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Uniqueness of Gibbs states of a quantum system on graphs
Vol. 59 (2007)
REPORTS ON MATHEMATICAL PHYSICS
No. 3
UNIQUENESS OF GIBBS STATES OF A QUANTUM SYSTEM ON GRAPHS DOROTA Kt~PA and YURI KOZITSKY* Instytut Matematyki, UniwersytetMarii Curie-Sldodowskiej,20-031 Lublin, Poland (e-mails: [email protected],[email protected]) (Received September 13, 2006) Gibbs states of an infinite system of interacting quantum particles are considered. Each particle moves on a compact Riemannian manifold and is attached to a vertex of a graph (one particle per vertex). Two kinds of graphs are studied: (a) a general graph with locally finite degree, (b) a graph with globally bounded degree. In case (a), the uniqueness of Gibbs states has been shown under the condition that the interaction potentials are uniformly bounded by a sufficiently small constant. In case (b), the interaction potentials are random. Here, under a certain condition imposed on the probability distribution of these potentials, the almost sure uniqueness of Gibbs states has been shown. Keywords: interacting quantum particles, random potentials, quantum Gibbs state, almost sure uniqueness.
1.
Introduction
Let (M, o-) be a compact Riemannian manifold with the corresponding normalized Riemannian volume measure ~r. Let also G = (L, E) be a (countable) graph with no loops, isolated vertices, and multiple edges. The model we consider in this article describes a system of interacting quantum particles, each of which is attached to a vertex ~ e k (one particle per vertex), its position is described by qe 6 M. The formal Hamiltonian of the model is h2 H --
'It"-'%
~m ~-~Ae -
TgC~L
~ vee,(qe, qe,), (e,e')~E
(1)
where m is the particle mass, Ae is the Laplace-Beltrami operator with respect to qe, and Vee, is a symmetric continuous function on M x M. By (e, e') we denote the edge of the graph which connects the named vertices. The Hamiltonian of the free particle h2 He Ae (2) 2m *Supported by the DFG under the Project 436 POL 113/115/0-1.
[281]
282
D. K/~PA and Y. KOZITSKY
is a self-adjoint operator in the physical Hilbert space 7-/e = L2(M, or). It is such that traceexp(-rHe) < oo for all r > 0. If one sets the periodic conditions at the endpoints of the interval [0,/3], /6 > 0 being the inverse temperature, then the semi-group e x p ( - r H e ) , r e [0,/3], defines a/%periodic Markov process, see [6], called a Brownian bridge, described by the Wiener bridge measure. More on the properties of such operators and measures can be found e.g. in [4]. A complete description of the equilibrium thermodynamic properties of an infinite particle system may be made by constructing its Gibbs states. In the quantum case, such states are defined as positive normal functionals on the algebras of observables satisfying the Kubo-Martin-Schwinger (KMS) conditions, which reflect the consistency between the dynamic and thermodynamic properties of the system proper to the equilibrium. Often, also in the case considered here, the KMS conditions cannot be formulated explicitly, which makes the problem of constructing the Gibbs states as KMS states unsolvable. Since 1975, an approach based on the properties of the semigroups generated by the local Hamiltonians has been developed. In this approach, in [1, 2] the Gibbs states of the model (1), with G being a simple cubic lattice Z d with the edges connecting nearest neighbours, were constructed in terms of probability measures on path spaces--the so-called Euclidean Gibbs measures. So far, this is the only possible way to define Gibbs states of such models and hence to give a mathematical description of their thermodynamic properties. The existence of Euclidean Gibbs measures by simple arguments follows from the compactness of the manifold M. In the small-mass limit, which can also be treated as the strong quantum limit, the uniqueness of Euclidean Gibbs measures at all 1~ was proven in [2]. The existence and uniqueness of the ground state, i.e. of the Euclidean Gibbs measure at /3 = +cx~, was proven in [1], also for small m. In both cases, /~ < + ~ and /3 = + ~ , the results were obtained by means of cluster expansions. The same method yields the proof of the uniqueness for small /3 (high temperature uniqueness), which is equivalent to the case of small sup sup [Ve,e,(qe, qe')l. £,e I q,~,qet
In the cluster expansion method, the properties of the lattice Z d are crucial. Thus, it would be of interest to extend the latter result to the case of more general graphs, especially those of globally unbounded degree. Another possible extension would be considering random potentials. In this article, we present two statements establishing the uniqueness of Gibbs states of the model (1), which occurs due to weak interaction. In the first one (Theorem 1), the graph G obeys a condition, by which vertices of large degree should be at large distance of each other. This is the only condition imposed on the graph--we do not suppose that it has globally bounded degree. In the second statement--Theorem 2--we suppose such a boundedness, however the functions Vee,, (£, g') e E, now are random. We claim that there exists a family of the probability
UNIQUENESS OF GIBBS STATES OF A QUANTUM SYSTEM ON GRAPHS
283
distributions on the space of these functions such that, for every member of this family, the Gibbs state of the model is almost surely unique. The proof of these statements is based on an extension of the method of [3], where the single-spin spaces were finite. Here we give its brief sketch--the only thing one can do in the frames of this article. A detailed presentation of this proof, as well as a more detailed explanation of the connection between the Gibbs states and Euclidean Gibbs measures of the model (1), will be given in a separate publication.
2.
Setup and results
Since in our study the role of the particle mass and the temperature will be trivial, for notational convenience we set m = 13 = h = 1. Throughout the paper, for a topological space Y, by /3(Y) we denote the corresponding Borel ~-field. By saying that /z is a measure on Y, we mean that /z is a measure on the measurable space (Y,/3(Y)). For a measurable function f • Y ~ ]K, which is #-integrable, we write P
#(f) = ] fd~. t /
Y
Given e ~ L, we denote Xe={xeEC([O,
1]~M)
I xe(0)=xe(l)},
(3)
which is a metric space with the metric O(x, y) =
sup d ( x ( r ) , y ( r ) ) , rE[0,1]
d being the Riemannian metric of M. This is our single-spin space. The Wiener bridge measure Xe is defined on Xe by its values on the cylinder sets Brl .....~n = {Xe E Xe I xe(z'l) E B1 . . . . , xe(rn) C Bn},
where IBi E ]3(M), i = 1 . . . . . n, and ri C [0, 1] are such that rl < r 2 - " < "On. For such a set,
x£(Brl ..... "On) =
f
P~2--~l(~1, ~2) X "'"
(4)
B 1x...×Bn
X prn_rn_l (~n-1, ~n)Pl+~l-r n (~n, ~ l ) ° ' ( d ~ l ) " ' " o ' ( d ~ n ) , where Pr(~, r/), r > 0 is the integral kernel of the operator e x p ( - r H e ) . Let ,~fin be the family of all finite nonempty subsets of l . As usual, for A C L, we use the notation A c = L \ A. For A c L, the Cartesian product X ~ = 1-I Xe,
(5)
is equipped with the product topology. Its elements are denoted by xA; we write X = XL and x = XL. Given (e,e') ~ E, let ~ee, be a copy of the Banach space
284
D. Kt~PA and Y. KOZITSKY
C(M x M ~ R) of continuous symmetric functions equipped with the supremum norm. Then we set f2 = 1-I ~ee,, (6) (e,e,)eE and equip this space with the product topology. Elements of f2 are denoted by v. For Vee, • f2ee, and given xe, Xe,, we set 1
Vee,(Xe, xe,) = [ Vee,(Xe (r), Xe, (r))dr.
(7)
0
Then Vee, is a bounded continuous symmetric function on Xe x Xe,. Obviously,
IIVee,II <_ IJvee,ll,
(8)
where I1" II stands for the supremum norm in both cases. For A C L, we set
X^ = @ Xe,
(9)
which is a probability measure on Xa. For such a A, we also set
E^ ={(£,£') • E I £,g' • A}, oqEA : {(6, 6') • E I 6 • A, 0LA = {6 t • A c I 36 • A :
(10)
g' • AC}, (6, 6') • E}.
The latter sets are called the edge and the vertex boundary of A, respectively. For A C A t, each element of Xa, can be decomposed xa, = xa x xa,\a. In particular, one has x = xa x xac. Given A • £fin, Y • X, and B • 13(X), we set 1
:rrA (BIy) -- Z^ (y~) f ]IB(XA x yAc) exp [VA(xAly)] XA(dXA),
(11)
t /
XA
where ]rB is the indicator of B,
VA(XAlY)=
~_,
Vee,(Xe,Xe,)+
(£,g')~E A
and
~_,
Vee,(xe, ye.,),
(12)
(e,e')~OEA
/,
Za(y) = [ exp [Va(x^ly)] x^(dxa).
(13)
t /
XA
Thus, for every v • f2, zra is a probability kernel from (X, 13(X)) into itself. The family {zra}^e~n is the local Gibbs specification of the model considered. DEFINITION 1. A probability measure /z on X is called a Euclidean Gibbs • ,~fin and any B • B(X),
measure of the model (1) if for every A /z(B)
f zra (B Ix)~ (dx). X
(14)
UNIQUENESS OF GIBBS STATES OF A QUANTUM SYSTEM ON GRAPHS
285
The set of all such Euclidean Gibbs measures will be denoted by G. If necessary, we write G(v) to indicate the dependence on the choice of the interaction potentials. For a topological space Y, by T'(Y) we denote the set of all probability measures on Y. We equip it with the usual weak topology defined by all bounded continuous functions f : Y --+ R, the set of which will be denoted by Cb(Y). One can show that for every f • Cb(X) and any A • ,~fin, zrA(fl') • C b ( X ) ; thus, the local Gibbs specification has the Feller property. By the latter property and by the compactness of M, one can show that for every x • X, the family {zrA(.Ix)}^e&n C P ( X ) is relatively compact in the weak topology and that its accumulation points are elements of G. Hence, the latter set is nonempty. For v = 0, the family {rrA}Ae~, is consistent in the Kolmogorov sense. Then by the Kolmogorov extension theorem, ~(0) is a singleton. We are going to prove that this property persists if v is nonzero but I[vl[ is small. Our results cover the following two cases. Given £ • L, by me we denote the degree of £, that is, me = #OE{~}. For £, £' • L, by p(£, £') we denote the distance between these vertices--the length of the shortest path connecting £ with £'. In the first case, the only condition imposed on the graph G is that for any ~, £' • L, p(£, £') > ~b(min{me, me,}), (15) where ~b : 1~ --+ [1, +c~) is a nondecreasing function, such that +~
2n
Z < c~. n=l ~b(n) Set
x(v) =
(16)
sup 16[exp(4llvee, l l ) - 1],
(17)
(e,e~)eE
and, for 6 > 0,
~ = {v E ~ t x ( v ) < 3}.
(18)
THEOREM 1. There exists ~ E (0, 1), depending on the choice of ~b, such that for every v E ~ , the set G(v) is a singleton. Our second result describes the case where the potential v • ~ is random. Let Q be the family of all probability measures on ~ having the product form Q = (~
Qee,,
Qee, •79(f2ee,).
(19)
(e,e/)eE
Given k > 0 and 0 • (0, 1), we set Q()~, 0) = {Q • Q [ ¥(£, £') • E :
Qu,(llvu, II > )~) < 0}.
(20)
THEOREM 2. Let the graph G be such that supme < cx~.
(21)
eeL
Then there exist )~ > 0 and 0 • (0,1) such that, for every Q • Q(k,O), the set G(v) is a singleton for Q-almost all v.
286 3.
D. KI~PA and Y. KOZITSKY
Comments on the proof
To prove the above results we will use the method developed in [3], where similar statements were proven for a model of classical spins with finite single-spin spaces. Thus, the only generalization we need is to extend this method to the case of the single-spin space Xe as introduced above. Given A e £fin, we set
vA (dxA) = ~
Vet, (Xe.Xe) XA (dx^),
exp
(22)
(t, ) where 1/ZA is a normalization factor. Thus, vA is a probability measure on X ~ - - t h e local Euclidean Gibbs measure. For A C A C L, we set EA(A) = 0EA A E~,
(23)
i.e. E~(A) is the smallest subset of EA along which one can cut out A from A. Then, for A C A e £fin, we have
vA (dxA) : vA (dx~\^ IxA) v~ (dx^),
(24)
where v~ is the projection of vA onto B(XA) and l exp[ vzx(dxA\A IxA) -- N^ (x^~
Vet'(Xt, Xt')]XAxA(dx~\^),
Z
(25)
(£,et)EEA\AUEA(A)
XA\A
(t,e)eEA\AUEA(A)
Given ~, ~' e L, by 0(~, ~') we denote the path which connects these vertices. That is, O(~, ~') is the sequence of vertices ~0 . . . . . ~,, n e N, such that: (a) ~0 = ~ and ~n = ~'; (b) for all j = 0 . . . . . n - 1, (~j+l, ~j) e E. The path O(~, ~') will be called admissible if: (c) met _> 2 for all k = 1 . . . . . n - 1; (d) if ~i = ~j for certain i < j , then there exists a k, i < k < j, such that reek >_mei = mtj. The length of a path 10(~, ~')1 is the number of vertices in it, i.e. IO(& ~')1 = n. For (~, ~ ' ) e E, we set, el. (17),
xee, = 16 [exp (411vte,ll) - 1].
(26)
For a path O(& ~'), we define n--1
R[O(& ( ) ] = 4 ;(t)+;(e') 1 7 Xee,,
(27)
i=0
where 5"(0 = - 1 if me = 1 and 5"(0 = 0 if me _> 2. Finally, for A C L, we set
S/,(e, ( ) = Z
R[O(e, ( ) ] ,
(28)
UNIQUENESS OF GIBBS STATES OF A QUANTUM SYSTEM ON GRAPHS
287
where the summation is performed over all admissible paths connecting ~ with ~'. The main element of the proof of both our theorems is the following lemma which is an adaptation of Theorem 1 of [3] to the model considered here. LEMMA 1. Given A ~ £~n, let f : X A ~ ~ be a bounded continuous function. Then for every A ~ £~n, such that A C A, and arbitrary ~' E A \ A, xe,, x~e, ~ Xe,, it follows that va(flxe,)
1 < ~
va(flx~,)
Sa(£, £').
(29)
e~A
Sketch of the proof" The proof of the lemma is based on the following estimates. Let (Y, 13(Y), P) be a probability space and a, b, c be positive measurable real-valued functions on Y. Then f a(y)b(y)P(dy) b(y) < sup , ycY c(y) - f a(y)c(y)P(dy) ycY c(y) inf b(y) <
f a(y)b(y)P(dy) b(y) f a ( y ) c ( y ) P ( d y ) - 1 < SUpycyc(y) Now let a , b , c be as above, let also a' : Y ~ conditions P(a) = P(a') = 1. Set
(30)
] --
1 .
(0+e~)
(31)
and this a obeys the
S(b, c) = max{supb(y); supc(y)},
I(b, c) = min{inf b(y); inf c(y)}
ycY
ycY
y~Y
y~Y
and suppose that b and c are such that I(b, c ) > 0. Then
I [a(y)b(y)P(dy) _ 1 < S(b, c) a'(y)c(y)P(dy)
1.
(32)
- I (b, c)
Suppose now that c is such that there exists Y0 6 Y for which c(yo)= infyey c(y). Let positive y, e, 3 be such that
a(y) a'(y) Then
1 < Y, -
_c(y) _ _ 1 < e,
b(y) - 1 < 3.
c(yo)
c(y)
-
f a (y)b(y) P (dy) < ~ + e y + 6ey. f a'(y)c(y)P(dy) -
(33)
By means of the estimates (30)-(33) the proof of the lemma follows by the same inductive scheme which was used in the proof of Theorem 1 in [3]. [] The proof of both theorems stated above follows from Lemma 1 by means of similar arguments the proof of Theorems 2, 3, and 4 in [3] was based on. REFERENCES [1] S. Albeverio, Yu. G. Kondratiev, R. A. Minlos and G. V. Shchepan'uk: Ground state Euclidean Gibbs measures for quantum lattice systems on compact manifolds, Rep. Math. Phys. 45 (2000), 419--429.
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D. KI~PA and Y. KOZITSKY
[2] S. Albeverio, Yu. G. Kondratiev, R. A. Minlos and G. V. Shchepan'uk: Uniqueness problem for quantum lattice systems with compact spins, Lett. Math. Phys. 52 (2000), 185-195. [3] L. A. Bassalygo and R. L. Dobrushin: Uniqueness of a Gibbs field with a random potential--an elementary approach, (in Russian) Teor. Veroyatnost. i Primenen. 31 (1986), 651-670; English translation: Theory Probab. Appl. 31 (1987), 572-589. [4] B. K. Driver: Heat kernels measures and infinite dimensional analysis, in P. Auscher, T. Coulhon, and A. Grigor'yan (editors), Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, pages 1-104, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003. [5] H.-O. Georgii: Gibbs Measures and Phase Transitions, Studies in Mathematics, 9, Walter de Gruyter, Berlin, New York 1988. [6] A. Klein and L. Landau: Periodic Gaussian Osterwalder-Schrader positive processes and the two-sided Markov property on the circle, Pacific J. Math. 94 (1981), 341-367.
REPORTS ON MATHEMATICAL PHYSICS
No. 3
UNIQUENESS OF GIBBS STATES OF A QUANTUM SYSTEM ON GRAPHS DOROTA Kt~PA and YURI KOZITSKY* Instytut Matematyki, UniwersytetMarii Curie-Sldodowskiej,20-031 Lublin, Poland (e-mails: [email protected],[email protected]) (Received September 13, 2006) Gibbs states of an infinite system of interacting quantum particles are considered. Each particle moves on a compact Riemannian manifold and is attached to a vertex of a graph (one particle per vertex). Two kinds of graphs are studied: (a) a general graph with locally finite degree, (b) a graph with globally bounded degree. In case (a), the uniqueness of Gibbs states has been shown under the condition that the interaction potentials are uniformly bounded by a sufficiently small constant. In case (b), the interaction potentials are random. Here, under a certain condition imposed on the probability distribution of these potentials, the almost sure uniqueness of Gibbs states has been shown. Keywords: interacting quantum particles, random potentials, quantum Gibbs state, almost sure uniqueness.
1.
Introduction
Let (M, o-) be a compact Riemannian manifold with the corresponding normalized Riemannian volume measure ~r. Let also G = (L, E) be a (countable) graph with no loops, isolated vertices, and multiple edges. The model we consider in this article describes a system of interacting quantum particles, each of which is attached to a vertex ~ e k (one particle per vertex), its position is described by qe 6 M. The formal Hamiltonian of the model is h2 H --
'It"-'%
~m ~-~Ae -
TgC~L
~ vee,(qe, qe,), (e,e')~E
(1)
where m is the particle mass, Ae is the Laplace-Beltrami operator with respect to qe, and Vee, is a symmetric continuous function on M x M. By (e, e') we denote the edge of the graph which connects the named vertices. The Hamiltonian of the free particle h2 He Ae (2) 2m *Supported by the DFG under the Project 436 POL 113/115/0-1.
[281]
282
D. K/~PA and Y. KOZITSKY
is a self-adjoint operator in the physical Hilbert space 7-/e = L2(M, or). It is such that traceexp(-rHe) < oo for all r > 0. If one sets the periodic conditions at the endpoints of the interval [0,/3], /6 > 0 being the inverse temperature, then the semi-group e x p ( - r H e ) , r e [0,/3], defines a/%periodic Markov process, see [6], called a Brownian bridge, described by the Wiener bridge measure. More on the properties of such operators and measures can be found e.g. in [4]. A complete description of the equilibrium thermodynamic properties of an infinite particle system may be made by constructing its Gibbs states. In the quantum case, such states are defined as positive normal functionals on the algebras of observables satisfying the Kubo-Martin-Schwinger (KMS) conditions, which reflect the consistency between the dynamic and thermodynamic properties of the system proper to the equilibrium. Often, also in the case considered here, the KMS conditions cannot be formulated explicitly, which makes the problem of constructing the Gibbs states as KMS states unsolvable. Since 1975, an approach based on the properties of the semigroups generated by the local Hamiltonians has been developed. In this approach, in [1, 2] the Gibbs states of the model (1), with G being a simple cubic lattice Z d with the edges connecting nearest neighbours, were constructed in terms of probability measures on path spaces--the so-called Euclidean Gibbs measures. So far, this is the only possible way to define Gibbs states of such models and hence to give a mathematical description of their thermodynamic properties. The existence of Euclidean Gibbs measures by simple arguments follows from the compactness of the manifold M. In the small-mass limit, which can also be treated as the strong quantum limit, the uniqueness of Euclidean Gibbs measures at all 1~ was proven in [2]. The existence and uniqueness of the ground state, i.e. of the Euclidean Gibbs measure at /3 = +cx~, was proven in [1], also for small m. In both cases, /~ < + ~ and /3 = + ~ , the results were obtained by means of cluster expansions. The same method yields the proof of the uniqueness for small /3 (high temperature uniqueness), which is equivalent to the case of small sup sup [Ve,e,(qe, qe')l. £,e I q,~,qet
In the cluster expansion method, the properties of the lattice Z d are crucial. Thus, it would be of interest to extend the latter result to the case of more general graphs, especially those of globally unbounded degree. Another possible extension would be considering random potentials. In this article, we present two statements establishing the uniqueness of Gibbs states of the model (1), which occurs due to weak interaction. In the first one (Theorem 1), the graph G obeys a condition, by which vertices of large degree should be at large distance of each other. This is the only condition imposed on the graph--we do not suppose that it has globally bounded degree. In the second statement--Theorem 2--we suppose such a boundedness, however the functions Vee,, (£, g') e E, now are random. We claim that there exists a family of the probability
UNIQUENESS OF GIBBS STATES OF A QUANTUM SYSTEM ON GRAPHS
283
distributions on the space of these functions such that, for every member of this family, the Gibbs state of the model is almost surely unique. The proof of these statements is based on an extension of the method of [3], where the single-spin spaces were finite. Here we give its brief sketch--the only thing one can do in the frames of this article. A detailed presentation of this proof, as well as a more detailed explanation of the connection between the Gibbs states and Euclidean Gibbs measures of the model (1), will be given in a separate publication.
2.
Setup and results
Since in our study the role of the particle mass and the temperature will be trivial, for notational convenience we set m = 13 = h = 1. Throughout the paper, for a topological space Y, by /3(Y) we denote the corresponding Borel ~-field. By saying that /z is a measure on Y, we mean that /z is a measure on the measurable space (Y,/3(Y)). For a measurable function f • Y ~ ]K, which is #-integrable, we write P
#(f) = ] fd~. t /
Y
Given e ~ L, we denote Xe={xeEC([O,
1]~M)
I xe(0)=xe(l)},
(3)
which is a metric space with the metric O(x, y) =
sup d ( x ( r ) , y ( r ) ) , rE[0,1]
d being the Riemannian metric of M. This is our single-spin space. The Wiener bridge measure Xe is defined on Xe by its values on the cylinder sets Brl .....~n = {Xe E Xe I xe(z'l) E B1 . . . . , xe(rn) C Bn},
where IBi E ]3(M), i = 1 . . . . . n, and ri C [0, 1] are such that rl < r 2 - " < "On. For such a set,
x£(Brl ..... "On) =
f
P~2--~l(~1, ~2) X "'"
(4)
B 1x...×Bn
X prn_rn_l (~n-1, ~n)Pl+~l-r n (~n, ~ l ) ° ' ( d ~ l ) " ' " o ' ( d ~ n ) , where Pr(~, r/), r > 0 is the integral kernel of the operator e x p ( - r H e ) . Let ,~fin be the family of all finite nonempty subsets of l . As usual, for A C L, we use the notation A c = L \ A. For A c L, the Cartesian product X ~ = 1-I Xe,
(5)
is equipped with the product topology. Its elements are denoted by xA; we write X = XL and x = XL. Given (e,e') ~ E, let ~ee, be a copy of the Banach space
284
D. Kt~PA and Y. KOZITSKY
C(M x M ~ R) of continuous symmetric functions equipped with the supremum norm. Then we set f2 = 1-I ~ee,, (6) (e,e,)eE and equip this space with the product topology. Elements of f2 are denoted by v. For Vee, • f2ee, and given xe, Xe,, we set 1
Vee,(Xe, xe,) = [ Vee,(Xe (r), Xe, (r))dr.
(7)
0
Then Vee, is a bounded continuous symmetric function on Xe x Xe,. Obviously,
IIVee,II <_ IJvee,ll,
(8)
where I1" II stands for the supremum norm in both cases. For A C L, we set
X^ = @ Xe,
(9)
which is a probability measure on Xa. For such a A, we also set
E^ ={(£,£') • E I £,g' • A}, oqEA : {(6, 6') • E I 6 • A, 0LA = {6 t • A c I 36 • A :
(10)
g' • AC}, (6, 6') • E}.
The latter sets are called the edge and the vertex boundary of A, respectively. For A C A t, each element of Xa, can be decomposed xa, = xa x xa,\a. In particular, one has x = xa x xac. Given A • £fin, Y • X, and B • 13(X), we set 1
:rrA (BIy) -- Z^ (y~) f ]IB(XA x yAc) exp [VA(xAly)] XA(dXA),
(11)
t /
XA
where ]rB is the indicator of B,
VA(XAlY)=
~_,
Vee,(Xe,Xe,)+
(£,g')~E A
and
~_,
Vee,(xe, ye.,),
(12)
(e,e')~OEA
/,
Za(y) = [ exp [Va(x^ly)] x^(dxa).
(13)
t /
XA
Thus, for every v • f2, zra is a probability kernel from (X, 13(X)) into itself. The family {zra}^e~n is the local Gibbs specification of the model considered. DEFINITION 1. A probability measure /z on X is called a Euclidean Gibbs • ,~fin and any B • B(X),
measure of the model (1) if for every A /z(B)
f zra (B Ix)~ (dx). X
(14)
UNIQUENESS OF GIBBS STATES OF A QUANTUM SYSTEM ON GRAPHS
285
The set of all such Euclidean Gibbs measures will be denoted by G. If necessary, we write G(v) to indicate the dependence on the choice of the interaction potentials. For a topological space Y, by T'(Y) we denote the set of all probability measures on Y. We equip it with the usual weak topology defined by all bounded continuous functions f : Y --+ R, the set of which will be denoted by Cb(Y). One can show that for every f • Cb(X) and any A • ,~fin, zrA(fl') • C b ( X ) ; thus, the local Gibbs specification has the Feller property. By the latter property and by the compactness of M, one can show that for every x • X, the family {zrA(.Ix)}^e&n C P ( X ) is relatively compact in the weak topology and that its accumulation points are elements of G. Hence, the latter set is nonempty. For v = 0, the family {rrA}Ae~, is consistent in the Kolmogorov sense. Then by the Kolmogorov extension theorem, ~(0) is a singleton. We are going to prove that this property persists if v is nonzero but I[vl[ is small. Our results cover the following two cases. Given £ • L, by me we denote the degree of £, that is, me = #OE{~}. For £, £' • L, by p(£, £') we denote the distance between these vertices--the length of the shortest path connecting £ with £'. In the first case, the only condition imposed on the graph G is that for any ~, £' • L, p(£, £') > ~b(min{me, me,}), (15) where ~b : 1~ --+ [1, +c~) is a nondecreasing function, such that +~
2n
Z < c~. n=l ~b(n) Set
x(v) =
(16)
sup 16[exp(4llvee, l l ) - 1],
(17)
(e,e~)eE
and, for 6 > 0,
~ = {v E ~ t x ( v ) < 3}.
(18)
THEOREM 1. There exists ~ E (0, 1), depending on the choice of ~b, such that for every v E ~ , the set G(v) is a singleton. Our second result describes the case where the potential v • ~ is random. Let Q be the family of all probability measures on ~ having the product form Q = (~
Qee,,
Qee, •79(f2ee,).
(19)
(e,e/)eE
Given k > 0 and 0 • (0, 1), we set Q()~, 0) = {Q • Q [ ¥(£, £') • E :
Qu,(llvu, II > )~) < 0}.
(20)
THEOREM 2. Let the graph G be such that supme < cx~.
(21)
eeL
Then there exist )~ > 0 and 0 • (0,1) such that, for every Q • Q(k,O), the set G(v) is a singleton for Q-almost all v.
286 3.
D. KI~PA and Y. KOZITSKY
Comments on the proof
To prove the above results we will use the method developed in [3], where similar statements were proven for a model of classical spins with finite single-spin spaces. Thus, the only generalization we need is to extend this method to the case of the single-spin space Xe as introduced above. Given A e £fin, we set
vA (dxA) = ~
Vet, (Xe.Xe) XA (dx^),
exp
(22)
(t, ) where 1/ZA is a normalization factor. Thus, vA is a probability measure on X ~ - - t h e local Euclidean Gibbs measure. For A C A C L, we set EA(A) = 0EA A E~,
(23)
i.e. E~(A) is the smallest subset of EA along which one can cut out A from A. Then, for A C A e £fin, we have
vA (dxA) : vA (dx~\^ IxA) v~ (dx^),
(24)
where v~ is the projection of vA onto B(XA) and l exp[ vzx(dxA\A IxA) -- N^ (x^~
Vet'(Xt, Xt')]XAxA(dx~\^),
Z
(25)
(£,et)EEA\AUEA(A)
XA\A
(t,e)eEA\AUEA(A)
Given ~, ~' e L, by 0(~, ~') we denote the path which connects these vertices. That is, O(~, ~') is the sequence of vertices ~0 . . . . . ~,, n e N, such that: (a) ~0 = ~ and ~n = ~'; (b) for all j = 0 . . . . . n - 1, (~j+l, ~j) e E. The path O(~, ~') will be called admissible if: (c) met _> 2 for all k = 1 . . . . . n - 1; (d) if ~i = ~j for certain i < j , then there exists a k, i < k < j, such that reek >_mei = mtj. The length of a path 10(~, ~')1 is the number of vertices in it, i.e. IO(& ~')1 = n. For (~, ~ ' ) e E, we set, el. (17),
xee, = 16 [exp (411vte,ll) - 1].
(26)
For a path O(& ~'), we define n--1
R[O(& ( ) ] = 4 ;(t)+;(e') 1 7 Xee,,
(27)
i=0
where 5"(0 = - 1 if me = 1 and 5"(0 = 0 if me _> 2. Finally, for A C L, we set
S/,(e, ( ) = Z
R[O(e, ( ) ] ,
(28)
UNIQUENESS OF GIBBS STATES OF A QUANTUM SYSTEM ON GRAPHS
287
where the summation is performed over all admissible paths connecting ~ with ~'. The main element of the proof of both our theorems is the following lemma which is an adaptation of Theorem 1 of [3] to the model considered here. LEMMA 1. Given A ~ £~n, let f : X A ~ ~ be a bounded continuous function. Then for every A ~ £~n, such that A C A, and arbitrary ~' E A \ A, xe,, x~e, ~ Xe,, it follows that va(flxe,)
1 < ~
va(flx~,)
Sa(£, £').
(29)
e~A
Sketch of the proof" The proof of the lemma is based on the following estimates. Let (Y, 13(Y), P) be a probability space and a, b, c be positive measurable real-valued functions on Y. Then f a(y)b(y)P(dy) b(y) < sup , ycY c(y) - f a(y)c(y)P(dy) ycY c(y) inf b(y) <
f a(y)b(y)P(dy) b(y) f a ( y ) c ( y ) P ( d y ) - 1 < SUpycyc(y) Now let a , b , c be as above, let also a' : Y ~ conditions P(a) = P(a') = 1. Set
(30)
] --
1 .
(0+e~)
(31)
and this a obeys the
S(b, c) = max{supb(y); supc(y)},
I(b, c) = min{inf b(y); inf c(y)}
ycY
ycY
y~Y
y~Y
and suppose that b and c are such that I(b, c ) > 0. Then
I [a(y)b(y)P(dy) _ 1 < S(b, c) a'(y)c(y)P(dy)
1.
(32)
- I (b, c)
Suppose now that c is such that there exists Y0 6 Y for which c(yo)= infyey c(y). Let positive y, e, 3 be such that
a(y) a'(y) Then
1 < Y, -
_c(y) _ _ 1 < e,
b(y) - 1 < 3.
c(yo)
c(y)
-
f a (y)b(y) P (dy) < ~ + e y + 6ey. f a'(y)c(y)P(dy) -
(33)
By means of the estimates (30)-(33) the proof of the lemma follows by the same inductive scheme which was used in the proof of Theorem 1 in [3]. [] The proof of both theorems stated above follows from Lemma 1 by means of similar arguments the proof of Theorems 2, 3, and 4 in [3] was based on. REFERENCES [1] S. Albeverio, Yu. G. Kondratiev, R. A. Minlos and G. V. Shchepan'uk: Ground state Euclidean Gibbs measures for quantum lattice systems on compact manifolds, Rep. Math. Phys. 45 (2000), 419--429.
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D. KI~PA and Y. KOZITSKY
[2] S. Albeverio, Yu. G. Kondratiev, R. A. Minlos and G. V. Shchepan'uk: Uniqueness problem for quantum lattice systems with compact spins, Lett. Math. Phys. 52 (2000), 185-195. [3] L. A. Bassalygo and R. L. Dobrushin: Uniqueness of a Gibbs field with a random potential--an elementary approach, (in Russian) Teor. Veroyatnost. i Primenen. 31 (1986), 651-670; English translation: Theory Probab. Appl. 31 (1987), 572-589. [4] B. K. Driver: Heat kernels measures and infinite dimensional analysis, in P. Auscher, T. Coulhon, and A. Grigor'yan (editors), Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, pages 1-104, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003. [5] H.-O. Georgii: Gibbs Measures and Phase Transitions, Studies in Mathematics, 9, Walter de Gruyter, Berlin, New York 1988. [6] A. Klein and L. Landau: Periodic Gaussian Osterwalder-Schrader positive processes and the two-sided Markov property on the circle, Pacific J. Math. 94 (1981), 341-367.