Modulated structures in the ANNNI model with an external magnetic field near criticality
Conditions for a phase transition from the paramagnetic state to the modulated structure are found in a class of anisotropic Ising models with an external magnetic field. The critical value of the external magnetic field is obt...
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Інститут фізики конденсованих систем НАН України
1999
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| Назва видання: | Condensed Matter Physics |
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| Цитувати: | Modulated structures in the ANNNI model with an external magnetic field near criticality / N.S. Gonchar, H.G. Hajduk // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 3-14. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-1199222017-06-11T03:02:59Z Modulated structures in the ANNNI model with an external magnetic field near criticality Gonchar, N.S. Hajduk, H.G. Conditions for a phase transition from the paramagnetic state to the modulated structure are found in a class of anisotropic Ising models with an external magnetic field. The critical value of the external magnetic field is obtained. Branching equations are derived and small branching theorems are proven for commensurate and incommensurate configurations. Знайдено умови фазового переходу із парамагнітного стану до модульованої структури в класі анізотропних моделей Ізінга з зовнішнім магнітним полем. Обчислено значення критичного магнітного поля. Отримані рівняння галуження та доведені теореми про малі галуження для співмірних та неспівмірних конфігурацій. 1999 Article Modulated structures in the ANNNI model with an external magnetic field near criticality / N.S. Gonchar, H.G. Hajduk // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 3-14. — Бібліогр.: 8 назв. — англ. 1607-324X DOI:10.5488/CMP.2.1.3 PACS: 05.50.+q, 64.60.Cn http://dspace.nbuv.gov.ua/handle/123456789/119922 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
Conditions for a phase transition from the paramagnetic state to the modulated structure are found in a class of anisotropic Ising models with an
external magnetic field. The critical value of the external magnetic field is
obtained. Branching equations are derived and small branching theorems
are proven for commensurate and incommensurate configurations. |
| format |
Article |
| author |
Gonchar, N.S. Hajduk, H.G. |
| spellingShingle |
Gonchar, N.S. Hajduk, H.G. Modulated structures in the ANNNI model with an external magnetic field near criticality Condensed Matter Physics |
| author_facet |
Gonchar, N.S. Hajduk, H.G. |
| author_sort |
Gonchar, N.S. |
| title |
Modulated structures in the ANNNI model with an external magnetic field near criticality |
| title_short |
Modulated structures in the ANNNI model with an external magnetic field near criticality |
| title_full |
Modulated structures in the ANNNI model with an external magnetic field near criticality |
| title_fullStr |
Modulated structures in the ANNNI model with an external magnetic field near criticality |
| title_full_unstemmed |
Modulated structures in the ANNNI model with an external magnetic field near criticality |
| title_sort |
modulated structures in the annni model with an external magnetic field near criticality |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119922 |
| citation_txt |
Modulated structures in the ANNNI model with an external magnetic field near criticality / N.S. Gonchar, H.G. Hajduk // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 3-14. — Бібліогр.: 8 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT goncharns modulatedstructuresintheannnimodelwithanexternalmagneticfieldnearcriticality AT hajdukhg modulatedstructuresintheannnimodelwithanexternalmagneticfieldnearcriticality |
| first_indexed |
2025-07-08T16:54:56Z |
| last_indexed |
2025-07-08T16:54:56Z |
| _version_ |
1837098543355002880 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No. 1(17), p. 3–14
Modulated structures in the ANNNI
model with an external magnetic field
near criticality
N.S.Gonchar, H.G.Hajduk
Bogolubov Institute for Theoretical Physics
of the National Academy of Sciences of Ukraine,
14 b Metrolohichna Str., 252143 Kyiv, Ukraine
Received December 3, 1997
Conditions for a phase transition from the paramagnetic state to the mod-
ulated structure are found in a class of anisotropic Ising models with an
external magnetic field. The critical value of the external magnetic field is
obtained. Branching equations are derived and small branching theorems
are proven for commensurate and incommensurate configurations.
Key words: anisotropic Ising model, phase transition, paramagnetic
phase, commensurate configuration, incommensurate configuration,
critical magnetic field
PACS: 05.50.+q, 64.60.Cn
Here we consider a class of anisotropic Ising models with an external magnetic
field and show a phase transition from the paramagnetic state to spatially inho-
mogeneous commensurate configuration or incommensurate one below the critical
magnetic field. To be more exact, we illustrate the method on the ANNNI model
[1,2], but one can apply it to models with an arbitrary number of interacting
neighbours.
Any N1-periodic configuration x can be expressed [3] as
x =
N
∑
l=1
(ale1(lq) + ble2(lq)) + a0ǫ, (1)
where N = [N1/2], [c] is an integral part of c, q = m/N1 is an irreducible quotient,
ǫ = {ai}
+∞
i=−∞, ai = 1 ∀i,
e1(q) = {cos i2πq}+∞
i=−∞, e2(q) = {sin i2πq}+∞
i=−∞ .
c© N.S.Gonchar, H.G.Hajduk 3
N.S.Gonchar, H.G.Hajduk
Then, the specific free energy for commensurate configuration (1) has the form [4]
Fh = −
1
2
(
1
2
N
∑
l=1
((
1 + δ
l,[N1+1
2 ]
)
a2l +
(
1− δ
l,[N1+1
2 ]
)
b2l
)
Φ(lq)
(2)
+ a20Φ(0)
)
+ ha0 +
1
2
T 〈ǫ, g (x)〉 ,
where function g is
g(x) = (1 + x) ln(1 + x) + (1− x) ln(1− x)
= lim
n→∞
2
(
x2
2
+
x4
3 · 4
+ · · ·+
x2n
2n · (2n− 1)
)
provided that
N
∑
l=1
(|al|+ |bl|) + |a0| < 1.
For the ANNNI model
Φ(q) = −2(J1 cos 2πq + J2 cos 4πq + 2J0),
and for the case when all the neighbours interact
Φ(q) = −2
(
+∞
∑
i=1
Ji cos 2πiq + 2J0
)
.
The free energy (2) is invariant under a discrete transformation group [5]
a′l = al cos 2πlsq + bl sin 2πlsq,
b′l = −al sin 2πlsq + bl cos 2πlsq,
(3)
q = m/N1, s = 0, 1, · · · , N1.
The free energy of incommensurate configuration (1) with irrational q and
arbitrary N is [6]
Fh = −
1
2
(
1
2
N
∑
l=1
(a2l + b2l )Φ(lq) + a20Φ(0)
)
+ ha0 +
1
2
T 〈ǫ, g (x)〉 . (4)
The free energy (4) is invariant of group G [3]
a′l = al cos lφ0 + bl sin lφ0,
b′l = −al sin lφ0 + bl cos lφ0,
(5)
φ0 ∈ [0, 2π].
4
Modulated structures in the ANNNI model
The necessary conditions of a free energy minimum are the equations
−alΦ(lq)
(
1 + δ
l,[N1+1
2 ]
)
+ T 〈ǫ, g′(x)e1(lq)〉 = 0,
−blΦ(lq)
(
1− δ
l,[N1+1
2 ]
)
+ T 〈ǫ, g′(x)e2(lq)〉 = 0, (6)
−a0Φ(0) + h+ T 〈ǫ, g′(x)〉 /2 = 0,
g′(x) = ln
1 + x
1− x
= 2
(
x+
x3
3
+
x5
5
+ · · ·+
x2n+1
2n+ 1
+ · · ·
)
,
for rational q and
−alΦ(lq) + T 〈ǫ, g′(x)e1(lq)〉 = 0,
−blΦ(lq) + T 〈ǫ, g′(x)e2(lq)〉 = 0, (7)
−a0Φ(0) + h+ T 〈ǫ, g′(x)〉 /2 = 0
for irrational q.
In the external magnetic field the systems (6) and (7) have a nonzero solution
in the set of vectors {(a1, b1), · · · , (aN , bN), a0} such that
N
∑
l=1
(|al|+ |bl|) + |a0| < 1.
The solution has the form {(0, 0), · · · , (0, 0), ā0} with ā0 solving the equation
−ā0Φ(0) + Tg′(ā0)/2 + h = 0. (8)
We assume that
x = x̄+ ā0ǫ,
x̄ =
N
∑
l=1
(ale1(lq) + ble2(lq)) + a′0ǫ,
g′(x) =
+∞
∑
k=1
g(k)(ā0)
(k − 1)!
x̄k−1 = g′(ā0)ǫ+ g′′(ā0)x̄+ ϕ(x̄),
g′(ā0ǫ) = g′(ā0)ǫ, x̄ǫ = x̄,
ϕ(x̄) =
+∞
∑
k=3
g(k)(ā0)
(k − 1)!
x̄k−1.
Then, system (6) becomes
al
(
T/
(
1− ā20
)
− Φ(lq)
)(
1 + δ
l,[N1+1
2 ]
)
+ T 〈ǫ, ϕ(x̄)e1(lq)〉 = 0,
bl
(
T/
(
1− ā20
)
− Φ(lq)
)(
1− δ
l,[N1+1
2 ]
)
+ T 〈ǫ, ϕ(x̄)e2(lq)〉 = 0, (9)
2a′0
(
T/
(
1− ā20
)
− Φ(0)
)
+ T 〈ǫ, ϕ(x̄)〉 = 0,
5
N.S.Gonchar, H.G.Hajduk
and system (7) becomes
al
(
T/(1− ā20)− Φ(lq)
)
+ T 〈ǫ, ϕ(x̄)e1(lq)〉 = 0,
bl
(
T/(1− ā20)− Φ(lq)
)
+ T 〈ǫ, ϕ(x̄)e2(lq)〉 = 0, (10)
2a′0
(
T/(1− ā20)− Φ(0)
)
+ T 〈ǫ, ϕ(x̄)〉 = 0.
A spatially inhomogeneous solution of systems (6), (7) exists if systems (9), (10)
have a nonzero solution. The first degeneration of the spectrum of the linearized
part of nonlinear operator (9) or (10) occurs when
T/ (1− ā20)− Φ(q) = 0,
−ā0Φ(0) + Tg′(ā0)/2 + h = 0.
(11)
If there exists a solution to system (11), then the zero solution to systems (9),
(10) bifurcates. It means that a transition from the paramagnetic phase to the
modulated structure occurs. As a result, system (11) gives a single equivalent
equation
T = Φ(q)− Φ(q)h2
/ [
Φ(0)−
1
2
T
(
1− T/Φ(q)
)− 1
2
g′
(
1− T/Φ(q)
)
1
2
]2
. (12)
From (12) it follows:
Lemma 1. The critical magnetic field for a phase transition from the paramag-
netic state to the modulated structure is equal to |Φ(0)|.
For the proof see [7].
Having in mind that the mean-field theory critical exponent β equals 1/2, we
suppose a deviation from the critical temperature to be standard. Therefore,
λ2(T, h) = Φ(q)− T/
(
1− ā20
)
,
where ā0 solves equation (8). If T0(h) solves equation (12), then λ2(T0(h), h) equals
zero.
In terms of new variables, system (10) for irrational q takes the form:
−λ2a1 + T 〈ǫ, ϕ(x̄)e1(q)〉 = 0,
−λ2b1 + T 〈ǫ, ϕ(x̄)e2(q)〉 = 0,
(13)
al
(
Φ(q)− Φ(lq)− λ2
)
+ T 〈ǫ, ϕ(x̄)e1(lq)〉 = 0,
bl
(
Φ(q)− Φ(lq)− λ2
)
+ T 〈ǫ, ϕ(x̄)e2(lq)〉 = 0, (14)
2a′0
(
Φ(q)− Φ(0)− λ2
)
+ T 〈ǫ, ϕ(x̄)〉 = 0.
We seek the solution of systems (13), (14) in the form
a1 = λã1, b1 = λb̃1, al = λ2ãl, bl = λ2b̃l, a′0 = λ2ã′0.
6
Modulated structures in the ANNNI model
Then we rewrite (13), (14) as
ã1 = F
(1)
1 (w̃) =
T
λ3
+∞
∑
k=3
g(k)(ā0)
(k − 1)!
λk−1
〈
ǫ, (·)k−1 e1(q)
〉
,
(15)
b̃1 = F
(2)
1 (w̃) =
T
λ3
+∞
∑
k=3
g(k)(ā0)
(k − 1)!
λk−1
〈
ǫ, (·)k−1 e2(q)
〉
,
ãl =F
(1)
l (w̃)=
T
λ2(λ2 +Φ(lq)− Φ(q))
+∞
∑
k=3
g(k)(ā0)
(k − 1)!
λk−1
〈
ǫ, (·)k−1 e1(lq)
〉
,
b̃l =F
(2)
l (w̃)=
T
λ2(λ2 +Φ(lq)− Φ(q))
+∞
∑
k=3
g(k)(ā0)
(k − 1)!
λk−1
〈
ǫ, (·)k−1 e2(lq)
〉
,
ã′0 =F
(0)
0 (w̃)=
T
2λ2(λ2 +Φ(0)− Φ(q))
+∞
∑
k=3
g(k)(ā0)
(k − 1)!
λk−1
〈
ǫ, (·)k−1
〉
, (16)
where
w̃ = {λ, z̃, ã1, b̃1}, z̃ = {ã, b̃} =
(
{ã2, b̃2}, · · · , {ãN , b̃N}, ã
′
0
)
,
(
·
)
=
(
λ
N
∑
l=2
(ãle1(lq) + b̃le2(lq)) + λã′0ǫ+ ã1e1(q) + b̃1e2(q)
)
.
Let’s consider the norm
|z̃| = max
26l6N
{|ãl|, |b̃l|, |ã
′
0|}
and the radius A ball-centred at the point
z̃0 =
(
{ã02, b̃
0
2}, · · · , {ã
0
N , b̃
0
N}, ã
′0
0
)
in a set of vectors z̃, where
ã0l =
T
λ2+Φ(lq)−Φ(q)
+∞
∑
k=3
g(k)(ā0)
(k − 1)!
λk−3
〈
ǫ, (ã1e1(q)+b̃1e2(q))
k−1e1(lq)
〉
,
b̃0l =
T
λ2+Φ(lq)−Φ(q)
+∞
∑
k=3
g(k)(ā0)
(k − 1)!
λk−3
〈
ǫ, (ã1e1(q)+b̃1e2(q))
k−1e2(lq)
〉
,
ã
′0
0 =
T
2(λ2+Φ(0)−Φ(q))
+∞
∑
k=3
g(k)(ā0)
(k − 1)!
λk−3
〈
ǫ, (ã1e1(q)+b̃1e2(q))
k−1
〉
.
Let F (w̃) be a nonlinear operator given by the right-hand part of system (16),
then this system turns into
z̃ = F (w̃) . (17)
7
N.S.Gonchar, H.G.Hajduk
Theorem 1. Let
min
l
|Φ(lq)− Φ(q)| = δ > 0,
l = 0, 2, 3, 4, · · · , N − 1, N,
and λ0 > 0 satisfy the inequality
max
l
T
|λ2 +Φ(lq)− Φ(q)|
(2N − 1)λ
+∞
∑
k=3
|g(k)(ā0)|
(k − 2)!
× λk−3((2N − 1)Aλ+ 2A)k−2 < 1,
then for all λ ∈ [0, λ0] and ã1, b̃1, h such that
ã21 + b̃21 6 A2, |h| < |Φ(0)|,
there exist critical temperature T0(h) and interval (T0(h) − λ2, T0(h)) on which
the solution of nonlinear equations (17) exists, being a continuously differentiable
function of λ ∈ [0, λ0] and ã1, b̃1 for ã21 + b̃21 6 A2.
Proof. Operator F maps radius A ball-centred at the point
z0 = F (λ, {0, 0}, ã1, b̃1)
into itself if the inequality
‖F (λ, {ã, b̃}, ã1, b̃1)− F (λ, {0, 0}, ã1, b̃1)‖ 6 αA, α < 1, (18)
holds. Let’s estimate the difference
‖F
(p)
l (λ, {ã, b̃}, ã1, b̃1)− F
(p)
l (λ, {0, 0}, ã1, b̃1)‖
6
T
|λ2 +Φ(lq)− Φ(q)|
+∞
∑
k=3
|g(k)(ā0)|
(k − 1)!
λk−3
×
∣
∣
∣
〈
ǫ,
(
(·)k−1 − (ã1e1(q) + b̃1e2(q))
k−1
)
ep(lq)
〉
∣
∣
∣
,
p = 0, 1, 2, l = 0, 2, N.
From this
‖F
(p)
l (λ, {ã, b̃}, ã1, b̃1)− F
(p)
l (λ, {0, 0}, ã1, b̃1)‖
6 max
l
T
|λ2 +Φ(lq)− Φ(q)|
+∞
∑
k=3
|g(k)(ā0)|
(k − 1)!
λk−3
×λ(2N − 1)A
k−2
∑
i=0
(λ(2N − 1)A)k−2−i (2A)i
6 max
l
T
|λ2 +Φ(lq)− Φ(q)|
λ(2N − 1)A
8
Modulated structures in the ANNNI model
×
+∞
∑
k=3
|g(k)(ā0)|
(k − 2)!
λk−3 (λ(2N − 1)A+ 2A)k−2 .
At last one can estimate the difference
‖F (λ, {ã′, b̃′}, ã1, b̃1)− F (λ, {ã, b̃}, ã1, b̃1)‖ 6 max
l
T
|λ2 + Φ(lq)− Φ(q)|
×
+∞
∑
k=3
|g(k)(ā0)|
(k − 2)!
λk−3λ(2N − 1) (λ(2N − 1)A+ 2A)k−2 ‖{ã′, b̃′} − {ã, b̃}‖.
Therefore, choosing a rather small λ we realize the inequality
max
l
T
|λ2 +Φ(lq)− Φ(q)|
(2N − 1)λ
+∞
∑
k=3
|g(k)(ā0)|
(k − 2)!
×λk−3((2N − 1)Aλ+ 2A)k−2 < 1,
that guarantees inequality (18) and contractivity conditions for nonlinear map F .
If ã1 and b̃1 are such that ã21 + b̃21 6 A2, then inequality (18) holds. This ends the
proof.
The similar theorem is valid for rational q.
Remark 1. Continuous differentiability with respect to the variables λ and ã1, b̃1
results from the continuous differentiability of the successive approximations and
their uniform convergence to the solution.
Theorem 2. For all the irrational q that guarantee the validity of
min
l
|Φ(lq)− Φ(q)| = δ > 0,
l = 0, 2, 3, 4, · · · , N − 1, N,
there is such λ1 > 0, that system (10) is solvable for 0 < λ < λ1,
|h| < |Φ(0)|. The action of group G (5) on this solution gives different solutions of
system (10) being continuously differentiable functions of λ.
Proof. Having calculated the averages,
Is,m =
〈
s
∏
k=1
e1(ikq)
m
∏
l=1
e2(jlq), ǫ
〉
entering equations (13) by formula [5]
Is,m =
0, m = 2M + 1, M = 0, 1, 2, · · · ,
(−1)M2−s−m
∑
t
k′
=±i
k′
τl=±jl
δt1+···+ts+τ1+···+τm,0(−1)k, m = 2M,
9
N.S.Gonchar, H.G.Hajduk
where k is the number of negative τl, we write system (13) as
ã1=T
(
g(3)(ā0)
2
( ã2ã1+b̃2b̃1
2
+ã′0ã1
)
+
g(4)(ā0)
16
(
ã31+ã1b̃
2
1
)
+λ2ϕ̃
(1)
1 (w̃)
)
,
b̃1=T
(
g(3)(ā0)
2
( b̃2ã1−ã2b̃1
2
+ã′0b̃1
)
+
g(4)(ā0)
16
(
b̃31+b̃1ã
2
1
)
+λ2ϕ̃
(2)
1 (w̃)
)
,
where
g(3)(ā0) =
4ā0
(1− ā20)
2
,
g(4)(ā0) =
4
(1− ā20)
2
+
16ā20
(1− ā20)
3
.
For the zero approximation, solutions for ã′0, ã2 and b̃2 are
ã
′0
0 =
Tg(3)(ā0)
(
ã21 + b̃21
)
8(λ2 + Φ(0)− Φ(q))
,
ã02 =
Tg(3)(ā0)
(
ã21 − b̃21
)
8(λ2 + Φ(2q)− Φ(q))
,
b̃02 =
Tg(3)(ā0)ã1b̃1
4(λ2 + Φ(2q)− Φ(q))
.
Due to the continuous transformation group G (5), the branching equation has
the form [7]
T
(
D
(
ã21 + b̃21
)
+ λ2ϕ̃0(λ, ã
2
1 + b̃21)
)
= 1, (19)
where
D =
T ā20
(1− ā20)
4
(
1
2(λ2 + Φ(2q)− Φ(q))
+
2
λ2 +Φ(0)− Φ(q)
)
+
ā20
(1− ā20)
3
+
1
4(1− ā20)
2
.
In terms of u = ã21 + b̃21 equation (19) becomes
u =
(
1/T − λ2ϕ̃0(λ, u)
)
/D, (20)
where ϕ̃0(λ, u) is a continuously differentiable function of all the variables in the
vicinity of
|λ| < λ0, 0 6 u 6 2A2.
The inequalities
sup
06u62A2
D−1
∣
∣T−1 − λ2ϕ̃0(λ, u)
∣
∣ < 2A2,
(21)
sup
06u62A2
λ2D−1 |∂ϕ̃0(λ, u)/∂u| < 1
10
Modulated structures in the ANNNI model
hold for a rather small λ and 2A2 > 1/TD. From inequalities (21) it follows that
equation (20) has the solution, which is a continuously differentiable function of λ
for rather small λ < λ1. This ends the proof.
Theorem 3. For those rational q, for which N1 6= 3, there is λ0 > 0 such that a
nonzero solution to system (9) exists for 0 < λ < λ0,
|h| < |Φ(0)|. The action of group (3) on this solution gives different solutions to
system (9) being continuously differentiable functions of λ.
Proof. Similarly to the case of irrational q, one can prove that the problem of the
existence of a nonzero solution to system (9) is equivalent to that of branching
equation [8]. To derive branching equations for different values of period N1 we
calculate the averages Is,m entering system (6) by formula [5]
Is,m =
0, m = 2M + 1, M = 0, 1, 2, · · · ,
(−1)M2−s−m
∑
r=0,±1,±2,···
∑
t
k′
=±i
k′
τl=±jl
δt1+···+ts+τ1+···+τm,|r|N1
(−1)k, m = 2M,
where k is the number of negative τl.
For N1 = 2 we obtain a single branching equation
ã1 = T
(
g(4)(ā0)ã
3
1/6 + λ2ϕ̃
(2)
1 (λ, ã21)ã1
)
/2. (22)
There are small solutions of equation (22) which are continuously differentiable
functions of λ.
For N1 = 3 the branching equations are
ã1 = T
(
ã1
(
ã21 + b̃21
)
((
g(3)(ā0)
4
)2
T
λ2 +Φ(0)− Φ(q)
g(4)(ā0)
16
)
+
g(3)(ā0)
8λ
(
ã21 − b̃21
)
+ λϕ
(3)
1 (λ, ã1, b̃
2
1)
)
, (23)
b̃1 = T
(
b̃1
(
ã21 + b̃21
)
((
g(3)(ā0)
4
)2
T
λ2 + Φ(0)− Φ(q)
+
g(4)(ā0)
16
)
−
g(3)(ā0)
4λ
b̃1ã1 + λϕ
(3)
2 (λ, ã1, b̃
2
1)b̃1
)
.
System (23) has no small solutions for a rather small λ. As a result, branching
from the paramagnetic phase to the commensurate one of period N1 = 3 does not
occur in the magnetic field.
For N1 = 4 (antiphase) the branching equations are
ã1 = T
[
ã1
(
g(3)(ā0)
2
)2
T
2
(
ã21 − b̃21
λ2 +Φ(2q)− Φ(q)
+
ã21 + b̃21
2(λ2 + Φ(0)− Φ(q))
)
11
N.S.Gonchar, H.G.Hajduk
+
g(4)(ā0)
12
ã31 + λ2ϕ
(4)
1 (λ, ã1, b̃
2
1)
]
, (24)
b̃1 = T
[
b̃1
(
g(3)(ā0)
2
)2
T
2
(
ã21 + b̃21
2(λ2 +Φ(0)− Φ(q))
−
ã21 − b̃21
λ2 + Φ(2q)− Φ(q)
)
+
g(4)(ā0)
12
b̃31 + λ2ϕ
(4)
2 (λ, ã1, b̃
2
1)b̃1
]
.
For N1 = 5 the branching equations are
ã1 = T
[
ã1
(
ã21 + b̃21
)
D̃ + λϕ
(5)
1 (λ, ã1, b̃
2
1)
]
,
(25)
b̃1 = T
[
b̃1
(
ã21 + b̃21
)
D̃ + λϕ
(5)
2 (λ, ã1, b̃
2
1)b̃
2
1
]
,
where
D̃=T
(
g(3)(ā0)
4
)2(
1
2(λ2+Φ(2q)−Φ(q))
+
1
λ2+Φ(0)−Φ(q)
)
+
g(4)(ā0)
16
.
If N1 > 6, then the branching equations are
ã1 = T
[
ã1
(
ã21 + b̃21
)
D̃ + λ2ϕ
(N1)
1 (λ, ã1, b̃
2
1)
]
,
(26)
b̃1 = T
[
b̃1
(
ã21 + b̃21
)
D̃ + λ2ϕ
(N1)
2 (λ, ã1, b̃
2
1)b̃
2
1
]
.
One essential distinction occurs, namely, the second equation allows a zero
solution, but the first one assumes a nonzero solution with respect to ã1. We
denote
ϕ̃
(N1)
1 (t, ã1) =
ϕ
(N1)
1 (t, ã1, 0)
ã1
, ϕ̃
(N1)
1 (t, 0) = 0.
Considering ã1 6= 0 and dividing by it, we obtain the equation
ã1 = ±
√
D̃−1
(
T−1 − λ2ϕ̃
(N1)
1 (λ, ã1)
)
. (27)
For A > 1/
√
D̃T and a positive sign we have
ã1 =
√
D̃−1
(
T−1 − λ2ϕ̃
(N1)
1 (λ, ã1)
)
. (28)
From the conditions
sup
ã16A
√
D̃−1
(
T−1 − λ2ϕ̃
(N1)
1 (λ, ã1)
)
6 A,
(29)
sup
ã1≤A
λ2T
∣
∣
∣
d
dã1
ϕ̃
(N1)
1 (λ, ã1)
∣
∣
∣
D̃
√
D̃−1
(
T−1 − λ2ϕ̃
(N1)
1 (λ, ã1)
)
< 1
12
Modulated structures in the ANNNI model
it follows that (28) has a unique solution for rather small 0 < λ < λ0.
Taking the minus sign in equation (27), one obtains an alternate nonzero so-
lution.
References
1. Villain J. La structure des substances magnetiques. // J. Phys. Chem. Solids, 1959,
vol. 11, p. 303–309.
2. Kaplan T.A. Classical spin-configuration stability in the presence of competing ex-
change forces. // Phys. Rev., 1959, vol. 116, p. 888–889.
3. Gonchar N.S., Kozyrski W.H. Hidden symmetry and small branchings in the
anisotropic model. // Ukr. Math. J., 1991, vol. 43, No 11, p. 1509–1516 (in Ukrainian).
4. Gonchar N.S., Hajduk H.G., Kozyrski W.H. Branching equations for commensurate
phase of the ANNNI model. Preprint ITP–95–2U, Kiev, 1995 (in Ukrainian).
5. Gonchar N.S., Kozyrski W.H. Incommensurate phases. Preprint ITP–90–51R, Kiev,
1990 (in Russian).
6. Gonchar N.S., Kozyrski W.H. One-parameter symmetry group of incommensurate
phases of the ANNNI model. // Ukr. Fiz. Zhurn., 1991, vol. 36, No 12, p. 1857–1864
(in Ukrainian).
7. Gonchar N.S., Hajduk H.G. Phase transition in the ANNNI model in the presence of
external magnetic field. Preprint ITP–94–31U, Kiev, 1994 (in Ukrainian).
8. Hajduk H.G. Branching equations for commensurate phases of the ANNNI model
in the presence of external magnetic field. Preprint ITP–95–22U, Kiev, 1995 (in
Ukrainian).
13
N.S.Gonchar, H.G.Hajduk
Модульовані структури поблизу критичності в
моделі ANNNI з зовнішнім магнітним полем
М.С.Гончар, І.Г.Гайдук
Відділ математичного моделювання Інституту теоретичної фізики
ім. М.М.Боголюбова НАН України,
252143 Київ, вул. Метрологічна 14 b
Отримано 3 грудня 1997 р.
Знайдено умови фазового переходу із парамагнітного стану до мо-
дульованої структури в класі анізотропних моделей Ізінга з зовнішнім
магнітним полем. Обчислено значення критичного магнітного поля.
Отримані рівняння галуження та доведені теореми про малі галужен-
ня для співмірних та неспівмірних конфігурацій.
Ключові слова: анізотропна модель Ізінга, фазовий перехід,
парамагнітна фаза, співмірна конфігурація, неспівмірна
конфігурація, критичне магнітне поле
PACS: 05.50.+q, 64.60.Cn
14
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