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0ВЕСТНИК ОШСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА МАТЕМАТИКА, ФИЗИКА, ТЕХНИКА. 2023, №2
МАТЕМАТИКА
УДК 517.51, 517.98
https://doi.org/10.52754/16948645 2023 2 180
GROUND STATES FOR THE SOS MODEL WITH COMPETING BINARY INTERACTIONS ON A CAYLEY TREE OF ORDER THREE
Rahmatullaev Muzaffar Muhammadjonovich., DSc, professor,
mrahmatullaev@rambler.ru Institute of Mathematics named after V.I. Romanovsky of the Academy of Sciences of the Republic of Uzbekistan,
Tashkent, Uzbekistan Abraev Bunyod Urinboevich, PhD student abrayev89@mail.ru Chirchik state pedagogical university, Chirchik, Uzbekistan.
Abstract: We consider a SOS (solid-on-solid) model with nearest-neighbor interaction J, prolonged next-nearest-neighbor interaction J and one level next-nearest-neighbor interaction J , where the spin takes values in the set Ф = {0,1,2} on a Cayley tree of order three. In the paper, we study translation-invariant and periodic ground states of the model SOS.
Keywords: Cayley tree, configuration, competing next-nearest-neighbor interactions, translation-invariant and periodic ground state.
ОСНОВНЫЕ СОСТОЯНИЯ ДЛЯ МОДЕЛИ SOS С КОНКУРИРУЮЩИМИ БИНАРНЫМИ ВЗАИМОДЕЙСТВИЯМИ НА
ДЕРЕВЕ КЭЛИ ПОРЯДКА ТРИ
Рахматуллаев Музаффар Мухаммаджанович, д.ф.-м.н., профессор,
mrahmatullaev@rambler. ru Институт Математики имени В.И. Романовского Академии Наук Республики Узбекистан, Наманган, Узбекистан Абраев Бунёд Уринбоевич, PhD докторант, Чирчикский государственный педагогический университет,
Чирчик, Узбекистан. abra yev89@mail. ru
Аннотация: Мы рассматриваем модель SOS (solid-on-solid) с взаимодействием ближайших соседей
J, длительным взаимодействием следующих ближайших соседей J2 и одноуровневым взаимодействием
следующих ближайших соседей где спин принимает значения в множестве Ф {0,1,2} на дереве Кэли порядка три. В статье исследуются трансляционно-инвариантные и периодические основные состояния модели SOS.
Ключевые слова: Дерево Кэли, конфигурация, конкурирующие взаимодействия ближайших соседей, трансляционно-инвариантное и периодическое основное состояние.
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, e.g., [1]). It leads us to investigate the problem of description of periodic and weakly periodic ground states. In this paper, we study periodic ground states for the SOS model with nearest-neighbor and competing binary interactions on a Cayley tree of order three.
Let Г = (V,L) be the Cayley tree of order k > 1, 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 Гк. It is known (see [2]) that there exists a one-to-one correspondence between the set V of vertices of the Cayley tree of order k > 1 and the group Gk of the free products of k +1 cyclic groups {e, ai }
, i = 1,...,k +1 of the second order (i.e. af = e, a-1 = щ) with generators ax,a2,...,ak+x.
For an arbitrary vertex x0 e V, we put
Wn= {x e V | d(x, x0) = n},V„ = {x e V | d(x, x0) < n},
where d(x, y) is a natural distance, being the number of nearest-neighbor pairs of the
minimal path between the vertices x and y . L denotes the set of edges in V . The fixed vertex
x0 is called the 0 -th level and the vertices in W are called the n -th level. For the sake of
n
simplicity, we put | x |= d(x, x0), x e V.
Two vertices x,y e V are called the next-nearest-neighbour neighbors if d(x,y) = 2. The next-nearest-neighbour vertices x and y are called prolonged next-nearest-neighbours if
| x y | and is denoted by ) x, y(. The next-nearest-neighbour vertices x, y eV that are not prolonged are called one-level next-nearest-neighbours since | x |=| y | and are denoted by )x, y(.
For each x e Gk, let S(x) denote the set of direct successors of x, i.e., if x e Wn then S(x) = {y eWB+1: d(x,y) = 1}.For each x e Gk, let S(x) denote the set of all neighbors of x, i.e., S(x) = {y e G : (x, y)e L}. The set S (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 Ф = {0,1,2,...,m}. A function c: x eV ^ c( x) e Ф is called configuration on V. The set of all configurations coincides with the set Q = ФV.
Consider the quotient group Gk / G* = {H,H2,...,Hr}, where G* is a normal subgroup of index r with r > 1.
Definition 1. A configuration c(x), x e V is said to be G* -periodic, if c(x) = < for all x e H. A Gk -periodic configuration is called translation invariant.
The period of a periodic configuration is the index of the corresponding normal subgroup. The Hamiltonian of the SOS model with competing nearest-neighbour and next-nearest-neighbour binary interactions has the form:
H(<) = - J X C(x) - c(y) | -J X C(x) -c(y) |,
(x ,y)eL )x,y<:
x, yeV
-J2 ZC(x) -<(y)| (1)
) x, y <: x, yeV
where (JUJ2,J3) el3.
We define the energy of the configuration cb on b by the following formula
u (cb, J2, J3)=-1 Ji X c(x)-c(y)| -J2 X c(x)-c(y)|
2 < x,y): ) x, y <:
x,yeb x, yeb
-J3 X (|c(x) -<(y)|), (2)
) x, y<: x, yeb
where (Jx, J2, J3) eM3.
In [4] we studied ground state for SOS model with competing nearest-neighbour and next-nearest-neighbour binary interactions on a Cayley tree of order two. We consider the case k = 3.
Let m = 2 . By (2) for any <Jb we have U(ab) e {U,U2,U3,...,U24}, where
U = 0, U = -1J - 3 J, U = -1J - J - 2J3, U = -J - 2J2 - 2J, U5 = - Ji - 6 J2,U6 = - Ji - 2 J - 4 J3,U7 = - 3 Ji - 3J2, U8 = - 3 Ji - J - 2 J3,
U = -2J - 4J - 4J U10 = -3J - 6J, Un = -3J - 2J - 4J,
Uu = -2J, - 4J2 - 2J3, U„ = -2J, - 2J2 - 4JS, U_4 = - f Ji - J - J
5 3 3
Ui5 = - Ji - 3 J2 - 4 J3 , Ui6 = - ^ Ji - 5 J2 - 2 J3 , Ui7 = - Ji - 3 J2 - 4 J3 ,
7 7
Ui8 = -2Ji, Ui9 = -4Ji, U20 = -^ Ji - 3 J2 , U2i = -Ji - J2 - 2 J3, U22 = - ^ Ji - 3 J2 , U23 = - ^ Ji - J2 - 2 J3 , U24 = - 3 Ji - 2 J2 - 2 J3 •
Definition 2. A configuration p is called a ground state for the Hamiltonian (1), if U (pb) = min{U ,U2 ,U ,-U} for Vb e M.
For / = i, 24 we put
C1={ab:U(ab) = Ui} and Am = {(JVJ2,J3) tR3\Um = mm{Uk}}.
1<£<24
Quite cumbersome, but not difficult calculations show that:
, 111
4={(J1,J2,J3)eM3| J1<0;J2<--J1;J3<--J1--J2},
0 4 2
. 1
A2 ={(Jj,J2,J3)eM3| Jx<0;J2=—J{,J3<J2),
6
1 1 1
4 ^{(J1,J2,J3)eM3| Jl<0-,J2<--Jl',J3=--Jl ~-J2),
6 4 2
A4 = A12 = A24 = {(JVJ2,J3) e IB:31 J, = 0;J2 = 0;J3 = 0},
, 111
A5 = {(Jl,J2,J3) e M| Jx <0;J2 >—J{,J3<—Jx+-J2),
6 4 2
1 1 1 A6={(J1,J2,J3)e K3| Jj < 0;J2 <- — J{,J3 >- — Jl- — J2),
Aj = A22 = {{Jx, J2, J3) e M31 J,= 0;J2 = 0\J3 < 0},
1
4 =A23 = {(J1,J2,J3)gR3\ Jl=0;J2<0;J3=--J2},
1
4, -{(/1,/2,/3)eM3| Jx = 0;/2 >0;/3 >-/2},
1 1 1
A10 ={(J1,J2,J3)eR3l Jx > 0;J2 > —J{,J3 < —Jl +-J2},
6 4 2
1 1 1
An ={(J1,J2,J3)gR3I J^Q-J^-J^ >-Jx ~-J2),
* 1
43={(J1,J2,J3)e№3| J,=0;J2<0;J3>--J2},
1 1
Al4={(Jl,J2,J3) ek3| Jx > 0;J2 > 0;J3 = —Jl+ — J2),
1
45 J2,J3)gM3I j1>0-J2=-J1-J3>J2},
, 1 1
= A ^ 0 J2 > 0;J3 = --7, +-J2},
1
4t = {(.A, J2,Js) G R I ^ 0;72 = --7i;73 > J2},
1
Au ={(J„J2,J3)gR3\ Jl=0;J2<0;J3<--J2},
1 1 1
Aig ={(J1,J2,J3)eR3l Jx > 0;J2 < —J{,J3 < —Jl -~J2},
6 4 2
1
A20 ={(J1,J2,J3)eR3l Jx > 0;J2 = —J{,J3 <J2},
6
1 1 1
4={(J„J2,J3)g13| Ji>0;J2<-J1;J3=-J1--J2},
6 4 2
24
and (J 4
Let HA = (x e G : (a) - even}, where A œ (1,2,3,...,k +1} and ax(a) is the number
ieA
of a in the word x. If | A |= k +1, then H = G2 = (x e Gk : | x | -even}, where |x| is length of the word x.
Note that HA is a normal subgroup of index two (see [2]). Let G / H = (H, G \ H} be the quotient group. Denote H0 = H, H = G \ H •
Now, we shall study H -periodic ground states. We note that each H0 -periodic configuration has the following form:
J(x) = {ai,lf x " H0, (3)
[ (, If x e H,
where j e$, i = 1,2.
Theorem 1. a) Let k = 3 and | A |= 1. Then for the model (1) the following statements
hold:
i) If (J, J2, J) e A then each translation invariant configuration is a ground state.
ii) If ( J, J, J) e A O A then each H0 -periodic configuration of the form (3) with J = ( ±1 Jis a ground state.
iii) If (J, J, J) e A O A then each H0 -periodic configuration of the form (3) with J = ( ± 2 J, ( eO, is a ground state.
b) Let k = 3 and | A |= 2. If ( J, J, J ) e A then each H0 -periodic configuration of the form (3) with j = j ±2 J,jeO, is a ground state.
c) Let k = 3 and | A |= 3. If ( J, J, J ) e A10 O An then each H0 -periodic configuration of the form (3) with J = J ±2 J,jeO, is a ground state.
d) Let k = 3 and | A |= 4.
i) If (J, J2, J) e As then each G^2) -periodic configuration of the form (3) with J = J ± 1 J,jeO, is a ground state.
ii) If (J, J2, J,) e Ag then each G(2 -periodic configuration of the form (3) with J = J ±2 J,jeO, is a ground state.
Remark 1.
1) Note that applying the methods of [3], one can construct some periodic ground states which are different from the ground states mentioned in Theorem 1.
2) Let k = 3 and A = l,l = 2,3. If J = J ± 1, J,jeO then the configuration (3)
is not an H -periodic ground state.
REFERENCE
1. Sinai Ya. G., Theory of phase transitions: rigorous results, / Ya. G. Sinai - Science, M. 1980.
2. Rozikov U.A., Gibbs measures on Cayley trees, / U.A. Rozikov - World Scientific, Singapore, 2013.
3. Rahmatullaev M.M., Abraev B.U., On ground states for the SOS model with competing interactions / M.M.Rahmatullaev, B.U. Abraev // Journal of Siberian Federal University, 2022,
Vol 15. No.2 pp. 1-14.
4. Abraev B.U. Ground states for the SOS model with competing binary interactions on a Cayley tree / Abraev B.U. // Uzbek Mathematical Journal 2022, Volume 66, Issue 3, pp.10-20.
