J Stat Phys (2011) 142: 314–321 DOI 10.1007/s10955-010-0106-6
A New Set of Limiting Gibbs Measures for the Ising Model on a Cayley Tree H. Akin · U.A. Rozikov · S. Temir
Received: 16 September 2010 / Accepted: 1 December 2010 / Published online: 16 December 2010 © Springer Science+Business Media, LLC 2010
Abstract For the Ising model (with interaction constant J > 0) on the Cayley tree of order k ≥ 2 it is known that for the temperature T ≥ Tc,k = J / arctan(1/k) the limiting Gibbs measure is unique, and for T < Tc,k there are uncountably √ many extreme Gibbs measures. In the Letter we show that if T ∈ (Tc,√k , Tc,k0 ), with k < k0 < k then there is a new uncountable set Gk,k0 of Gibbs measures. Moreover Gk,k0 = Gk,k0 , for k0 = k0 . Therefore if T ∈ (Tc,√k , Tc,√k+1 ), Tc,√k+1 < Tc,k then the set oflimiting Gibbs measures of the Ising model contains the set {known Gibbs measures}∪( k0 :√k
H. Akin Faculty of Education, Department of Mathematics, Zirve University, Kizilhisar Campus, 27260 Gaziantep, Turkey e-mail:
[email protected] U.A. Rozikov () Institute of Mathematics and Information Technologies, 29, Do’rmon Yo’li str., 100125 Tashkent, Uzbekistan e-mail:
[email protected] S. Temir Department of Mathematics, Arts and Science Faculty, Harran University, Sanliurfa 63120, Turkey e-mail:
[email protected]
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mechanics. The spins are arranged in a lattice or graph (in particular Cayley tree), and each spin interacts only with its nearest neighbors. It is well known that for 1-dimensional Ising model there is a unique Gibbs state (for positive temperature) and that in dimension 2, a phase transition occurs at some critical temperature above which there is only one state and below which 2 extreme translational invariant states coexist. It is natural to ask whether or not the Ising ferromagnetic also admits some Gibbs measures which fail to be translational invariant. In three or more dimensions this is indeed the case and was shown by Dobrushin [7]. In this paper we consider Ising model on a Cayley tree. Models on a Cayley tree were discussed in many papers (see for example [1–15, 17–20]). In the paper we recall all known Gibbs measures for the Ising model on the Cayley tree and show that under some conditions on the temperature for each known Gibbs measure on Cayley tree of order k0 one can constructed a new Gibbs measure on Cayley tree of order k, k > k0 . There are several classes of non translational invariant Gibbs measures on Cayley tree (see for example [3, 9, 15] which are analogues of the Dobrushin’s non translational invariant measures constructed in [7]). Our result gives one new class of such Gibbs measures on the Cayley tree.
2 Preliminaries The Cayley tree (Bethe lattice [1]) k of order k ≥ 1 is an infinite tree, i.e., a graph without cycles, such that exactly k + 1 edges issue each vertex. Let k = (V , L), where V is the set of vertices of k , L is the set of edges of k . Two vertices x and y are called nearest neighbors if there exists an edge l ∈ L connecting them, which is denoted by l = x, y . A collection of the pairs x, x1 , . . . , xd−1 , y is called a path from x to y. Then the distance d(x, y), x, y ∈ V , on the Cayley tree, is the number of edges in the “shortest” path from x to y. We consider Ising model where the spin takes values in the set := {−1, 1}, and is assigned to the vertices of the tree. A configuration σ on V is then defined as a function x ∈ V → σ (x) ∈ ; the set of all configurations is V . The (formal) Hamiltonian of ferromagnetic Ising model is σ (x)σ (y), (2.1) H (σ ) = −J x,y ∈L
where J > 0 is a coupling constant and x, y stands for nearest neighbor vertices. As usual, one can introduce the notions of Gibbs measure of the Ising model on the Cayley tree [9, 16]. Now we present a number of known results related to the description of a general structure of extreme Gibbs measures on Cayley tree. Our exposition is based on [3, 4, 9, 11, 12, 14, 15, 17]. If an arbitrary edge x 0 , x 1 = l ∈ L is deleted from the Cayley tree k , it splits into two components—two semi-infinite trees 0k and 1k . Theorem 2.1 [3, 9] A Gibbs measure μ on k is an extreme measure if and only if there exist extreme Gibbs measures μ0 , μ1 on 0k , 1k respectively, such that μ = μ0 μ1 Z −1 exp{(J /T )σ (x 0 )σ (x 1 )}, where Z > 0 is the normalizing constant.
(2.2)
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This theorem reduces the description of extreme Gibbs measures on k to the semi-infinite tree 0k . Let V 0 be the set of vertices of 0k and L0 its set of edges. For a fixed x 0 ∈ V we set Wn = {x ∈ V 0 |d(x, x 0 ) = n},
Vn =
n
Wk .
k=1
Denote Sk (x) = {y ∈ Wn+1 : d(x, y) = 1},
x ∈ Wn ,
this set is called a set of direct successors of x. On the tree 0k one can introduce a partial ordering, by saying that y > x if there exists a path x = x0 , x1 , . . . , xn = y from x to y that “goes upwards”, i.e., such that d(xm , x 0 ) = d(xm−1 , x 0 ) + 1, m = 1, . . . , n. The set of vertices Vx0 = {y ∈ V 0 |y ≥ x} and the edges connecting them from the semi-infinite tree xk “growing” from the vertex x ∈ V 0 . Theorem 2.2 [3, 9] Let n ≥ 1. In order for μ to be an extreme Gibbs measure on 0k , it is necessary and sufficient that there exist extreme Gibbs measures μx on xk , x ∈ Wn such that μx , (2.3) μ = Zn−1 exp{(1/T )Hn (σ )} x∈Wn
where Hn (σ ) = −J
x,y :x,y∈Vn
σ (x)σ (y).
Denote by E the set of extreme Gibbs measures on 0k and by F the set of Gibbs measures μ on 0k such that there exists Gibbs measures μx on 0k , x ∈ V 0 , such that for each n ≥ 0 the factorization (2.3) is true. From Theorem 2.2 it follows that E ⊂ F . Define a finite-dimensional distribution of a measure μ ∈ F in the volume Vn as hx σ (x) , (2.4) μn (σn ) = Zn−1 exp −βHn (σn ) + x∈Wn
where β = 1/T , is the inverse temperature, Zn−1 is the normalizing factor and {hx ∈ R, x ∈ V 0 } is a collection of real numbers. We say that the probability distributions (2.4) are compatible if for all n ≥ 1 and σn−1 ∈ Vn−1 : μn (σn−1 ∨ ωn ) = μn−1 (σn−1 ). (2.5) ωn ∈Wn
Here σn−1 ∨ ωn is the concatenation of the configurations i.e. ⎧ ⎨σn−1 (x) if x ∈ Vn−1 (σn−1 ∨ ωn )(x) = ⎩ ωn (x) if x ∈ Wn . In this case there exists a unique measure μ on V such that, for all n and σn ∈ Vn , μ({σ |Vn = σn }) = μn (σn ). Such a measure is called a splitting Gibbs measure corresponding to the Hamiltonian (2.1) and function hx , x ∈ V . The following statement describes conditions on hx guaranteeing compatibility of μn (σn ).
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Theorem 2.3 [3, 9, 19] Probability distributions μn (σn ), n = 1, 2, . . ., in (2.4) are compatible iff for any x ∈ V 0 the following equation holds:
hx =
(2.6)
f (hy , θ ).
y∈Sk (x)
Here, θ = tanh(Jβ), f (h, θ ) = arctan(θ tanh h) and Sk (x) is the set of direct successors of x on Cayley tree of order k. From Theorem 2.3 it follows that for any h = {hx , x ∈ V } satisfying (2.6) there exists a unique Gibbs measure μ and vice versa. However, the analysis of solutions to (2.6) is not easy. It is known periodic (in particular translational invariant) and an uncountable set of nonperiodic solutions of (2.6), and Gibbs measures corresponding to these solutions are also known. The following theorem summarizes the known Gibbs measures of the Ising model on the Cayley tree of order k ≥ 2. Denote Tc,k =
J , arctan(1/k)
Tc,√k =
J
√ . arctan(1/ k)
Theorem 2.4 For the Ising model on the Cayley tree of order k ≥ 2 the following statements are true (1) [9, 14] If T ≥ Tc,k then there is unique Gibbs measure μ0 . (2) [4, 9, 14] If T ∈ (Tc,√k , Tc,k ) then there are 3 extreme Gibbs measures μ− , μ0 , μ+ . (μ0 is called disordered Gibbs measure) (3) [4, 12] If T ∈ (0, Tc,√k ) then the measures μ− , μ0 , μ+ still exist, μ− and μ+ are extreme but the measure μ0 is not extreme. (4) If T < Tc,k then there are the following uncountable sets of Gibbs measures: (4.a) [3] There are uncountably many extreme Gibbs measures μt , t ∈ [0, 1] in which “half” of the Cayley tree is occupied by the plus-phase μ+ and “half” by minusphase μ− . (4.b) [9] There are uncountably many extreme Gibbs measures ν t , t ∈ [−h∗ , h∗ ], h∗ > 0, which are different from μt . (4.c) [9] There are two periodic extreme Gibbs measures μ± , μ∓ for antiferromagnetic (J < 0) Ising model. (4.d) [15] There are 7 weakly periodic Gibbs measures and uncountably many nonperiodic Gibbs measures μtnp constructed by (weakly) periodic Gibbs measures. In this Letter we show that there are another classes of Gibbs measures which are different from the measures mentioned in Theorem 2.4.
3 New Measures In this section we are going to describe new Gibbs measures of the Ising model on the Cayley tree of order k ≥ 2. The main result of the Letter is
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Fig. 1 The function h˜ x on Cayley tree of order k = 3 for k0 = 2. The bold subtree has the set of vertices V k0 = V 2
Theorem 3.1 For the ferromagnetic Ising model (i.e. J > 0) on the Cayley tree of order √ k ≥ 3, if T such that Tc,√k < T < Tc,k0 , with k < k0 < k then there is uncountable set Gk0 ,k of extreme Gibbs measures which are different from the measures mentioned in Theorem 2.4. √ Proof For given k0 ( k < k0 < k) denote by Gk0 the set of all known extreme Gibbs measures for the Ising model on Cayley tree of order k0 . By Theorem 2.4 we know that Gk0 is an uncountable set. Now for any μ ∈ Gk0 , μ = μ0 we shall construct a Gibbs measure ν = ν(μ) which is measure on the Cayley tree of order k > k0 . By Theorem 2.3 to each measure μ ∈ Gk0 corresponds unique function hx = hx (μ) which satisfies (2.6) on k0 . Construct function h˜ x = h˜ x (ν) on k as follows. Let V k be the set of all vertices of the Cayley tree k . Since k0 < k one can consider V k0 as a subset of V k . Define the following function ⎧ ⎨hx (μ), if x ∈ V k0 h˜ x = (3.1) ⎩0, if x ∈ V k \ V k0 . This function on the Cayley tree of order k = 3 for k0 = 2 is shown in Fig. 1. Now we shall check that (3.1) satisfies (2.6) on k . Let x ∈ V k0 ⊂ V k . We have h˜ x = f (h˜ y , θ ) y∈Sk (x)
=
f (h˜ y , θ ) +
y∈Sk (x)∩V k0
f (h˜ y , θ ) =
y∈Sk (x)∩(V k \V k0 )
f (hy , θ ) = hx ,
y∈Sk0 (x)
here we used f (0, θ ) = 0. If x ∈ V k \ V k0 then it is easy to see that Sk (x) ⊂ V k \ V k0 . Therefore we have h˜ x =
y∈Sk (x)
f (h˜ y , θ ) =
f (0, θ ) = 0.
y∈Sk (x)
Thus h˜ x , x ∈ V k satisfies the functional equation (2.6) and we denote by ν = ν(μ) the Gibbs measure which by Theorem 2.3 corresponds to h˜ x . By the construction one can see that ν(μ1 ) = ν(μ2 ) if μ1 = μ2 and the measure ν is different from measures mentioned in Theorem 2.4. Denote by Gk0 ,k the set of all Gibbs measures defined on k which are constructed by Gibbs measures defined on k0 .
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We shall prove that ν is an extreme Gibbs measure. Let us decompose ν into extreme Gibbs measures ν = ν¯ (ω)λ(dω). By definition, ν ∈ F and by Theorem 2.2 ν¯ ∈ F . We use the factorization (2.3) Zn−1 exp{−βHn (σ )} νx = Zn−1 (ω) exp{−βHn (σ )} νx (ω)λ(dω), x∈Wn
from which it follows that
x∈Wn
νx =
(Zn /Zn (ω))
x∈Wn
νx (ω)λ(dω).
x∈Wn
Integrating this equality with respect to σz = {σ (t), t ∈ Vz0 }, z = x, we obtain νx = Ln (ω)νx (ω)λ(dω),
(3.2)
where Ln (ω) > 0. For x ∈ V k \ V k0 we have νx = μ0 (where μ0 is the disordered Gibbs measure, which corresponds to solution hx ≡ 0 to (2.6)), since Tc,√k < T < Tc,k0 , by Theorem 2.4 μ0 is an extreme Gibbs measure, consequently νx (ω) = μ0 for almost all ω with respect to the measure λ(dω). Hence if hy (ω), y ∈ Vx0 corresponds to νx (ω) then hy (ω) = h˜ y (μ) = 0,
for all y ∈ V k \ V k0
for almost all ω.
(3.3)
From (3.3) and (3.2) we get that measure νx is extreme for any x ∈ V k \ V k0 . Now we shall prove that measure νx is extreme for any x ∈ V k0 . Here we use the argument used in the proof of Theorem 3 of [11]. Let |t| denote the distance between x and 0 t ∈ Vx0 ∩ Vn = Vn,x , ω ∈ Vx , s ∈ Vx0 and n ≥ |s| put:
ω (σ ) = −J Hn,s
σ (x)σ (y) − J
x,y : x,y∈Vn,s
ω (ε) = Wn,s
σ (x)ω(y),
x,y : x∈Vn,s , y∈∂Vn,s
ω exp −βHn,s (σ ) − Jβεω(t) ,
ε = ±1,
σ ∈Vn,s :σ (s)=ε ω ω (+1)/Wn,s (−1), Rn,s (ω) = Wn,s
where t is the unique point such that s, t . We shall use the following Lemma 3.2 [11]. Let P be a Gibbs measure of the Hamiltonian (2.1), then (1) P is an extreme measure iff lim Rn,s (ω) = rs a.s.(P )
n→∞
for each s ∈ Vx0 with |s| ≥ N for some N > 0, where rs is a constant depending only on s.
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(2) The limit limn→∞ Rn,s (ω) = rs (ω) exists almost surely with respect to P for every s ∈ Vx0 . Denoting G(x) = (1 + e2Jβ x)/(e2Jβ + x), we have (see formula (22) of [11]) rt (ω) = lim Rn,s (ω) n→∞
=
u∈V κ0 :t,u
G(ru (ω))
v∈V k \V κ0 :t,v
G(rv (ω)) =
G(ru (ω)),
(3.4)
u∈V κ0 :t,u
here we used rv (ω) = 1 for all v ∈ V k \ V k0 which follows from (3.3). Since on V k0 the measure μ is extreme by definition of h˜ x (see (3.1)) and Lemma 3.2 from (3.4) we get rt (ω) = rt a.s. (νx ), x ∈ V k0 . Thus by Theorem 2.2 we conclude that the measure ν is extreme. Theorem is proved. Corollary 3.3 Let Ising model be ferromagnetic on Cayley tree of the order k. If Tc,√k < the √ √ T < Tc, k+1 then Gibbs measures of the set Gk = k0 : k
Acknowledgments UAR thanks the Scientific and Technological Research Council of Turkey (TUBITAK) for support and Zirve and Harran Universities for kind hospitality. This work was completed in the Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy and UAR thanks ICTP for providing financial support and all facilities and IMU/CDE-program for travel support. He also thanks the TWAS Research Grant: 09-009 RG/Maths/As-I; UNESCO FR: 3240230333. We thank all three reviewers for their useful suggestions.
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