with long-range interaction
Azer KerimovCitation: J. Math. Phys. 40, 4956 (1999); doi: 10.1063/1.533009 View online: http://dx.doi.org/10.1063/1.533009
View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v40/i10 Published by the American Institute of Physics.
Additional information on J. Math. Phys.
Journal Homepage: http://jmp.aip.org/
Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors
Uniqueness of Gibbs states in one-dimensional
antiferromagnetic model with long-range interaction
Azer Kerimov
Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey
共Received 28 May 1998; accepted for publication 2 April 1999兲
Uniqueness of Gibbs states in the one-dimensional antiferromagnetic model with very long-range interaction is established. © 1999 American Institute of Physics. 关S0022-2488共99兲03309-5兴
I. INTRODUCTION
We study a model on the lattice Z1 with the Hamiltonian
H共共x兲兲⫽
兺
x,y苸Z1;x⬎y
U共x⫺y兲共x兲共y兲⫺
兺
x苸Z1
共x兲, 共1兲
where the spin variable(x) takes the values 0 and 1,is a chemical potential. The antiferro-magnetic potential U(x)⬎0 satisfies the following conditions:
共1兲 U(x⫹y)⫹U(x⫺y)⬎2U(x);x,y苸Z1,x⬎y.
共2兲 The function U(x) can be extended to a twice continuously differentiable function such that U(x)⬃A(x)⫺␥, U
⬘
⬃⫺A␥x⫺␥⫺1 and U⬙
(x)⬃A␥(␥⫹1)x⫺␥⫺2 at x→⬁; where ␥⬎1, and A is a strong positive constant.The first convexity condition plays a significant role for the structure of the set of all ground states of the model共1兲. The second condition determines the character of the potential’s decrease at infinity and is important in further calculations.
The hypothesis on the uniqueness of the Gibbs states in the model共1兲 was stated by Sinai in 1983共see Ref. 1, Problem 1兲.
It is well known that the condition⌺x苸Z1,x⬎0xU(x)⬍⬁ automatically implies the uniqueness
of the Gibbs states.2–4We investigate the phase transition problem in the model 共1兲 in the alter-native case, when U(x)⬃Ax⫺␥, where␥⫽1⫹␣, 0⬍␣⬍1.
The ferromagnetic version of this model关when the potential U(x) is negative兴 was considered by Dyson in his well-known papers.5,6He proved the existence of two extreme limit Gibbs states P⫹and P⫺corresponding to the ground states(x)⫽⫹1 and(x)⫽⫺1 at low temperatures.
A series of papers has been devoted to the investigation of the antiferromagnetic model 共1兲.1,7–13
The validity of Sinai’s hypothesis for rational values of the density共for almost each value of the external field兲 at low temperatures was proved in Ref. 13.
The main purpose of this paper is to extend the result of Ref. 13 to all values of the external field and to all values of the temperature.
Theorem 1: The model (1) has a unique limit Gibbs state at all values of the temperature ⫺1.
Let us introduce necessary definitions. The set of all periodic configurations we denote by ⌽per. For every苸⌽per, we define q⫽⌺
y⫽x⫹1 x⫹p
(x), where p is the period of. It is obvious that q does not depend on x. Therefore, the density of each periodic configuration is⫽q/p. It is more convenient to work with the reciprocal of the density,((x))⫽p/q, which represents the aver-age distance between neighboring points at which (x)⫽1. For every configuration苸⌽per the mean energy h() is defined as follows:
4956
h共共x兲兲⫽1 py⫽x⫹1
兺
x⫹p 共x兲兺
z⬎0 U共z兲共y⫹z兲. The last expression does not depend on x.The following definition is useful for describing the zero temperature phase diagram of the model 共1兲.
We fix a positive rational number p/q.
A configuration0(x)苸⌽per with(0(x))⫽p/q is called a special ground state 1
if h共共x兲兲⫽ inf
苸⌽per,共兲⫽p/q
h共兲.
Hubbard’s criterion (Refs. 1 and 7): Let苸⌽perand ri(x;) denotes the distance between a
particle placed at x苸Z1 and ith particle on the right. If for each x and i 关i兴⭐ri共x;兲⭐关i兴⫹1,
共the square brackets denote the integer part of the enclosed number兲 then is a special ground state.
The existence of configurations satisfying Hubbard’s criterion 共the special ground states兲 is shown in Ref. 1. The remarkable elegant formula for the special ground states was offered by Aubry. Here we give the construction of the special ground states for each fixed rational value of the density.1
Every rational number p/q has a unique decomposition into a finite continued fraction: p/q⫽关n0,n1,...,ns兴, this means that
n0⫹ 1
n1⫹ 1
n2⫹...⫹ 1 ns
.
The ground state for a configuration with⫽关n0,n1,...,ns兴 will be constructed by recursion. 共1兲 ⫽n0⭓1, n0 is an integer. The periodic configuration with equally distant x at which (x)⫽1 satisfies Hubbard’s criterion i.e., is a special ground state. In this case ri(x;)⫽in0, i
⬎0.
共2兲⫽n0⫹1/n1, where n0 and n1 are integers, n0⭓1, n1⬎1. Then the (n0n1⫹1) periodic configuration
also satisfies Hubbard’s criterion and is a special ground state.
共3兲 ⫽关n0,n1,...,ns兴, where n0,n1,...,ns are integers, n0,n1,...,ns⭓1. For s⫽0 and s
⫽1 the required configurations are already constructed. Suppose we have already constructed a
ground state with s⫽m and ⫽关n0,n1,...,nm兴. Then the following configuration with s⫽m
⫹1 and⫽关n0,n1,...,nm⫹1兴 is constructed as
Here,(n0,...,n j ), j⫽m⫺1, m,m⫹1, are the blocks from which the ground states for ⫽关n0,...,nj兴 are obtained by periodic continuations.
The constructed configuration satisfies Hubbard’s criterion and therefore is a special ground state for⫽关n0,n1,...,nm,nm⫹1兴.
The following explicit expression for the mean energy of the special ground state follows from Hubbard’s criterion:1
h⫽
兺
i⫽1 ⬁
U共mi兲i⫹U共mi⫹1兲共1⫺i兲, 共2兲
where mi⫽关i兴, i⫽1⫹mi⫺i.
This formula shows that the function of mean energy as a function of the density is continuous on the set of all rationals and can be extended to a continuous function defined on whole segment 关0, 1兴.
Theorem 2: (Refs. 9 and 1.)共1兲 The function h is convex.
共2兲 In each rational point the function h has a left-hand derivative ⫺ and a right-hand derivative⫹, with⫹⬎⫺.
共3兲 The Lebesgue measure of the complement of the set 艛(⫺,⫹) in the real line R is zero.
The following theorem gives the full description of the set of all special ground states of the model 共1兲 at rational densities.
Theorem 3: (Ref. 12.) Suppose that the value of the external fieldof the model (1) belongs to the interval (⫺,⫹) for some number⫽q/p. Then the special ground state of the model 共1兲 is unique up to translations.
Following Theorem 4 generalizes the main result of Ref. 13 for all values of the temperature and is a special case 共rational densities兲 of Theorem 1.
Theorem 4: Suppose that the value of the external field of the model 共1兲 belongs to the interval (⫺,⫹) for some number⫽q/p.
Then the model共1兲 has a unique limit Gibbs state at all values of the temperature⫺1. Suppose that the value of the external field of the model 共1兲 belongs to the interval (⫺,⫹) for some number⫽q/p.
Let us consider an arbitrary configuration(x). We say that(关a,b兴); a,b苸Z1is a preregu-lar phase, if there exists a special ground state, such that the restriction of this configuration to 关a,b兴 coincides with(关a,b兴). We say that(关c,d兴); c,d苸Z1 is a regular phase, if there exists a preregular phase(关a,b兴); a,b苸Z1, such that c⫺a⬎d
0p and b⫺d⬎d0p. Thus, right and left d0p extensions of a regular phase are ground states.
Let us consider a set A⫽艛i关ai,bi兴, where(关ai,bi兴) is a regular phase and supp PB is the complement of A in Z1. The connected components of supp PB defined in such a way are called supports of precontours and are denoted by supp PK: supp PK⫽艛i苸Indsupp PKi.
For each fixed rational densitythe constant d0 satisfies some technical conditions.13In this work we do not need the explicit value of d0.
Definition 1 (Ref. 13): The pair PK⫽(supp PK,
⬘
(supp PK)) is called a precontour. The set of all precontours is called a preboundary PB of the configuration⬘
(x). Two precontours PK1 and PK2 are said to be connected if dist(supp PK1,supp PK2)⬍Nb. The set of precontours( PKi;i苸Ind) is called connected if for any two precontours PKcand PKd;c,d苸Ind there exists
a collection ( PKj1⫽PKc,..., PKji,..., PKjn⫺1, PKjn⫽PKd); ji苸Ind, i⫽1,...,n; such that any two
precontours PKji and PKji⫹1, i⫽1,...,n⫺1 are connected. Let 艛i⫽1 n
PKi be some maximal
con-nected component of the preboundary PB. Suppose that supp PKi⫽关ai,bi兴 and bi⬍ai⫹1; i
⫽,...,n⫺1.
The pair K⫽(supp K,
⬘
(supp PK)), where supp K⫽关a1,bn兴 is called a contour. The set ofall contours is called a boundary B of the configuration
⬘
(x). In this work we do not need the exact value of the constant Nb.12
From Ref. 12 it becomes clear that limp→⬁Nb⫽⬁. Thus, for irrational values of the density Nbis not defined, but as will
be seen below, we do not need to define Nb for irrational densities. Note that supp K⫽(艛in⫽1supp PKi)艛(关a1,bn兴⫺(艛i⫽1
n
The sets supp1K and supp2K will be, respectively, called the essential and regular parts of the support supp K.
Let the boundary conditions¯ (x)⫽关(x),x苸(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁)兴 be fixed. The set of all configurations(x); x苸关⫺V,V兴 we denote via ⌽(V).
It is obvious that for each contour K, such that supp K苸关⫺V⫹(d0⫹1)p,V⫺(d0⫹1)p兴, there exists a configuration K(关⫺V,V兴) such that the boundary of the configuration
K(关⫺V,V兴) includes the contour K only:
B共K共关⫺V,V兴兲兲⫽K.
Let supp K⫽关a,b兴. It is obvious that the restrictions of the configurationK(关⫺V,V兴) to the segments关⫺V,a⫺1兴 and 关b⫹1,V兴 coincide with two ground states1(x) and2(x).
A contour K is called an interface contour, if1(x)⫽2(x).
Note that,1(x) can be obtained by some shifting of the configuration2(x). An interface contour will be denoted as IK.
Let K be a usual contour 共not an interface contour兲 K,supp K傺关⫺V,V兴 and K共x兲 ⫽([⫺V,V]) if x苸关⫺V,V兴, and ¯ (x) if x苸(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁); IK,supp IK傺 关⫺V,V兴 be an interface contour and IK(x)⫽(关⫺V,V兴) if x苸关⫺V,V兴, and ¯ (x) if x苸
(⫺⬁,V⫺1兴艛关V⫹1,⬁); and ¯1(x)⫽1(x), if x苸关⫺V,V兴, and ¯ (x) if x苸(⫺⬁,⫺V ⫺1兴艛关V⫹1,⬁).
Below the configuration¯1(x) defined for usual contours will be denoted by¯(x). The weights of the usual contour K and interface contour IK will be calculated by the follow-ing formulas:
␥共K兲⫽H共K共x兲兲⫺H共¯共x兲兲, 共3兲
␥共IK兲⫽H共IK共x兲兲⫺H共¯
1共x兲兲. 共4兲
The proof of Theorem 4 is based on the following idea. Let the boundary conditions¯ (x) ⫽关(x),x苸(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁)兴 be fixed. The set of all configurations (x); x苸 关⫺V,V兴 we denote via ⌽(V). Suppose a configurationmin(x)苸⌽(V) be a configuration with the minimal energy:
H共min共x兲兲⫽min共x兲苸⌽共V兲H共共x兲兲 .
Then the configurationmin(x) almost coincides with a special ground state of the model共1兲 共Lemma 1 in Sec. II兲. This fact allows us, based on special ground states, to define a common 共for all boundary conditions兲 contour model and after that by using well-known trick14 共this trick, which was introduced in Ref. 14 for some special extensions of Pirogov–Sinai theory, is directly applicable to one-dimensional models with long-range interaction兲 to come to noninteracting clusters from interacting contours. Consider an arbitrary segment I, a sufficiently large volume V, two arbitrary boundary conditions 1(x) and 2(x). It turns out that the dependence of the expression P1(
⬘
(I))/P2(⬘
(I)) on the boundary conditions1(x) and 2(x) can be estimated through the sum of statistical weights of super clusters connecting the segment I with the boundary and this sum is negligible. Thus, two arbitrary extreme Gibbs states are relatively continuous and hence coincide. In Ref. 13 we developed this method 关the estimation of dependence of the ex-pression P1(⬘
(I))/P2(⬘
(I)) on the boundary conditions through the sum of statistical weights of super clusters connecting the segment I with the boundary兴 at low temperatures. It turns out that after some modification the method works at all temperatures.The contents of this paper are as follows. In Sec. II we prove Theorem 4, in Sec. III we complete the proof of Theorem 1.
II. UNIQUENESS OF GIBBS STATES: THE DENSITYISp/q Let us now introduce some necessary facts.
Suppose that the value of the external field of the model 共1兲 belongs to the interval (⫺,⫹) for some number⫽q/p.
Let the boundary conditions1(x)⫽关1(x),x苸(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁)兴 be fixed and
H共共x兲兩1共x兲兲⫽⫺
兺
x苸Z1,x苸关⫺V,V兴 共x兲⫹兺
x,y苸Z1,x⬎y;x,y苸关⫺V,V兴 U共x⫺y兲共x兲共y兲 ⫹兺
x,y苸Z1,x⬎y;x苸关⫺V,V兴;y苸关⫺V,V兴 U共x⫺y兲共x兲1共y兲 ⫹兺
x,y苸Z1,x⬎y;x苸关⫺V,V兴,y苸关⫺V,V兴 U共x⫺y兲1共x兲共y兲. 共5兲Lemma 1: Letmin(x)苸⌽(V) be a configuration with the minimal energy: H共min共x兲兩1共x兲兲⫽min共x兲苸⌽共V兲H共共x兲兩1共x兲兲 . Then the configurationmin(x) has the following structure.
The restriction of the configurationmin(x) on the set关⫺V⫹Nb,V⫺Nb兴 contains at most
p⫺1 contours, moreover, all of them are interface contours IKi,, i⫽1,...,m, where m⬍p⫺1 and
兩supp IKi兩⬍3d0p⫹Nb.
Lemma 1 was proved in Ref. 13关see Lemma 12 共Ref. 13兲 and Sec. 5 of Ref. 13兴.
Let H((x)兩1(x),min(x)) denote the relative energy of a configuration(x)关with respect to min(x)]:
H共共x兲兩1共x兲,min共x兲兲⫽H共共x兲兩1共x兲兲⫺H共min共x兲兩1共x兲兲.
Consider the Gibbs distribution P1 on ⌽(V) corresponding to the boundary conditions 1(x)⫽关1(x),x苸(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁)兴:
P1共
⬘
共x兲兲⫽ exp共⫺共H共⬘
共x兲兩1共x兲,min共x兲兲兲兲
⌺共x兲苸⌽共V兲exp共⫺共H共共x兲兩1共x兲,min共x兲兲兲兲. 共6兲 Let (x)苸⌽(V) be an arbitrary configuration, the boundary of the (x) includes a finite number of usual contours Ki; i⫽1,...,n, and a finite number of interface contours IKi; i⫽n
⫹1,...,n⫹m. Let Ki⫽Ki; i⫽1,...,n; Ki⫽IKi; i⫽n⫹1,...,n⫹m. The set of all contours of the
boundary conditions1(x) will be denoted by K0.
The statistical weights of contours and interface contours are
w共Ki兲⫽exp共⫺␥共Ki兲兲. 共7兲
The following equation is a direct consequence of the formulas共3兲, 共4兲, and 共7兲
exp共⫺H共共x兲兩1共x兲,min共x兲兲兲⫽Q1
兿
i⫽1 n⫹m
w共Ki兲exp共⫺G共K0,K1,...,Kn⫹m兲, 共8兲
where the multiplier G(K0,K1,...,Kn⫹m) corresponds to the interaction between contours共usual
and interface兲, and with the boundary conditions1(x)
G共K0,K1,...,Kn⫹m兲⫽
兺
i, j⫽0;i⬍ j n⫹m G共Ki,Kj兲⫽兺
i, j;i⬍ j 共x,y兲苸Int共K兺
i,Kj兲 f共x,y,兲 共9兲and the multiplier Q1⫽Q1(V,(x),1(x)) is uniformly bounded from below and above: 0 ⬍const1⬍Q1⬍const2. The factor Q1 appears due to the facts that the configuration min(x) not necessarily coincides with a special ground state and is bounded due to Lemma 1.
Now we write down the value of the interaction between the contours Kiand Kj, the value of
the interaction between the interface contours IKiand IKjand the value of the interaction between
contour Ki and interface contour IKj.
Suppose supp Kl⫽关al,bl兴; supp IKl⫽关al,bl兴.
Let
supp IKi⫹⫽关bi,ai⫹1兴 and supp IKi⫺⫽关bi⫺1,ai兴,
where b0⫽c, if there exists K苸B(
⬘
(x)), such that supp K⫽关⫺⬁,c兴 and b0⫽⫺⬁ otherwise; am⫹1⫽d, if there exists K苸B(⬘
(x)), such that supp K⫽关d,⬁兴 and am⫹1⫽⬁ otherwise.共1兲 The contour Ki苸B(
⬘
(x)) interacts with the contour Kj苸B(⬘
(x)) through all pairs(x, y ), such that (x, y )苸Int(Ki,Kj) and f
⬘
(x, y ,)⫽0 whereInt共Ki,Kj兲⫽关共x,y兲:x,y苸Z1;x苸supp Ki, y苸supp Kj兴.
The value of the interaction
f
⬘
共x,y,兲⫽U共x⫺y兲共⬘
共x兲⬘
共y兲⫺Ki共x兲Ki共y兲⫹ ¯ i共x兲¯ i共y兲 ⫺Kj共x兲Kj共y兲⫹¯ j 共x兲¯ j共y兲兲.
共2兲 The interface contour IKi苸B(
⬘
(x)) interacts with the interface contour IKj苸B(
⬘
(x)) 共let aj⬎bi) through all pairs (x,y ), such that (x, y )苸Int(IKi,IKj) and f⬙
(x, y )⫽0,where
Int共IKi,IKj兲⫽Int1共IKi,IKj兲⫹Int2共IKi,IKj兲⫹Int3共IKi,IKj兲⫹Int4共IKi,IKj兲,
Int1共IK
i,IKj兲⫽关共x,y兲:x,y苸Z1;x苸supp IKi and y苸supp IKj兴,
Int2共IKi,IKj兲⫽关共x,y兲:x,y苸Z1;x苸supp IKi and y苸supp IK⫹j 兴,
Int3共IKi,IKj兲⫽关共x,y兲:x,y苸Z1;x苸supp IKi⫺ and y苸supp IKj兴,
Int4共IKi,IKj兲⫽关共x,y兲:x,y苸Z1;x苸supp IKi⫺ and y苸supp IK⫹j 兴.
The value of the interaction
f
⬙
共x,y,兲⫽ f1⬙
共x,y兲⫽U共x⫺y兲共⬘
共x兲⬘
共y兲⫺IKi共x兲IKi共y兲⫹¯ i共x兲¯ i共y兲⫺ IKj共x兲IKj共y兲⫹¯j共x兲¯j共y兲兲
if (x,y )苸Int2(IKi,IKj),
f
⬙
共x,y兲⫽ f2⬙
共x,y兲⫽U共x⫺y兲共⬘
共x兲⬘
共y兲⫺IKi共x兲IKi共y兲⫹¯ i共x兲¯ i共y兲兲
if (x,y )苸Int2(IK
i,IKj),
f
⬙
共x,y兲⫽ f3⬙
共x,y兲⫽U共x⫺y兲共⬘
共x兲⬘
共y兲⫺IKj共x兲IKj共y兲兲⫹¯j共x兲¯j共y兲)if (x, y )苸Int3(IKi,IKj),
f
⬙
共x,y兲⫽ f4⬙
共x,y兲⫽U共x⫺y兲共⬘
共x兲⬘
共y兲⫺¯1,i共x兲¯1,i共y兲⫺¯2,j共x兲¯2,j共y兲兲 if (x,y )苸Int4(IKi,IKj).共3兲 The contour Ki苸B(
⬘
(x)) interacts with the interface contour IKj苸B(⬘
(x)) through allpairs (x,y ), such that (x,y )苸Int(Ki,IKj) and f
(x,y )⫽0, whereInt共Ki,IKj兲⫽Int1共Ki,IKj兲⫹Int2共Ki,IKj兲,
Int1共Ki,IKj兲⫽关共x,y兲:x,y苸Z1;x苸supp Ki and y苸supp IKj兴,
Int2共Ki,IKj兲⫽关共x,y兲:x,y苸Z1;x苸supp Ki and y苸supp IK⫹j 兴
if aj⬎bi, and
Int2共Ki,IKj兲⫽关共x,y兲:x,y苸Z1;x苸supp Ki and y苸supp IK⫺j 兴
if ai⬎bj.
The value of the interaction
f
共x,y兲⫽ f1
共x,y兲⫽U共x⫺y兲共⬘
共x兲⬘
共y兲⫺Ki共x兲Ki共y兲⫹¯ i共x兲¯ i共y兲⫺ IKj共x兲IKj共y兲兲⫹¯j共x兲¯j共y兲)
if (x,y )苸Int1(Ki,IKj),
f
共x,y兲⫽ f2
共x,y兲⫽U共x⫺y兲共⬘
共x兲⬘
共y兲⫺Ki共x兲Ki共y兲⫹¯ i共x兲 ¯ i共y兲兲
if (x,y )苸Int2(Ki,IKj).
For simplicity Ki, i⫽1,...,n⫹m will be denoted by Ki, i苸Ind, where the statistical weights
w(Ki) are defined by the formulas共7兲, 共3兲, and 共4兲. Thus, the formula 共8兲 has the form
exp共⫺H共共x兲兩1共x兲,min共x兲兲兲⫽Q1
兿
i苸Ind
w共Ki兲exp共⫺G共K0,K1,...,Kn⫹m兲兲. 共10兲
The set of all pairs共x,y兲 in the double sum 共9兲 will be denoted by Y⫽Y(K0,K1,...,Kn⫹m). Write共10兲 as follows:
exp共⫺H共共x兲兩1共x兲,min共x兲兲兲⫽Q 1
兿
i苸Ind
w共Ki兲
兿
共x,y兲苸Y 共1⫹exp共⫺f共x,y,兲⫺1兲. 共11兲 From 共11兲 we get exp共⫺H共共x兲兩1共x兲,min共x兲兲兲⫽Q1
兿
G⬘傺Gi苸Ind兿
w共Ki兲兿
共x,y兲苸Y⬘; f共x,y,兲⫽0 g共x,y兲, 共12兲where the summation is taken over all subsets Y
⬘
共including the empty set兲 of the set Y, and g(x, y ,)⫽exp(⫺f(x,y,))⫺1.Consider an arbitrary term of the sum共12兲, which corresponds to the subset Y
⬘
傺Y. Let the bond (x,y )苸Y⬘
. Below, contours and interface contours will be called contours. Consider the set K of all contours such that for each contour K傺K, the set supp K艚(x艛y) contains one point. We call any two contours from K connected. The set of contours K is called Y⬘
connected if for any two contours Ka and Kb there exists a collection (K1⫽Ka,K2,..., Kn⫽Kb) such that any twocontours Ki and Ki⫹1, i⫽1,...,n⫺1, are connected by some bond (x,y)苸Y
⬘
.The pair D⫽关(Ki,i⫽1,...,s);Y
⬘
兴, where Y⬘
is some set of bonds, is called a cluster providedthere exists a configuration (x) such that Ki苸B((x)); i⫽1,...,s; Y
⬘
傺Y; and the set (Ki, iw共D兲⫽
兿
i⫽1 s w共Ki兲兿
共x,y兲苸Y⬘ g共x,y,兲. 共13兲Two clusters D1 and D2are called compatible provided any two contours K1 and K2 belong-ing to D1 and D2, respectively, are compatible and not connected. A set of clusters is called compatible provided any two clusters of it are compatible.
If D⫽关(Ki,i⫽1,...,s);Y
⬘
兴, then we say that Ki苸D; i⫽1,...,s.The following lemma is a direct consequence of the definitions.
Lemma 2: Let the boundary conditions 1(x)⫽关1(x),x苸(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁)兴 be fixed.
If关D1,...,Dm兴 is a compatible set of clusters and 艛i⫽1 m
supp Di傺关⫺V,V兴, then there exists a
configuration(x) which contains this set of clusters. For each configuration(x) we have
exp共⫺H共共x兲兩1共x兲,min共x兲兲兲⫽Q1
兺
Y⬘傺Y
兿
w共Di兲,
where the clusters Diare completely determined by the set Y
⬘
. The partition function is ⌶共1共x兲兲⫽Q兺
w共D1兲¯w共Dm兲,
where the summation is taken over all nonordered compatible collections of clusters and the factor Q⫽Q(V,1(x)) is uniformly bounded: 0⬍const⬍Q⬍const2.
Lemma 2 shows that we come to noninteracting clusters from interacting contours.
Let P1and P2be two Gibbs states of the model共1兲 corresponding to the boundary conditions 1(x) and2(x), respectively.
The following lemma has a key role in the proof of Theorem 4.
Lemma 3: Suppose that the value of the external field of the model (1) belongs to the interval (⫺,⫹) for some number⫽q/p.
Then the measures P1 and P2 are absolutely continuous with respect to each other. Proof: Let I⫽关a,b兴 be an arbitrary segment and
⬘
(I) be an arbitrary configuration. In order to prove the lemma we show that there exist two positive constants s and S not depending on I,1(x), 2(x) and⬘
(I), such thats⭐P1共
⬘
共I兲兲/P2共⬘
共I兲兲⭐S. 共14兲 Let PV1 and PV2 be Gibbs measures corresponding to the boundary conditions 1(x), and 2(x), x苸Z1⫺I V, respectively, where IV⫽关⫺V,V兴. Therefore, lim V→⬁ PV1⫽P1 and lim V→⬁ PV2⫽P2,where by convergence we mean weak convergence of probability measures.
In order to establish the inequality 共14兲 it will be proved that for each fixed interval I, I傺关⫺M,M兴 there exists a number V0( M ), which depends on M only, such that
s⭐PV1共
⬘
共I兲兲/PV2共⬘
共I兲兲⭐S 共15兲 if V⬎V0.PV1共
⬘
共I兲兲⫽兺共IV兲:共I兲⫽⬘共I兲exp共⫺H共共IV兲兩
1共x兲,min共x兲兲兲O共共I兲,V,1兲 兺共IV兲exp共⫺H共共IV兲兩
1共x兲,min共x兲兲兲O共共I兲,V,1兲
⫽ ⌶共IV⫺I兩1共x兲,
⬘
共I兲,min共x兲兲O共共I兲,V,1兲兺⬙共I兲⌶共IV⫺I兩
1共x兲,
⬙
共I兲,min共x兲兲O共共I兲,V,1兲where⌶(IV⫺I兩1(x),
⬘
(I),min(x)) denotes the partition function corresponding to the bound-ary conditions 1(x), x苸Z1⫺IV, ⬘
(I), x苸I andO共共I兲,V,1兲⫽exp(⫺
兺
x,y苸Z1;x苸Z1⫺IV,y苸I
U共x⫺y兲共1共x兲共y兲⫺1共x兲min共x兲兲).
We can express PV2(
⬘
(I)) in just the same way.In order to prove the inequality共15兲 it is enough to establish inequality 共16兲 and inequality 共17兲:
1/2⬍O共共I兲,V,i共x兲兲⬍2, i⫽1,2 共16兲 关where the inequalities in 共16兲 are held uniformly with respect to(I) andi: for each I there exists V, not depending on(I) andi] and
1/S⭐⌶共IV⫺I兩
1共x兲,
⬙
共I兲,min共x兲兲 ⌶共IV⫺I兩1共x兲,⬘
共I兲,min共x兲兲冒
⌶共IV⫺I兩2共x兲,
⬙
共I兲,min共x兲兲⌶共IV⫺I兩2共x兲,
⬘
共I兲,min共x兲兲⭐1/s 共17兲for arbitrary
⬙
(I).Indeed, if the inequality共17兲 holds, then ⌶共IV⫺I兩1共x兲,
⬘
共I兲,min共x兲兲兺⬙共I兲⌶共IV⫺I兩1共x兲,
⬙
共I兲,min共x兲兲冒
⌶共IV⫺I兩2共x兲,
⬘
共I兲,min共x兲兲兺⬙共I兲⌶共IV⫺I兩2共x兲,
⬙
共I兲,min共x兲兲⫽AV 1共
⬘
共I兲兲/AV 2共
⬘
共I兲兲⫽1
冒 冉
兺⬙共I兲⌶共IV⫺I兩1共x兲,⬙
共I兲,min共x兲兲⌶共IV⫺I兩1共x兲,
⬘
共I兲,min共x兲兲冒
兺⬙共I兲⌶共IV⫺I兩2共x兲,
⬙
共I兲,min共x兲兲⌶共IV⫺I兩2共x兲,
⬘
共I兲,min共x兲兲冊
⫽1
冒
共兺⬙共I兲⌶共IV⫺I兩1共x兲,⬙
共I,min共x兲兲兲⌶共IV⫺I兩2共x兲,⬘
共I兲,min共x兲兲共兺⬙共I兲⌶共IV⫺I兩2共x兲,
⬙
共I兲,min共x兲兲兲⌶共IV⫺I兩1共x兲,⬘
共I兲,min共x兲兲.
Therefore,
1/共1/s兲⭐AV1共
⬘
共I兲兲/AV2共⬘
共I兲兲⭐1/共1/S兲 since the quotient of兺in⫽1ai/兺ni⫽1bilies between min(ai/bi) and max(ai/bi).Thus, if in addition, the inequality共16兲 holds, then
2⫺4s⬍PV1共
⬘
共I兲兲:PV2共⬘
共I兲兲⬍24S. Now we start to prove the inequalities共16兲 and 共17兲.It can be easily shown that共16兲 is a direct consequence of the condition U(x)⬃Ax⫺␥, at x →⬁; where␥⬎1, and A is a strong positive constant.
So, in order to complete the proof of Lemma 3 we must establish the following inequality 关which is just transformed inequality 共17兲兴:
1/S⭐⌶共IV⫺I兩
1共x兲,
⬙
共I兲,min共x兲兲)⌶共IV⫺I兩2共x兲,
⬘
共I兲,min共x兲兲⌶共IV⫺I兩 2共x兲,
⬙
共I兲,min共x兲兲)⌶共I V⫺I兩 1共x兲,⬘
共I兲,min共x兲兲⫽ ⌶1,⬙⌶2,⬘ ⌶2,⬙⌶1,⬘⭐1/s. 共18兲 Consider ⌶1,⬙⌶2,⬘⫽⌶共IV⫺I兩1共x兲,
⬙
共I兲,min共x兲兲⌶共IV⫺I兩2共x兲,⬘
共I兲,min共x兲兲.The following generalization of the definition of the compatibility allows us to represent ⌶1,⬙⌶2,⬘as a single partition function.
A set of clusters is called super compatible provided any of its two parts coming from two partitions sums is compatible. In other words, in super compatibility an intersection of supports of two clusters is allowed.
The following lemma is an analogue of Lemma 2.
Lemma 4: Let boundary conditions 1(x)⫽关1(x),x苸(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁)兴 and 2(x)⫽关2(x),x苸(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁)兴 be fixed.
If关D1,...,Dm兴 is a super compatible set of clusters and 艛i⫽1 m
supp Di傺关⫺V,V兴, then there
exist two configurations 3(x) and 4(x) which contain this set of clusters. For each two con-figurations3(x) and4(x) we have
exp(⫺H共3共x兲兩1共x兲,min共x兲兲exp(⫺H共4共x兲兩1共x兲,min共x兲兲⫽Q1
兺
G⬘傺G,G⬙傺G
兿
w共Di兲,
where the clusters Di are completely determined by the sets G
⬘
and G⬙
. The super partitionfunction is
⌶1,⬙,2,⬘⫽⌶1,⬙⌶2,⬘⫽Q
兺
w共D1兲¯w共Dm兲,
where the summation is taken over all nonordered super compatible collections of clusters and the factor Q⫽Q(V,1(x),2(x)) is uniformly bounded: 0⬍const1⬍Q⬍const2.
Lemma 4 is a direct consequence of the definitions.
An arbitrary connected component of an arbitrary super compatible set of clusters will be called a super clusters. A super cluster SD⫽关(Ki,i⫽1,...,r);G
⬘
兴 is said to be long if theinter-section of the set (艛im⫽1supp Ki))艛G
⬘
with both I and Z1⫺IV⫽(⫺⬁,⫺V⫺1兴艛关V⫹1,⬁) isnonempty. In other words, a long super cluster connects the boundary with the segment I. A set of super clusters is called compatible provided the set of all clusters belonging to these super clusters are super compatible.
It turns out that in our estimates long super clusters are negligible.
Lemma 5: For each fixed interval I, there exists a number V0(I), which depends on I only, such that if V⬎V0(I)
1/2⌶1,⬘,2,⬙⬍⌶1,⬘,2,⬙,共n.l.兲⫽
兺
w共SD1兲¯w共SDm兲⬍3/2⌶1,⬘,2,⬙,where the summation is taken over all nonlong, nonordered compatible collections of super clusters 关SD1,...,SDm兴, 艛i⫽1
m
supp(SDi)傺IN⫺I corresponding to the boundary conditions
1(x),2(x), x苸Z1⫺I
V;
⬘
(x) and⬙
(x), x苸I.Consider a collection of contours K0,K1,...,Kn. The value of the interaction of the contour
K0 with the contours K1,...,Kn we denote by G(K0兩K1,...,Kn):
G共K0兩K1,...,Kn兲⫽
兿
B苸IG共0兩1,...,n兲共1⫹exp共⫺
where IG(0兩1,...,n) is the set of all interaction elements intersecting the support of the contour K0. Lemma 6: G共K0兩K1,...,Kn兲⫽
兿
B苸IG共0兩1,...,n兲兩共1⫹exp共⫺ f共B兲⫺1兲兲 ⭐const共dist共0兩1,...,n兲兲⫺␣共兩supp共K0兲兩兲1⫺␣, 共20兲 where dist(0兩1,...,n) is the distance between the support of K0 and the union of the supports of contours K1,...,Kn.In other words, the interaction of K1,...,Kn on K0 tends to zero when the distance between them increases, and value of the interaction increases with a rate less than the length of the support of K0.
The technical Lemma 6 follows from the decreasing conditions of the potential U(x). For the rigorous proof see Ref. 13, Lemma 4.
The following lemma is an analogue of Lemma 5 for clusters共not super clusters兲.
Lemma 7: For each fixed interval I, there exists a number V0(I), which depends on I only, such that if V⬎V0(I)
1/2⌶1,⬘⬍⌶1,⬘,共n.l.兲⫽
兺
w共D1兲...w共Dm兲⬍3/2⌶1,⬘,where the summation is taken over all nonlong, nonordered compatible collections of clusters 关D1,...,Dm兴, 艛i⫽1
m
supp Di傺IN⫺I corresponding to the boundary conditions 1(x), x苸Z1
⫺IV;
⬘
(x), x苸I.Proof:
⌶1,⬘⫽⌶1,⬘,共n.l.兲⫹共⌶1,⬘⫺⌶1,⬘,共n.l.兲兲⫽⌶1,⬘,共n.l.兲⫹⌶1,⬘,共l.兲,
where the summation in ⌶1,⬘,(l.) is taken over all nonordered compatible collections of clusters 关D1,...,Dm兴 containing at least one long cluster, 艛i⫽1
m supp D
i傺IN⫺I corresponding to the
boundary conditions1(x), x苸Z1⫺IV;
⬘
(x), x苸I.By dividing both sides of the last equality by⌶1,⬘, we get
1⫽⌶1,⬘,共n.l.兲/⌶1,⬘⫹⌶1,⬘,共l.兲/⌶1,⬘. 共21兲 Now we are going to show that the second term共which is not necessarily positive兲 is negli-gible, that is the absolute value of it is less than 1/2共actually we can show that the absolute value of the second term is less than any fixed positive number at sufficiently large values of V).
The term⌶1,⬘,(l.)/⌶1,⬘can be interpreted as a ‘‘probability’’ P共Long兲 of the event that there exists at least one long cluster.
We show that the absolute value of this ‘‘probability’’ is less than 1/2 by the following method. We estimate the density of long clusters: the probability that a given segment belongs to the support of some long cluster. Since some statistical weights of clusters are positive and some negative, we estimate the absolute values of these ‘‘probabilities.’’ We show that for a fixed segment the ‘‘probability’’ that this segment belongs to the support of some long cluster with positive ‘‘probability’’ minus the ‘‘probability’’ that this segment belongs to the support of some long cluster with negative ‘‘probability’’ is less than one. Since the density is less than one, by the law of large numbers a ‘‘typical’’ long cluster has not very long support, and therefore has long bonds. When V tends to infinity, the total length of bonds tends to infinity, and the impact of these bonds tends to zero.
Now we replace a statistical weight w(Di) of each cluster Dibelonging to the configuration
becomes positive兲 and the expression ⌶1,⬘,(l.)/⌶1,⬘ transfers into ⌶1,⬘,(l.abs)/⌶1,⬘,(abs). It can be easily shown that, without loss of generality we can suppose that⌶1,⬘,(l.)⭓0. Obviously,
兩⌶1,⬘,共l.兲/⌶1,⬘兩⭐⌶1,⬘,共l.abs兲/⌶1,⬘,共abs兲.
Now the expression⌶1,⬘,(l.abs)/⌶1,⬘,(abs) can be interpreted as a ‘‘absolute probability’’ Pabs 共Long兲 of the event that there is at least one long cluster.
Now our aim is to estimate the ‘‘absolute probability’’ Pabsof the event that a given segment belongs to the support of long cluster. In other words, we are going to estimate the statistical weights of long clusters after replacing of the values of all negative bonds in configurations containing at least one long cluster with their absolute values.
Let(IV⫺I) be an arbitrary subconfiguration which contains contours K1,...,Kl, belonging to long clusters, K⫽艛1lsupp1K
i, K1⫽K艚关⫺V,⫺(兩I兩/2)兴 and K2⫽K艚关兩I兩/2,V兴.
Put C1((I
V⫺I))⫽兩K1兩 and C2((IV⫺I))⫽兩K2兩. We have
兩P共Long兲兩⫽兩⌶1,⬘,共l.兲/⌶1,⬘兩 ⭐Pabs共Long兲
⫽
兺
wabs共D1兲¯w共Dm兲/⌶1,⬘,共abs兲⫽
兺
p,1
wabs共D1兲...wabs共Dm兲/⌶1,⬘,共abs兲⫹
兺
p,2
wabs共D1兲...wabs共Dm兲/⌶1,⬘,共abs兲 ⫽Pabs共Long,⬎p兲⫹Pabs共Long,⭐p兲,
where wabs(Di)⫽兩w(Di)兩 for all clusters belonging to the configuration containing at least one
long cluster and wabs(Di)⫽w(Di) for other clusters 关note that the statistical weight wabs(Di) of
fixed cluster in one configuration can be positive, in other negative兴, last two summations are taken over all nonordered compatible collections of clusters关D1,...,Dm兴 containing at least one
long cluster, 艛im⫽1supp Di傺IV⫺I corresponding to the boundary conditions 兵1(x),x苸Z1
⫺IV;
⬘
(x),x苸I其, the summation in 兺p,1 is taken over all configurations (IV):(I)⫽⬘
(I);2C1((IV⫺V))/(兩IV兩⫺兩I兩)⬎p; 2C2((IV⫺V))/(兩IV兩⫺兩I兩)⬎p, the summation in 兺p,2is taken
over all configurations (IV):(I)⫽
⬘
(I); 2C1((IV⫺V))/(兩IV兩⫺兩I兩)⭐p; 2C2((IV⫺V))/(兩IV兩⫺兩I兩)⭐p. It means that the density of contours belonging to long clusters in each
configuration from兺p,1(兺p,2) in both segments关⫺V,⫺(兩I兩/2)兴 and 关兩I兩/2,V兴 is greater than p 共is not greater than p兲.
We fixed the value of p as 1⫺q/2l, where the values of q and l will be defined in the proof of Lemma 9.
It turns out that the long clusters are negligible.
Lemma 8: For each fixed interval I there exists a value of V0, such that if V⬎V0
Pabs共Long兲⫽Pabs共Long,⬎p兲⫹Pabs共Long,⭐p兲⬍1/2. 共22兲 Lemma 8 is a consequence of the following two lemmas.
Lemma 9: For each fixed interval I there exists a value of V0, such that if V⬎V0 Pabs共Long,⬎p兲⬍1/4.
Lemma 10: For each fixed interval I there exists a value of V0, such that if V⬎V0 Pabs共Long,⭐p兲⬍1/4.
Proof of Lemma 9: Consider the partition of Z1 into segments Tk⫽Tk(l p), where Tk(l p) is
the segment with the center at x⫽(lp/2)⫹klp and with the length lp (Tkconsists of l segments Ik
with the length p, where p is the period of the special ground state兲. The value of l will be defined later. Let us consider an arbitrary configuration (x). We say that a segment Ik is regular, if Ik
does not belong to the support of some long cluster. We say that a segment Tkis super-regular, if
Tkcontains at least one regular segment.
Let PVbe a Gibbs measure corresponding to the boundary conditions1(x), x苸Z1,
⬘
(I), x苸I.Let the segment IV⫺I consist of n segments Tk; k⫽1,...,n.
We define a sample space⍀ consisting of 2n elementary events Aj⫽关(1),...,(n)兴, where (k), k⫽1,...,n takes two values: (k)⫽0 corresponds to the case when the segment Tk is
super-regular and(k)⫽1 corresponds to the case when the segment Tkis not super-regular. On
the sample space⍀ we define two different probability spaces (⍀,P1) and (⍀,P2) by the follow-ing formulas:
P1共Aj兲⫽P1关共1兲,...,共n兲兴⫽PV关共1兲,...,共n兲兴,
where PVis the Gibbs distribution PV, corresponding to the boundary conditions 1(x), x苸Z1,
⬘
(I), x苸I andP2共Aj兲⫽P2关共1兲,...,共n兲兴⫽qn⫺s共1⫺q兲s,
where s denotes the total number of 1 entries of the vector Aj⫽关(1),...,(n)兴.
We define a random vector ((1),(2),...,(n)) on the probability space (⍀,P1) and, respectively, a random vector ((1),(2),...,(n)) on the probability space (⍀,P2) by the for-mulas:
共k兲共Aj兲⫽共k兲 and 共k兲共Aj兲⫽共k兲 .
The random variables(k) and(k) are defined on the same sample space but on different probability spaces.
Due to the definitions, the random variables(k) are dependent, and the random variables (k) are independent and identically distributed.
Consider the two sums兺kn⫽1(k) and兺kn⫽1(k). Suppose that
P共共m兲⫽1兩any conditions outside Tm兲⭐1⫺q. 共23兲 Note that P((m)⫽1兩any conditions outsideTm)⭐1⫺q⫽P((m)⫽1) and therefore the fol-lowing natural lemma holds.
Lemma 11:
P
冉
兺
k苸K 共k兲⭓l
冊
⭐P冉
k兺
苸K共k兲⭓l冊
for all natural values of l.
The proof of the probabilistically clear Lemma is omitted. For the detailed proof see the Proposition in Ref. 15.
The random variables (k) are independent and identically distributed. The mathematical expectation of (k) equals 1⫺q.
Now we show that
Let PV be a Gibbs measure corresponding to arbitrary boundary conditions and Tk be an
arbitrary segment. Consider the set of all configurations on the interval Tkand the restriction of the
measure PVon this set. We show that at some value of l the ‘‘absolute probability’’ Pabsthat in Tk
there is at least one regular segment Ikis greater than q⬎0 for some constant q not depending on
k. The event(k)⫽1 means that all segments belonging to Tk are nonregular.
Suppose that a fixed configuration
⬘
(Tm) does not coincide with the ground state at all Ii苸Tm.
The Peierls argument method directly imply that for some positive constant t0 Pabs共
⬘
共Tm兲兩conditions outside Tm are gr共x兲兲⭐exp共⫺t0l兲.Note that when we increase the value of l the influence of the conditions outside Tmon the
configuration in Tmincreases with the rate less than l and therefore at some value of l and for some
positive constant t we have
Pabs共
⬘
共Tk兲兩any conditions outside Tm兲⭐exp共⫺tl兲⭐1⫺q0.Thus, the probability Pabs((m)⫽1兩any conditions outside Tm) as a union of at most 2l p
events with probabilities less than 1⫺q0, is bounded by some number 1⫺q. The inequality 共24兲 is proved.
Now Lemma 9 is a direct consequence of the strong law of large numbers for(k) and the Lemma 11. Indeed, consider independent Bernoulli trials when the probability of success at each trial is 1⫺q. According to the law of large numbers, the probability of the event that the density of successes exceeds 1⫺q
⬘
; 0⬍q⬘
⬍q, is less than 1/4, when V tends to infinity. It means that the ‘‘absolute probability’’ of the event that the density of non-super-regular segments Tk is greaterthan 1⫺q
⬘
is less than 1/4. Due to Lemma 11, this probability is greater than the Pabsprobability of the event that the density of non-super-regular segments Tm is greater than 1⫺q⬘
. In otherwords, the Pabsprobability of the event that the density of super-regular segments Tmis less than
1⫺q
⬘
is less than 1/4. Thus, the Pabsprobability of the event that the density of super-regular segments Tmis greater than 1⫺q⬘
is greater than 1/4. Taking into account that each super-regularsegment Tm contains at least one regular segment, one can see that the last statement implies the
Lemma 9 if the parameter p is chosen from the open interval (1⫺q
⬘
/l,1). We choose the value of p as 1⫺q/2l.Lemma 9 is proved.
Proof of Lemma 10: Let us consider the set of all long clusters Diwith the density of supports
less than p. Let supp(D)⫽艛ir⫽ jsupp(Kj). These supports Ki are connected between themselves
and with the boundary. Since the density of supports is not greater than p⬍1, the sum of the lengths of bonds in both halves 关⫺V,⫺兩I兩/2 and 关兩I兩/2,V兴 is not less than (V⫺兩I兩/2)(1⫺p). When V goes to infinity the sum of lengths of bonds of any long cluster with the density less than p tends to infinity. As it becomes apparent from the proof of Lemma 8 Pabs(Long,⬎p) does not exceed one. And it does not exceed one, if we omit the factor g(x, y ) corresponding to the long bond and since g(x, y ,)⫽exp(⫺f(x,y,))⫺1 关see 共12兲兴 the impact of these bonds tends to zero. By choosing the appropriate value of V we complete the proof of Lemma 10.
Lemma 10 is proved.
We omit the huge proof of Lemma 5 since it is absolutely analogous to the proof of Lemma 6. The only difference is the fact that in⌶1,⬘,2,⬙overlapped clusters are allowed, so the density of nonregular segments of typical configurations in Lemmas 8,9 instead of p will be a number less than 1⫺(1⫺p)(1⫺p).
Partition functions including only non-long-super clusters satisfy the following key lemma which has a geometrically-combinatorial explanation.
Lemma 12:
where the factor Q⫽Q(1(x),2(x),
⬘
(x),⬙
(x)) is uniformly bounded: 0⬍const1⬍Q ⬍const2.The factor appears due to the fact that configurations with minimal energy corresponding to the different boundary conditions do not coincide everywhere 共they coincide to within shifts, everywhere but finite area兲.
Proof of Lemma 12: Due to the constant Q without loss of generality we assume that the configurations with minimal energymin for both boundary conditions coincide.
According to the definitions and Lemma 4
⌶1,⬙,2⬘,共n.l.兲⫽Q
⬘
兺
*w共SD1兲¯w共SDm兲,
where the summation is taken over all nonlong, nonordered compatible collections of super clus-ters.
According to the definition of the super cluster
Q
⬘
兺
* w共SD1兲¯w共SDm兲⫽Q⬘
兺
1,⬘,* w共D1兲¯w共Dk兲兺
2,⬙,* w共D1兲¯w共Dl兲in 兺1,⬘,* and 兺2,⬙,* the summation is taken over all nonordered collections of clusters w(D11,⬘)¯w(Dk1,⬘) and w(D2,1⬙)¯w(Dl2,⬙) such that their product belongs to 兺*.
Similarly, ⌶1,⬙,2,⬘,共n.l.兲⫽Q
⬙
兺
** w共SD1兲¯w共SD m兲 ⫽Q⬙
兺
1,⬙,** w共D1兲¯w共Dk兲兺
2,⬘,** w共D1兲¯w共Dl兲.In order to prove Lemma 12 we put one-to-one correspondence between 兺*w(SD1)...w(SDm) and兺**w(SD1)...w(SDm).
Let us consider an arbitrary term U⫽w(SD1)...w(SDa) of兺*. By definitions
U⫽w共D11,⬘兲¯w共Dm1,⬘兲w共D12,⬙兲¯w共Dk2,⬙兲, where the factor w(D1
1,⬘
)¯w(Dm
1,⬘
) belongs to the兺1,⬘,* and the factor w(D1 2,⬙
)¯w(Dk
2,⬙ ) be-longs to the 兺2,⬙,*.
A cluster D⫽关(Ki,i⫽1,...,r);G
⬙
兴 is said to be basic, if the set((艛im⫽1supp Di)艛G
⬙
)艚((Z1⫺IN)艛I)) is not empty. In Fig. 1 all clusters are basic.
Consider the set of all clusters W(U) of the term U: W(U)⫽艛im⫽1Di1,⬙ 艛ik⫽1Di2,⬘ and four subsets of W(U):
W
⬘
⫽冋
D⬘
⫽关(Ki,i⫽1,...,r);G⬘
兴苸⌶2,⬘:冉冉
艛 i⫽1k
supp Di
冊
艛G⬘
冊
艚I is not empty册
,W
⬙
⫽冋
D⬙
⫽关共Ki,i⫽1,...,r兲;G⬘
兴苸⌶1,⬙:冉冉
艛 i⫽1m
supp Di
冊
艛G⬘
冊
艚I is not empty册
,W1⫽
冋
D1⫽关共Ki,i⫽1,...,r兲;G⬘
兴苸⌶1,⬙:冉冉
艛 i⫽1m
supp Di
冊
艛G⬘
冊
艚共Z1⫺IN兲 is not empty册
,W2⫽
冋
D2⫽关共Ki,i⫽1,...,r兲;G⬘
兴苸⌶2,⬘:冉冉
艛 i⫽1k
supp Di
冊
艛G⬘
冊
艚共Z1⫺IN兲 is not empty册
.Note that the subsets W
⬘
,W⬙
,W1,W2contain only basic clusters and the union of them contain all basic clusters of the term U.Let us consider an arbitrary term U⫽w(SD1)¯w(SDb) of⌺**. By the definitions U
⬘
⫽w共D11,⬙兲¯w共Dl1,⬙兲w共D12,⬘兲¯w共Dn2,⬘兲,where the factor w(D11,⬙)¯w(Dm1,⬙) belongs to the ⌺1,⬙,** and the factor w(D12,⬘)¯w(Dk2,⬘) belongs to the ⌺2,⬘,**.
Consider the set of all clusters W(U
⬘
) of the term U⬘
:W(U⬘
)⫽艛im⫽1Di1,⬙艛ik⫽1Di2,⬘. In just the same way we can define four subsets of W(U⬘
).Consider a term U⫽w(D1)¯w(Dk)苸⌺*, containing only basic clusters. By definition
艛i⫽1 k D i can be represented as 艛i⫽1 k D i⫽(艛i⫽1 m D i)艛(艛i⫽m⫹1 k D
j), where the clusters 艛i⫽1
m D i ⫽W1艛W
⬘
; and艛 i⫽m⫹1 k Dj⫽W2艛W⬙
.From the definition of nonlong clusters and W
⬘
,W⬙
,W1,W2 it easily follows that there exists the same term U⬘
⫽w(D1)...w(Dk)苸⌺**, such that艛i⫽1k Di⫽(艛i⫽1 m Di)艛(艛i⫽m⫹1 k Dj), where
the clusters艛im⫽1Di⫽W1艛W
⬘
; and艛i⫽m⫹1 kDj⫽W2艛W
⬙
.Figure 1 shows four collections of clusters COL1⫽关D11,⬙,D21,⬙,D31,⬙,D41,⬙兴, COL2 ⫽关D52,⬘
,D62,⬘,D72,⬘,D2,8⬘兴, COL3⫽关D11,⬘,D61,⬘,D1,7⬘,D41,⬘兴, COL4⫽关D52,⬙,D2,2⬙,D32,⬙,D82,⬙兴.
Two coincident terms U⫽U
⬘
⫽⌸i8⫽1w(Di) belonging to the sums ⌺* and ⌺** arecon-structed by the Cartesian product of the collections COL1, COL2, and COL3, COL4, respectively. We see that between terms U苸⌺*and U
⬘
苸⌺**containing only basic clusters we easily can put a one-to-one correspondence.Consider a term U⫽w(D1)¯w(Dk)w(Dk⫹1)¯w(Dn)苸⌺*, containing basic clusters
D1¯Dk and not basic clusters Dk⫹1¯Dn.
It can be easily shown that there exists a term U
⬘
⫽w(D1)¯w(Dk)w(Dk⫹1)¯w(Dn)⫽w(D1)¯w(Dk)w(Dk⫹1)¯w(Dn)苸⌺**coinciding with the term U苸⌺*. Then, according to the
definition of the long clusters, we directly get that, the term U contains long super cluster, which contradicts the definition of⌺*.
Lemma 12 is proved.
Remark: The essential point of the proof of the important Lemma 12共therefore, of this paper兲 is the amusing fact that⌺*w(SD1)¯w(SDm) and⌺**w(SD1)¯w(SDm) coincide.
Now the demanded inequality共18兲 is a direct consequence of Lemmas 5 and 12. The inequal-ity共18兲, therefore Lemma 3 is proved.
Let P1 and P2 be two different extreme Gibbs states of the model共1兲 corresponding to the boundary conditions1(x) and2(x), respectively.
Theorem 5:共Ref. 16.兲 P1 and P2are singular or coincide.
Proof of Theorem 4: Let P1 and P2 be two different extreme Gibbs states of the model共1兲 corresponding to the boundary conditions1(x) and2(x) respectively. According to Lemma 3 P1 and P2 are not singular. Therefore, according to Theorem 5 P1 and P2coincide, which con-tradicts the assumption. Theorem 4 is proved.
III. UNIQUENESS OF GIBBS STATES
In this section we prove the main Theorem 1.
The statement of Theorem 1 for rational densities coincides with Theorem 4. Thus, in order to complete the proof of Theorem 1, we have to prove the following theorem, which covers the case when the density of the special ground state is irrational.
Theorem 6: Suppose that the value of the external fieldof the model (1) belongs to the set Cir⫽R1⫺艛(⫺,⫹). Then the model (1) has a unique Gibbs state at all values of the tem-perature⫺1.
It can be easily shown that the special ground states of the model共1兲 are not stable when the density is irrational. In other words, the Peierls constant t for the special ground state tends to zero, when p→⬁. The essence of this fact is the following.
For the fixed irrational number ⫽关n0,n1,...,ns,...兴 consider the corresponding special
ground state(x) and its arbitrary perturbation
⬘
(x). The configuration⬘
(x) is not a special ground state, therefore for some pair of points, say x and y苸Z1;
⬘
(x)⫽⬘
(y )⫽1, we have a violation of Hubbard’s criterion. Let x and y be closest points with this property. When the distance between x and y tends to infinity, the Peierls constant tends to zero.In the irrational case the special ground states are not stable, but this fact is not crucial for our method. Since the essence of our method is the estimation of long super clusters connecting the boundary with the segment I, small clusters not satisfying Peierls condition cannot ‘‘help’’ to connect the boundary with I, and it turns out that big clusters satisfy the Peierls stability condition and the method works. One can say that the special ground states in the irrational case are ‘‘stable in general.’’
Below we give the mathematical details of the last observation. Consider(s)⫽关n0,n1,...,ns兴.
Lemma 13: Suppose that the value of the external field of the model (1) belongs to the interval ((s)⫺ ,(s)⫹ ) for some number (s)⫽(s)⫺1. Let
⬘
(x) be an arbitrary finite pertur-bation of the special ground state(s)(x) such that the boundary B of the configuration⬘
(x) includes a unique contour K. Then there exists a positive constant ts depending only on theHamiltonian (1), such that
H共
⬘
共x兲兲⫺H共共s兲共x兲兲⭓ts兩supp B兩where兩supp B兩 is the total area of the support of the boundary.
Lemma 13 was proved in Ref. 13关see Lemma 1 and Sec. 5 共Ref. 13兲兴.
Thus, for each nonnegative integer s the number tsis defined. Suppose that a positive number
t less than t1 is fixed. Let s be the maximal number meeting the condition ts⬎t.
Let us consider an arbitrary configuration(x). Let C⫽艛i苸Ind关xi,yi兴, where xi,yi苸Z1and
xi⫽yihas the following properties:
共1兲 For each segment 关ai,bi兴 from the set Z1⫺C there exists a special ground state, such
that the restriction of this configuration on关ai,bi兴 coincides with(关ai,bi兴).
共2兲 For any C
⬘
傺C; C⬘
⫽C the property 1 is not held.It can be easily shown that the set C⫽C((x)) is not uniquely defined. Suppose that, some rule uniquely determines the set C for each configuration(x). Let Z1⫺C⫽艛i关ai,bi兴. We say
that(关ai,bi兴); is a preregular phase. Consider any segment 关xi,yi兴 belonging to C. The segment
关xi, yi兴 is said to be t-negligible, if for each segment 关vi,wi兴 covering 关xi,yi兴,wi⫺vi⫽p 关p is the
numerator of (s)] there exists a special ground state (s), such that the restriction of this configuration on 关vi,wi兴 coincides with (关vi,wi兴). Let C⫽艛i苸Ind关xi, yi兴⫽(艛i苸Ind(t)
⫻关xi,yi兴)艛(艛i苸Ind-Ind(t)关xi,yi兴), where Ind(t) means that the union is taken over all t-negligible
segments. The support of the preboundary supp PB of the configuration(x) will be defined as supp PB⫽(艛i苸Ind(t)关xi, yi兴)艛(艛i苸Ind-Ind(t)关xi⫺d0p,yi⫹d0p兴)⫽supp PB共main兲艛supp PB(t). Each segment belonging to the union supp PB will be called a support of a precontour and is denoted by supp PK. The support 关xi,yi兴 of a precontour is said to be t-negligible, if 关xi, yi兴
belongs to supp PB(t).
We define contours as in the Definition 1. The constants p,d0 and Nb for irrational density
⫺1will be constants defined for rational density (s)⫺1.
The pair PK⫽(supp PK,
⬘
(supp PK)) is called a precontour. The set of all precontours is called a preboundary PB of the configuration⬘
(x). Two precontours PK1 and PK2 are said to be connected if dist共supp PK1,supp PK2)⬍Nband at least one of them is not t-negligible. The set of precontours ( PKi;i苸Ind) is called connected if for any two precontours PKc and PKd;c,d 苸Ind there exists a collection (PKj1⫽PKc,..., PKji,..., PKjn⫺1, PKjn⫽PKd); ji苸Ind, i⫽1,...,n; such that any two precontours PKji and PKji⫹1, i⫽1,..., n⫺1 are connected. Let
艛i⫽1
n PK
i be some maximal connected component of the preboundary PB. Suppose that
supp PKi⫽关ai,bi兴 and bi⬍ai⫹1; i⫽,...,n⫺1.
The pair K⫽(supp K,
⬘
(supp PK)), where supp K⫽关a1,bn兴 is called a contour. The set ofall contours is called a boundary B of the configuration
⬘
(x). A contour is said to be t-negligible, if its support is t-negligible.By the definitions, the distance between the supports of two t-negligible contours exceeds p, where p is the numerator of(s) and the length of the support of any t-negligible contour is one.
The following lemma is reformulation of Lemma 13 for irrational densities.
Lemma 14: Suppose that the value of the external fieldof the model (1) belongs to the set Cir⫽R1⫺艛(⫺,⫹). Let
⬘
(x) be an arbitrary finite perturbation of the special ground state (x) such that the boundary B of the configuration ⬘
(x) includes a unique contour (not t-negligible contour) K. Then there exists a positive constant tsdepending only on the Hamiltonian(1), such that
H共
⬘
共x兲兲⫺H共共x兲兲⭓ts兩supp B兩where兩supp B兩 is the total area of the support of the boundary.
Suppose that the value of the external field of the model 共1兲 belongs to the set Cir⫽R1 ⫺艛(⫺,⫹). Let t,0⬍t⬍t1 is fixed and ts is chosen as above.
Lemma 15: Letmin(x)苸⌽(V) be a configuration with the minimal energy:
H共min共x兲兩1共x兲兲⫽min共x兲苸⌽共V兲H共共x兲兩1共x兲兲.
Then the configurationmin(x) has the following structure:
The restriction of the configurationmin(x) on the set关⫺V⫹Nb,V⫺Nb兴 contains t-negligible
