|
Дифференциальные уравнения и родственные проблемы анализа, Бухара-2021
1
PERIODIC GROUND STATES CORRESPONDING TO SUBGROUPS OF
INDEX THREE FOR THE ISING MODEL ON THE CAYLEY TREE OF
ORDER THREE
Egamov D. О.
V.I. Romanovskiy Institute of Mathematics, 4-b, University str, 100174, Tashkent,
Uzbekistan
dilshodbekegamov87@gmail.com
The Cayley tree
Γ
k
of order
k
≥
1
is an infinite tree, i.e., a graph without cycles, from
each vertex of which exactly
k
+ 1
edges issue(see [1]). Let
Γ
k
= (
V, L, i
)
, where
V
is the
set of vertices of
Γ
k
,
L
is the set of edges of
Γ
k
and
i
is the incidence function associating
each edge
l
∈
L
with its endpoints
x, y
∈
V
. If
i
(
l
) =
{
x, y
}
, then
x
and
y
are called
nearest neighboring vertices
, and we write
l
=
h
x, y
i
. The distance
d
(
x, y
)
, x, y
∈
V
on the
Cayley tree is the shortest path from
x
to
y
.
For the fixed
x
0
∈
V
(as usual,
x
0
is called a root of the tree) we set
W
n
=
{
x
∈
V
|
d
(
x, x
0
) =
n
}
.
We write
x < y
if the path from
x
0
to
y
goes through
x
and
|
x
|
=
d
(
x, x
0
)
,
x
∈
V
.
It is known 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
G
k
of the free products of
k
+ 1
cyclic
groups
{
e, a
i
}
,
i
= 1
, ..., k
+ 1
of the second order (i.e.
a
2
i
=
e
,
a
i
6
=
e
) with generators
a
1
, a
2
, ..., a
k
+1
.
At first, we give main definitions and facts about the Ising model. We consider models
where the spin takes values in the set
Φ =
{−
1
,
1
}
. For
A
⊆
V
a spin
configuration
σ
A
on
A
is defined as a function
x
∈
A
→
σ
A
(
x
)
∈
Φ
; the set of all configurations is denoted
by
Ω
A
= Φ
A
. Put
Ω = Ω
V
,
σ
=
σ
V
and
−
σ
A
=
{−
σ
A
(
x
)
, x
∈
A
}
.
Define a
periodic
configuration
as a configuration
σ
∈
Ω
which is invariant under cosets of a subgroup
G
∗
k
⊂
G
k
of finite index.
The index of a subgroup is called the
period of the corresponding periodic configuration
.
A configuration that is invariant with respect to all cosets is called
translation-invariant
.
Let
G
k
/G
∗
k
=
{
H
1
, ..., H
r
}
be a family of cosets, where
G
∗
k
is a subgroup of index
r
≥
1
.
We consider model which its spins take values in the set
Φ =
{−
1
,
1
}
.
The Ising model with competing interactions has the form
H
(
σ
) =
J
1
X
h
x,y
i∈
L
σ
(
x
)
σ
(
y
) +
J
2
X
x,y
∈
V
:
d
(
x,y
)=2
σ
(
x
)
σ
(
y
)
,
where
J
= (
J
1
, J
2
)
∈
R
2
are coupling constants and
σ
∈
Ω
.
Let
M
be the set of unit balls with vertices in
V
. We call the restriction of a configuration
σ
to the ball
b
∈
M
a
bounded configuration
σ
b
.
Define the energy of a ball
b
for configuration
σ
by
U
(
σ
b
)
≡
U
(
σ
b
, J
) =
1
2
J
1
X
h
x,y
i∈
L
σ
(
x
)
σ
(
y
) +
J
2
X
d
(
x,y
)=2
σ
(
x
)
σ
(
y
)
, x, y
∈
b,
where
J
= (
J
1
, J
2
)
∈
R
2
.
1
Дифференциальные уравнения и родственные проблемы анализа, Бухара-2021
2
We consider periodic ground states on the Cayley tree of order three, i.e.,
k
= 3
. Let
B
0
=
{
3
,
4
}
, B
d
=
{
d
}
, d
∈ {
1
,
2
}
, i.e.,
m
i
=
i, i
∈ {
1
,
2
}
. Now, we consider functions
u
B
1
B
2
:
{
a
1
, a
2
, a
3
, a
4
} → {
e, a
1
, a
2
}
and
γ
:
< e, a
1
, a
2
>
→ {
e, a
1
, a
2
}
u
{
B
1
}
,
{
B
2
}
(
x
) =
e, if x
=
a
i
, i
= 3
,
4
a
i
, if x
=
a
i
, i
= 1
,
2
,
γ
(
x
) =
e if x
=
e
a
1
if x
∈ {
a
1
, a
2
a
1
}
a
2
if x
∈ {
a
2
, a
1
a
2
}
γ
(
a
i
a
3
−
i
...γ
(
a
i
a
3
−
i
))
if x
=
a
i
a
3
−
i
...a
3
−
i
, l
(
x
)
≥
3
, i
= 1
,
2
γ
(
a
i
a
3
−
i
...γ
(
a
3
−
i
a
i
))
if x
=
a
i
a
3
−
i
...a
i
, l
(
x
)
≥
3
, i
= 1
,
2
.
Let
H
1
:=
=
1
B
1
B
2
(
G
3
)
. Then
H
1
=
{
x
∈
G
3
|
γ
(
u
B
1
B
2
(
x
)) =
e
}
.
Note that
H
1
is a
subgroup of index 3 (see[2]).
G
3
/H
1
=
{
H
1
, H
2
, H
3
}
where
H
2
=
{
x
∈
G
3
|
γ
(
u
B
1
B
2
(
x
)) =
a
1
}
, H
3
=
{
x
∈
G
3
|
γ
(
u
B
1
B
2
(
x
)) =
a
2
}
.
H
1
-periodic configurations have the following forms
σ
(
x
) =
σ
1
x
∈
H
1
,
σ
2
x
∈
H
2
,
σ
3
x
∈
H
3
,
where
σ
i
∈
Φ
, i
∈ {
1
,
2
,
3
}
.
Note that if
σ
1
=
σ
2
=
σ
3
then this configuration is
translation-invariant
and the full
details of such configuration are given (see[1]).
Theorem 1.
Let
k
= 3
.
1) If
(
J
1
, J
2
) =
{
(
J
1
, J
2
) :
J
2
=
−
1
2
J
1
, J
1
≤
0
}
, then there are six
H
1
-periodic ground
states which corresponding to the following configurations
σ
(
x
) =
±
σ
1
if
x
∈
H
1
,
σ
2
if
x
∈
H
2
,
σ
3
if
x
∈
H
3
,
where
(
σ
1
, σ
2
, σ
3
)
∈ {
(
−
1
,
1
,
1)
,
(1
,
−
1
,
1)
,
(1
,
1
,
−
1)
}
.
2) If
(
J
1
, J
2
)
∈
R
2
\{
(
J
1
, J
2
) :
J
2
=
−
1
2
J
1
, J
1
≤
0
}
, there are not
H
1
-periodic (except
for translation-invariant) ground states.
REFERENCES
1. U. A. Rozikov:
Gibbs measures on a Cayley tree,
World Scientific Publishing, Singapore
2013.
2. U. A. Rozikov, F. H. Haydarov:
arXiv:1910.13733
, Invariance property on group
representations of the Cayley tree and its applications.
2
Do'stlaringiz bilan baham: |
