CC BY
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NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2016, 7 (3), P. 401-404
Functional equations for the Potts model with competing interactions on a Cayley tree
G. I. Botirov
Institute of Mathematics, National University of Uzbekistan botirovg@yandex.ru
PACS 05.50.+q, 05.70.Fh, 02.30.-f, 02.50.Ga DOI 10.17586/2220-8054-2016-7-3-401-404
In this paper, we consider an infinite system of functional equations for the Potts model with competing interactions of radius r = 2 and countable spin values 0,1,..., and non-zero-filled, on a Cayley tree of order two. We describe conditions on hx guaranteeing compatibility of distributions ^(n)(<rn).
Keywords: Cayley tree, Potts model, Gibbs measures, functional equations.
Received: 23 March 2016 Revised: 13 April 2016
1. Introduction
The Potts model is related to and generalized by several other models, including the XY model, the Heisenberg model and the N-vector model. The infinite-range Potts model is known as the Kac model. When the spins are taken to interact in a non-Abelian manner, the model is related to the flux tube model, which is used to discuss confinement in quantum chromodynamics. Generalizations of the Potts model have also been used to model grain growth in metals and coarsening in foams. A further generalization of these methods by James Glazier and Francois Graner, known as the cellular Potts model, has been used to simulate static and kinetic phenomena in foam and biological morphogenesis. In this model, introduced by Askin and Teller (1943) and Potts (1952), the energy between two adjacent spins at vertices i and j is taken to be zero if the spins are the different and equal to a constant Jj if they are same.
In [1], the Potts model with countable set $ of spin values on Zd was considered and it was proved that with respect to Poisson distribution on the set of limiting Gibbs measure is not empty. In [2], the Potts model with a countable set of spin values on a Cayley tree was considered and it was shown that the set of translation-invariant splitting Gibbs measures of the model contains at most one point, independent of parameters for the Potts model with countable set of spin values on the Cayley tree. This is a crucial difference from models with a finite set of spin values, since those may have more than one translation-invariant Gibbs measures.
The work initiated in [4] was continued in [3] and a model was considered with nearest-neighbor interactions and local state space given by the uncountable set [0,1] on a Cayley tree rk of order k > 2. The translationinvariant Gibbs measures are studied via a non-linear functional equation and we prove the non-uniqueness of translation-invariant Gibbs measures in the right parameter regime for all k > 2 and not only for k g {2, 3} as in [3]. In [5], models (Hamiltonians) with-nearest-neighbor interactions and with the set [0,1] of spin values, on a Cayley tree rk of order k > 1 were studied.
In this letter, we consider Potts model with competing interactions and countable spin values and we derive an infinite system of functional equations for the Potts model on a second order Cayley tree.
2. Preliminaries
The Cayley tree (Bethe lattice) rk of order k > 1 is an infinite tree, i.e., a graph without cycles, such that exactly k + 1 edges originate from each vertex. Let rk = (V, L) where V is the set of vertices and L the set of edges. Two vertices x and y are called nearest neighbors if there exists an edge l e L connecting them and we denote l = (x,y). A collection of nearest neighbor pairs (x, xi), (x1,x2),..., (xd-1,y) is called a path from x to y. The distance d(x, y) on the Cayley tree is the number of edges of the shortest path from x and y.
For a fixed x0 € V, called the root, we set:
Wn = {X G V|d(x, X0) = n}, Vn = Q W„
m= 1
and denote:
S(x) = {y G Wn+i : d(x, y) = 1}, x G W„,
the set of direct successors of x.
The vertices x and y are called next-nearest-neighbor (NNN) which is denoted by }x,y(, if there exists a vertex z G V such that x, z and y, z are nearest-neighbor. We consider NNN }x, y(, for which there exists n such that x g Wn and y G Wn+2, this kind of NNN is considered with the three states Potts model (see [6]).
We consider a Potts model with competing nearest-neighbor and prolonged next-nearest-neighbor interactions on a Cayley tree where the spin takes values in the set $ := 0,1, 2,.... A configuration a on V is then defined as a function x G V ^ a(x) G the set of all configurations is .
The Hamiltonian for the Potts model with competing interactions has the form:
H(a) = -J X/ ^(xMy) - J1 X/ S^(x)^(y), (2.1)
(x,y) )x,y{
where J, J1 G R are coupling constants and S is the Kroneker's symbol.
Let A be the Lebesgue measure on [0,1]. For the set of all configurations on A, the a priori measure Aa is introduced as the |A| fold product of the measure A. Here and subsequently, |A| denotes the cardinality of A. We consider a standard sigma-algebra B of subsets of Q = [0,1]V generated by the measurable cylinder subsets. A probability measure ^ on (Q, B) is called a Gibbs measure (with Hamiltonian H) if it satisfies the Dobrushin-Lanford-Ruelle (DLR) equation, namely for any n =1, 2,..., and an G QVn:
H ({a G Q : j|vn = a„}) = J
Vn
where is the conditional Gibbs density:
vViW„+i (an) = Z7-1-) exP { - (a„ 1 |^|W„+1 ^,
1 "+1 Zn("|Wn+1) ^ J
and ft = 1, T > 0 is the temperature.
Let Ln = {(x, y) G L : x, y G Vn} and QVn is the set of configurations in Vn (and QWn that in Wn). Furthermore, j|v^ and "|W denote the restrictions of configurations j, w G Q to Vn and Wn+1, respectively. Next,
an : x G Vn ^ an(x) is a configuration in Vn and H (an|MW„+1) is defined as the sum H(an)+U ^an, "|Wri+1 j where:
H (jn) = -J ^ ^(x^n^
(x,y)£Ln
U(jn,"|w„+J =-J ^ e
CT„(x)w(y)-
(x,y): xev„, yeWn+i
Finally, Zn +i j represents the partition function in Vn, with the boundary condition
Zn (HWn+J = J eXp{ - (5n || HWn+J }AVn (d?n).
"I w :
IWn + 1
„0
We write x < y if the path from xu to y goes through x. We call vertex y a direct successor of x if y > x and x,y are nearest neighbors. We denote by S(x) the set of direct successors of x and observe that any vertex x = x0 has k direct successors and x0 has k + 1.
Let h : x G V ^ hx = (ht,x,t G [0,1]) G R[0,1] be a mapping of x G V \ {x0}. Given n =1,2,..., consider the probability distribution ^(n) on QVn defined by:
M(n)(an) = Z—1 exp J - ftH(an) + £ hff(x),J. (2.2)
I xEWn J
Here, as before, an : x G Vn ^ a(x) and Zn is the corresponding partition function:
Zn = exp < - ftH(?n)+ h5(x),x MVn(dan). (2.3)
o{ I x£W„ J
Functional equations for the Potts model with competing interactions on a Cayley tree 403
The probability distributions ^(n) are called compatible if for any n > 1 and <7n-1 e :
J M(n) (a„_i V Wn) Aw„ (d(wn)) = m("-1) K-1). (2.4)
Here, <rn-1 V wn e is the concatenation of <rn-1 and wn. In this case, because of the Kolmogorov extension theorem, there exists a unique measure ^ on such that, for any n and an e , = °"n j^j =
M(n)(ffn). Such a measure is called a splitting Gibbs measure corresponding to Hamiltonian (2.1) and function
x i ^ hx, x x .
The following theorem describes conditions on hx guaranteeing compatibility of distributions ^(n)(an). 3. Functional Equations
Theorem 3.1 Probability distributions ^(n)(an), n = 1,2,..., in (2.2), for a Cayley tree of order two, are compatible iff for any x e V \ {x0} the following equation holds:
h!,x = Fi(hy ,h^,J), i = 1, 2,..., (3.1)
where S(x) = {y, z}, hX = ( h1,x - ho,x + ln ^^, h2,x - ho,x + ln ^^, .. . ) and
V v (0) v(0) J
1 + Eo exp {J(<Sip + ) + J^, + hp,y + hq,^} ^¿(hy,hz)=in ^
1+ ^ expjJ(¿op + ¿oq) + Ji^^pq + +
^pq
p,q = 0 ^
p + q = 0
Proof. Necessity Assume that (2.4) holds; we will prove (3.1). Substituting (2.2) in (2.4), obtain that for any configurations an-i : x G Vn—i ^ an-i(x) G
1 " exp^ - ^H„(a„)+ V hCT(x),x }> x II v (a(y))
Z^ ^ exPS - ^H„(a„)+ ^ hff(x),x > x H
exM - ^H„_i(a„_i)+ hCTn-i(x),x f.
---1" I r--n — i v - n — i / i / , '"^n
n 1 I xeWn-i
^ ex^ i(a„—i)+J^ ^ (¿<x(xMy) + ¿„(xWz)) +Ji^ E
n I xGWn-i xGWn-i xGWn J
y,z<=S(x) y,z<=S(x)
X n v(a(y)) = exp < - i(a„— i) + ^ hffn-i(x),x f.
After some abbreviations, we obtain:
Zn
Zn
n XI exM + ¿CT(x)CT(z)) +hCT(z),z +lnv(a(y))+lnv(a(z))
xeWn- „(n)={ff(y)jff(z)}
n ex^ h*n-i(x),x} .
xeWn-i
Consequently, for any i G
exp {ho,y + ho,z + 2ln v(0)) + ] exp { J^(¿¿p + ¿¿q) + Ji^pq + hp,y + hq,z + ln v(p) + ln v
^ ^ p,q = 0 ^
p+q=0
exp j ho,y + ho,z + 2ln v(0) j + J] exp { J^op + ¿oq) + Ji^pq + hp,y + hq,z + ln v(p) + ln v(q)|
P+q=0
= exp j hi,x - ho,x},
1
such that:
1+ ^ expj Jft^ip + Siq ) + JiftSpq + hp,y + hq,z|
7 * i p+q=0
hi,x = ln-5S-;-r
1+ ex^ Jft(^0p + ¿0q)+ JiftSpq + hp,y + hq,z
V (i)
where:
ht x = hi,x - ho,x + ln -
v (0)'
Sufficiency. Let (3.1) is satisfied we will prove (2.4).
to
^ exp { Jft(Sip + Siq) + J^Spq + hp,y + hq,z + ln v(p) +ln v(q)} = a(x)exp { hj,x}, (3.2)
p,q=0
here i = 0,1,.... We have:
LHS of (2.4) = Z-exp{ - ^H„_1(a„_1^ fl v(a(x))x
^ exP { (Sa(x)a(y) + ¿CT(x)CT(z)) + J1 ft^j(y)j(z) + hj(y),y + hj(z),zj • (3-3)
xe wn-1 y ,«ES(i)
Substituting (3.2) into (3.3) and denoting An = ^Q a(x), we get:
xeWn -1
RHS of (3.3) = Az1 exp{ - ftHn_i(jn-i^ n hjn - i(x),x. (3.4)
n x£Wn-i
Since M(n), n > 1 is a probability, we should have:
££M(n) (an-i,J(n-i)) =1. jn-1 j
Hence, from (3.4) we obtain Zn-iAn-i = Zn, and (2.4) holds. References
[1] Ganikhodjaev N.N. The Potts model on Zd eith countable set of spin values. J. Math. Phys., 2004, 45, P. 1121-1127.
[2] Ganikhodjaev N.N., Rozikov U.A. The Potts model with countable set of spin values on a Cayley tree. Letters in Math. Phys., 2006, 75, P. 99-109.
[3] Eshkabilov Yu.Kh., Rozikov U.A., Botirov G.I. Phase transition for a model with uncountable set of spin values on Cayley tree. Lobachevskii Journal of Mathematics, 2013, V.34 (3), P. 256-263.
[4] Eshkobilov Yu.Kh., Haydarov F.H., Rozikov U.A. Non-uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree. J. Stat. Phys., 2012, 147 (4), P. 779-794.
[5] Rozikov U.A., Eshkabilov Yu.Kh. On models with uncountable set of spin values on a Cayley tree: Integral equations. Math. Phys. Anal. Geom., 2010, 13, P. 275-286.
[6] Ganikhodjaev N.N., Mukhamedov F.M., Pah C.H. Phase diagram of the three states Potts model with next nearest neighbour interactions on the Bethe lattice. Phys. Lett. A, 2008, 373, P. 33-38.
