptations
of the notions
of [6] Take a c a n o n i c a l
configuration
A "perpendicular" restriction
of
p
.X
.X
= { {%
...
,'/-.~-_,, ( . ) ) }
on the c y l i n d e r
h a s the f o l l o w i n g p r o p e r t y : ~X~-
Cp
after a suitable
~(_+~ I ~ 0~
is c a l l e d ~.o~orrect one if the
{[Sl,...,S ~)
=
vertical
; ~
I$.,:-'t~2 / ~J¢~ }
s h i f t of
for all
4
~
for all
e E (]p
.X
, either
C~
or
(30)
I×~-~t~l (Accordingly, Note.
We
(30), and
p
such
say
that
is needed
Definition
as in part
~._+ and
a
any
such
I. We
~+_
Remove
O~
be "clearly
p
~'*_
).
is large but not
distinguishable
in
determined.
is a standard
one if no shift
definition.
from . J ~
remainder
component
~(_+ or
recall that
would
perpendicular
in the preceding
Split the
is of the type
shift would be uniquely
correct
of a wall.
perpendicular. on
~
that
a possible vertical
We of
we say that a perpendicular
use the same
"too large"
-~ o~
all the points belonging
into connected
W
is called
components.
a wall,
denoted
to some correct A
by
restriction
of
W
is
.~
called the support of kA/ . The union of all correct perpendiculars components, of the
which
ceiling
are
called
cylinder with ~
"flat" (as in the discrete be
clearly
ceiling
assigned
. The
spin case),
to them,
as
a
can be also splitted into connected
cylinders. ceilings
but,
A
ceiling
nevertheless,
height
is
an
intersection
are not necessarily
of some
absolutely
a unique
"height" can
perpendicular
intersecting
~hem.We do not specify any configuration on a ceiling . External plane
walls.
Taking. the
~_~'-4 C
projection
7fw" one can
("shadow")
of ~
define the notion of an
into
external
the
wall
horizontal analogously
as we did it for contours in part I. The
only
difference
included
into ext
"exterior have
is that F'
of a
wall
the same
type
while
all the
(for a contour W
[~
+ components
.X_+ or
~±
prescription
for ext
) and
(supp
of the type +), we
"only those ceilings cylinders
(
of
include
"touching
also the same
f'~
height
~
) were into the " which
as the "truly
external" one. This whether
one wall
is "inside"
W
and
another
int W
one.
will be used w h e n
In particular
specifying
this will be used w h e n
defining the notion of an external wall. Standard
walls.
We
external
ceiling
cylinder.
The
Figure
8 shows
define a height of a wall Standard
two standard
ceiling but does not change
walls
walls
are W
its type, while
as the height of the surrounding those ones , W
W
having
. The wall
changes
W
the height 0. elevates the
the type of the ceiling.
69
r e type ~
y ~
~ / j
N
Reconstruction s t a t e m e n t . Call the a
~/
u n i o n of a l l w a l l s of -/'2.
"compatible"
collection
of w a l l s
W
as the " t r u n c a t e d i n t e r f a c e " ../~w . _.O_~ where
c o m p a t i b i l i t y means
is
a nonconflicting
p r e s c r i p t i o n on i)
the types of the s u r r o u n d i n g c e i l i n g s ( " h o r i z o n t a l " c o m p a t i b i l i t y )
ii) the h e i g h t s of them . We claim t h a t there is a .... one to one correspondence between t r u n c a t e d i n terfaces a n d h o r i z o n t a l l y compatible collections of s t a n d a r d w a l l s .
§. 2. Outline of the further strategy. Wall and aggregate models. Fix some finite A and some truncated interface.~=J'~(~A).Assume that dist ( ~ , A ¢" )~ 2 for all walls W of .f~_~ and, analogously, for all contours of XA . Assume e.g. that the partition function (~! _~w) = f
~A
exp
is "externally of the type
.~_+ " and take
& H ( . X A I(.X~)~A¢ ) ~ ~ A
(31)
the integral being taken over ~II D(A described above, with a fixed _ ~ . (Notice that we integrate "through" the ceilings, too). Very roughly this partition function can be expressed as In ~ _ ( ~ i
_~_w) =
const
-
( ~ ICI t ~ I ~ l
+ ~
~(W,g))(32)
where const depends only on A , H ; ~ resp. ~ are the "mean interface energies of X_+ resp. ~± and where I C I (analogously, I ~I ) is the cardinality of the projection, to ~ " ~ - of all the ceiling points of the type J(_* . The quantities ~ (V~) are some "wall energies" assigned to each wall W of _~_~ . (They are defined similarly as the contour energies ~ + ~)(P) in part I, § 2). We know (from the reconstruction statement above) that . ~ W (and ~ W~g~ ) will be uniquely determined if we prescribe the "standard vertical shifts" of W,~ only. Thus, the behaviour of _ ~ is roughly-modelled by some "standard walls model" with a hamiltonian (assigned to any collection of standard walls { W 4 ~ ) of the following type:
This is
apparently
a Pirogov S i n a i type
contour" being obviously needed
model
(7)
(the " a b s t r a c t
notion of a
there - even if one "factorizes" the notion of
70
a wall by taking its shadow to ~_q'-'~ ). The model is a V'-"I dimensional one and we can, therefore, expect phase transition in it for ~" ~,- ~ .(This will be the phase transition in the type ( J(+ or ~_* ) of the external ceiling). The basic question now is about an exact substitute for (32). Note. We will formulate the result only. The methods of its proof are essentially those of
[I]
needed).
notion of an in part
§2..3.8
,
Perhaps
(as in the proof of (2~), with some generalizations
the only really new
approximating
I. Namely
gaussian
to avoid
to have
an
"above"
"below"
the interface
arises there is that the be defined
so simply as
problems in the expansions
approximating
is the horizontal translation
which
measure cannot
technical
is desirable and
problem
measure which is translation
(at least far from
invariancy
like (27-) it
-/~L~
invariant
). Also desirable
around the ceilings where the approxi-
mating measure changes "vertically from OA~*
to j
". Clearly,
re cannot be realized by a finite range quadratic hamiltonian.
such a measuIt can be, ho-
wever, realized by an infinite range hamiltonian of the type (we write ~ = A " _ ~ )
H appr°x
t x.o)
=
.
N,f: Nn M = { ~ } where ~UN (analogously, ~ N ) decay as
Io.,NI d
~
(N) being
~
<
,I
(35)
the lenght of a shortest tree on
in detail in Using now
or, , ~ ( ~ ) , [ 2 ]
, developing
N,~:,s:Nnt1={s,t~ orit}
further
~/
(This will be explained
some simple estimates of §2.2j[I]).
some adaptation of the techniques of § 2.3.8 [I ]
of (27); this will be explained in detail in
E 2 3
(as in the proof
), we obtain the following
result. Theorem. In
-
"~z_{'/~1-0-~ ) = const(A)
÷ ~J~,l -I"
~lgl *
+ T_
+ F_ where to
~
~ (~V)are
the wall energies,
~ (
. )
/~t ~/
)
I X = A ¢ and
(36)
are some "counterparts"
which are linear quadratic functions of Wn(~-t)~ where W = ( ~
~t'Ll~l~: and-
) + I
WZ)~)
~i[, are some quantities defined by means of the
+
contour models. They satisfy the following estimates:
i)
I ~('E,~/)l
ii)
I ,~- T
The quantities is true for
~'[W)
I
, ~(~,
/~rr, (if also,,~~'O ~
Note. The quantities
~
~ Ol~'¢('(~t~E) ~
g ~
) and
(37)
~
(38)
are translation invariant and the same are shifted).
disappear in the discrete spin case.
71
Another expression of ~ ( A, -O-~).
Fix _0.~ and ~
= AlL ~
A
and consider the limit
A ~' ( A
~, _.~.= )
i
(see Figure 9 ). Writing
A we obtain,
in the limit,
the following consequence of (36):
z
w,^
T~mf~¢
where only those c e i l i n g p o i n t s are counted which are "above/A~ "; a n d where
some modified
~
must be t a k e n
for those
~"
~
C
/J~
which i n t e r s e c t
AI,~. This c a n be w r i t t e n , with a n o t h e r
~7'
(see Figure 9)
~1~/
.~
Figure 9
' as follows:
~,
~
"l~"
a , ' $ ~ ( T % , Tx,) >~ 9
A
Y~
e+
¢
A
GIueing of
W
and
T
. Aggregates.
We
will ignore the quantities
as
,~t).
= exp ( - ~ Having
)
~
, for the simplicity of notations.
- i, the quantities
~
(lqriting
can be handled much the same way
the idea of a "wall model" in mind,
the relation (£0) can be rewritten
as follows: Write T cv ~V if the "shadows" of 1. Split the graph ~ ted components.
The
T
and ~
on
(on the set of all f W x ]
components
are called aggregates
~v"
have a distance
and i T X }
into connec-
and we can write (40),
u s i n g the notion of an a g g r e g a t e weight
as
4~m} the sum b e i n g t a k e n over a l l a g g r e g a t e s walls which are contained in .J'l~ .
whose u n i o n c o n t a i n s e x a c t l y those
72
Aggregate model. Notice that all the external walls of the system of all walls of some aggregate have
the same
height.
exterior of an
This is called the height of the aggregate
aggregate
secting the aggregate,
~
. The
is the union of all correct perpendiculars not inter-
having
the same
type and a same height as the "true
exterior". It is easily seen that the aggregate model with a hamiltonian
= is
oC l Cl
+ 7_.
a model of the type o u t l i n e d in
for the i d e a of a w a l l can be u n i q u e l y
)
.,C
§ 1, p a r t
( 31
1 . This is the exact s u b s t i t u t e
model. Namely any " c o m p a t i b l e " c o l l e c t i o n of aggregates
represented as a " h o r i z o n t a l l y c o m p a t i b l e " c o l l e c t i o n of s t a n d a r d
aggregates. We will not go into the details of this model, and conclude our exposition by few remarks on this model only: I) If we show that an aggregate model behaves like an "infinite ceiling of the type
~_+
"islands" by
(resp.
~
surrounded
the P.S.
theory)
) with
by
external
small
and
aggregates
rare
(but uniformly
distributed)
(a characteristic picture obtained
then a similar picture is obtained when
looking
on the
behaviour of the system of all walls. (External walls can be reconstructed from external aggregates). 2) We
have
to check
a Peierls type condition for
~(~)
. Such a condition
follows from (38) and the following inequality: Peierls type condition for ~ W ) .
~(W)
1> "c I W l
,
~
large
(/~f+)
(This is an znequality of the same type as (9). It follows from (2)). The condition for
where
~
~(~)
is then as follows:
I exp(-
~C~))I_~ ~
= ( ~. k~V~')L' ((~.'~,
Notice that while walls It turns out. however, replace
the
~(/~)
I ~ ~ (~)
assumption _> T
I~ I
~
, IE small ,
= ~ VV~'} ~
,
%<4
(/~5)
{ ~" ~
are connected this is no longer true for aggregates. that formulating the "abstract "contours ,
are
~" large,
connected holds
for
P. S. setting" one can
and
the
them"
by
Peierls condition a
more
general
assumption "contours are some objects, whose energy satisfies a condition I exp ( - ~ ( / " ) ) 1 ~
~., I/PI G[/'~('/'P{
~. small ,
~
< 4
,
"
(The notion of an e x t e r n a l contour must be g e n e r a l i z e d a c c o r d i n g l y ) . 3 ) The q u a n t i t y
exp ( - ~ [ ~ ) )
may be n e g a t i v e , because ~
This is u n u s u a l but cause no d i f f i c u l t i e s . (39) with n e w , positive ~ ,
and some new
may be n e g a t i v e .
(One can also take a v a r i a n t of (36), ~
, ~
).
73
4)
The
degeneracy
~, , 9, I
removing
the coexistence
condition
(4) assures
that for suitable values
of both the aggregate
(The proof of an existence of such
~ I kt
of
models really takes place.
is some elaboration of the simple
observation that a coexistence of both types of an interface is mainly controlled by a change in the mean ceiling energies Concluding
We tried to explain that it would general
version
useful in m a n y also
other
~ ,
remark. of the
Pirogov
applications.
interesting
be really desirable to have Sinai
theory,
Besides the applications
fields which
should
be
some reasonably
as this approach
proved
to be
sketched there, there are
investigated
from
this point of
view : i) Infinite range models (with rapidly decaying interactions) 2) Wetting phenomena 3) Impure models (e.g. with an external random field). While I) seems to be accesible without much change the problem 3) is apparently very
involved
as
some of its partial solutions
(Fr~Shlich, Fisher,
Spencer
and
Imbrie - see EIO] show. See the contributions of J. Bricmont
and J. Slawny
for some further information
about the various aspects of the theory [in particular, about 2), and also about some
discrete
spin models with infinitely m a n y
tion that m a n y
authors (Lebowitz, Bricmont, Pfister, Kuroda,
buted to the study of interfaces. type"
ground states). We want to men-
line of study.
Finally
we
and others) contri-
See CII J for the references along this "Peierls note that there
are m a n y
other
advances
of
the theory e.g. [12 ~ ) which are not reflected in our text. References. 1
Dobrushin R.L., Zahradnlk M., Phase diagrams for continuous spin systems; to appear in "Math. Problems of Star. Phys. and Dyn., ed. by R.L. Dobrushin, D. Reidel (1985)
2
Holick~ P., Koteck~ R., Zahradnlk systems; to appear
3
Zahradnlk M., Rigid interfaces; Seminar in Crit. Phenomena, R a n d o m Systems, Gauge Theories, Les Houches Summer School 8~, ed. by K. Osterwalder and R. Stora, North Holland, to appear (1986)
4
Sinai Ya.G.,
5
Zahradnlk M., An Alternate Version of P.S. Theory; Comm. 93,559-581 (1984)
6
Dobrushin R.L., Gibbsian state which describes coexistence of phases for a three dimensional Ising model; Theor. Prob. and AppI. 17,612-639 (1972) in Russian
7
Imbrie J.Z., Phase Diagrams and Cluster expansions for Low Temperature ~(~)2Models; Comm. Math. Phys. 82,261-304 and 305-393 (1981)
8
M.,
Rigid interfaces for continuous
Theory of Phase Transitions; New York, Pergamen
spin
Press 1982
Math. Phys.
Malyshev V.A., Cluster exPansions in lattice models of statistical physics and the quantum theory of fields; Uspechi Mat, Nauk 35,212 (1980) (in Russian. English translation in Russ.Math.Surv. 35, 1-62 (1980)
74
9
Minlos R.A., Sinai Ya. G. "Phase separation" phenomenon at low temperatures, in some lattice models of a gas, I, Matem. Sbornik, 72, No 3, 1967, 1967, 375-~48; II Trudy Mosk. Matem. Obsch. 19,1968, 113-178 in Russian
iO
FrDhlich ]., Disordered Systems; Lectures in Crit. Phenomena, Random Systems, Gauge Theories, Les Houches Summer School 84, ed. by K. Osterwalder and R. Stora, North Holland, to appear (1986)
ii
Pfister C.E., in Birkhauser 1983
12
Dinaburg E.I. Sinai Ya. G., An analysis of ANNI model by Peierls contour method;Comm. Math. Phys. 98,119-144 (1985) See [L i ]J ,
Sc~iling and Selfsimilarity in Physics,
[L 4 ]J , [ 1L1 ]
J
ed. by ~. FrDhlich,
for a more complete list of references
SPIN GLASSES,
E F F E C T I V E D E C R E A S E OF L O N G - R A N G E
INTERACTIONS
A e r n o u t C.D. van Enter SFB 123 Im N e u e n h e i m e r Feld 294 6900 Heidelberg,
Abstract:
FRG
L o n g - r a n g e s p i n - g l a s s m o d e l s w i t h random H a m i l t o n i a n s
H = -.Z J(i,j) li- j I-US.S. ~3 i 3 where the J(i,j)
are independent,
random variables with mean conditions,
identically distributed
E J(i,j)
= 0, s a t i s f y i n g some m o m e n t
in some respects b e h a v e like n o n r a n d o m models w i t h
Hamiltonians
H = - Z li- jl-2~SiSj.
This result is o b t a i n e d through a study of the t h e r m o d y n a m i c limit in any dimension,
and the absence of long range order and
the a s y m p t o t i c c o r r e l a t i o n decay in d i m e n s i o n one and two. Hence the e f f e c t i v e d e c r e a s e of the i n t e r a c t i o n tends to be twice as fast as the o r i g i n a l one.
§i
Introduction
Spin glasses are m a g n e t i c m a t e r i a l s as impurities
(like iron) w h i c h are b r o u g h t
into a n o n - m a g n e t i c e n v i r o n m e n t
(like gold). The
m a g n e t i c p r o p e r t i e s of the iron atoms in their m e t a l l i c host are s u p p o s e d l y d e s c r i b e d by the dilute R K K Y - H a m i l t o n i a n
76
H = -
E i,j
~.e. cos 2 K F I i - J l i 3
S.S. i 3
(1)
li -Jl 3 where
the
e. are 1
(site)
random variables
describing
the dilution.
The s y s t e m
randomness
(due to the site dilution)
long-range
interaction.
w i t h values
0 and
is c h a r a c t e r i z e d
i,
both by
and by an o s c i l l a t i n g
Experimentally
there occurs
a transition
in the s u s c e p t i b i l i t y . According glasses [1,2].
to Edwards
by p u t t i n g
r a n d o m bond
The H a m i l t o n i a n
H =-
with
and A n d e r s o n
(EA),
one can d e s c r i b e
interactions
on a regular
spin lattice
then b e c o m e s
E J(i,j)S.S. i,j 1 3
the J(i,j)
a distribution
(2)
as r a n d o m v a r i a b l e s which
only depends
w h i c h has zero average.
This
gives
which
are independent,
on the d i s t a n c e
with
li- Jl and
a r a n d o m bond m o d e l
for a
spin glass. These
are the m o s t g e n e r a l l y
physicists
there
and the results to p r e s e n t p robl e m s
is still which
sufficiently
Anderson
long-range there
has gone down
A spin-glass
rigorously
but
first
Among
how they b e h a v e are modest.
I will mention
there exists
(non-mean-field)
in a s u f f i c i e n t l y
at w h i c h
bour model
is to show that
for some
lattice
dimension
one can prove
models.
I plan some
are still open.
The first p r o b l e m
regular
spin-glass
a lot of c o n t r o v e r s y
some of these results,
which
transition
studied
transition
order parameter
r a n d o m bond m o d e l
high d i m e n s i o n
interaction.
last year,
a
lowest
for the n e a r e s t
from four to three
is c h a r a c t e r i z e d
on a
or w i t h
The c o n j e c t u r e d
is a t r a n s i t i o n
[1,5]
a spin-glass
by a p o s i t i v e
neigh-
[3,4]. Edwards-
77
qEA
= lim sup ~ {mi(Si)} 2 A÷~ ~ iEA
(3)
>
IAt Here the ~ denote the set of Gibbs states ditions).
N o n a n a l y t i c behaviour,
(or b o u n d a r y con-
due to G r i f f i t h s s i n g u l a r i t i e s
is e x p e c t e d to occur in a m u c h larger domain A second p r o b l e m is the following. (equivalent neighbour)
The S h e r r i n g t o n - K i r k p a t r i c k
m o d e l has p r e s u m a b l y a transition. W h a t
happens at low t e m p e r a t u r e s ?
Is it p o s s i b l e to give m a t h e m a t i c a l
m e a n i n g to the Parisi h y p o t h e s i s c o m p l i c a t e d low T structure? yes
[6-9].
[10-12]
and the c o r r e s p o n d i n g
In some respects the answer is
[13], though in a rather a r t i f i c i a l model.
Has this a n y t h i n g
to do w i t h the b e h a v i o u r in the physical d i m e n s i o n d = 3? The latest guess
[14,39]
is: P r o b a b l y not that much.
W h a t one can prove at p r e s e n t is that some of these q u e s t i o n s are w e l l - d e f i n e d dynamic limit
(for example,
there is a w e l l - d e f i n e d thermo-
[15,16] and a w e l l - d e f i n e d EA order p a r a m e t e r
[1,5].) and that there is no
spin-glass
phase for some m o d e l s
w h e r e the c o r r e s p o n d i n g ferromagnets do have a t r a n s i t i o n [7-9,17-24].
In the next section I w i l l p r e s e n t some of these
results. As a side remark,
I w a n t to m e n t i o n that there are some spin-
glass models w h i c h one can treat in a m a t h e m a t i c a l l y m o r e d e t a i l e d way. T h e y are o b t a i n e d by w e a k e n i n g the EA assumption. E i t h e r one can c o n s i d e r r a n d o m site models b o n d m o d e l s on n o n - r e g u l a r
(hierarchical)
[25-29] or r a n d o m lattices
[30,31].
78
§2
Lon@-range
interactions,
effective
decrease:
results
The models we will treat have the following H a m i l t o n i a n H = -
Z J(i,j) li- jI-~ds.s. i,j 1 ]
where the J(i,j)
are independent,
variables with mean
~
(4)
identically
distributed
random
J(i,j)
= 0 and a cumulant expansion with
finite radius of convergence.
The S i denote either classical or
quantum spins. The results which we have obtained are compared both with the w e l l - k n o w n
in the next table,
n o n - r a n d o m case and with previous
which have been obtained with the help of Bernstein's
results
inequality
for large deviations. SG [16c,20-23,38]
A) Thermodynamics
~>~
1
Non-random [32-35]
SG
(Bernstein)
[7, 16a, 17-19]
~>i
=>~"1
~>2
~>~
exist B) Unique Gibbs
e>l
3
(+C~property)
state d = 1 C) NO broken rotation symmetry
in n-vec-
>3
~>i
tor models d = 2 D) M c B r y a n - S p e n c e r bounds
~>i
3
a>2
in n-vector
models ..l<S0Sn>I < (d = i)
(d = 2)
Cst n- (e-½)
"exp{- (in n)l-e},,
Cst n -(~-I
n
-T
Cst n
-cst
exp{ oIE}
79
Except for the result in quotation marks which holds all the spin-glass
results are true almost surely with respect t o
the random variables optimal
§3
in L2-sense,
{J(i,j)}~
These results are supposedly
[24,36].
Proofs
A) Existence
of thermodynamics
To prove the existence of the thermodynamic energy density one uses subadditive
ergodic
For notational
[16c]
(for free boundary
theorem of A k c o g l u
simplicity we consider
a Gaussian distribution
limit of the free
of the J(i,j)
conditions)
and Krengel
the
[37].
Ising spins in d = 1 with with unit variance.
We write
~(i,j)
= J(i,j)li
To apply the theorem, subadditive a uniform
j I -(~
(5)
"
we have to check that the free energy is
and that the averaged
lower bound
Subadditivity Griffiths
--
free energy density
satisfies
(stability).
of the free energy follows
from an argument of
[40] w h i c h applies to all even spin models.
A stability bound for the averaged
free energy is derived as
follows
-~F A =
~
~
in Z(HA)
tr exp(-HA)
=
~
in tr exp(-HA) < (by Jensen's
= in tr
inequality)
~ • exp~(i,j)SiS j = Z li-j1-2~ i,j£A i,j.EA
(which is of order 0({A[)) In the last equality we used the fact that the J(i,j)
(6) have a
80
Gaussian
distribution
there
occur
alter
the
some
extra
argument conditions.
Take
A the
~i
= +i Then - "
the S i are of h i g h e r
This
Now we
interval the
that
terms
[16c].
boundary for
and
proves
add
energy
order,
spins. but
the r e s u l t
an a r b i t r a r y
[-IAI + 1,0]
surface
Ising
and
In g e n e r a l
they
for
do not
free
boundary
condition.
take ~ = {oi }
between
A and
with i6 IW the r i g h t h a l f - l i n e
is
~A =
Z ~(i, j) Si~ j : E W~.. i6 A i6A z j 61~
We estimate
(7)
its n o r m by
½
: i6A
9 61~
--
J
E i6A
E j 61~
] i - j [ -2~
½ :
3
= 0(I^I By the
law of l a r g e
it f o l l o w s
numbers
for n o n - i d e n t i c a l
)
random
(8)
variables
that
lim
= 0
J-almost
surely
(9)
A÷~ T ~ T
whenever
s > ½ and hence
f = lim
(zo)
FA
A÷- T ~ T
exists
and
the p r o o f
is i n d e p e n d e n t of A).
of b o u n d a r y
conditions.
This
finishes
81
Remark:
If {~l(i,j)} and {~2(i,j)}
interactions,
are two i n d e p e n d e n t sets of
an e s t i m a t e of the form
(6), w i t h tr r e p l a c e d by ~j. 1 the Gibbs state for H I = - Z ~l(i,j)SiSj, shows that the energy density is c o n t i n u o u s as a function of the interactions
I f ( H I + H 2) - f(Hl) I ~< Z ~ J
(ii)
~2(i,j)
B) U n i q u e n e s s of the Gibbs state in one d i m e n s i o n W h e n we prove the absence of transitions, f l u c t u a t i o n s suppress e n e r g e t i c a l l y If we c o n s i d e r the
W =
formula
(formal)
it turns out that thermal
"bad" configurations.
quantity
Z ~(i,j)SiS j i~0 j>0
(12)
(8) shows that IIWII = -
c o n f i g u r a t i o n appropriate. This looks p l a u s i b l e from
for
3 ~ ~ ~ by c h o o s i n g the spin
It turns out that IIwII < ~
for
(8) and was p r o v e n by K h a n i n
e > ~3 .
[17]. This
fact is basic for the d e r i v a t i o n of the results in the last (Bernstein)
column of §2 and restricts "its d o m a i n of validity.
However,
if we c o n s i d e r the c o r r e s p o n d i n g free energy expression,
w h i c h is
(6) w i t h H A r e p l a c e d by W, we see that this is finite
w h e n e v e r ~ > i. The a p p r o p r i a t e free e n e r g y - l i k e q u a n t i t y to c o n s i d e r is a r e l a t i v e e n t r o p y phase transition"
[33b,34b]. A w e a k v e r s i o n of "no
is the absence of s p i n - f l i p s y m m e t r y breaking.
One proves this by showing that the free energy cost of f l i p p i n g an infinite h a l f - l i n e almost surely
(which is a r e l a t i v e entropy)
is finite, J-
[20]. Let ~ be an extremal Gibbs state and ~' the
state w h i c h one obtains by flipping all the spins on a half-line. Then,
for the r e l a t i v e e n t r o p y it holds
0 ~ S(~ I ~' ) ~ S d~ ind~/d~ ' = -2~(W)
(13)
82
We write ~ = ~HI + H 2 + W right half-line
' with H I and H 2 living on the left and
respectively.
Then the relative entropy can be
rewritten:
S(~I~' ) = -~HI + H 2
(2W expW)
(14)
Z (H 1 + H 2) Z (H I + H 2 + W)
Both factors
in this product can be shown to be finite J-almost
surely by an estimate of the form the fact that
IKeKI ~ exp(2~)
we use the fact that
Z(H 1 +H2)
~
+ exp(-2K)
(variational
exp
(6). For the first term we use and for the second term
principle)
(15)
-~HI + H 2 ( W ) .
Z (H 1 + H 2 + W)
The important point to note is that one may interchange
the
average over the J's which occur in W with the m o d i f i e d
thermal
~H1 + H2 w h i c h d o e s n o t c o n t a i n
expectation
By translating
these
J's
anymore.
the origin one shows that ~ and the spinflipped
have a finite relative entropy with respect to each other, hence are absolutely
continuous w i t h respect
to each other and thus
identical because they are extremal Gibbs. notion of absolutely equivalent Remark:
continuous measures
For q u a n t u m spins the
gets replaced by quasi-
states.
To prove uniqueness
of the Gibbs state,
one applies a
similar estimate on a large
(finite)
shows that the interaction
energy between A and the far outside
is uniformly
small
region surrounding
A, and
[38]. The fact that in an operational
the Gibbs state is unique,
because all J - i n d e p e n d e n t
c o n d i t i o n s give rise to m u t u a l l y absolutely states was commented upon in [21].
sense
boundary
continuous
Gibbs
83
C) Unbroken rotation symmetry in two dimensions To prove a result of Mermin-Wagner type, one applies to a state a constant rotation in an inner region AI, keeps all the spins outside a large region A 2 m A 1 fixed and rotates the spins in A 2 - AI, over angles 0 i which linearly interpolate. We write ~Ai,A2for this transformed state (this idea has been applied by Fr~hlich and Pfister [34b] in the nonrandom case). The relative entropy S(aI~AI,A 2) is now bounded by a sum of terms of the form
{I -cos(0 i - 0j)}~(SiS j ~(i,j)) =
{i - cos(0 i - 0j)}~i,j(SiS j ~(i,j) expSiS j ~(i,j))
=
~i,j (expSiSj ~(i,j))
{ I - cos(0 i - 0j)}~i,j(SiS j ~(i,j) exp SiS j ~(i,j))
(16)
1 + ~i,j{exp(SiSj ~(i,j)) - i}
Short range contributions can be treated separately, along the lines of [34b], hence without loss of generality we can assume that the ~(i,j) are small. In fact, here we need a distribution for the J(i,j) with bounded support. However, more general distributions like Gaussians can also be treated. Thus we can make a convergent Taylor expansion of (16) in ~(i,j). The modified thermal average ~i,j' in which the term SiS j ~(i,j)
is subtracted
from the original Hamiltonian does not contain the random variable J(i,j) and hence can be interchanged with the expectation over J(i,j). The result is
m(16)
= { I - cos(0 i - 8j)}0( m ~(i,j)2).
(17)
84
In this way we derive Fr~hlich
and Pfister
(by reducing the estimates
to those of
[34b])
S(~I~AI,A 2) < ~
uniformly
(18)
in AI, if A 2 (dependent on A I) is chosen appropriately.
This proves the rotation D) M c B r y a n - S p e n c e r
invariance
of ~ [21].
bounds
To prove M c B r y a n - S p e n c e r for n-vector models,
I<SoSn> I ~ Z(H')
upper bounds on correlation
one uses the following
functions
inequality
[35a,b]
exp - (a n - a 0)
(19)
Z(H) where
H' = - Z ~'(i,j)SiS j
(20a)
~' (i,j) = cosh(a i - aj)~(i,j)
(20b)
and
By the variational
Z(H[!
principle
< e x p - ~ H , ( H ' - H).
(21)
Z(H)
The expression
~H,(H' - H) is estimated
relative entropy Following
Picco
in case C)
aj = a'~11Jl
for d = 1 a n d
[22].
[].9] and Messager,
we take the following
in a similar way as the
choices
Miracle-Sole
and Ruiz
[35c]
for the aj:
k and aj - aj_ 1 = ~ , j = i, 2 . . . .
(22)
85
- 1
alj I - alj
=
k
,
lJl
=
1 ....
N
(23)
j(in+j)l -
for d = 2. Again,
the r e w r i t i n g of
(21) in terms of m o d i f i e d thermal expecta-
tions allows us to reduce the e s t i m a t e s to the n o n - r a n d o m case of [35c]. Acknowledgements:
Most of this w o r k was done t o g e t h e r w i t h J.L. van
H e m m e n and J. Fr~hlich, w h o m I thank for p l e a s a n t collaborations. w a n t to thank the o r g a n i z e r s
I
for their i n v i t a t i o n to present it in
a p l e a s a n t conference.
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is not broken
88
are treated in a) N.D. M e r m i n and H. Wagner: b) J. Fr~hlich
Phys.
and C.E. Pfister:
Rev. Lett. 17,
Comm. Math.
1133
Phys. 81,
(1966). 277
(1981). and references
mentioned
[35] M c B r y a n - S p e n c e r a) O. M c B r y a n
there.
estimates
are given in
and T. Spencer:
b) J. Glimm and A. Jaffe: New York, Heidelberg, c) A. Messager, Poincar~ 40, d) K.R.
Comm. Math.
Quantum Physics Berlin
S. M i r a c l e - S o l e 85
[36] G. Kotliar,
299
(1977).
§16.3, Springer-Verlag
(1981). and J. Ruiz: Ann.
Inst. Henri
(1984).
Ito: J. Stat. Phys. 29,
e) S.B. Shlosman:
Phys. 53,
Theor. Math.
P.W. A n d e r s o n
747
(1982).
Phys.
and D.L.
37, 1118
Stein:
(1978).
Phys. Rev. B 27,
602
(1983). [37] a) M.A. A k c o g l u and U. Krengel:
J. Reine Angew. Math.
323, 53
(1981). b) U. Krengel: York
Ergodic
theorems
§6.2, de Gruyter,
Berlin,
New
(1985).
[38] M. Campanino, [39] J. Fr~hlich:
A.C.D.
van Enter and E. Olivieri:
private communication.
[40] R.B. Griffiths:
Phys.
Rev.
176, 655
(1968).
in preparation.
ANALYTICITY IN SOME MODELS OF QUANTUM STATISTICAL MECHANICS
Huzihiro ARAKI Research Institute for Mathematical Sciences Kyoto University, Kyoto 606, JAPAN
Abstract
Application of C*-algebra approach to proof of analyticity
of correlation functions for some models of quantum statistical mechanics
is reviewed.
Analyticity of correlation functions of ground
states with respect to the asymmetry parameter magnetic field strength
~
¥
and the external
for the XY-model on one dimensional lattice
and that of equilibrium states with respect to the inverse temperature and the coupling constants
Jl
and
J2
for the Ising model on two
dimensional lattice are obtained for all real values of parameters except the singularity at the critical lines. The same method also gives an exact decay rate for the exponential clustering of the correlation functions of the latter model as a function of parameters.
§i.
Introduction
The C*-algebra approach is by now a well-established mathematical framework for quantum statistical mechanics of spin lattice systems. Recent study shows that somewhat abstract theory of C*-algebras
can
actually be used in finding physically interesting quantitative properties of some (exactly soluble) models of spin lattice systems such as the one-dimensional XY-model and the two-dimensional model.
Ising
In the present article, we review the results in [9] and [5]
about analytic properties of the correlation functions. For the one-dimensional
lattice with its sites lebelled by
integers
j E g, we consider the C*-algebra
algebras
~j
different
of
2×2
matrices,
jE g
j's), where elements of
~.
identity matrix and Pauli spin matrices J is the dynamical
system
( ~ ,a t )
~
generated by the
(mutually commuting for are linear combinations of the a ' a=x,y,z. ~(J)
defined by
The
X-Y model
go
~(A)=lim eitH(N)Ae -itH(N),
A~
(i.l)
N-~oo
(i.2) where
H(N) denotes H with summation restricted to ~ j , -N<j<__N. All ground states of the dynamical system (~,~t) are obtained in [8]. For the (finite-dimensional) subalgebra ~(N) generated by ~j, -N<~<_N, there is a unique eigenstate ~ N of H(N)E~(N) with the lowest eigenvalue. The limit ~l,,'(A)=limN÷ ~ ~N(A),
AGU~(N)N
(1.3)
exists and determines a state ~X,T of ~ . If either ]X]~I or y:0, ~k,y is the unique ground state of (~,at). If [k]
A £6@~,
t £& ~.
(1.4)
Theorem 1.1. The correlation functions (1.4). are real analytic functions of the parameters (k,y) for ~=~k,y except for the critical lines X=±l and Y = 0 , and for ~=~k,y ± in the re~ion Ik]
(1.5)
~,j=~ being the spin variable at the lattice site (i,j) and J's being nonzero real parameters, it is known [1,10] that there is a unique equilibrium state ~B for IB]<~ while there are exactly two extremal (and ergodic) equilibrium states @S,+ and @S,- for [B]>~, where ~ is the unique solution of (K~)(1/2)log{coth]~l[}=~IJ2](~K2). The correlation functions are
(1.6)
91
n ¢((i,j)~l
~i,j
(1.7)
) t
for a finite subset Theorem 1.2. functions of
I
of lattice sites.
The correlation functions (1.7) are real analytic
~' Jl
and
J2
for
~=~B' IBl<~c
and for
~=~B,±'
IBI>c. §2.
The CAR algebra and the spin algebra The Jordan-Wigner transformation cj=Tj(g~J)-i~ (Jy))/2, c~=Tj(o~J)+ig (jy))/2, (jEg),
bridges the spin algebra
~
(2.1)
and the CAR algebra grgCAR generated by
c's and c*'s satisfying the canonical anticommutation relations (CAR's), where T. is supposed to serve the role of the product of ) J , j ran~ing from the left end of the ene-dimensional lattice up to the site j-1. The difference with the case of a finite lattice is that T's are no longer an element of ~. Following the method introduced in [4], we extend the algebra ~ to
a~J
= ~+T~
(2.2)
by adding T=T I satisfying T=T*=T -I and TAT=@_(A) Here @_ is the automorphisms of ~ satisfying
O_(cY(j)) =
for
a=x,y,z
c j)
for
j>0
O(c (j) )
for
j<0
I
and
@ ( ~ J ))=~J ),
@
for all
AE~.
(2.3)
is another automorphism of @(c~J ))=-c~J ),
@ ( ~ J ) )=-~J )
~
satisfying (2.4)
for all j EL. The C*-algebra ~ is the crossed product of ~ by @_-action of the group Z 2 of two elements. The operator Tj in (2.1) is T multiplied by ~(j)...~(j-l) for j>l and by Z Z ~0)...~(j) for j
92
The automorphism 9 can be extended to ga such that 9(T)=T. It maps ~CAR onto itself and counts the even-oddness of c's and c*'s: G(cj)=-cj, 9(c~)=-c~.j The algebra ~ and ~CAR are split into a sum of
e-even
and
O-odd
parts (denoted by suffices ±)
with
the following relations. ~+= ~AR~
~
=T~CAR
(2 5)
In particular, this gives a specific isomorphism of the @-even part ~+ of the spin algebra ~ onto the even part ~_CAR of the CAR + algebra ~CAR. --CAR . Therefore the limit Since H(N) is 9-even, H(N)G ~+= ~+ (i.I) for
A~
CAR
makes sense within
~CAR.
exists and can be described as follows. {gj;j~g} For each
be in ~2(g), and h, we define
h=(f)
Let
be in
In fact the limit f~{fj; j ~ L }
and
B(h) = )~ (fjc~+gjcj). j~Z It satisfies
(2.6)
B(hl)*B(h2)+B(h2)B(hl)*=(hl,h2 )
r(f)=[~], and generates
~CAR.
r ~[Z)induces an automorphism
~(U)B(h)=B(Uh), The dynamics
st
of
~(U)
Any unitary of
~CAR
U
and on
B(h)*=B(Fh), X
satisfying
(h 6~'~). ~CAR
(2.7)
is given by
satisfying
Kr=-FK.
Combining exists for dynamics
(2.8)
K=K(~,y)
K*=K,
st
(2.9) for
~
A=T=(Cl+C~)a(1)x st
where
commuting with
~t=~(e iKt) for some
g--
~ = ~ 2 ( Z ) $ g 2 ( Z ) = ~ 2 ( Z ) ~ 2.
on ~.
and
~CAR,
we see that the limit (i.i)
and hence for any
AG~
defining a
93
§3.
Ground states of the XY model
A method of obtaining all ground states of (~,~t) from the knowledge of ground states of ( ~ C A R , ~ t) via ground states of (~+, ~t ) has been developed in some generality [8,7]. ~n the present model, the operator K above does not have an eigenvalue 0. This implies that ( ~ C A R ,st) has the unique (Fock) ground state
~E
characterized by
~E(B(hl)*B(h2))=(hl,Eh2
where
E=E(~,Y)
(3.1)
)•
is the spectral projection of
K=K(~,Y)
for
(0,+~).
If (~,y)#(0,+l), K has a continuous spectrum. This implies that the restriction of ~E to aL+-CAR_-~+ is the unique ground state of (~+, ~t )
and hence
~k, ~(A)=~E((A+®(A) )/2)
(3.2)
is the unique 0-invariant ground state of (~,~t), necessarily coinciding with (1.3). The equality of (1.3) with (3.2) persists for (~,y)
=(0, ±l). By the associated
equivalence
criterion
with
Fock
Z2-index
)
it was
shown
in
state
of
(~,~t)
(and
hence
is pure)
there
exist
disjoint
pure
the
states
for
(in terms [8]
cyclic
representations
of the
Hilbert-Schmidt
that
~
for
A,y I~I>__I
ground
states
'
mutually
is the and
for
~l,y,±
I~l
§4.
Strategy for the proof of analyticity
~+
norm
unique
~,y(A)=(~A,y,+(A)+~,y,_(A))/2
for
of
and
ground
y=0,
while
satisfying
(3.3)
~0.
For any two projections Ej satisfying E~=E.=E~j ~ J and Ej+FEjF=I (j=l,2), there exists a unitary UI2 commuting with F and satisfying UI2E2U~2=EI and hence satisfying
(4.1)
~E 2 (A)--~E1 ( ~(U12 )A ) for all
A E ~ cAR.
Since
o=~(-I)
always commutes with
~(U), ~(UI2)
94
is an automorphism of
w+--CAR = ~ + .
We use the following criteion for
the extendability of
a(U)
~+
Theorem 4.1. from
~+
unitary
from
(Evans and Lewis)
to an automorphism u
in
~+
~
to an automorphism The automorphism
o__ff ~
~
(U)
of ~ . extends
if and only if there exists a
satisfying
[~(U)@_~(U*)@_](A)=uAu* for all
AE~CAR'
Then
u
(4.2) can be chosen to satisfy
u@ (u)=l
and
^
is given by ~(A++A_T)=~(U)(A+)+~(U)(A_)uT A + ~~ +~
for
CAR
(4.3)
(and hence for any
A=A++A_TE~).
The strategy for the proof of analyticity is as follows. given
(~,~)
(~',y')
not on the critical lines
l~I=l
and
in a sufficiently small neighbourhood of
Theorem 3.1 to show that, for EI=E(~,y) and has an extension from ~ + to an automorphism
For
y=0, and for (~,¥), we use
E2=E(~',y') , ~(UI2) a of ~ . The formulas
(3.2) and (4.1) imply
(4.4)
~k, y,(A):~k,y(~(A)) • We show that of An .
A
~(A)
including
is real analytic in A=~tl(Al)'''~tn (An)
(~',¥')
for a certain class
with strictly localized
Then we obtain the real analyticity of (1.4) for all
(X,y)
AI,..., not
on critical lines. The real analyticity of Step i.
E~,y
analytic in (~,y)
~(A)
is proved in 3 steps:
is explicitly given (see Section 4) and is real (except on critical lines).
is a canonical construction of
UI2
For
IIEI-E2D < i, there
given in [9] (based on the angle
operator between 2 projections, see Appendix of [3]), such that is real analytic in (k,y) and (~',y') for the present case. Step 2. j~0.
Let
Let
(e_f)j=~jfj
I°:
(8_h)= e
g
for
with
:
h=().
Ej=I Then
for
j>0
o_=~(e_)
and
Ej=-I
UI2 for
and the condi-
tion (4.2) amounts to the condition that a(V) for v=u12e_u~2 e_ is inner and its implementer is in ~ +CAR . The criterion for this given in Section 8 of [8] is that
~-l
is in the trace class and
det V=l.
95
The trace class condition will be discussed determinant
condition
the property
rVF=V
multiplicity IIEz-E211
is automatically which implies
of the eigenvalue
is sufficiently
When
the trace class, trace class.
H
Then
det V=(-I) v
-I
small
I!V-III< 2, V-l=e iH
for
(<~/2) for
V.
if
in
with
(B,HB)
v
(~,Y)
The
due to
being the
is satisfied
if
EIUI2-111 < I.
J]Hll<w and
if
V-I
is in
with values in the
in Theorem 4.1 is given by
with the bilinear Hamiltonian
IIV-III < 2
ILV-III< 2
so that
H=H*,
is real analytic u
in the next section.
satisfied
exp i(B,HB)/2
([3]) and is real analytic in
(~,Y). Step 3. B(Uh) nomial ~(A)
If
U
is real analytic
is real analytic A
of
in
the size of localization with sufficiently
with
Bn
supported
sufficiently at(A)
in
(l,y)
region,
for any poly-
exponential [-n,n]
A.
Hence A.
of local algebra with
this real analyticity
in the interval p.
~(U)(B(h)) =
for strictly local
of the dimension
fast decreasing
large positive
then
~(U)(A)
all strictly localized
given by (4.3) is real analytic increase
in (~,y),
and so is
B's, which includes
Due to the exponential A
(X,Y)
tail,
and
extends to all i.e.
A=[B n
IiBnll ~ Cpexp-pn
(See Lemma 3.3 of [5].)
is shown to be such an operator with any choice of
for
In [2], p.
(cf.
Theorem 4.2 of [2].)
§5.
Trace class estimate We introduce
the Fourier series expansion
for
fGZ2(Z)
and
h G Z2(Z)SZ2(Z)
~(e)= [
einOfn,
(5.1)
h(8)=[~((881]
nEL
The operator
and
K
in (2.8) is then the multiplication
Icos8-1
-iysin@]
K(O)=2 [iysine
-cosS+lJ
E=E(~,y)
is the multiplication
of
(5.2) of
E(e)=(2~(e))-l(K(e)+~(e))
(5.3)
p(8)=2[(cose-l)2+y2sin2e] 1/2
(5.4)
96
where
D(e)#o
for all real
0
if
(l,y)
is not on critical lines.
The crucial property is that E(e) is holomorphic in ~, y, and z=e ie in the neighbourhood of real CI,y) not on critical lines and Izl=l.
This property persists for UI2 which is again a multiplica~ eie ). tion of UI2(e)=UI2( By a shift of integration contour yr={Z;IzI=r} for Ul2(z)zJ-k-ldz , ^
(Ul2)j,k=(2~i)-l~
(5.5)
YI we obtain the exponential decrease of where j>0
(Ul2h)j=~(Ul2)j,kh k. and for
Since
(UI2)j, k
q±z(l±8_)/2
for large lJ-kl, are projections for
j ~0, the identity
Ul2e_u~2e_-l=U12(q+-q_)U~2(q+-q_)-Ul2(q++q_)U~2(q++q_)
(5.6)
=-2(Ul2q+u~2q_+Ul2q_u~2q+) and the exponential decrease of
(UI2)j, k
(and hence that of
(U~2)j, k) for large lj-kl imply that q+U~2q , q U~2q + (5.6) are in the trace class due to the following Lemma.
and hence
Lemmla 5. I. lIXiltr:inf [ [(e£,Xe~) i [ [(ez,Xem) Z,m £,m
(5.7)
where the infimum is over all orthonormal bases
{e£}
and
{el}.
The inequality follows from the following estimate for the polar decomposition X=ulXI: HXlltr=trIXl=II(e~, m
due to
u*Xel)[=[l[(el,u*ez)(e£,Xel)l~ m i
[ I(ez,Xel) l
£~m
l(e~,u*eg)lal.
The trace class estimate above also much simplifies the HilbertSchmidt class estimate of E(l,y)-e_E(l,y)e_=2(q+E(l,y)q
+q E(l,y)q+)
in [5] necessary for judging equivalence of
~E
and ~e Ee "
97
§6.
The Isin$ model By the transfer matrix method, the correlation function
(1.7) can
be expressed as Schwinger functions of a ground state of a C*-dynamical (~,~) with the same spin algebra g~ as before and the system dynamics
~
given in [63:
¢(~(kl,ll)...6(kn,ln))= where
~(k,l)= H 6k,j'
~(aBik I (~x(ll)
°x(1)= H °(j)
jEI
-..~Bik n (~x(In)))
(6.1)
and the formula holds for
jEI x
¢=@B and ¢B,± with ~ = ~ B and ~B,+. Here ~B is the unique @-invariant ground state of (~,~), which is pure for IBI>Bc and is an average of mutually disjoint pure ground states ~B,± for IBI>Bc
([6]). The identification [6]
and for
~B,±
and
(6.i) has been attained for ~,±
~B
and
B
in
by using the preceding~ result along
with the cluster property in [5]. The operators ~k(~x(1)) and their products are of the type discussed in Step 3 of Section 4 (with any p) Thus the machinery explained for the ground states of the XY model works also for the present case and Theorem 1.2 follows. Details are refered to [5]. By using the exact information on the spectrum of the time translation generator (in the representation associated with 's), an exact decay rate for the exponential clustering of the correlation functions can be obtained. A typical result is as follows (Proposition 6.5 (2) and (3) of [5]): Theorem 6.1. lim where
(i)
For any polynomial
FI
~(~,0)
is a lattice translation,
~=4(IK2[-K{)
~=gB i_ff IBI
~÷+~ e£(6+e)I~(F1,T(£,0)(F2))-~(F1)9(F2)I=~
lim
6
F2
o_~f ~'s,
eI~I61¢(F1T(~,0)(F2))-¢(F1)¢(F2)I=0
i__ff IBI>B c and 6=2(K{,IK21) and (2) There exists polynomials given s>0 such that
where
and
and
~
are the same as (I).
(6.2) and
¢=gB,±
for any
(6.3)
98
References [I] [2] [3] [4] [5]
[6] [7]
Aizenmann, M., Co~amun. Math. Phys. 73, $3(1980). Araki, H., Commun. Math. Phys. 14, ~ 0 ( 1 9 6 9 ) . Araki, H., Publ. RIMS, Kyoto Univ. 6, 384(1970). Araki, H., Publ. RIMS, Kyoto Univ. ~0, 277(1984). Araki, H., Analyticity of correlation fuctions of the ising model in two dimensions, Prepreint RIMS-525, submitted to Commun. Math.
Phys. Araki, H., and Evans, D.E., Commun. Math. Phys. 91, 489(1983). Araki, H., and Matsui, T . , P r o g r . i n Phys. 1~0, 17-~1985).
(Statistical Physics and Dynamical Systems, eds. Fritz, J., Jaffe, A., and Sz~sz, D., Birkh~user). [8] Araki, H., and Matsui, T., Commun. Math. Phys. i01, 213(1985). [9] Araki, H., and Matsui, T., Analyticity of ground states of the XY-model, Preprint RIMS-522. Lett. Math. Phys. in press. [i0] Higuchi, Y., p.517 in Random Fields I, ed. Fritz, J., Lebowitz, J.L., and Sz~sz, D., Colloquia Societatis Janos Bolyai, voi.27 (North-Holland, Amsterdam-Oxford-N.Y., 1981). [ii] Matsui, T., Explicit formulas for correlation functions of ground states of the i dimensional XY model, Ann. Inst. H. Poincar~ Sect. A, in press.
K-TXEORY OF C*-ALGEBRAS IN SOLID STATEPHYSICS
Jean BELLISSARD Univeriit6 de Provence et Centre de Physique Th6ot'ique
MARSEILLE(FRANCE)
I)- C'algebras : an appealing but Incomolete tool ? The subject I would like to present here today pretends to use C"-algebras in concrete problems of modern physics, a mathematical tool which have been liable to controversies among the mathematical physicists for the last twenty years. When D. Testard, R. Lima and I started our program about solving some mathematical problems in the study of disordered systems, five years ago, we naively believed that using C"-algebras was just like using any other too] of mathematical physics like functional analysis or probability theory. However when we began to explain to the experts what we were able to do, not very much I confess, we realized that before trying to convince them, it was necessary to get really hard results. Since we knew perfectly well the limitations of these technics, we understood quite soon that what could be solved with it, was not as spectacular as was implicitly demanded by both the believers and the sceptics in the community. We spontaneously adopted what can be qualified as a schizophrenic attitude : on the one hand we presented an external work as reliable as possible, and on the other hand we kept unpublished for a very long time our results always present in the back of our mind as a guide in performing our program. What I will explain today have been presented almost in essence at an informal meeting in Marseille during the spring of 1981, apart from the chapter on the quantum Hall effect. Actually some traces of this work can be found in the recent literature. Several results were partially quoted in seminars, conferences [14], reviews [94], or even in some papers [17,56]. We finally published the first part very recently [19]. Reading this last paper will explain the way it has been received by the scientific community. At first sight, the scheme seems very simple and so efficient that it looks miraculous. And when miracles appear the world is immediately divided into two irreducible components : those who are true believers and those who just reject the claims. Fortunately, or perhaps unfortunately depending upon the side one prefers to choose, there is no miracle, at least in physics. In our problem, even though the scheme looks very simple, when one starts the proofs, one realizes immediately how heavy they are : all the machinery of C~-algebras and algebraic geometry is needed, and one looses quite rapidly the original physical intuition! Moreover many technical details remain unsolved or requlre very complicated and winding proofs. Clearly C*-algebras
100 are very appealing mathematical tools but they must be used together with other technics to become efficient. One can say that C"-algebras started their career in mathematical physics in the early sixties with the Haag-Kastler axioms [48] of what is known today as the Structural Field Theory. The hope was that the miraculous aspect of these objects would solve the mysteMous difficulties encountered in Relativistic Quantum Field Theory. Twenty years later it is quite clear that the flu went the other way : more have been learned from the physics to understand the mathematical tool than the converse. Moreover, very few came from the original motivation namely the understanding of the particle physics. On the contrary, like for many advances in physics during the l a s t thirty years, the concrete understanding came from Melds connected with Condensed Matter Physics. For example, the KMS condition, one of the major step in the Tomita-Takesaki theory [1"129]and in the classification of type III factors by A. Connes[25], was introduced in the field by R. Haag, H. Hugenholtz & ~ Winnink [47]. This improvement came after several mathematical physicists who started there work with the axiomatic Quantum Field Theory, had changed their mind and decided to study the Statistical Mechanics [1"124]. The attempt of J. Glimm and A., Jaffe [45] in using C~-algebras in constructive quantum field theory was soon abandoned and replaced by the probabilistic approach[M10,1,127]. However eight years ago a breakthrough was performed by A. Connes [26,27,30] who associated to any foliated manifold a canonical C~-algebra, playing the role of the basic object needed to apply the methods of algebraic geometry and to get topological invariants as well as a cohomotogy theory. He contributed in giving a concrete content to the K-theory, and created the cyclic cohomology, to take into account this new situation [28-34]. He also emphasized that non commutativity of the algebras was closely related to the ergodicity of the foliation, or equivalently to the fact that the set of leaves was not endowed with a non trivial topology. At about the same time, algebraic geometry entered in quantum field theory with the contribution of mathematicians to the classification of instantons in gauge field theories [M15]. Later on, they also interpreted the anomalies (as Adler's one) in term of topological obstructions. More recently the supersymmetries offered the mathematicians a formal tool for calculating simply the formulae appearing in the Index theory. Nevertheless the previous works are all performed on the ground of a semi classical quantization by mean of hypothetical Feynman path integralS: the algebraic geometry concerns usual fibre bundles, on usual manifolds. Nothing in this approach is essentially non commutative. The reason is that nobody knows how to built a theory for describing gauge fields in four dimensions, and one has the hope that something of the topological obstructions will remain in the final theory if any. However literally speaking there is an apparent contradiction between the fact that topological obstructions require smooth
I01
configurations of gauge fields, whereas it is well known in constructive field theory that almost surely no configuration is smooth. Topological invaMants actually reflect the properties of the observable algebra, more than those of the configurations. The Quantum Hall Effect (QHE) may be a good laboratory to test such an idea. What will be presented here concerns only the ordinary QHE, namely a one particle theory. The quantum mechanics of such a system is well understood. Moreover its has been demonstrated that, when the Fermi energy belongs to a gap, and at zero temperature, the Hall conductivity can be interpreted (in e21h units) as the Chern character of some fibre bundle above the Brillouin zone, at least when the magnetic flux through a unit cell, correctly normalized, is rational [99,9]. When it is irrational, the C"-algebra of observables becomes non commutative in a non trivial way : it is type II as was remarked a long time ago by A. Grossmann [46]. It is no longer possible to speak about Brillouin zone, and the previous interpretation of the Hall conductivity becomes meaningless. Nevertheless, thanks to the A. Connes proposal, the C"-algebra of observable can be interpreted as the algebra of continuous functions on an hypothetical Brillouin zone which does not exist as a geometrical object but has sufficientl~/many continuous "non-commutative" functions to be studied by the technics of the algebraic geometry. In particular, the Chern character still exists and varies continuously with the magnetic field, in such a way that the semiclassical approach gives the correct answer. It is widely admitted that the fractional QHE comes from many bodies interactions. A correct mathematical theory of it should require a Fock space, and a second quantized algebra of observable. It is therefore a (non relativistic) quantum field theory. Nevertheless there is a structural question here related to the calculation of the Hall conductivity : somewhere there must be some topological invariant. Is this related to the differential algebra introduced by Arveson [4] and used in the definition of the cyclic cohomology [32,33]? Is the vacuum a closed current in the sense of Connes ? Is the Hall conductivity given by a Chern character in the many body problem ? Besides the internal interest of the physical problem, which goes far beyond the analysis I will give here, it seems that investigating easier situations which are better understood in the physical set up than the intricate questions in particle physics, is an efficient way of improving our understanding oI~ the mathematical tool, and gives the hope.that in the next future, paradoxes will be solved. For this reason I feel that mathematical physicists should take more seriously the mathematical study of simple , realistic models of condensed matter physics where topological invariants exist in order to understand much better what to do in more ambitious questions like the structure of gauge field theories.
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 = <~i I ~uj> = 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
-->
<~iI~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)
<@lH~i>-- E <@l~i> + ~'j,t <¢lv( .-Rxj)~/i> = 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)
<
I H(z)q}j > = 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 <~ where :
(10)
II g II -- T.~ qD-1 supkl~q I g (x)l
Then one can show that (9) implies sup=lv=(g)l ~ const.llgll, and therefore, since such a g can be approximated in this norm by continuous functions with compact support, v~(g) is also continuous with respect to co. Let B be the Banach space of such g's. Now let us choose v in L=(RD) and such that II v II = = v(°(f.v) is continuous in co. Again since II v II is finite, v~ ls essentla]ly bounded, which shows that It ls weakly continuous with respect to co. As we already quoted, the flow on ~ can be restricted to E by mean of the "first return map" introduced by H. Poincar~ [1't21] in the 19m century to study the existence and the stability of perlodic orbits of the planets. In our many dimensional framework, the first return map does not exist, but is replaced by the notion of groupoid of the transversal _= [27]. Let r ( _=) be the (closed) subset of £2x1~° made of the elements 7 : (co,x) such that co E =_ and
111
T_,co E = It is a locally compact groupold in the sense that ),=(co,x)and 7"=(co',x ') can be multiplied provided co' =T_,co; in this case we get 7o7'= (co,x+x'). A *algebra can be constructed through continuous functions on r(.=.) with compact support with the following rules" (11 )
a. b(co,x) = ~ ~(e) a(co,y)b(Twco,x-y)
a*(co,x) = a(T xco,-x)"
As in the previous section, let C~(~) be the completion of this algebra with respect to the C* norm Ilall--sup~_~ II%(a)ll where TI~ is the "representation on the space 12(L(co)) of square summable sequences indexed by L(co) and defined by : Tl(o(a) U/(X) = T. y~.((o)a(T-xr"°,Y-X) ~(Y)
({2)
These representations satisfy also a covariance condition of a different kind. Namely, for ),=(co,x) in r(~), let u()') be the unitary operator from 12(L(T_xco)) into ]2(L(co)) given by U(y) ~u(y) = ~(y-x).Then we get" (13)
U(y) ~r-Xco(a) = %(a)U()')
The main application of this formalism is given byProposition 5- Let h(z) be the function on r(=_) defined from (7) b y (14)
h(z;co,x) = < ~ I Hco(z) U(x)~ >
Then h(z) belongs to C*(E) for any z for which H(z) is defined. Moreover the matrix of H~o(z) in the basis {~i } and of Tl~o(h(z)) in the basis {x i} coincide. 0
We will not give the proof of this fact, for (14) is just another way of writing H(o(z). To flnish this sectlon let us glve one example of such reductlon, whlch Is sometime called a "lattlce approxlmatlon" or also In $olld 5tare Physlcs a "tlght binding representation" because when (4) is valid, the electrons are essentlally locallzed near the bottoms of the potential and are tlghtly bound to the atoms. Our flrst example wlll be a quasi periodic 5chrOdlnger operator In one dimension [see In particular 16]; let v be a smooth functlon on R decreasing rapidly to zero at infinity. The potential is given by :
112
(15)
V~(x) = ~ nez v(X-Xn)
xn = n + g(()-g((-ncO
where g is a continuous function of one variable ~ periodic of period one and a is an irrational number. Then g can be seen as a continuous function on the one dimensional torus T = RIZ. To built the hull we define on Txl~ the map F(~,t) = (~+o{,L+f({~)), where f(~)= l+g(~+a)-g(~). Then the function V(~,x) = V~(x) given by (15) is F invarlant and continuous on TxR. Therefore it defines a continuous function on the quotient space Q=TxI~/F. A flow on ~ is defined by taking the quotient of the translation Ty:(~,x)->(~,x+y) (which commutes to F), and we get V~(x-y)= V(T~(~,x)). This shows that ~ is actually the hull. The set T = {(~,x)e Q ; x=O } is a transversal, and the first return map is nothing but the rotation R a by c(.Then it is easy to check that the C * algebra of this transversal C*(T, R a) is generated by two unitaries U and V such that U V = ei 2~a V U as was firstly proposed by M. RJeffel [86]. The reduced hamiltonian H(z) has a matrix indexed by Z which can be seen as an element of C*(T, Ra). A special case is provided if we can approximate H(z) by a tridiagonal matrix, and if in addition only the first Fourier coefficients are big, one gets: (16)
h(z) = a (U + U* ) + b(V + V* )
a,b e I~
This operator is called the Almost MathJeu operator and has many interesting properties [Mi6,1"128,16,18,25,41,53,94], as we shall se later on. Our second example is given by the recently discovered quasicrystals [91]. The potential seen by an electron in a quasicrystal can be described in much the same way. However the precise structure of the hull is usually quite complicated: the quaslcrystal of A1Mn originally discovered had a symmetry of order five. In the orlglnal work of D. Sched~tman, I. Blech, D. Gratias, J.V. Cahn [91], it was recognized that 6 integers were necessary to label the positions of the peaks of intensity in the X-ray picture. A model has been proposed recently [39,40,65,70] which describes the diffraction paltern observed experimentally. Even though the method leading to its construction was already known before [64], this model have the advantage of being very simple and efficient. Slnce this crystal exhiblts a diffraction pattern having the icosahedral symmetry, and since 6 integers are needed to label the positions of what one believes to be the atoms (there is still a problem in interpreting the nature of the white spotson the pictures), we wlll look at a 6 dimensional representation of the icosahedral group. This group leaves the 6 dimensional hypercublc iattlce Z 6 invariant. Moreover it has two 3-dimensional invariant subspaces. Let 11 and 11' be the corresponding
113
eigenprojections. Of course since there is no trans]ation invariant 3-dimensiona] ]attice having a symmetry of order 5, these eigensubspaces are not rationa]]y oriented with respect to the lattice Z 6 . Let A be the 6-dimensiona] c]osed hypercube of unit ]ength, and Q -- Fi'& It has been shown that Q is a po]yhedron ca]led a triacontahedron [40] which has axis of order 2, 3 and 5. The mode] proposed by Duneau & Katz consists in se]ecting those points m E Z 6 such that ~'m E Q, and to project them on the orthogona] subspace, name]y x m = ]'[m. Since Im(]'l) is 3-dimensional, we wil] identify it wlth IR3 and the x m's with the positions of atoms. By construction the icosahedral group commutes with 1] and [l', and therefore the set of atoms is a (non trans]ation invariant) lattice invaMant by this group. If we want to take into account a generic translation in IR6 we get the fol]owing e]ectronic potential for the quasi crysta]: (17)
V~(x) = ~ m~Z 6 v(x - ]'l(l~+m)) X o (]]'(~+m))
where Xo is the characteristic function of Q. Using the previous method, the Hull is the 6-dimensional torus [e=I~6/Z6. Then V~(x) is given by a discontinuous function on this torus (because of the characteristic function of ~) as a function of ~, and by an action of I~3 in the Fi direction. The dynamica] system that one obtains is an abelian virtual group. The tight binding representation of our hami]tonian wili give rise to the C'algebra of the transversal given by the image of the set ~x[O} in this torus. In principle the construction is so genera] that any kind of flow on a compact manifold cou]d be described. The question ls to know whether other examples are to be found in experimenta] physics. The proposa] of 5adoc and Mosseri [76], recently studied in detai] by D. Ne]son [see 78], seems to indicate that indeed other kinds of flows appears in the description of the short range order in ]iquids and glassy mateMals.
4)- Th.e. densit_u of states and Shobin's formula : The resu]t of the previous ana]ysls is that a disordered system ls described by a se]f adjoint operator which is affiliated to the C'algebra of a dynamical system given by some compact space ~, the hull, on which there is an action of IR° . As we showed in the previous section, this continuous dynamical system can be replaced by a dlscrete one, which, In many cases reduces to the action of Z° on some compact space -C. Before proceeding, let us note that even in physlca] situations, the groups IR° or Z° can be replaced by a more genera] locally compact group. For instance in describing the local structure of glassy systems, Kleman & Donnadleu[61] found that a description
114
through hyperbolic geometry is relevant. In this case we have a SL(2,1R)action or even the action of a Fuchsian group [M8]. On the other hand, theoretical physicists like to use a special lattice called the "Bethe lattice" [67], to describe the thermodynamics of lattice models in the limit where the dimension goes to infinity. It is not difficult to convince oneself that the Bethe lattice can be seen as a sublattice of the Poincar~ disc, and is generated from one point by a free group with a finite number of parameters. Therefore it appears also as a piece of hyperbolic geometry. Another example is given by disordered systems in which spin effects are taken into account, or systems having more symmetries or quasi symmetries than just the translation one as it is the case for quasicrystals. For this reason let us consider a compact space • on which acts a locally compact group G, In much the same way it is easy to construct a C* algebra for this system. We will avoid the construction (see [M23]) for it is formally the same as before. Let us simply remark that it has an identity operator if and only if G is discrete. There is however one property of G which is essential in what follows : it must. be amenable. Amenability means that in G, there is an increasing sequence (An)n~o of open sets of finite Haar measure (llke hypercubes in R o) the union of which covering the group, the frontier of which has a negligible measure compare to their volume in the limit n->oo In other words surface e f f e c t s are negligible compare to volume effects. More precisely, if x~ 6, & denotes the symmetric difference, and IAI the volume of A with respect with a left Haar measure one demands : (I)
limn_~= IAnA(X.An)IIIAnl = 0
Such a family of open sets of finite volume is called a "Felner sequence" [MI2]. As a matter of fact this condition precisely eliminates the hyperbolic geometry, for it is known that In the Bethe lattice,surface effects are not negligible compare to volume ones ! Given one self adjoint operator on L2(G) affiliated to the C'algebra of (~,G), one defines its integrated density of states (if it exists) as follows : (2)
No~(E) = llmn_~, IAn14 #[elgenvalues of H=IA ~ E } n
Let us mention that the first deMvative dN=(E)/dE of this quantity is accessible to the experiment. This is done either around the Fermi level by measuring the current-voltage characteMsttcs of a diode in which the sample has been put, or by mean of X-ray scattering, which permits to investigate the band structure at smaller energies. For this definition to be meaningful, it is obviously necessary that the
115
restriction of H to bounded regions exists and have a discrete spectrum. Actually if H is affiliated to C"(~,G), this is always true. Indeed if a ~ CN(~,G), and if X is the operator of multiplication by the characteristic function of some bounded open set A on L2(G), then for any co ~ Q, X~(~(a)x is compact. Thus, the restriction of the reso]vent of H on X is compact. Thls ls enough to show that the resolvent of the restriction is also compact. However the restriction of the resolvent ls not the resolvent of the restriction. The main difficulty comes from the definition of the restriction of H to A. If H is a 5chr~dinger operator, this requires to specify the boundary conditions. In general, if G ls not discrete, then H must be unbounded and bounded from below, at least locally, in order that its restriction have only finitely many eigenvalues near -~. On the other hand, the domain of H must at least contain "sufficiently many" functions supported by A : the set D(A) of functions in the domain of H supported by A must be dense in L2(A). Moreover It is also necessary that IinD(A n) be a core for H In order to recover H as the llmlt of the family of its restrictions in the strong resolvent sense. If this is true, since H is bounded from below locally, its restriction on D(A) has self adjoint extensions (the Friedrichs one for instance). A boundary condition is just the assignment of one self adjoint extension. The independence with respect to the boundary conditions is related to the fact that boundary effects are negligible compare to volume effects, namely to the amenability of G. Yet, we do not know what are the necessary and sufficient conditions on H to .insure the existence of the density of states in general. However, there are many concrete cases where this has been proved (see conditions (A) below). When the integrated density of states of H exists, it is a G-invariant Borel function of co. In particular, it is almost surely constant with respect to an ergodlc G-invariant probability on 9. On the other hand, it is obviously increasing with respect to E. It has been proved by F. Delyon & B. $ouillard [36] that for Schr6dinger operators on a lattice ZD, this function is continuous. Their argument is again related to the amenability of ZD, and we suspect that for reasonable hamiitonians, the integrated density of states is indeed continuous with respect to the energy. Let us note however that there are C* algebras having projections the kernel of which being continuous with compact support (see the appendix) and therefore it is possible to exhibit examples of hamiltonian in these algebra, having a discontinuous density of states. There are also models studied by physicists, like the laplacian on a Serpinsky lattice [1,85], which admit a discontinuous density of states. Thus we ask the following questions : Question 1 : What are the necessary and sufficient conditions on a serf adjolnt operator H affl]lated to C"(~,6)to get the convergence of the Integrated denslty of states ?
116
Question 2 : What are the self adjoint operators H affiliated to C*(~,G), having an integrated density of states continuous with respect to the energy, when G is amenable ? Before giving a more precise result, let us introduce another too]. A G-invariant probability measure I~ on Q deflnes a trace on C"(Q,G) in the following way (where "e" is the neutral element in G): (3)
"[tJ (a) = [ dp(co) a(co, e)
a e Cc(flxG)
where Cc(~XG) ls the space of continuous functions with compact support on ~xG. For indeed, It is a linear map, such that ~ ( a * a ) ~ O
and that
•[~ (ab)= ~p (ha). It can be extended to a (weakly dense)ideal J of C"(fl,G), which is often denoted by Li (C*(Q,G),~ v ). By the Birkhoff ergodic theorem, if is G-ergodic, then for any F~lner sequence we get" (4)
~ (a) = lim,_~,,
IAn1-1J^ dx a( x-l.co, e)
for I1 a.e. co
n
Using the covariance condition, this is also equivalent to" (5)
"~ (a) = l i m ~ , .
IAn1-1T r ( x ^
~'~(a))
for !1 a.e. o~
Hence such a trace is nothing but a trace per unit volume, and the limit converges for I1 almost all co. Let us note the following properties of the trace' Proposition 6 : (i) If G is amenable, there is always at least one G-invariant ergodic probability measure 11on ~. (ii) For any G-invariant ergodic probability measure il on £~, and any a ~ Cc(~XG), the limit (5) exists for I~ almost all co in ~, and defines a trace on C"(~,G). (iii) If the support of I1 is dense in fl, this trace is faithful. (iv) If G is discrete, this trace is bounded and normalized, i.e., '~(i ) = 1 if I ls the identity operator. (v) If the dynamical system (~,G) is uniquely ergodic, namely if there is a unique G-invariant ergodic probability measure 1~ on ~, then the limit in (5) is uniform with respect to co. o That (i) ls true comes from a characterization of amenable groups by the existence of invariant means [H12]. Namely, on the C'algebra of continuous bounded functions on G, there is a G-invariant state I"1, if and only if G is amenable. Such a state is called an invariant mean. Therefore, if f is a
117
continuous function on D, the function x e G-> 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) =
a,b,c ~ at
We denote by i~I, T the W'algebra generated by TI(A) i.e.il(at)".Then ~: extends to ~T. Moreover, if h is a self adjoint element of at, then ~(h) has a spectral decomposition and the spectral projections belong to M,~. Then we have : Shubin's formula • Let H be a self adjoint operator on L2(G) affiliated to C*(~,G), and li be a G-invariant ergodic probability measure on ~. We denote by x(H~E) the spectral projection of H on energies smaller than or equal to E. Then, under one of the condition (Ai)
118
below, the density of states exists and for all E, and I~ almost all co one gets" (7) N~(E) = ~p(X( H ~ E )) O
From what we already explained, this formula is intuitively obvious (see eq. 2 & 5). However we need somewhere to control the restriction of H to a finite volume : as we said, if f is a Borel function on the spectrum of H, the restriction of f(H) is not equal to the image through f of the restriction of H This is hopefully true only asymptotically in the infinite volume limit. For this reason there is no general proof of this formula yet. However it is true under one of the conditions below : (AI) G is discrete and H is bounded (see the appendix).
(A2) G = FI° and H is a pseudodifferential operator with quasi periodic coefficients (see Shubin [93], an extension of this work to the case where ~ is any smooth manifold should be easy)) (A3)
G = R D and H is a
SchrSdinger operator (see Bellissard, Lima,
Testard [ lg]) The first case corresponds to the "lattice approximation", while the others correspond to the real continuum case. Treating the general case ls the object of question 1.
119
5)- Cantor snectra.~ao labelling theorem and Ko:groul2 : Using the covariance property of a homogeneous operator, one gets some elementary informations about its spectrum. proposition 7 : Let (£~,IRD,T) (resp. (D,ZD,T)) be a dynamical system, and 11 be !aT-invarlant ergodic probability measure on Q. Let H = {He ;o~g } Fbe a covariant ~-measurable family of self adjoint operators on a Hilbert space L2( I~I)) (resp. 12( Z 0 )). Then : (i) there is a closed subset Y. of I~ such that for ll-almost all co in D, the spectrum of H~ is Y.. (ii) If E is an isolated point of E it is an eigenvalue of infinite multiplicity of He, ~-almost surely. (iii) If co->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< i
{Z÷ Zp ÷ Z7(2) +...+ Zy(n(f)) }n [O,l]
If 0 does not belong to the spectrum of the llnearlsation H(F), the amount of rotation or the map F on the Cantorus M(p) belongs also to this set. <>
134
7)-
The ouantum Hall effect :
The Hall effect was discovered and discussed 1879 by E.H. Hall [49], and has been one of the most striking and useful phenomena in solid state physics for a century, since it was one of the first effect giving informations on the microscopical properties of metals. For instance it gives the sign of the charge of the particles carrying the current. This was the way one discovered that the electric current can be carried either by electrons or by holes depending on the mateMal used. It gives also a method to measure the velocity of the carriers [21] and the charge carriers density. On the other hand, it is nowadays commonly used in several kind of devices in electronic, computer; electronic typewriters, automobile engines, etc. In 1980, a century after the discovery of this effect, a new breakthrough was performed w i t h the discovery [62] of its quantum counterpart which leads to such a precise experiment that the standard for the Ohm, the unit of resistance, is to be defined from it quite soon. Undoubtely, there will be other applications of this effect when the technology w111be controlled in order to produce cheap devices. Let us briefly recall the nature of the classical Hall effect, in order to go step by step toward the understanding of the quantum one. If a thin strip of a metal is submitted to a uniform magnetic field perpendicular to it, one observes a potential difference in the direction perpendicular to the electric current and the magnetic field (see fig. 2). HeasuMng this voltage leads to the value of the Hall resistance given by the ratio of this Hall voltage and the intensity of the current. To compute it, let us assume that the permanent regime is established. Two forces act on the unit volume of charged particle.
:<
L
Fin. 2: The classical Hell effect.. The slmle must be a thin strip of pure metal. The magnetic field B is perpendiculr to the strip. The electric c~-e~-~L j is stationary e~l oriented along the si~ip in the directJen of t ~ longest, side. The mecjnetic forces push the charges as tndic~.ed. In the si.~enry state, they a-este an electric field E perpendicular to the direcUons of t.he magnetic field md the cur'rent. One measures the Hell voltaQe in the E-direcLien.
135
The first one ls the Lorentz force, perpendicular to the magnetic field and to the current density. Its intensity is given by: (I)
F ~ . - jB/c
Here c is the speed of light, j is the current density namely the ratio [/~,6, where ! is the intensity of the current, 6 the thickness and ~, the width of the strip. B denotes the magnitude of the magnetic field in the direction perpendicular to the strip. This force is oriented in the direction opposite to the electric field (see Mg.2). The other force is created by the charges accumulated on the edges of the strip. They create an electric field E and the force is given by : (2)
Fel~. = nqE = nqV H/~
where n ls the density of charge carriers, q the electric charge of them, and V H ls the Hall voltage. If we equal the two forces, we get the Hall resistance and the Hall resistivity : (3)
Hall resistance
RH = VH/I -~B/(n6qc)
Hall resistivity
PH= E/j = B/(nqc)
We see that a high resistance is obtained when the strip is very thin, and the carrier density is low enough. Moreover the sign of the Hall voltage is determined by the sign of the charges carrying the current. They are negative for the gold or the copper, but positive for the iron. This was noted by Hall in 1880. Another consequence is that the speed v of the carriers is obtained from the remark that j = nqv which implies v = cE/B (L. Boltzmann 1880 [2I]). The only characteristic of the sample enteMng in this formula is the charge carrier density. This explain why this effect is so universally used. Let us also remark that in two dimensions, namely in the limit where 6 tends to zero, we do not define the resistivity nor the current density in the same way in order to take into account the divergence when 6->0. 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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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~< ~ <~ ~(xi)>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
<~ ~ (xi) ~ (Yi) >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 ~ << A the cutoff-
dependence of H E~FF(A) is very small. Thus positive coupling ~ A is a perfectly good
effeotive field theory
describing "physics"
for a wide range of energy-scales. From the recursion
(33) we see, that formally the theory for
negative ~ has an unstable direction, and thus is a candidate for continuum theory. Of course the problem then is the stability of the theory. In fact it is very easy to produce a stable cutoff theory by analytic continuation from positive I[13]. In practical terms this is achieved by
taking
(32) with ~ negative and
rotating the fields in complex plane to e i(± Then in the functional integral
~ ) ~ E C-.
(in a finite box V)
_H A D ~ e v
(34)
169
both the 44 part and the kinetic part have positive real parts: Re 144 = ll~41 cos e (35) Re(V~) 2 = IVy] 2 sin ~. (35) allow us to carry out the RG analysis of this theory as in the positive coupling case° The upshot is, that one may prove the existence of the continuum limit of the Euclidean Green functions of the negative coupling theory. These functions are non-trivial, they have the formal renormalized perturbation asymptotic
series as an
expansion and satisfying the axions ~mospt reflection
positivity.
In fact we expect a violation of this property.
the two different theories with C = C ± in (34) correspond different analytic continuations l-plane.
The difference
Namely,
to two
in the upper and lower complex
of these theories is the imaginary part
of the Green functions and is given formally by an instanton contribution.
Thus heuristically
we expect the Green functions to
have a nonperturbative
0 (e-c/gR)
reflection positivity.
One might still ask, whether this theory
imaginary part violating
at least formally would represent any Minkowski
theory with a
decaying vacuum° A standard instanton computation surprising answer to this question.
Namely,
part of the ground state energy density,
gives a
formally the imaginary
i.e. the decay rate per
unit volume of the false vacuum is given by -i m(j ~ ~ lim | ~
-4 p
-SCp)/I(O) e
(36)
A-I where p is the scale of the scale of the instanton, dimensional
factor for energy density,
p
-4
the
S(p) the action for
170
instantons of scale p and l(p) the running coupling at length scale p. The numbers are
S(p) = 16~ 2
Thus
,
i l(p) -I ~ IR
(36) behaves as A as A +~.
3 log po 16 2
(37)
The lifetime of the false vacuum
tends to zero as the cutoff is removed, and this unstability UV effect, contrast,
even though the theory is asymptotically in QCD m = 0 and
(36) diverges
free!
is a
(For a
in the IR) o
REFERENCES
[ i]
Jo Glimm, A. Jaffe, Quantum Physics,
[ 2]
M. Aizenman,
Phys. Rev. Lett. 47 (198!)
[ 3]
J. Fr0hlich,
Nuclo Physo B200, 281
[ 4]
J. Feldman,
J. Magnen, V. Rivasseu,
Lett. 54,
1479
Springer
(1981)
(1982) R. Seneor, Phys. Rev.
(1985)
[ 5]
G. 't Hooft,
Commun. Math. Phys. 86, 449
[ 6]
V. Rivasseu,
Preprint,
[ 7]
Go Felder, Preprint ETH-Z0rich
[ 8]
K. Wilson, Rev. Mod. Phys. 55, 583
[ 9]
K. Gawedzki,
[10]
D. Gross, A. Neveu, Phys. Rev. DI0, 3235
[11]
K. Gawedzki,
(1982)
Ecole Polytechnique
(1984)
(1985) (1983)
A. Kupiainen, Phys. Rev. Lett. 54, 2191
A. Kupiainen,
(1985)
(1974)
Helsinki Preprint
(1985) and
Physo Rev. Lett. 55, 363 (1985) [12]
K. Gawedzki,
A. Kupiainen,
Commun. Math° Physics I00 (1985)
[13]
K. Gawedzki,
A. Kupiainen,
Nuclear Physics B257, 474
[14]
K. Gawedzki,
A. Kupiainen,
Harvard Preprint
appear in Proceedings
(1985)
(1985), to
of Les Houches Ecole de l'~t4 1984.
RENORMALIZATION GROUP METHODS IN RIGOROUS QUANTUM FIELD THEORY
K. Gaw@dzki CNRS, IHES 91440 Bures-sur-Yvette France
One of the fundamental problems of Quantum Field Theory (QFT) is the existence of non-trivial models of local quantum fields. For many years the opinion dominated that only perturbatively renormalizable (or super-renormalizable) models have a chance to exist. However, in the course of time, especially with the development of the Renormalization Group (RG) ideas, the evidence began to accumulate that p erturbatiyerenorma!izability
.is neither a sufficient no r a
necessary condition for the existence of a quantum field. The key to this realization lies in K.G. Wilson's description [I] of (euclidean) sequence of effective statistical-mechanical Hamiltonians to the momentum scales
corresponding
H(~)
to the unit cut-off regime. The Hamiltonians
are related by the RG transformations
(I)
for
H(~)
< p . For easy comparison of the theory on various scales,
it is convenient to rescale each H(~)
QFT through the
H(~)
RL
> H(~IL)
L > 1 , consisting (on the level of the Gibbs states associated to the
Hamiltonians) of integrating out the degrees of freedom with momenta between ~/L
and
~
(and of rescaling the result to the unit cut-off regime). Wilson's
description allows to translate the properties of the field theory into the properties of the flow of the semi-group
RL
in a space of the unit cut-off
Hamiltonians. For example, the (non-perturbative) renormalizability of a family H(g i)
of interactions described by local (cut-off) Hamiltonians becomes equivalent
to the existence of functions
(2)
bare, ren A~ gi ~g ,n)
such that
Heff(gren,~) = lim H(gbare(gren,A)) A_>~ 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)
forallz}.
It is easy to see that when a commuting pair is associated as above with a circle mapping, the rotation number for the commuting pair defined in this way agrees with the usual rotation number of the circle mapping. It is also easy to extend many of the elementary properties of the ordinary rotation number to this more general situation. Thus, for
182
exaanple, a commuting pair (~, q) has a rationa~ rotation number p/q if and only if ~P o I/¢ has a fixed point. To define a renormalization operator, we work, as in the first section, in the space ~4 of normalized critical commuting pairs with rotation number strictly between zero and one. The operator will be defined for all pairs ~ in this space for which the rotation number is rot the reciprocal of an integer. The construction will be guided by the objective of making a "canonical" (i.e., coordinate independent) operator T with the property that p(Tf) = f r a c ( A ). This property will imply immediately that the action of T on commuting pairs induces an action on rotation numbers which, in the continued fraction representation, is simply the left shift. More explicitly, a ~ in 3i is in the domain ot T ff and only the continued fraction representation of p(f) has length at least two, and, if we write p(f) as [rl, r~,...], then p(T f) = [r~,...]. The idea of the construction is as follows: Let (~, ~/) be a normalized commuting pair with rotation number p > 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
, ~=
~<
1,2,.
~
(
satisfy the ( ~ , ~ ) _h
% = (~ ~ ) ~ e b y e
sum ru-
length;
In the Sine-Gordon formalism used here, the sum rules follow from the fact that the Gaussian measure with covariance given by the inverse of the Coulomb potential
(the L a p l a c i a n - ~
translations by harmonic functions
) is formally invariant under
q~ (X)
nically, one has first to approximate
~
(~}
with ~ ) =
O
. Tech-
by a smooth function
with compact support, the resulting boundary terms do not contribute as a consequence
of exponential clustering which is known to hold in the
Debye regime [12]. Alternatively, brium equations
the ( 4, ~ ) sum rules can be deduced from the BGY equili(assuming now that the state exists in the thermodynamic
limit and verifies the BGY hierarchy) whenever the rate of clustering is sufficiently fast r13, 8, 14~. For an homogeneous neutral state, the BGY hierarchy has the form
192
~" ~ 9(9,q,]= f ~1~£j)p(9~'"9~) + I ~ F'(~9)(?(q'"'q~q ) - ~(q){ ~''''q~)) (3) F(9~q~]= _e~,e~=?~,(~-xjis the force, and ¢ (X) is the Coulomb potential regularized at the origin. Introducing the fully truncated correlations ~ (9~ ''" ~ ) defined in the usual way, we have : Proposition 2
[ 8 ,14]
If the correlations satrisfy Eq.
for some
~)O ,
(3) and
m=
sum rules hold for
s.~.PlXi'xjl # Zj--4,'.'ta;f[=Z...~+Zthen the ( £ ~ ) % ~ _~ ~o ~ h -~ ~ o "
The main argument is simple : written in terms of truncated functions
(3)
is equivalent to rl.
j=z + [~9 F(q,q) ~ ( q , q @) (m-..q,),
with the natation
;
(a) z.
,
and
pcq, Q} - p ~q,) p (9) p(q, qQ) - p(q) pcq,~) - p(q,} (p(q Q)- p(~)p(Q)) (5) In Eq, (4)
E(X,)
t r i c field at Xt We integrate Eq. let ~Xl]---~ eo
~X
r(~,-x) --2--e~ ~(~Xl ~ )
(4) on a ball of fixed radius centered around
so decays faster than sion, we conclude that d(X]
X~
and
• Then, the cluster assumption implies that all terms
except the first one in the r.h.s, are
Let
is precisely the elec-
generated by the excess charge density ~ e ~ ( ~ x l
Introducing its multipolar expan-
IXII"~).I ÷ {o ~e,~(~(I~
---- Z3 ~ J ' ~ ( ~ - ~ j )
~
P~
O- (IXII'V*I+~°). Hence, E(W~) al-
)
has no multipoles up to order ~o. be the charge density and
(6) be truncated charge charge correlation function . In an homogeneous neutral state ~ X ) >
= ~-@~m~
+~
= o
193
S(X~,X%) =
and
Then, with
~(~-x~,o)
~(Xl-X~)
~(x)
~(~)
vanishes
is spherically symmetric,
der sum rule is the dipole sum rule: n :
consequence that
0
S&)
(~:l,
.
(6), the (0,1)-sum rule has the important
the total integral of
Since
~
the first non trivial higher or-
when two particles are fixed
2)
foE9 e ~ x fCql q.q~) = o
(8)
III. COMPLETE SHIELDING The
(~=0,n)
charge sum rule, which holds under the condition
grable clustering,
of inte-
can be interpreted as the shielding of test particles
of the same species as those which constitute the system itself. The situation is however which are different
not the same
when we introduce test charges
from the system's charges
: they may be shielded or
not according to the plasma or dielectric nature of the phase E15,4]. We say that-the system has the complete shielding property if any external distribution is screened by the system's charges. In this case, the charge-charge
correlation has to satisfy the additional
constraint
l×J
=
(9)
which can easily be derived by expressing the shielding of an infinitesimal charge in the linear response theory. An alternative
form of (9) is the second moment Stillinger~ovett
condi-
tion [16] li,~
£"(~)
= I< --> o
"1 + where ~ Ok) and
gC~)
Ikl ~
~v~ 2V
('~cx I×l ~',g(~) l
is the Fourier transform of is the dielectric
function.
=
o
(i0)
~(×) The property
h~
~'~(k)= o
cha-
194 racterizes
a perfect
conductor.
The equivalence
of an integration by parts and of the Poisson
of (9) and (i0} result equation
Zk~
=-~v~C×)
writing I
We show i n t h e n e x t p r o p o s i t i o n occurs under slightly
stronger
that
clustering
for the simple electroneutrality the dipole sum rule Proposition
the complete
(@=0,n)
shielding
conditions sum rule
relation
(9)
than those needed : one needs also that
(8) is satisfied.
3 [17]
Assume that
If the (0,i) charge and (i~2) dipole sum rules are verified, complete shielding relation
(9) holds.
In the case n:2 (
), Eq.
~ ~ ~a
and using translation
invariance,
(4) can be written,
lar product
= o
and the antisymmetry
of (ii) with
e~
integration by part the l.h.s,
setting
the
~I=(~I,O)
as
(Notice that under rotations
then
by the i n v a r i a n c e o f the s t a t e
of the force). We take now the sca-
X~
and sum over ~
. After an
of (ii) equals
where the charge sum rule has been used. It is easy to check that the dipole sum rules
(8)implies
also for the truncated
functions
(13) Hence, exchanging the q and q~ integrals the contribution of th~ last term of (ll) vanishes. Finally, introducing the definition (6) in the first term of the r.b.s,
of (ll) gives with
(12),
(13)
195
After an integration by part, this gives (9). This result implies that in a phase where arbitrary test charges are not screened and (9) does not hold (as in the Kosterlitz-Thouless phase EIS, 4]), the dipole sum rule (8) cannot be true. It then follows from Prop.2 that the clustering cannot be faster than
~|-(~I)
in a dielectric
phase. IV.
CHARGE FLUCTUATIONS
A direct consequence of screening is the reduced growth of the charge fluctuations
~18]. Heuristically, a free charge carries always its neut-
tralizing cloud (of radius of the order ~
), constituting with it a
neutral entity. In a macroscopic region
A
(of volume | A [ ~
those entities cutting the boundary ~ A
of A
net charge
CA
in A
only
at random contribute to the
. The f l u c t u a t i o n s ~
pected to be proportional to the surface
~)
--~)~ I~AI
of ~
may then be ex, and not to its
volume as are the usual fluctuations of extensive quantities. The correlations between two subdomains functions
where
~AaCx)
~^
~^.a~) = __~ ~ ^ C ~
tion of
~
Cx)
~
=
~A~CX+~)
A~
=
and
A~
with characteristic
are
is the volume of the intersec-
with the x-translate of
Choosing first
~^^ (~
and
~I
~
~
~ cube of side L, we have
v
=
L~ -
:;'- ax'J L ~''
+
CCL ~'~)
With (7), the volumic contribution vanishes, and we find that of the order of the surface
L + ' ~ ~ - - " L" ~ '
=
-
~
r.,
~ >
is
196 We consider now the charge fluctuations of volume L ~
~^,^~¢~
=
in two adjacents
cubes
centered at (0,0...0) and (L, 0...0). In this case
ix~iL~"
~ O ( L ~'~)
-L
,
~
w
~
o
,
and thus
I;,,,
g~^, C^~)
The negative
,
Ida, ix"l E(~)
correlations
:
- '--
of adjacent regions
M
reflects
(16)
again the fact
that the system behaves as built of neutral entities. More generally,
let ~ 9 ~
me L ~
on a simple cubic lattice,
centered
~
be divided
the appropriately
normalized
charges
in
next proposition,
these random variables
riance given by the finite difference Proposition
4
into disjoint
cubes of~volube
and let C A ; ~ % v l v " ] "~ ~ ~
. Then, as stated in the
are jointly Gaussian with cova-
Laplacien
-- ~
on
Z v.
[19]
Assume that
(i)
I~x~...~x~.
I p.r(~o.q~...q,ol
4
,: -
( £ "clustering)
for n = 2,3
(iii)
the
(~=0,n)
sum rules hold for n = i, 2, 3.
Then the joint distribution covariance
~
C~g~ - ~
of the
C^~
C_~jg.~bI ) =
To obtain the covariance
~]
-~
~V
sults from the fact that, with
(i),
L -~
with
~.
(in addition to (15) and ~A,^ ~x)=
~(~v-~);
The Gaussian distribution
(ii) and the sum rules
(iii) all n
rethe
except when n = 2.
POTENTIAL AND FIELD FLUCTUATIONS
For particles with short range interaction ge ~
E
one has always
cubes are not correlated.
order cumulantsvanish V.
~- ~]g~
, one notes that
(16)) when the cubes are not adjacents, thus non adjacent
is Gaussian as
), the potential
at X
the part of the particle
is a local quantity,
configuration
around X • Then the potential
: here
i.e. it depends
V(x)
m
only on
which is in a sphere of radius
and force fluctuations
soon as the state has a thermodynamic Coulomb potential
(for instance with finite ran-
are well defined as
limit. This is not the case for the [~
~-~i~-"| ~ ( ~
is genuinely
non
197
local and particles
far away will contribute to the fluctuations
Our main point is that whenever the potential fluctuations thermodynamic
at X •
exist in the
limit, they can as well be computed as limit of averages
of strictly local functions in the infinitely extended state der here the three dimensional situation We first split the potential at X
V=
3).
into a local
part
V~(~) due to the
particles inside a sphere
~-R
bal outside part
due to the particles in the exterior of this
V~(X)
of radius ~
(we consi-
centered at X
, and a glo-
sphere
VCx]= with
V~ C~) +
V~r~Q(].
•
I~
-R
JX-~|
.D
I
We introduce moreover the spatial average of the total potential on this sphere
%.c:,o = r~ I :~~ vc,~ ..
vx~.J = ~.""~:'..,
Then the formula
IdN I~IS
ix-~i =
-~i~, ~ + z ~
R
~
J~I
-
yields the identity
with
Proposition 5 Let
[20]
~/A (M,~) =
X,
~I
be the potential fluctuations thermodynamic
Ix,-xI
J~,-~J
~ A (XI'~')
in the finite volume system.
If (i) the
limit ~V(~-~} = A / ~ %VA(2,~ ) exists and ( i i ) i ~ l ~ W(',) = O ,
198 then the potential
fluctuations
can be computed
locally in the infinite
state by
(2o) 9RC,~I =
with
The p r o p o s i t i o n
V~a('~) results
9l~(~)-
from the fact that the spatially
tial
V~cx)
does not contribute
to the fluctuations.
ption
(i) implies that for each fixed I~
i~--~.i~
^-~
averaged poten-
Indeed,
the assum-
[c1~4 ~¢x,-~,) I~,-xl ~ ~ I ~ , - ~ 1 . ~
l~al.~l -I~,1_< I and by
(ii) this expression tends to zero as
We therefore infinity -- V~(x)
conclude
to the fluctuations
expresses
correlation
Y~(~)
can be calculated
, and the d e c o m p o s i t i o n
ween the contribution position
that the contribution
~--) of the particles
from the local function
(17) provides
a natural distinction
of the nearby and far away particles.
these fluctuations
at
bet-
The next pro-
with the help of the charge-charge
and shows that the potential
from distant
regions has a Gaus-
sian distribution. Proposition
6
[20] exists
holds,
and the
(0, 0 )-charge
sum rule
one has
and as
Moreover, i f the state is ~ c l u s t e r i n g , ~-@
q~,
with covariance
IXI -9 ~
~C~)
•
are jointly
(21)
Gaussian as
199 It is interesting
to note that particles
for half of the total potential To obtain the asymptotic the spherical
symmetry
fd~
fluctuations.
behaviour
XrXs
clustering
~'~" ~ ( ~ ) =
~gfs
is exponentially
(i0) holds,
universal
fd~|~l ~ ~ C ~ )
everywhere
fluctuations
fast. Moreover,
the asymptotic
and independent
The electric
by
In two dimensions, nite; however,
is always
(21). We see
slow, even if the
when the complete
behaviour of
- ~TVC~}
~v¢~)
shielding
is ~ " T~, ! 1.e. "
can be treated in the same way, replacing ,
the fluctuations
the potential
[20~.
see
of the potential
differences
at a point are infi-
have finite fluctuations
to the ones which occur in three dimensions
VI.
, we obtain
of the short range part of the interaction.
field fluctuations V(x)
(21) we use the charge sum rule and
f~
that the decay of the potential
relation
in ~ ' ~ x )
to write
3 Since
at infinity contribute
similar
[21].
DECAY OF CORRELATIONS
Not much is known on the asymptotic systems,
form of the correlations
except in the Debye regime where it is bounded by an exponential
[2}. However,
because of the local relation between the charge density
and the field generated by it (due to the harmonicity tential),
all types of asymptotic
relations
are integrable
viour of a neutral ~{~) =
cay monotonously, We formulate mogeneous
behaviour
and monotonous
faster tha~ any inverse power.
tential
in charged
$~
exactly as
are not allowed
at infinity,
: if the cor-
they have to decay
This is in sharp contrast with the beha-
fluid of particles
~|xI'S~
of the Coulomb po-
interacting with an integrable
: here, at low activity,
po-
the correlation
|~1-3 [22].
our result for the case of the two point function of the ho-
OCP in three dimensions.
the spherically
symmetric
Denoting
p~(r}=
truncated two point function,
pc~-p~j (~j=r)j Eq.
(4) reduces
in this case to
~ p ~ ( ~ ) = e E ( ~ ) + 7~ p~(~j + ~ ix-~ where ~(P}
de-
is the radial component
F(x-~) p~C~,~,o)
of the electric
(22)
field determined by
200 the Poisson equation
• _ I,'~ r~ d,"
l
,- c£(x) )
p
(23)
(In the OCP a short range regularization of the Coulomb potential is not needed). Proposition 7 Assume that ro as t = li~
[2~
~(~,o]
r-~
is integrable and p ~ c r }
. Moreover, for
l~i4(JX-~!~ I~ I) r P
~m(?)
:
IXl
~Ji~,~.~-
and
tends monotonously to ze-
large enough,J~(x,~,o]l ~ ( ~ ] J ~ ( ~ J ,
~ (~) = 0
0
for all
then
p)O
The assumed bound on the three point truncated function is slightly stronger than in Prop. 2 in the sense that some joint decay is required as a second particle is sent to infinity. This bound is compatible with known exact results and bounds indicated by perturbative expansions ~ 3 ~ . Let us suppose first that
~,'~
r P p~(r) =
A
Integrating the Poisson equation (23) gives for
The dominant contribution to ECo)
~3
for some
(24)
r ~o
, which would be ~ ; ~ (
~
*('1
~('1),
vanishes because of the charge sum rule. Moreover, the condition on the three point function implies that (lemma I of [23])
(26) Hence, inserting (25) and (26) in (22) gives
( l'~ ~r l°,r cr~ :
a'rreZ
A
ffp'~) rP"
÷
G
l ('~'~r-,)
(27)
and therefore, by integratio n
Thus, we conclude from (24) that The a priori assumption
~=0.
(24) can be removed to exclude all kinds of mono-
201 tonous non algebraic decays which are not faster than inverse powers g23]. However,
the monotonicity hypothesis
plays
and we cannot exclude an oscillatory behaviour as
an essential role, ¢o~XP
because
~P
then the field and the charge density are of the same order at infinity. These arguments ticomponent
can be extended to higher order correlations and to mul-
systems.
VII. TIME DISPLACED CORRELATIONS It is of interest to investigate to what extent the screening sum rules remain
valid in a dynamical situation.
correlations particles
of the homogeneous at time ~ i o
• L=C~p) coordinate
k
j#...~j.
denotes the position
and momentum o f a p a r t i c l e ,
are sets of coordinates;
andVz(~j...t~)
(X~(~)~pl~}}
~(6~,
are the
of the i th particle at time ~ under the time evolution a n d ~ . . ~
is the thermal average on the initial conditions set~
particles at time t and
, formally defined by
~,~...,iM "V'. ( ~ . . . ~ )
We consider the time dependent
0CP involving
is empty,
~(~)=
~g
. When the
the correlations reduce to their equilibrium value k
The existence of the dynamics in the infinite system has so far not been proven
: we assume here that the correlations
namic limit and obey the BBGKY equation
(29) exist in the thermody-
(for ~ | )
The time dependent generalization of the static excess particle density (i) is *)
*)In the sequel, we simply suppress momentum arguments functions when they have been integrated out.
in the correlation
202
and reduces to (i) at time ~ = cluded in the definition
O
(notice that self correlations are in-
(29), but not in the definition of the static
correlations). We introduce the truncated correlations ~(~,~%t~,~]~
)
p~[~l,~/')
and
defined as in (5). Then, under some reasonable clus-
tering assumptions, the charge and dipole sum rule still hold for all times in the homogeneous OCP. Proposition 8
[I~
Assume that (i)
I P(X,P ,61v) I ~
p$~
, Ipl--~ ~ , P
tence of the kinetic energy density) and (iii)
~X,~X,
(ii)
(insuring the exis•
--
I~I~
p~×,p,61v~
IXIIIp,(X.,~,,~I%F) I ~ -
.
If the static
(O,~)
and ( I , ~ ) sum rules hold~ then
zI~
~C×,67%r) - o
(32)
(33)
for all
X,... X,, and "V'=
...
x,,,p,).
Using the definition of the truncated functions as well as (3) to cancel the contribution of time independent terms, we find the equivalent of (30) written for the Zruncated functions
:
The evolution of a general momentum independent function by integrating out the momentum in (34), the
%-
~Cx] is obtained
terms giving no contri-
bution
Setting now
t(x)=1
,
the r.h.s, of (55) vanishes in view of (ii). Taking
into account the initial condition (the ( o . ~ )
--.-[~P~(~J%f)lb.~ o.P~)[~x~(xJ61~r)It-- • =_ 0
static sum rule) leads to (32).
To derive the dipole sum rule, it is useful to consider the second time derivative obtained from (34), (35) and partial integrations on momentum
203
- -J Ia~, Iap, cp,.v~,~r~,)l(p,.v., p. rx,,r,,~lv~)
(37
+ ~ Ie.,Io., cv~.¢c~,.~l, rc.,..,j m(~,,~.,olv)
(38~
The first term of the r.h.s, and ~ p
=
the terms
( ~ I TI~/P~)- -
results
(37) and (38) vanish,
and ( 3 8 ) b e c a u s e
the initial
equation.
conditions
_~ Ie~ × p,,,(x,el v )
ae
¢:o
Choosing now
(37) being the integral
of the antisymmetry
dinary differential
of Poisson equation~Tx.F(x)=~ITd(a)
is the plasmon frequency.
of the force.
of a gradient
So (36) becomes
From the (1,~)-static
are found to be (with ~=(~(,,p~ ...X~,p~)
-
--' ~
an or-
sum rules and (35) )
ei ) f ~v~
-:
and the solution is thus
~(,J"---~ = PS p(I#').~;~(-~p'~ Averaging
(39) on initial momenta
There is also a time dependent relation
(9)
(39)
gives the dipole sum rule
generaliTation
of the complete
(33). shielding
:
where ~rX,*] ; < C O X , e) ¢ ( o , o } > =
ced charge-charge c o r r e l a t i o n .
e~(p(~,6]o)p~) is the time d i s p l a The r e l a t i o n (40) is w e l l known in the
frame work of linear response,
our point being here to derive it from
the microscopic ~=
(o,p)
equations.
in (36) and integrate
sis as in Prop. of the sum rules quat ion
dynamical
8, the terms (32) and
For this, we set
~Ex)=
l~(l~
,
over p . Then, under the same hypothe-
(37) and (38) vanish again as a consequence
(3~).
Thus
(36) reduces
to the differential
e-
204 The
|
initial conditions-
from (35))gives
(9) and __~ ~'XlXl ~ ~(X,e)[6:, = O (which follows
(40).
After the change of variables invariance
the term (38) becomes
We have also used
"ptv1elV)= --""
narity of the equilibrium imply the same relations thus
~
(42) vanishes.
, ~m
-X
(exchanging
pLv~-el?J)~" ' "
state. for
~-~
the ~
which follows
for them. The validity
instead of
systems
(32) and (33)
~,~1~0)
,
(32) remains true
of (33) and (40) for the OCP is due to the special
feature here that the electric
VIII.
from the station-
show a much more com-
: only the charge sum rule
current
is proportional
mentum, which is not affected by interparticle
linked
integrals)
It is easily checked that
p~(~l~)
It should be stressed that multicomponent
to the total mo-
collisions.
Even in the OCP
@ - s u m rules are not expected to hold, a fact which is
to the existence
correlations
and ~
The term (37) is treated in the same way.
plex dynamical behaviour
the higher order
and using translation
of long range time dependent momentum-momentum
of the type [ i ~
QUANTUM SYSTEMS
Some properties tum mechanical
of the classical situation.
correlations
Let ~ I ~
be the reduced density matrices
can be extended to the quan-
, ~=~q4...q~)
(RDM) formally
, ~I=~...~i~
defined by
('cl.,).'.C~ ('qe) ~('q~)... O..(q,) ~'
:
'
'
/
where
~ql),
~(~)
ing the canonical
are the creation and annihilation commutation
riance Of the state if
e~
~
the excess particle
(charge
~o¢~-
or anticommutation conservation)
operators
relations.
implies
Gauge inva-
~C~]~]]= ]
. We define the quantum mechanical
density by
satisfyo
analog
of
205 When
~
(~s (diagonal
~[@IQ)
RDM),
(44) is identical
and has the same interpretation.
to (i) up to the factor
Then the
~-sum
(45) can be derived from the quantum BBGKY hierarchy cluster assumptions, cluster properties
rules are
under appropriate
following the same method as in Prop.
2 524]. The
of the RDM needed to carry out the proof have so far
not been rigorously
established,
even in the Debye-H[ckel
regime.
In fact,
different behaviours will occur depending on the type of observables which are considered. In the plasma phase, one may expect that the correlation of the charge with any other local observable has a fast (at least integrable decay). However, up to second order in Plank's constant, the current-current correlations ~ ~ ~ integrable screening
decay,
Q~
important
QJ
to distinguish
, and the off-diagonal
gonal sum rules will hold for all correlation because
classical
have a non
state shows Deb~e
e.g. for the OCP ~25]
It is therefore rules
even when the corresponding
) ~ s (o) ~
sum rules
~ ~ ~
, as in the classical
(46), the off-diagonal
with the method of [24]
sumably not true when A generalization tum mechanical
sum
. The diacase, if the
of the charge decays faster than any inverse power.
of the long range
established
~
in (45) between the diagonal
However,
sum rules can only be
for the case ~ o
, and they are pre-
~ ~ I
of the complete
shielding relation
(9)-(10)
to the quan-
OCP is
= -
~('X).=~C¢X) C(o)>
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>
~ <'A*Ab,
A being a local observable.
~,"~/'"r-~'>"
balance inequalities
~26~
(48)
The proof is based on the remark that in the
206
OCP, the local dipole ~ region A Setting pairs
and current
~
(a%tached to a cylindrical
) behave as a quantum oscillator of frequency ~ p A~
=
I~ ~ ~ ~
A~
( ~
)
~
and
as
~--~ ~ ,
, we write the inequality (48) for the
(AA~ A;)
(49)
Under the assumption that the diagonal sum rules (45)(resp. off-diagonal) ~ =
holds for
~
~
, (resp. for
~m 0
) one finds removing the cut-
off that
(5o)
[AI~
[A|
IA I-->,~
[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
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