ON GROUND STATES FOR THE SOS MODEL WITH COMPETING INTERACTIONS Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2022-15-2-162-175 УДК 517.98

On Ground States for the SOS Model with Competing Interactions

Muzaffar M. Rahmatullaev*

Institute of Mathematics Academy of Sciences of Uzbekistan Toshkent, Uzbekistan Namangan State University Namangan, Uzbekistan

Bunyod U. Abraev^

Chirchik state pedagogical institute Chirchik, Uzbekistan

Received 30.05.2021, received in revised form 17.09.2021, accepted 14.12.2021 Abstract. We study periodic and weakly periodic ground states for the SOS model with competing interactions on the Cayley tree of order two and three. Further, we study non periodic ground states for the SOS model with competing interactions on the Cayley tree of order two.

Keywords: Cayley tree, SOS model, periodic and weakly periodic ground states.

Citation: M.M. Rahmatullaev, B.U. Abraev, On Ground States for the SOS Model with Competing Interactions, J. Sib. Fed. Univ. Math. Phys., 2022, 15(2), 162-175. DOI: 10.17516/1997-1397-2022-15-2-162-175.

Introduction

It is known that a phase diagram of Gibbs measures for a Hamiltonian is close to the phase diagram of isolated (stable) ground states of this Hamiltonian. At low temperatures, a periodic ground state corresponds to a periodic Gibbs measure, (see [1,2]). It leads us to investigate the problem of description of periodic and weakly periodic ground states. For the Potts model with competing interactions on the Cayley tree of order k = 2 periodic ground states are studied in [3] (see also [4]). The notion of a weakly periodic ground state is introduced in [5]. For the Ising model with competing interactions, weakly periodic ground states are described in [1,5]. Such states for the Potts model for normal subgroups of index 2 are studied in [6,7]. For the Potts model with competing interactions, such states for normal subgroups of index 4 are studied in [8] and in this work also studied periodic ground states for normal subgroups of index 4 (see also [9]). In [10] for the Potts model, with competing interactions and countable spin values, on a Cayley tree of order three periodic ground states are studied.

In [11] finite-range lattice models on Cayley trees with two basic properties: the existence of only a finite number of ground states and with a Peierls type condition are considered and the

* mrahmatullaev@rambler.ru tabrayev89@mail.ru © Siberian Federal University. All rights reserved

notion of a contour for the model on the Cayley tree is defined. Also using a contour argument the existence of different Gibbs measures is shown.

A q-component models on a Cayley tree is investigated in [12] and using a contour argument the existence of q different Gibbs measures for several q-component models is shown.

In [13] for the SOS model with m = 2 on the Cayley tree order of k = a + b + 2 the existence of at least two non periodic Gibbs measures is proved. In [14] an infinite system of functional equations for the Ising model with competing interactions and countable spin values 0,1,... and non zero field on a Cayley tree of order two is investigated. In [15] the authors proved the existence of weakly periodic Gibbs measures for the Ising model on the Cayley tree of order k = 2 with respect to a normal divisor of index 4.

In this paper, we study periodic and weakly periodic ground states for the SOS model with competing interactions on a Cayley tree of order k = 2 and k = 3. Moreover, in the case k = 2 the existence of a countable set of non periodic ground states is proved.

1. Preliminaries

Let rk = (V,, L) be the Cayley tree of order k, i.e., an infinite tree such that exactly k + 1 edges are incident to each vertex. Here V is the set of vertices and L is the set of edges of rk. Let Gk denote the free product of k + 1 cyclic groups {e; ai} of order 2 with generators ai, a2, a3, . . . ak+1, i.e., let a2 = e (see [4]).

The group of all left (right) shifts on Gk is isomorphic to the group Gk. Each transformation S on the group Gk induces a transformation S on the vertex set V of the Cayley tree rk. In the sequel, we identify V with Gk.

The following assertion is quite obvious (see also [4]).

Theorem 1.1. The group of left (right) shifts on the right (left) representation of the Cayley tree is the group of translations.

By the group of translations we mean the automorphism group of the Cayley tree regarded as a graph. Recall (see, for example, [4]) that a mapping ^ on the vertex set of a graph G is called an automorphism of G if ^ preserves the adjacency relation, i.e., the images ^(u) and ^(v) of vertices u and v are adjacent if and only if u and v are adjacent.

For an arbitrary vertex x0 e V, we put

Wn = {x e V\d(x, x0) = n}, Vn = {x e V\d(x, x0) < n},

where d(x, y) is the distance between x and y in the Cayley tree, i.e., the number of edges of the path between x and y.

For each x e Gk, let S(x) denote the set of immediate successors of x, i.e., if x e Wn then

S(x) = {y e Wn+i : d(x,y) = 1}.

For each x e Gk, let S1(x) denote the set of all neighbors of x, i.e., S1 (x) = {y e Gk : (x,y) e L}. The set S1 (x) \ S(x) is a singleton. Let x± denote the (unique) element of this set.

Let us assume that the spin values belong to the set $ = {0,1,2,.. .m}. A function a : x e V ^ a(x) e $ is called configuration on V. The set of all configurations coincides with the set Q = $V.

Consider the quotient group Gk/G*k = {H1, H2,... ,Hr}, where G*k is a normal subgroup of index r with r ^ 1.

Definition 1.1. A configuration a(x) is called G*k-periodic, if a(x) = ai for all x G Gk with x G Hi. A Gk -periodic configuration is called translation invariant.

The period of a periodic configuration is the index of the corresponding normal subgroup.

Definition 1.2. A configuration a(x) is called G^-weahly periodic, if a(x) = aij for all x G Gk with x| G Hi and x G Hj.

The Hamiltonian of the model SOS model with competing interactions has a form:

H(a) = -Ji £ \a(x) - a(y)\- J £ \a(x) - a(y)\, (1)

(x,y)eL x,yyV :

X 'y/ d(x,y)=2

where (J1J2) G R2.

2. Ground states

In this section, we study ground states for the SOS model on a Cayley tree. For a pair of configurations a and p which coincide almost everywhere, i.e., everywhere except finitely many points, we consider the relative Hamiltonian H (a, p) describing the energy differences of the two configurations a and p :

H(a,p) = - Ji £ (\a(x) - a(y)\-\p(x) - p(y)\)-

(x,y)€L

^ (2) - J2 YV (\a(x) - a(y)\-\p(x) - p(y)\),

d(',,y) = 2

where (J1,J2) G R2.

Let M be the set of all unit balls with vertices in V, i.e. M = {{x} U 51(x) : Vx G V}. A restriction of a configuration a to the ball b G M is a bounded configuration and it is denoted by ab.

We define the energy of the configuration ab on b by the following formula

U(ab) = U(ab, Ji, J2) = -2 Ji £ \a(x) - a(y)\- J2 £ \a(x) - a(y)\, (3)

(x,y) : x,yyb:

x,yyb d(x,y)=2

where (J1J2) G R2.

The following assertion is known (see [4]).

Lemma 2.1. Relative Hamiltonian (2) has the form:

H (a, p) = £ (U (ab) - U (pb)).

beM

The existence of a countable set of non periodic ground states on the Cayley tree of order two

We consider the case k = 2.

Let m = 2. It is easy to see that U(ab) G {Ui : i = !,..., 10} for ab, where

1 3

Ui =0, U2 = - 2 Ji - 2J2, U3 = -Ji - 2J2, U4 = - 2 Ji, U5 = - Ji - 4J2,

3 5

U6 = -2Ji - 4J2, U7 = -3Ji, U8 = --Ji - 4J2, U9 = -2Ji - 2J2, Ui0 = --Ji - 2J2.

Definition 2.1. The configuration ^ is called the ground state for the Hamiltonian (1) if U (Vb) = min[Ui, U2, U3,..., Uw} for any b G M.

Let

It is easy to check that

Am = {J, J2) G R | Um = min {Uk}}.

1<fc<10

A! = {(J1J2) G R2

A2 = {(J1J2) G R2 A3 = {(J1J2) G R2 A4 = {( Ju-h) G R2 A5 = {(J1J2) G R2 Ae = {(J1J2) G R2

Ar = {(J1J2) G R2 As = {(J1J2) G R2 A9 = {(J1J2) G R2

J1 < 0; J2 < - 4 J1}, J1 < 0; J2 = -1J1}, J1 =0; J2 =0}, J1 =0; J2 < 0}, J1 < 0; J2 > - 4 J1}, J1 > 0; J2 > 4 J1}, J1 > 0; J2 < 4 J1},

J1 =0; J2 > 0},

J1 = 0; J2

0}}, 1

A10 = {J J2) G R2 | Ji > 0; J2 = 4 Ji}

10

and |J Ai = R2.

i=i

In [16] periodic ground states are studied for SOS model on Cayley tree order of 2. In this subsection we shall prove the existence of a countable set of non periodic ground states on the Cayley tree of order two. The next subsection we study periodic and weakly periodic ground states for the model (1) on the Cayley tree of order three. Let cb denote the center of a unit ball b. We put

Ci = {ab : U(ab) = Ui},i = 1,10,

B(i) =| {x G S1(cb) : pb(x) = i} I, for i = 0,1, 2 and Di = ii U li, where

Hi = {ab : &b(cb) =0, Ix G b\{cb} : ab(x) = 2I = i; Ix G b\{cb} : ab(x) = 1I = 0},

Hi = {ab : a(x) - a(x

For Ai,Aj,i = j we have

= 2, Ix G b\{cb} : ab(x) = 1I =0,x G b}, i = 0,1, 2, 3, i.e.,

2, if a(x) = 0 0, if a(x) = 2 '

a(x)

Ai n Aj =

A2 if i = 1,j = 5, A4 if i = 1,j = 7, As if i = 5,j = 6, A10 if i = 6,j = 7.

(4)

Fix J = (J1,J2) e R2 and denote

Nj<) = \{j : ab e Cj}|.

Using (4) one can prove Lemma 2.2. For any b e M and <b we have

10, if J = (0; 0)

Nj(<b) = { 3, if J e Ai \{(0, 0)}, i = 22, 4, 8,10 . (5)

1, otherwise

Let GS(H) be the set of all ground states of the Hamiltonian (1).

Theorem 2.1. (i) If J = (0; 0) then GS(H) = Q.

(ii) If J e Ai \ {(0, 0)}, i = 22, 8,10 then there exists a countable set of non periodic ground states.

Proof. The assertion (i) is trivial. Prove (ii):

a) if J e A2 \ {(0,0)} then the minimum points of U(<b) would belong to the classes C1, C2 and C5;

b) if J e A8 \ {(0,0)} then the minimum points of U(<b) would belong to the classes C5, C6 and C8;

c) if J e Aio \ {(0,0)} then the minimum points of U(<b) would belong to the classes C6, C7 and C10.

Below we define the configurations of classes C1,C5,C6 and C7 which satisfying the condition

\x e b\{cb} : <b(x) = 1\ =0,

ab°\cb)=0, \x e b\{cb} : ab0)(x) = 2\ =0 and

^(cb) = 2, \x e b\{cb} : ab°\x) = 0\ =0, < (0),<(0) e Ci,

<{b](cb) = 0, \x e b\{cb} : <(1](x) =2\ = 1 and

ab1](cb) = 2, \x e b\{cb} : <b1)(x) = 0\ = 1, <b 1),<(1) e C5,

. (6)

<{b\cb) =0, \x e b\{cb} : of\x) = 2\ = 2 and

~ab?\cb) = 2, \x e b\{cb} : <b2)(x) = 0\ = 2, <2),<(2) e C6,

<b3\cb) =0, \x e b\{cb} : <b3)(x) = 2\ = 3 and 43)(cb) = 2, \x e b\{cb} : *b2)(x) = 0\ = 3, <(3), a(3) e C

Thus any ground state p e Di must satisfy

Pb e{abi),abi),<bi+1),<bi+1)}, i = 0,1, 2, b e M. (7)

Now we shall construct ground states p e Di which satisfying (7).

Note that the configurations <b and (b, b' e M) are the same up to a motion in Gk so we shall omit b. Thus configuration < (i) is the configuration such that on any unit ball b e M the condition (6) is satisfied.

Suppose two unit balls b and b' are neighbors, i.e., they have a common edge. We shall then say that the two bounded configurations ob and ob' are compatible if they coincide on the common edge of the balls b and b'. Denote by B(b) the set of all neighbor balls of b.

Denote tti = {o(i), a(i), o(i+1), a(i+1)}, i = 0,1, 2. For any w,v G tti denote by n(w,v) the number of possibilities to set up the configuration v as a compatible configuration (with w) around (i.e., on neighboring balls of the ball on which w is given ) the configuration w. Clearly n(w, v) G {0,1, 2, 3}, for any w,v G tti,i = 0,1, 2.

Denote

Ni =

( n(o(i), o(i)) n(a(i), o(i))

n(o(i), a(i)) n(o(i),o n(a(i), a(i)) n(a(i),o

n(o(i+1), o(i)) n(o(i+1), a(i)) n(o(i+1), o

\ n(a(i+1),o(i)) n(a(i+1),a(i)) n(a(i+1),o

It is easy to see that

No

n(o(i), a n(a(i), a n(o(i+1), a

n(a(i+1), a

3 0 3 0 2 1 2 1

0 3 0 3 , N1 = 1 2 1 2

2 0 2 1 1 2 1 2

0 2 1 2 2 1 2 1

\

N2 =

1202 2 12 0 0303 3030

Consider 3 sets Qi = {Q}, (i = 0,1,2) of matrices Q = {q(u, v)}uve^li such that q(u, v) G {0,1, ..., n(u, v)}, ^^ q(u, v) = 3, G tti.

veil i

q(u, o(i))+ q(u, o(j)) = n(u, o(i)), q(u, ) + q(u, a(j)) = n(u, a(i)), and q(u, v) = 0 if and only if q(v, u) = 0, u,v G tti.

Using matrices Ni we have

Qo

Q

fa 0 3 - a 0 \

0 b 0 3 - b

c 0 2 - c 1

\ 0 d 1 2 - d J

here a,b G {0,1, 2, 3}; c,d G {0,1, 2}; a = 3 iff c = 0; b = 3 iff d = 0. For i = 1 we get

a1 b1 2 - a1 1 - b1

b2 a2

Q1 = Q =

1 - b2 2 - a2 c1 d1 1 - c1 2 - d1

d2 c2 2 - d2 1 - c2

here a1, a2, d1, d2 G {0,1, 2}; b1, b2, c1, c2 G {0,1}; a1 =2 iff c1 = 0; a2 = 2 iff c2 =0; b1 =0 iff b2 = 0; b1 = 1 iff d2 = 0; b2 = 1 iff d1 = 0; d1 = 2 iff d2 = 2.

For i = 2 we obtain

1a 0 2 —a

b 1 2 — b 0

0 c 0 3 —c

d0 3 — d 0

= 0 iff b = 0; a = 2 iff d

Q2 = < Q =

here a,b e {0,1, 2}; c,d e {0,1, 2, 3}; a = 0 iff b = 0; a = 2 iff d = 0; b = 2 iff c = 0; c =3 iff d =3.

For a given £ e Qi and Q = {q(u,v)}uv€Q. e Qi we recurrently construct a ground state by the following way: fix a ball b e M and put on b the configuration pQ'^ := £. On balls taken from B(b) we set exactly q(£,w) copies of w for any w e Qi. Thus configurations pQ'^, b' e B(b) are defined. Using these configurations, we define configurations on the balls B(b') \ {b}, (b' e B(b)) putting q(pQ'^,v) copies of v e Qi \ £ and q(pQ'^,£) — 1 copies of £ which are compatible with . Further, on the balls B(b'') \ {b'}, (b'' e B(b)),b' e B(b) we set q(pQfi,t) copies of t e Qi \ {pQ'^} and q(pQfi,pQ'^) — 1 copies of pwhich are compatible with p(Q^. Repeating this construction one can obtain a ground state pQ'^ such that

pb

Q'Z e Qi, \{b' e B(b): pQ'Z = w, pQ* = v}\ = q(w, v),

for any b e M and w,v e Qi.

In general, the ground state pQ^ is non periodic (see example below). It is easy to see that

pb

Qi,vu) — (j) Qi, — ~(j) ■ • • , 1 • n 1 o = <(j), pb ' = a(j), j = i, i + 1, i = 0, 1, 2,

where

Qi =

3— i i 0 0

i 3 — i 0 0

0 0 2 — i i + 1

\ 0 0 i + 1 2 — i J

(8)

Now using the ground states pQ^ we shall construct an infinite set of ground states by the following way: one can choose £ = n, £,n e Qi and Q1,Q2 e Q1 such that for configurations pQi,i,pQ2,v there are infinitely many b e M on which pQl'^ and pQ,2'v are compatible for some b' e B(b). Indeed it is sufficient to take £ = n such that q1(£,n)q2(£,n) = 0 (see example below).

Denote

M1 = M?(Q1, Q2) = {b e M : pQ1,e is compatible pQ2'v for some b' e B(b)};

N1 = {n e {0,1, 2,... } : 3b e M1 such that \cb\ = n};

V(y) = {z e V : y <z}.

Fix m e N1 and denote

Wm = {x e Wm : 3b e M1 such that cb = x}.

Consider the configuration

oQi-«(x) if x e Vm u {V(y),y e wm\Wm}

p%>Q2*>n (x)

(pQ1, \ pQ2'

pQ2n(x) if x e V(y),y e Wm

Clearly , m G H1 is a ground state and the number of such ground states is infinite,

since |H1| = to. This finishes the proof of Theorem 2.1. □

Remark 2.1 Let J G A4\(0,0). tt3 = {o(0), a(0), o(3), a(3)} are periodic ground states such that on any b G M the bounded configurations o(0), a(0) G C1 and o(3), a(3) G C7, i.e., o(0), a(0) are translation-invariant and o(3), a(3) are periodic with period 2. tt3 = {o(0), a(0),o(3), a(3)} and Q3 contains the unique matrix

I 3 0 0 0 \ 0300 0003 0030

Example. Take matrices

Q3 =

Q2

( 1 1 0 1 1 2 0 0

1 1 1 0 1 1 1 0

0 1 0 2 , Q'2 = 0 1 0 2

V i 0 2 0 0 0 3 0

and £ = o-(2), n = o"(3)- The configurations , and are represented in Fig. 1

a), b) and c), respectively.

Fig. 1. Ground states

Periodic and weakly periodic ground states on the Cayley tree of order three

We consider the case k = 3.

Let m = 2. By (3) for any <b we have U(<b) e {U1, U2, U3,..., U15}, where

U1 =0, U2 = — 2 J1 — 3J2, U3 = —J1 — 4J2, U4 = —J1 — 6J2, 3

U5 = — ^ J1 — 3J2, UG = —2J1 — 8J2, Ur = —3J1 — 6J2, Us = —2J1 — 6J2,

5 3

U9 = — - J1 — 7J2, U10 = — - J1 — 7J2, U11 = —2J1, U12 = —4J1,

7 5

U13 = — ^ J1 — 3J2, U14 = — ^ J1 — 3J2, U15 = —3J1 — 4J2.

Definition 2.2. The configuration p is called the ground state for the Hamiltonian (1), if U (pb) = min{U1, U2, U3,..., U15} for Vb e M.

Let Am = {(J1, J2) e R2 \ Um = min {Uk}}. It is easy to check that

1<k<15

A1 = {(J1, J2) e R2

A2 = {(J1, J2) e R2 A3 = {(J1, J2) e R2 A4 = {(J1, J2) e R2 A5 = {(J1, J2) e R2 Ae = {(J1, J2) e R2 Ar = {(J1, J2) e R2 As = {(J1, J2) e R2 A9 = {(J1, J2) e R2

A10 = {(J1J2) e R An = {(J1J2) e R2 A12 = {(J1J2) e R2

A13 = {(J1J2) e R2

A14 = {(J1J2) e R2 A15 = {(J1J2) e R2

J1 < 0; J2 < —-J1} 6

J1 < 0; J2 = —1J1}

6

J1 = 0; J2 = 0},

J1 < 0; — - J1 < J2 < — - J1}, 62

J1 = 0; J2 = 0},

J2 > 2\J1\},

J1 > 0;1J1 < J2 < 1J1}, 62

J1 = 0; J2 = 0}, J1 > 0; J2 = 2 J1},

2 \ J1 < 0; J2 = — 2 J1}, J1 =0; J2 < 0},

J1 > 0; J2 < 1J1}, 6

J1 > 0,J2 = 1J1}, 6

J1 = 0; J2 = 0}, J1 = 0; J2 = 0}

15

and U Ai = R2.

i=1

Let cb be the center of a unit ball b. We put

Ci = {<b : U(<b) = Ui},i = 1,15

and

B(i) = |{x G S\(cb): <fb(x) = i}|,

for i = 0,1, 2.

Let HA = {x G Ok '-Y1,-ieA wx(ai) — even}, where wx(ai) is the number of ai in the word x. Note, that HA is a normal subgroup of index two (see [4]). Let Ok/HA = {HA,Ok\HA} be the quotient group. Denote H0 = HA, Hi = Ok\HA.

Periodic Ground States for the case k = 3

In this section, we shall study H0-periodic ground states. We note that each H0 periodic configuration has the following form:

i , if x G H0

o(x) = 1 TT , (9)

t o2, if x G H1 y '

where o* G $ = {0,1, 2}, i = 1, 2.

Theorem 2.2. Let k = 3. The configuration (9) is Ho-periodic ground state iff one of the following conditions holds:

a) |A| = 1.

i) I oi — o2 1= 0, and (Ji, J2) G Ai.

ii) | oi — 02 I= 1, and (Ji, J2) G A2.

iii) | oi — o2 ^ 2, and (Ji, J2) G A4.

b) |A| = 2.

i) If | oi — o2 ^ 1,then there is not a H0-periodic ground state;

ii) | oi — 02 ^ 2, and (Ji, J2) G A@.

c) |A| = 3.

i) If | oi — o2 ^ 1, then there is not a H0-periodic ground state;

ii) | oi — o2 ^ 2, and (Ji, J2) G A7.

d) |A| = 4.

i) | oi — 02 ^ 1, and (Ji, J2) G An.

ii) | oi — o2 ^ 2, and (Ji, J2) G Ai2.

Proof: a) i) Let us consider the following configuration

i, if x G H0

¥>(x) =

ii,,

if x G Hi

where i = 0,1,2. We denote the center of b G M by cb. Let cb G H0, then we have

<fb(cb) = i, B(i) = 4.

Hence, <^b(x) G Ci, i.e. if (Ji, J2) G Ai then the corresponding configuration is a ground state. ii) Now we consider the following configuration

^ = {j ;

if x G H0 j, if x G Hi '

where | i — j ^ 1.

1) Assume that cb G H0

pb(cb) = i,B(i) = 3,B(j) = 1.

Hence, pb(x) G C2.

2) Let cb G Hi, then one has

pb(cb) = i,B(i) = 3,B(j) = 1.

Hence, pb(x) G C2.

We conclude that, if (Ji,J2) G A2 then the corresponding periodic configuration p(x) is a Ho-periodic ground state.

iii) Let us consider the following configuration

A*) = {j i

if x G Ho j, if x G Hi '

where \ i - j \=2.

1) Assume that cb G Ho

pb(cb) = i,B(i) = 3,B(j) = 1.

Hence, pb(x) G C4.

2) Let cb G Hi, then one has

pb(cb)= j,B(j) =3,B(i) = 1.

Hence, pb(x) G C4.

We conclude that if (Ji,J2) G A4 then the corresponding periodic configuration p(x) is a Ho-periodic ground state.

The proofs of assertions b), c) and d) of Theorem 2.2 are similar to the proof of assertion a). This finishes the proof of Theorem 2.2. □

Remark 2.2 In the case c), the H0 periodic ground states coincides with the G^-periodic ground states, where G^ = {x G Gk : \x\ is even}.

Weakly Periodic Ground States for the k = 3

In this section, we describe HA-weakly periodic ground states, where HA is a normal subgroup of index two. Due to the definition of weakly periodic configuration, we infer that each HA-weakly periodic configuration has the following form:

a(x)

aoo, if xi G Ho x G H

a01, if xi G Ho, x G H

aio, if xi G Hi x G H

an, if xi G Hi x G H

(10)

where aj G i,j = 0,1.

In the sequel, we write a = (a00,a0i,al0,all) for such a weakly periodic configuration a(x), x G Qu-

Theorem 2.3. Let k = 3 and = 1. Then for the SOS model there is no HA-weahly periodic (non periodic) ground state.

o

Proof. Consider (10). If o00 = o0i = oi0 = oii, then corresponding configurations are translation-invariant. Translation-invariant ground states for this case are studied in Theorem 2.2. It is easy to see that in the case o00 = oi0 and o0i = oii the HA-weakly periodic configurations (10) are periodic configurations which are studied in Theorem 2.2.

Now we consider the cases o00 = oi0 or o0i = oii.

Let

¥>(x)

0, if x| G Ho, x G Ho

0, if x| G H0,x G H1

1, if x| G H1, x G Ho 0, if x4 G H1,x G H1

Let cb G H0, we have the following possible cases:

a) cb4 G Ho and pb(cbi) = 0, then <^b(cb) = 0, B(0) = 4, pb(cb) G C1,

b) cb4 G Ho and Mcb.i) = 1, then pb(cb) = 0, B(o) = 3, B(1) = 1, pb(b G C2,

c) cb4 G H1 and pb(cb^) = 1, then there is not any HA-weakly periodic ground state,

d) cb^ G H1 and pb(cbi) = 0, then pb(cb) = 1, B(o) = 4, pb(cb) G Cn.

Let cb G H1 , we have the following possible cases:

a) cb^ G Ho and pb(cbi) = 0, then pb(cb) = 0, B(o) = 4, pb(cb) G C1,

b) cb4 G Ho and ^b(cb^) = 1, then there is not any HA-weakly periodic ground state,

c) cb4 G H1 and ^b(chi) = 0, then ^bO = 0, B(o) = 3, B(1) = 1, ^fa) G C2. We conclude that the configuration p is a ground state on the set

A1 n A2 n An = {(J1, J2) G R2 : J1 = J2 =0}.

Therefore, if J1 = 0 and J2 = 0 then the weakly periodic configuration p is not a weakly periodic ground state.

By similar way we can prove that all HA-weakly periodic (non periodic) configurations are not ground states.

This finishes the proof of Theorem 2.3. □

Remark 2.3. 1) Theorem 2.3 shows that for the SOS model with competing interactions, every HA-weakly periodic ground state is either HA-periodic or translation-invariant.

2) The fact that for k = 3 there exists a set of countable non-periodic ground states can be proved in the same manner as in Theorem 2.1.

3) For the k > 3 by the same manner as in Theorem 2.1 periodic (and weakly periodic) ground states could be studied.

The authors thank Professor U. A. Rozikov for useful discussions. The authors are grateful to the referee's helpful suggestions.

References

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Основные состояния для модели SOS c конкурирующими взаимодействиями

Музаффар М. Рахматуллаев

Институт Математики АН РУз Ташкент, Узбекистан Наманганский государственный университет Наманган, Узбекистан

Бунёд У. Абраев

Чирчикский государственный педагогический институт

Чирчик, Узбекистан

Аннотация. В работе для нормального делителя индекса 2 изучены слабо-периодические основные состояния для модели SOS с конкурирующими взаимодействиями на дереве Кэли порядка 2 и порядка 3. Далее изучены непериодические основные состояния для модели SOS с конкурирующими взаимодействиями на дереве Кэли второго порядка.

Ключевые слова: дерево Кэли, SOS-модель, периодические и слабо-периодические основные состояния.