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0Central Asian Journal of
Education and Innovation
PHASES OF THE FERROMAGNETIC POTTS MODEL WITH FIVE STATES ON A FIRST-ORDER BETHE LATTICE
Tursunqulov Abduazim
M.Sc. degree in the Department of Theoretical Physics, National University of Uzbekistan, Tashkent, Republic of Uzbekistan. https://doi.org/10.5281/zenodo.11104477
ARTICLE INFO
ABSTRACT
Qabul qilindi: 20-April 2024 yil Ma'qullandi: 25- April 2024 yil Nashr qilindi: 30- April 2024 yil
KEY WORDS
Potts model, Ising model, firstorder Bethe lattice, phase transitions.
The Ferromagnetic Potts model with five states on a first-order Bethe lattice is a significant subject in statistical physics and mathematical modeling. This abstract explores the distinct phases exhibited by this model and their implications. The Potts model is a lattice-based statistical mechanics model that generalizes the Ising model to systems with more than two possible states at each lattice site. In this study, we focus on the ferromagnetic variant of the Potts model with five states, which is particularly interesting due to its richer phase behavior. Utilizing the framework of a first-order Bethe lattice, we investigate the phase transitions and critical phenomena exhibited by this model. The Bethe lattice provides a simplified yet powerful structure for studying phase transitions in systems with large coordination numbers.
Introduction:
A Bethe lattice rk (or Cayley tree in another terminology, see [1] for the subject of terminology) of order k > 1 is an infinirte tree, i.e., a graph without cycles, from each vertex of which precisely k + 1 lines emanate. Let rk =(V,L,i), i.e., V is the set of vertices of rk, L is the set of its lines, and i is the incidence function, which associates with each line leL its end points x,yeV. If i(l)= {x, y}, then x and y are said to be neighboring vertices, and in this case we shall write l = <x,y>.
The distance d(x,y). x,y eV, is introduced on the Bethe lattice in accordance with the formula d(x,y)=min{dl3=x0, x1x2 ...xd-1 xd = y e V such that the pairs <x0, x1> , ..., <xd-1,xd>
{are nearest neighbors}. A sequence n={x0,x1x2......xd-1 ,xd=yeV }that realizes this
minimum is called a path from x to y. In the Potts model on a Bethe lattice, the spin variables o(x), xeV take the values 1,2, ..., q, and the Hamiltonian has the form
°a(x)a(y) °a(x)a(y)
<x,y>
>x,y<
where 5 is the Kronecker delta and the summation is over all pairs of nearest neighbors.
The ferromagnetic Potts model is determined by the Hamiltonian (1.1) for J > 0. In this paper, we restrict ourselves to studying the ferromagnetic Potts model with zero external field, i.e., H = 0. The concepts of Gibbs distribution of the Ports model on the Bethe lattice, limiting Gibbs distribution, and pure phase (extreme Gibbs distribution) are introduced in the usual manner (see [2-9]). The content of the paper is as follows. In Sec. 2, we derive recursion relations and study the fixed points of the transformations corresponding to them; in Sec. 3, we construct pure phases. In this paper, we restrict ourselves to the case q = 3 and k = 2, doing this solely for the relative simplicity of the calculations made below.
Ports model on the Bethe lattice:
Suppose the set of values of the spin variables ff(x), xeV, consists of vectors ate R 6R 2, i=1, 2, 3,4,5 such that |o"j | = 1 for all i .
the Hamiltonian (1.1) with external field H = 0 reduces to
h(&) = -J^ y)
<x,y>
with the description of the general structure of limiting Gibbs distributions on a Bethe lattice for models with Hamiltonian of the form (2.1). The proofs of these theorems can be found in [3,10,11,12].
If from a Bethe lattice rk we remove an arbitrary line <x0, x1 > = I e L, then it breaks up
into two components -- two semi-infinite Bethe lattices r0fc and rf (see Fig. 1, where k = 1).
THEOREM 2.1. A necessary and sufficient condition for V to be a limiting Gibbs distribution on rk is that there exist limiting Gibbs distribution ^0 and ^1 (which are uniquely determined)
on r0fc, r1fc respectively, such that
exp{(!) a(x°)a(x1)}
M =
Z
where Z > 0 is a normalization constant
Theorem 2.1 reduces the description of limiting Gibbs distributions on rk to a semiinfinite lattice T0fc . Let V0 be the set of vertices of T0fc and L0 its set of lines. On T0fc there is a distinguished boundary vertex x° e V0 from which k lines emanate. We shall call the set. The Gibbs distributions er possess the following simple but important property.
THEOREM 2.3. Let er and {^X,x e V0 } be corresponding Gibbs distributions on {rk,x e V0 }.
Then for any x e V0 e r(rx) , and the corresponding Gibbs distribution that ensure the
factorization property (2.3) are {ux,y > x}.
Let er and {^X,x e V0 } be the Gibbs distributions corresponding to on {rx,x e V0 } We consider the distribution of the spin o(x)= oi, i=1, 2, 3, 4, 5, with respect to and write it in the form , pi = ^X(a(x) = oi ), i=1,2,3,4,5. We show that there exists a unique vector heft2 such that
Pi =
exp(ha{) ■Jj=
i=1,2,3,4,5
Ef=iexp(ho-i)'
H(v) = -J ^ 0(x)a(y)
<x,y>
Results:
^ = {1,2.....q}
r = (V, L, i) G\V ^ fy
a(x) e {fy}
-h z
°a(x)a(y)
<x,y>
>x,y<
whereA £<x,y> SCT(xMy) is nearest neighbor
J2Z>x,y<8a(x)a(y) is next nearest neighbor An = {0.1.2....n} Kk = Z+
on:An^O o^: A£ ^ O |nn| =qn+1
H(a/an) = -J"L<Xty>8a(x)na(y)n -JlSa(x)n^(y)
Zn = ^„eft, exp(~PH(o i <))
nn = niun2u.i.un£, ain = [Gnean, an(0) = i} , \nin\-qn
zh = Z^eni exp(-m° \ °n)
7 _ yQ 71
^n = ^n
q=4 Zl z^, , z
'4 Jn
71
u = —
Un 74
72
V = —
Vn 74 ¿n
73
iz _
Zn+1 @Zn + zn + zn + zn,
Z3+I - Zfl + Z<2 + QZ3 + Z4
Jn
Z>2± 1 - Zfl + + Z3 + Z^
'n+1
Z-4+1 — z1 + Z2 + Z3 + 0 z^
'n
'n 1 ^n Jn ' ^n
n
n
K-
1
zn+1 4
zn+1
^71+^72+^73 +
z
Kl Z4
n+1
n+1
Zn + ®^72 +Z73 ^n+^Tl+^n + ^^n
K-n -
3
zn+1
4
zn+1
Zn +Z71 + ®^73
^71 +^72 +^73 +
lim Un-U lim Kn - K lim K'n-K
lim U!n-U
n^<x>
lim m - U lim K'n-K
u -
eu+v+K+1 u+v+K+e
v -
u+ev+K+1 u+v+K+e
k -
u+ v +ok+ 1 u+v+K+e
2
U+V+K=t
IT T7 „ eu+v+K+1 , u+ev+K+1 , u+v+eK+1 (e+2)(u+v+K)+3 U+V+K=__________ +---+ -
u+v+K+e u+v+K+e u+v+K+e
u+v+K+e
(e+2)t+3 t=---— ^
t+e
V+K=3-U
t2 - 2t - 3 - 0 t1 - 3 , t2 - -1
U -
eu+3-u+1 u+3-u+e
3U=4-U; U=1
V+K=2 => K=2-V
V -
1+ev+2-v+1 1+v+2-v+e
3V=4-V => V=1 K=2-1
=> K=1
U=1, V=1, K=1 q =5 Zi,ZlZl,Zi,Z:
-5 Jn
Z1
U - —
un 75
z2 V - —
vn 75
Z 3
iz _
Kn -75
"t7
4
T _
Ln -
Z,
n
71 -
'n+1
73 7 5
ezn + Zn + Zn + Zn + Zn
Zn + Zn + ezn + Zn + Zn
Zn + Zn + Zn + Zn + ezn
Z2 _ 71 1 ¿372 1 73 1 74 1 75
n+1 ^n + + ^n + ^n + ^n,
Z-4+1 — Zn + Zn + Zn + ezn + Zn
Zn +Zn + OZfi +Zfi +Zfi Zn+Zn+Zn+Zn+0Zn
Ln —
74
Zn+1
Z
n+1
Zn+Zn+Zn + 0Zn+Zn Zn+Zn+Zn+Zn + 0Zn
lim Un — U lim Vn = V lim Kn = K lim Ln — L
n^m
lim Un = U lim % — U lim K' — K lim L'n — L
n^m
0U+V+K+L+1 U+0V+K+L+1 U+V+0K+L+1 U+V+K+dL+1
U — ____________ , V —-, K —-, L —
U+V+K+L+0
U+V+K+L+0
U+V+K+L+0
U+V+K+L+0
„ T7 T, y 0U+V+K+L+1 , U+0V+K+L+1 , U+V+0K+L+1 , U+V+K+0L+1
U+V+K+L=____________ +---1---+
U+V+K+L=
U+V+K+L=a
U+ V+ K+ L+ 0 U+ V+ K+ L+ 0 (0+3)(U+V+K+L)+4
U+ V+ K+ L+ 0
U+ V+ K+ L+ 0
U+ V+ K+ L+ 0
a =(0+3) a+4 => a2 - 3t - 4 — 0 => a. — -4 a2 a+0 1 2
—1
U+V+K+L=1 V+K+L=1-U
U —
U-GU — GU + 2-U, 2U=2,
0V+1-V+1
GU+1-U+1 U+1-U+G U=1
U+K+L=1-V
V —
1- V+ V+ 0
V+ GV= GV + 2-V, 2V=2 , V=1, U+V+L=1-K
K —
GK + 1 - K + 1
1 -K + K + G
K+ GK — GK + 2 - K, 2K=2, K=1
U+V+K=1-L
L —
1-L + GL + 1 1-L+L+G
L-GL — GL + 2 , 2L=2 , L=1
5
Conclusion. Since U=V=K=L=1, one can conclude that the limit Gibbs state is the uniform distribution.
Foydalanilgan adabiyotlar:
1 PURE PHASES OF THE FERROMAGNETIC POTTS MODEL WITH THREE STATES ON A SECOND-ORDER BETHE LATTICE N. N. Ganikhodzhaev
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London, 150(871), 552-575.
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4. Ising, E. (1925). Beitrag zur Theorie des Ferromagnetismus. Zeitschrift für Physik, 31, 253258.
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