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9
УДК 517.98
New Periodic Gibbs Measures for q-state Potts Model on a Cayley Tree
Rustamjon M. Khakimov*
Institute of Mathematics, National University of Uzbekistan, Do'rmon Yo'li, 29, Tashkent, 100125, Uzbekistan
Received 06.03.2014, received in revised form 29.04.2014, accepted 26.05.2014 In this paper under some conditions on parameters of the q-state Potts model on a Cayley tree of order k we prove existence of the periodic (non translation-invariant) Gibbs measures. Also we give a result about the number of such measures.
Keywords: Cayley tree, configuration, Potts model, Gibbs measure, periodic Gibbs measures, translationinvariant measures.
Introduction
The main problem for a given hamiltonian is the description of all corresponding limiting Gibbs measures (see f.e. [1,3]). This problem was fully studied for the Ising model on the Cayley tree. For example, in [4] an uncountable set of extremal Gibbs measures is constructed and in [5] a necessity and sufficient condition of extremity of unordered phase for Ising model on a Cayley tree is found.
The Potts model is a generalization of the Ising model. The Potts model is not studied to the same extent as the Ising model. For example, in [6] a ferromagnetic Potts model with three-states on a second-order Cayley tree was considered and it was proved that there exists a critical temperature Tc > 0 such that for T < Tc, there are three translation-invariant and uncountably many not translation-invariant Gibbs measures. The results of [6] on the Potts model with finitely many states were generalized to a Cayley tree of an arbitrary (finite) order in [7].
It was proved [8] that the translation-invariant Gibbs measure of the antiferromagnetic Potts model with an external field is unique. In [9] the Potts model with a countable number of states and nonzero external field on a Cayley tree was considered. It is proved that this model has a unique translation-invariant Gibbs measure.
Other properties of the Potts model on a Cayley tree were studied in [10, p. 105-121]. In [11] it were showed that the Potts model (with an external field a e R) admits only periodic Gibbs measure of period two; it was considered the case a = 0, and on the base of the same invariants, is was proved that all periodic Gibbs measures are neccesarily translation-invariant; it were found conditions under which the Potts model with a nonzero external field admits periodic (non translation-invariant) Gibbs measures. In [12] it was fully describe the set of translationinvariant Gibbs measures for the ferromagnetic q-state Potts model and it is proved that the number of translation-invariant measures can be up to 2q — 1. In [13] for q-state Potts model (with an external field a e R) on the Cayley tree of order k = 3 and k = 4 under some conditions on parameters it was proved existence of periodic (non translation-invariant) Gibbs measures of period two. In [14] a ferromagnetic Potts model (with zero external field a e R) on a Cayley
* [email protected] © Siberian Federal University. All rights reserved
tree of order k > 3 was studied and it was proved that there exists a critical temperature Tc such that for T < Tc, there exist at least two of periodic (non translation-invariant) Gibbs measures.
In this paper under some conditions on parameters of the q-state Potts model on a Cayley tree of order k > 2 we shall prove existence of the periodic (non translation-invariant) Gibbs measures, and we give a lower bound for number of these measures.
1. Definitions and known facts
The Cayley tree 9fc 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 9fc = (V, L, i), where V is the set of vertices 9fc, L the set of edges and i is the incidence function setting each edge l G L into correspondence with its endpoints x,y G V. If i(l) = {x, y}, then the vertices x and y are called the nearest neighbors, denoted by l = (x,y). The distance d(x,y),x,y G V on the Cayley tree is the number of edges of the shortest path from x to y:
d(x, y) = min {d|3x = xo, xi,..., xd_i, x^ = y G V such that (xo, xi),..., (xd_i, x^)}.
For a fixed x0 G V we set Wn = {x G V | d(x, x0) = n},
V„ = {x G V | d(x,x0) < n}, Ln = {l = (x, y) G L | x, y G K}. (1)
It is known that there exists a one-to-one correspondence between the set of vertices V of the Cayley tree 9fc and the group Gk that is the free product of k +1 cyclic groups of second order with the generators a1, a2,..., ak+1.
We consider the model in which the spin variables take values in the set $ = {1,2,..., q}, q > 2 and are located at the tree vertices. A configuration a on V is then defined as a function x G V ^ a(x) G the set of all configurations coincides with Q = . The Hamiltonian of the Potts model is defined as
H(a) = -J S^(x)a(y), (2)
(x,y)eh
i 0 if i = j
where J G R, (x, y) are nearest neighbors and Sj is the Kronecker symbol: Sj = < 1, if ^ = ^ Define a finite-dimensional distribution of a probability measure ^ in the volume Vn as
Mn(an) = Z-1 exp < -^Hn(an) + ^ , (3)
I xeWn J
where p = 1/T, T > 0 is temperature, Z—1 is the normalizing factor, {hx = (h1x,..., hq x) G Rq, x G V} is a collection of vectors and
Hn(an) = -J S^(x)ff(y)
(x,y)eLn
is the restriction of Hamiltonian on Vn.
We say that the probability distributions (3) are compatible if for all n > 1 and an-1 G :
y^ v ^n) = Mn-i(ffn-i)- (4)
Here an_i V ivn is the concatenation of the configurations. In this case, there exists a unique measure ^ on such that, for all n and an e $Wn
M^Wn = an}) = Vn(Vn)-
Such a measure is called a splitting Gibbs measure corresponding to the Hamiltonian (2) and vector-valued function hx,x e V.
The following statement describes conditions on hx, guaranteeing compatibility of ^n(an).
Theorem 1 ( [8]). The probability distributions ^n(&n), n = 1, 2,... in (3) are compatible for Potts model iff, for any x e V the following equation holds:
hx = £ F(hy,0), (5)
yes(x)
where F : h = (hi,..., hq_i) e Rq_i ^ F (h, 0) = (Fi,..., Fq_i) e Rq_i is defined as
/ q_i \
(0 - l)ehi + ^ ehj +1 __
q_i
o + y; eh \ j=i /
Fi = ln
and 6 = exp(Jß), S(x) is the set of direct successors of x and hx = (h\ x,..., hq-i,x) with
hi,x hi,x hq,x, ^ . . . , q 1
Let Gk be a subgroup of the group Gk.
Definition 1. The set of vectors h = {hx, x £ Gk} is said to be Gk-periodic if hyx = hx for all x £ Gk,y £ Gk.
The Gk-periodic sets are said to be translation-invariant.
Definition 2. The measure ^ is said to be Gk-periodic if it corresponds to the Gk-periodic set of vectors h.
The following theorem characterizes periodic Gibbs measures.
Theorem 2 ( [11]). Let K be a normal divisor of finite index in the group Gk. Then for the
(2)
Potts model, all K-periodic Gibbs measures are either Gyk -periodic or translation-invariant,
(2)
where Gk ) = {x £ Gk : the length of x is even}.
2. Periodic Gibbs measures
We consider case q ^ 3, i.e. a : V ^ $ = {1, 2, 3, ...,q}. By Theorem 2, we have only Gk2)-periodic Gibbs measures corresponding to the sets of vectors h = {hx £ Rq-1 : x £ Gk} of the form
h, if |x| is even, I, if |x| is odd.
hx =
Here h = (hi, h2,..., hq-i), l = (li, •••, lq-i). From equality (5), we then obtain
h, = k ln -
l, = k ln ■
q-i
9 - 1) exp(lj) + £ exp(lj ) + 1 _j=i_
q-i ,
Eexp(lj ) + 9 _
j=i q-i i = 1,9 - 1.
- 1) exp(hj) + E exp(hj) + 1
j=i
q-1
£exp(hj ) + 9
j=i
We introduce the notations exp(hj) = x^ exp(Zj) = y». We can then rewrite the last system of equations for i = 1, q — 1 as
y,
q-1
- 1)y, + £ yj + 1 _j=i
q-i
£ yj + 9
j=i
q-1
(9 - 1)x, + £ xj + 1
j=i
q-i
£ xj + 9
j=1
(6)
Remark 1. 1. In the case q = 2, the Potts model coincides with the Ising model which was studied in [8].
(2)
2. In the case k = 2, q =3 and J < 0, it was proved that all G\ -periodic Gibbs measures on base of invariant I = {(x1,x2,y1 ,y2) G R4 : x1 = x2, y1 = y2} are translation-invariant (see [11]).
(2)
3. In the case k > 1, q =3 and J > 0, it was proved that all G\ -periodic Gibbs measures are translation-invariant (see [11]).
For q > 3, 0 < 9 < 1, k > 3, we define
Im = {z = (u, v) G Rq x Rq : x, = x, y, = y, i = 1, m; x, = y, = 1, i = m + 1, q - 1},
i.e. u = (x, x,..., x, 1,1,..., 1), v = (y, y,..., y, 1,1,..., 1) and
Im = {z = (u, v) G Rq x Rq : x, = x, i = 1, m; x, = 1, i = m + 1, q - 1 - m;
; = y, i = q - m, q - 1; y, = y, i = 1, m; y, = 1, i = m + 1, q - 1 - m; y, = x, i = q - m, q - 1},
i.e. u = (x, x,..., x, 1,1,..., 1, y, y,..., y), v = (y, y,..., y, 1,1,..., 1, x, x,..., x). Here 2m ^ q - 1.
k
k
x
We consider the map W : Rq 1 x Rq 1 ^ Rq 1 x Rq 1, defined as
7-1
q— 1
( q—1 \
- i)yi + E yj + 1 _j=1
q—1
E yj + o
j=1
q—1
- l)Xi +Y, Xj + 1
j=1
q—1
E Xj + o
j=1
/
We note that the system (6) is the equation z = W(z). Solving the system (6) is therefore equivalent to finding fixed points of the map z = W(z), where z = (u,v), z = (u ,v ).
Lemma 1. Sets Im and Im are invariant subsets relatively to the map W.
The proof is similar to that of Lemma 2 in [11]. The case Im. In the case we rewrite the system (6) as
Oy + (m - 1)y + (q - m)
O + my + (q - m - 1) oX + (m - 1)X + (q - m) O + mx + (q - m - 1)
or
x = C (y), where f (x) = Ox +(m - 1)X + (q - m)
y = f k(x),
(7)
(8)
0 + mx + (q — m — 1) and fk (x) is k-power of function f (x).
Remark 2. Let n e Sq_i be a permutation. We shall define the action of n to the vector
(x1, x2,..., xq—1) as n(x) = (xn(1),
t(2),
T(q—1)). Then n(A) = {(nx,ny) : (x,y) G A},
where A = Im or Im is also invariant subset relatively to the map W but in cases n(Im) and n(Im) corresponding system of equations coincides with (7) and (9) (see below), respectively. Therefore without loss of generality, we can consider sets Im h Im.
Proposition 1. Let k ^ 3, 3 ^ q < k + 1, Oc
k - q + 1 k + 1
< 1. Then system of equations (6)
on Im has at least three solutions for 0 < 8 < 8cr, it has at least one solution for 0 = 0cr and it has only one solution for 0 > 0cr.
Proof. By (8) we obtain
We have
f '(x)
= g(x) = fk (f k(x)). (O - 1)(O + q - 1)
■) + my + q — m — 1)2 '
g'(x) = k2 f k_i(f k (x))f '(f k (x))f k_i(x)f '(x).
Consequently, for 0 < 0 < 1 the function f (x) decreases monotonically and the equation f (x) = x
01
has a unique solution x = 1 such that f'(1) = ---. We note that g(x) is increasing
O + q - 1
and x = 1 is a solution to g(x) = x. If g'(1) = k2(f'(1))2 = I k
O - 1 O + q - 1
> 1 then this
k
x=
k
y
k
x=
k
y
x
x
x
solution is not unique, because in this case for x > 1 the graph of the function g lies above the bisector and the function g is bounded. Thus a critical value for 0 can be found by the equation
k—0-—1— ) = 1 which for 0 < 1 gives 0cr = k , q + 1. Hence it follows that for 0 < 0 < 0cr 0 + q- 1) k + 1
the equation g(x) = x has at least three solutions x* < x* = 1 < x*, i.e. the equation g(x) = x
has at least two roots, which are distinct from roots of the equation f (x) = x. For 0 = 0cr the
graph of the function g tangents to the bisector in x = 1. This means that in this condition the
equation g(x) = x has at least one solution. Besides it is clear that for 0 > 0cr the equation
g(x) = x has a unique solution x* = 1, which it is solution to equation f (x) = x. □
The case Im. We consider the set Im. We rewrite the system of equations (6) on this set as
(0 — 1)y + my + (q — 2m — 1) + mx + 1 x k
x = '
0 + mx + my + (q — 2m — 1) 7 (9)
1 — 1)x + mx + (q — 2m — 1) + my + 1 x k
y = 1 "-
0 + mx + my + (q — 2m — 1) where exp(hj) = xj, exp(lj) = yj.
Remark 3. 1. For m = 0 we obtain u = (1,1,..., 1), v = (1,1,..., 1), which corresponds to the
translation-invariant Gibbs measure. Thus we consider the case m > 1.
' (2)
2. In the case k = 2, q = 3, m =1 on Im it was proved that all Gk -periodic Gibbs measures
are translation-invariant (see [11]).
In the last system substituting kx = z, tyv = t, we obtain
(0 + m — 1)tk + mzk + q — 2m 0 + mzk + mtk + q — 2m — 1 ' l (0 + m — 1)zk + mtk + q — 2m 0 + mzk + mtk + q — 2m — 1
From the first equation of (10) we find tk, t:
k mzfc+1 — mzk + (0 + q — 2m — 1)z — q + 2m 0 + m — 1 — mz '
b
t / mzk+1 — mzk + (0 + q — 2m — 1)z — q + 2m \ 0 + m — 1 — mz
and substitute to the second equation of (10). Then we obtain
f (z) = [(0 + 2m — 1)zk — mzk+1 + mz + q — 2m]k (0 + m — 1 — mz) — — (mzk + q — m — 1 + 0)k [mzk+1 — mzk + (0 + q — 2m — 1)z — q + 2m] = 0.
(10)
(11).
We consider the function f (z). We note that f (0) = (q-2m)k(0+m-1)+(q-2m)(0+q-m-1) > 0 for 2m < q. Besides f (1) = 0 and f (z) ^ —to for z ^ Consequently it is clear that if f'(1) > 0, then the equation (11) has at least three solutions. Therefore we consider
f'(1) = (k2 — 1)s2 — 2qs — q2 = (k2 — 1) (s — T) (s + > 0,
where s = 0 — 1 < 0. Consequently, if s + q < 0, i.e. 0 < 0 < 1 - q = 0cr then f '(1) > 0.
k +1 k +1
Thus we proved the following
k — q + 1
Proposition 2. Let k ^ 3, 3 ^ q < k + 1, 0cr = —;--— < 1. Then the system of equations
k +1
(6) on Im:
1) for 0 < 0 < 0cr has at least three solutions;
2) for 0 = 0cr has at least one solution;
3) for 0 > 0cr has only one solution.
Remark 4. It is clear that in Propositions 1 and 2 one of measures corresponds to the so-
(2)
lution x* = 1 which is translation-invariant, the remainning measures are Gk -periodic (nontranslation-invariant), and in case 0 > 0cr the measure corresponding to the unique solution
x* = 1.
Similarly as in [12, p. 6], it is easy to show that for 0 < 0 < 0cr on each Im and Im, where m = 1,2, ...,q, the number of Gk2)-periodic (non-translation-invariant) Gibbs measures is not less than 2 • (m) and 2 • (m) • (qmm), respectively. Consequently, the number on |Jm=1 Im and Um=i Im is not less than
q /x [q/2] b \ /
2 • V (q) = 2q+1 — 2, and 2 • V ( q) • q — m \mj \mj V m
m=1 v y m=1 v y v
respectively.
Thus we have the
Theorem 3. For k > 3, 3 < q<k +1 and 0 < 0 < 0cr for the Potts model exist at least
[q/2] , \ b
2 • — 1+ Ef^ • fq — "
^—' \ to/ V to
m=1 v y v (2)
Gk -periodic (non translation-invariant) Gibbs measures.
Remark 5. In [12] the number and the description of all translation-invariant Gibbs measures for the Potts model were given.
Acknowledgments. The author is grateful to Professor U. A. Rozikov for useful discussions.
References
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Новые периодические меры Гиббса для модели Поттса с д-состояниями на дереве Кэли
Рустамжон М. Хакимов
В данной статье изучается модель Поттса с д-состояниями на дереве Кэли порядка к и показано существование периодических (не трансляционно-инвариантных) мер Гиббса при некоторых условиях на параметры этой модели. Кроме того, указана нижняя граница количества существующих периодических мер Гиббса.
Ключевые слова: дерево Кэли, конфигурация, модель Поттса, мера Гиббса, периодические меры, трансляционно-инвариантные меры.
