Novel phase transition in two-dimensional xy-models with long-range interaction
The purpose of this article is to give an overview of results concerning ordering and critical properties of two-dimensional ferromagnets including the dipolar interaction. We investigate a two-dimensional xy-model extended by the dipolar interaction. Describing our system by a nonlinear σ-model...
Збережено в:
| Дата: | 2005 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
2005
|
| Назва видання: | Condensed Matter Physics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/119389 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Novel phase transition in two-dimensional xy-models with long-range interaction / P.G. Maier, F. Schwabl // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 103-111. — Бібліогр.: 20 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-119389 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1193892017-06-07T03:05:10Z Novel phase transition in two-dimensional xy-models with long-range interaction Maier, P.G. Schwabl, F. The purpose of this article is to give an overview of results concerning ordering and critical properties of two-dimensional ferromagnets including the dipolar interaction. We investigate a two-dimensional xy-model extended by the dipolar interaction. Describing our system by a nonlinear σ-model and using renormalization group methods we predict a phase transition to an ordered state. This transition is due to the long-range dipolar interaction. The ferromagnetic phase is governed by a low temperature fixed-point with infinite dipolar coupling. In the critical regime we find exponential behavior for the correlation length and the order parameter in contrast to the usual power laws. The nature of the transition shows a striking similarity to the Kosterlitz-Thouless transition. We show that there is a whole class of longrange xy-models leading to such non-standard behavior. Parameterizing the divergencies in terms of the correlation length we are able to calculate the critical exponents. These exponents are correct in any loop order. Метою цієї статті є дати огляд результатів, що стосуються впорядкування і критичних властивостей двовимірних феромагнетиків, які включають дипольні взаємодії. Ми досліджуємо двовимірну xy-модель розширену дипольними взаємодіями. Описуючи нашу систему нелінійною σ-моделлю і використовуючи ренормалізаційногрупові методи ми передбачаємо фазовий перехід до впорядкованого стану. Цей перехід з’являється в результаті далекосяжної дипольної взаємодії. В критичному режимі ми знаходимо експоненційну поведінку для кореляційної довжини та параметра порядку на відміну від звичайних степеневих законів. Природа переходу виявляє разючу подібність з переходом Костерліца-Таулеса. Ми показуємо, що існує цілий клас далекосяжних xy-моделей, які приводять до такої нестандартної поведінки. Параметризуючи розбіжності в термінах кореляційної довжини, ми здатні обчислити критичні показники.Ці показники є точними в будь-якому петлевому порядку. 2005 Article Novel phase transition in two-dimensional xy-models with long-range interaction / P.G. Maier, F. Schwabl // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 103-111. — Бібліогр.: 20 назв. — англ. 1607-324X PACS: 05.70.Jk, 75.40.Cx, 75.70.Ak DOI:10.5488/CMP.8.1.103 http://dspace.nbuv.gov.ua/handle/123456789/119389 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The purpose of this article is to give an overview of results concerning
ordering and critical properties of two-dimensional ferromagnets including
the dipolar interaction. We investigate a two-dimensional xy-model extended
by the dipolar interaction. Describing our system by a nonlinear σ-model
and using renormalization group methods we predict a phase transition to
an ordered state. This transition is due to the long-range dipolar interaction.
The ferromagnetic phase is governed by a low temperature fixed-point with
infinite dipolar coupling. In the critical regime we find exponential behavior
for the correlation length and the order parameter in contrast to the usual
power laws. The nature of the transition shows a striking similarity to the
Kosterlitz-Thouless transition. We show that there is a whole class of longrange
xy-models leading to such non-standard behavior. Parameterizing
the divergencies in terms of the correlation length we are able to calculate
the critical exponents. These exponents are correct in any loop order. |
| format |
Article |
| author |
Maier, P.G. Schwabl, F. |
| spellingShingle |
Maier, P.G. Schwabl, F. Novel phase transition in two-dimensional xy-models with long-range interaction Condensed Matter Physics |
| author_facet |
Maier, P.G. Schwabl, F. |
| author_sort |
Maier, P.G. |
| title |
Novel phase transition in two-dimensional xy-models with long-range interaction |
| title_short |
Novel phase transition in two-dimensional xy-models with long-range interaction |
| title_full |
Novel phase transition in two-dimensional xy-models with long-range interaction |
| title_fullStr |
Novel phase transition in two-dimensional xy-models with long-range interaction |
| title_full_unstemmed |
Novel phase transition in two-dimensional xy-models with long-range interaction |
| title_sort |
novel phase transition in two-dimensional xy-models with long-range interaction |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119389 |
| citation_txt |
Novel phase transition in two-dimensional xy-models with long-range interaction / P.G. Maier, F. Schwabl // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 103-111. — Бібліогр.: 20 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT maierpg novelphasetransitionintwodimensionalxymodelswithlongrangeinteraction AT schwablf novelphasetransitionintwodimensionalxymodelswithlongrangeinteraction |
| first_indexed |
2025-07-08T15:47:22Z |
| last_indexed |
2025-07-08T15:47:22Z |
| _version_ |
1837094297479938048 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 1(41), pp. 103–111
Novel phase transition in
two-dimensional xy-models with
long-range interaction
P.G.Maier, F.Schwabl
Institut für Theoretische Physik,
Physik-Department der Technischen Universität München,
James-Franck-Straße, D–85747 Garching, Germany
Received January 13, 2005
The purpose of this article is to give an overview of results concerning
ordering and critical properties of two-dimensional ferromagnets including
the dipolar interaction. We investigate a two-dimensional xy-model extend-
ed by the dipolar interaction. Describing our system by a nonlinear σ-model
and using renormalization group methods we predict a phase transition to
an ordered state. This transition is due to the long-range dipolar interaction.
The ferromagnetic phase is governed by a low temperature fixed-point with
infinite dipolar coupling. In the critical regime we find exponential behavior
for the correlation length and the order parameter in contrast to the usual
power laws. The nature of the transition shows a striking similarity to the
Kosterlitz-Thouless transition. We show that there is a whole class of long-
range xy-models leading to such non-standard behavior. Parameterizing
the divergencies in terms of the correlation length we are able to calculate
the critical exponents. These exponents are correct in any loop order.
Key words: critical phenomena, magnetic properties, low-dimensional
systems
PACS: 05.70.Jk, 75.40.Cx, 75.70.Ak
Dedicated to Reinhard Folk on the
occasion of his 60th birthday
1. Introduction
Magnetic order in two dimensions has been investigated from various authors.
In a two-dimensional ferromagnet with isotropic exchange interaction the order pa-
rameter has to vanish. On the other hand, a finite order parameter is possible when
rotational invariance is broken, in case the interaction contains uniaxial contributi-
ons. Even for the magnetic dipolar interaction, which does not favor any particular
direction, magnetic fluctuations are sufficiently reduced, leading to a finite order
c© P.G.Maier, F.Schwabl 103
P.G.Maier, F.Schwabl
parameter below a transition temperature. In this article we want to describe the
critical properties in the ordered phase [1].
The low temperature phase of the two-dimensional xy-model shows critical be-
havior, but long-range order is suppressed by strong fluctuations. The absence of
spontaneous order for isotropic, short-range systems was rigorously proven by Mer-
min and Wagner [2]. The high temperature phase with exponentially decaying corre-
lations is, according to Kosterlitz and Thouless, due to the existence of free vortices
[3–5]. On the contrary, for low temperatures, these topological excitations are bound
closely in pairs of charge zero. Introducing a long-range force alters the low temper-
ature properties of the xy-model in a crucial way. Due to the long-range interaction
the fluctuations are reduced and the system orders at a certain temperature, thereby
showing a very special critical behavior.
Situations described by a dipolar xy-model also appear in experimental systems.
For example consider a two-dimensional layer of magnetic moments. Such layers are
created by depositing magnetic atoms on a non-magnetic substrate (e.g. [6–8]). In
the simplest picture the magnetic properties are described by a 2d Heisenberg-model
with isotropic short-range interaction. But for a more realistic model one also has to
be aware of the dipolar interaction between the magnetic moments and the existence
of magneto-crystalline anisotropies due to spin-orbit coupling [9,10]. Under certain
conditions, when the spins prefer to order in the plane of the magnetic atoms, it is
sufficient to concentrate on the isotropic exchange and the dipolar interaction and
to deal with a two-component spin (for details see [1]). In this case it is possible to
describe the large scale properties of the system by a continuum model
H[S]
t
=
1
2t
∫
d2x (∂µS(x))2− H
t
∫
d2xSx(x)+
G
4πt
∫
d2x d2x′ (∂S(x))(∂ ′S(x′))
|x − x′| ,
(1)
where S is a two-component spin constrained by S2 = 1, t is an effective tempera-
ture, H is the external field and G measures the strength of the dipolar interaction.
Due to the dipolar interaction this model no longer becomes trivial by introducing
polar coordinates. So we prefer to renormalize (1) in the coordinates of the nonlin-
ear σ-model [11,13,14,16]. As the dipolar interaction violates the O(2)-symmetry the
proof of renormalizability of Brézin, Zinn-Justin and Le Guillou is no longer appli-
cable [12]. The model described by the Ginzburg-Landau functional (1) and which
we call dxy-model is instead invariant only if we rotate spin and coordinate-vectors
simultaneously. With the help of this invariance we show in [1] that the dxy-model
is renormalizable.
The next step is to express the spins by independent coordinates of the nonlinear
σ-model
(Sx, Sy) = (σ, π), σ =
√
1 − π2, (2)
and to expand the functional in terms of π. According to symmetry considerati-
ons it is not necessary to renormalize each arising term. Instead, it is sufficient to
renormalize the field π, the temperature t and the dipolar coupling G. Usually one
renormalizes the nonlinear σ-model at the lower critical dimension. By including
104
Two-dim xy -models with long-range interaction
the long-range forces this lower critical dimension becomes dL = 1 [15]. One of the
constraints arising from the form of the dipolar interaction in (1) is the restriction
for the number of components N to N = d, where d is the dimension of the lattice.
Hence working at the lower critical dimension reduces the number of components to
N = dL = 1. As one component is eliminated via the condition S2 = 1 one effectively
deals with a zero component field. Therefore an ε-expansion about the lower critical
dimension extrapolates from a trivial situation. To avoid this problem we work in
the fixed dimension d = N = 2 and refrain from using an ε-expansion. Moreover,
we use a cutoff regularization and work out the renormalization-group equations for
the non-renormalized correlation functions following the ideas of Zinn-Justin [17].
Using this method one reduces the two-dimensional integrals to a single integration.
For more details see [1]. Finally we derive the Callan-Symanzik equations
(
Λ
∂
∂Λ
+ βt(t, g)
∂
∂t
+ βg(t, g)
∂
∂g
+
n
2
ζ(t, g) + ρ(t, g)H
∂
∂H
)
G(n)({qi} , t, g, H, Λ) = 0,
(3)
where Λ is the cutoff, g is the dimensionless dipolar coupling G/Λ and G(n) are the
n-point correlation functions. Doing a one loop calculation and expanding the flow
functions βt and βg in the vicinity of the xy-line (t = t0, g = 0) one obtains for the
linear order
βt(t, g) =
5t20
32π
g, (4a)
βg(t, g) =
(
t0
2π
− 1
)
g . (4b)
Obviously any point on the xy-line is a fixed point. But the stability properties
change at tc ≡ 2π. While the xy-line is stable for t > tc it becomes unstable for
t < tc due to the dipolar interaction1. At the critical point (tc = 2π, gc = 0) the linear
order vanishes and the critical behavior is determined by the nonlinear contributions
to (4b).
The global flow predicted by the renormalization group is depicted in figure 1.
One observes that for temperatures below tc, where the xy-line is unstable, there
is a flow towards a low temperature fixed point with infinite dipolar coupling. This
fixed point is attractive for the regions I and II of the flow diagram and determines
the asymptotic properties. In region III the long-range interaction is irrelevant and
the system shows classical xy-behavior. As we used the low temperature form of the
Hamiltonian and did not consider any topological excitations, the xy-line persists
to be stable in region III. Our physical predictions are therefore restricted to the
low-temperature and to the critical part of the flow diagram. In the vicinity of the
critical point the flow diagram shows a striking similarity to the flow diagram of
the Kosterlitz-Thouless transition [4]. But while in the Kosterlitz-Thouless model
the vortices destabilize the high-temperature part of the xy-line here it is the low
temperature part which becomes unstable due to the long-range interaction.
1The change of stability at t = 2π is in agreement with the results of [15,18]. Our general scaling
form (13) reduces to the results of [15,18] in the limit of vanishing g.
105
P.G.Maier, F.Schwabl
0 0,2 0,4 0,6 0,8 1
0
0,2
0,4
0,6
0,8
1
II
I III
η
θ
Figure 1. Renormalization group flow of the dxy-model with θ = t/(tc+t) and
η = g/(1+g). The trajectory crossing the critical point (tc = 2π, gc = 0) divides
the diagram into three different regions. In region III the xy-line (g=0) is stable
against the perturbation from the long-range interaction, while it is unstable in
region I. For region I and II the low temperature fixed point with infinite dipolar
coupling is attractive. Bold printed is the critical trajectory of the system.
In order to find an interpretation for the flow diagram we solve the Callan-
Symanzik equations for the correlation length ξ and the spontaneous magneti-
zation M0. From the numerical solution one obtains a finite correlation length ξ
that diverges by approaching the xy-line or the critical trajectory. Hence as in the
Kosterlitz-Thouless transition, the destabilization of the xy-line leads to a finite cor-
relation length. On the other hand, the evaluation of the spontaneous magnetization
shows that there exists a finite order parameter in the regions I and II of the dia-
gram. This order parameter vanishes at the critical trajectory and on the xy-line.
Therefore it is well justified to call the bold printed curve in figure 1 the critical
trajectory. At a first glance it is surprising that the critical point has the fixed point
value gc = 0, although it is just the long-range interaction which is responsible for
the ferromagnetic phase. But one has to bear in mind, that in this description every
non-vanishing value of g produces a finite magnetization. As the order parameter
has to vanish at the critical point, the fixed point has to be located at the xy-line.
In the vicinity of the critical point it is possible to obtain some analytic results.
Expanding around the critical point, the lowest orders of the flow equations are
s
dϑ(s)
ds
= λg(s), (5a)
s
dg(s)
ds
= −µg2(s) +
1
2π
g(s)ϑ(s), (5b)
where ϑ = t − tc and λ and µ are non-universal constants. In order to label the
different trajectories we define τ to be the value of the trajectory at t = tc. As
concerns the explicit form of the trajectories g(t, τ) a positive value of τ means that
the trajectory lies in region II, while negative values indicate region I and III. Using
106
Two-dim xy -models with long-range interaction
this convention an explicit solution of (5) is given by
g(t, τ) = gc(t) + τ exp
[µ
λ
(t−tc)
]
. (6)
Here gc(t) is the trajectory traversing the critical point and the integration constant
τ measures the distance from the critical trajectory according to the above definition.
In this approximation the diverging parts of the correlation length are
ξ ∝ exp
[
∫ tc
t0
dt′
1
βt(t′, g(t′, τ))
]
. (7)
Expanding (6) around the critical point one can determine the asymptotic behavior
of the correlation length to be
ξ(τ) ∝ exp
(
b√
τ
)
(8)
with the non-universal constant b =
√
π3/λ. Following Kosterlitz we interpret τ as
the relative temperature. Hence the correlation length diverges exponentially.
Solving the Callan-Symanzik equations for the spontaneous magnetization one
derives for the behavior of M0 in the vicinity of the critical point
M0(τ) ∝ exp
[
−1
2
∫ tc
t0
dt′
ζ(t′, g(t′, τ))
βt(t′, g(t′, τ))
]
. (9)
As the ζ function takes the finite value ζ? = tc/(2π) at the critical point the asymp-
totic behavior is given by
M0(τ) ∝ ξ−ζ?/2 , (10)
defining the critical exponent β̃ ≡ β/ν to equal ζ?/2.
The asymptotic forms of the critical correlation function and the magnetization
on the critical isotherm can be calculated in the usual manner to give
G(2)(q) ∝ q−2+ζ?
, (11)
M(H) ∝ H
ζ?
4−ζ? , (12)
which yields the exponents η = ζ? and δ = (4−ζ?)/ζ?, in accordance with the results
of the short-range xy-model. In order to find the exponents α and γ we solve the
Callan-Symanzik equations (3), obtaining the general scaling form of the correlation
functions
G(n) ({qi} , t, g, H, Λ) = Mn
0 ξ2(n−1) g(n)
(
{qiξ} , H
M0 ξ2
t
, τ
)
. (13)
For the following we assume that the critical behavior is contained in the first two
arguments, while we can use τ = 0 in the vicinity of the critical point for the last
argument. Then the above scaling form yields for the free energy, which is just G(0)
F (ξ, H) = ξ−2f
(
Hξ2−ζ?/2
)
. (14)
107
P.G.Maier, F.Schwabl
Comparing this with the usual scaling form
F ∝ τβ(1+δ)f(Hτ−δβ) ∝ ξ−β̃(1+δ)f(Hξδβ̃) (15)
one obtains the two equations β̃(1+ δ) = 2 and δβ̃ = 2− ζ?/2 which are satisfied by
the exponents resulting from (10) and (12). The specific heat can be calculated by
transforming the derivative with respect to τ into a derivative with respect to the
correlation length by means of equation (8). Neglecting logarithms it follows
CH ∝ ξ−2 , (16)
predicting an exponent α̃ ≡ α/ν = −2, which is the same as the exponent of the
Kosterlitz-Thouless transition. The exponent γ can be obtained by considering the
scaling form of the uniform susceptibility χ
χ ∝ ξ2−ζ?
χ̂(Hξ2−ζ?/2) . (17)
Comparing this result with the usual scaling form leads to γ̃ ≡ γ/ν = 2 − ζ?.
In the framework of the φ4-model it is known that nonanalytic interactions like
the dipolar interaction renormalize trivially, that is to say that there are no contri-
butions from perturbation theory [15]. Within the nonlinear σ-model the situation is
less clear, because there exist vertices of a nonanalytic structure. However, a careful
analysis shows that the dipolar interaction behaves trivially also in the nonlinear
σ-model. The same is true for a generalized isotropic long-range interaction of the
form
gΛ2−σ
∫
q
|q|σS(q)S(−q) . (18)
As long as σ < 2, this interaction is long-ranged and nonanalytic. For the considered
models the flow function βg is due to the trivial renormalization given by
βg(t, g) = −g
(
2 − σ − ζ(t, g)− βt(t, g)
t
)
. (19)
with σ = 1 for the dxy-model. Using the known properties of the short-range 2d
xy-model 2 and expanding the flow functions around the critical point one again
obtains the flow equations (5) 3. Therefore, the basic structure of the flow equations
(5) and the location of the critical point remain unchanged in any loop order. For
the interaction (18) the critical fixed point is now given by (tc = 2π(2− σ), gc = 0),
which is shifted towards the Polyakov fixed point [19] (tc = 0, gc = 0) for σ → 2.
Since the fixed point is located at the xy-line one can use the value [11]
ζ? = ζ(tc, 0) =
tc
2π
(20)
2One has to use the properties βt → c1tg and ζ → t/(2π) + c2g in the vicinity of the xy-line.
3For the generalized long-range isotropic interaction this can be verified by comparing with the
results of Sak [20]. Although he excluded the case N = d = 2 his flow functions are still valid and
result in flow equations having the structure (5) with µ = 0. But also for µ = 0 renormalization
group leads to the exponential behavior (8) of the correlation length.
108
Two-dim xy -models with long-range interaction
Table 1. Critical exponents for the generalized isotropic long-range xy-model
(first row). The fixed point value of ζ is given by ζ? = tc/(2π) = 2 − σ. The
second and the third row show the exponents for the dipolar xy-model (tc = 2π)
and the Kosterlitz-Thouless transition (tc = π/2). Exponents marked by a tilde
parameterize the divergence in terms of the correlation length.
β̃ γ̃ α̃ δ η
long-range xy-model ζ?/2 2 − ζ? −2 (4 − ζ?)/ζ? ζ?
dipolar xy-mode 1/2 1 −2 3 1
Kosterlitz-Thouless 7/4 −2 15 1/4
for the ζ function, which is exact in the absence of topological excitations. Hence
we conclude that the analytic structure of the divergence (8) and the derived values
for the critical exponents are correct in any loop order.
The calculated exponents for the different models are summarized in table 1. As
for the Kosterlitz-Thouless transition the exponents fulfill the usual scaling laws if
one uses ν = ∞ [4]. The exponents for the dxy-model agree with the isotropic long-
range model for σ = 1. The value of σ = 1 corresponds to an interaction that has the
same scaling behavior as the dipolar interaction. Therefore, the anisotropy of the
dxy-model is not important here for the critical behavior. Surprisingly the exponents
of the Kosterlitz-Thouless transition fit into the structure of the generalized long-
range model, as one can see by using ζ? = 1/4. The reason is that the asymptotic
properties are determined by the nature of the critical point. As for the long-range
models the fixed point of the Kosterlitz-Thouless transition is located at the xy-
line and this enforces the same nonlinear behavior. This can also be verified by
comparing (5) with the flow equations of the Kosterlitz-Thouless transition [4]. The
value ζ? = 1/4 is in accordance with the critical temperature tc = π/2 of the
unbinding transition. As concerns the critical temperature and the exponents, the
vortex interaction behaves similar to a long-range force scaling like q7/4.
Finally we want to make some remarks concerning the role of topological excita-
tions. As the transition temperature of the Kosterlitz-Thouless transition (tv = π/2)
is lower than the temperature of the ferromagnetic transition (tf = 2π) in the dxy-
model, one might speculate about an unbinding transition taking place in the dipolar
system. A detailed investigation of the influence of the dipolar interaction on the
behaviour of topological excitations is beyond the scope of this contribution. But
staying in the vortex picture 4 one can try to estimate the main effect of the dipolar
interaction on the behavior of the vortices. Using some approximations this leads to
4Taking the dipolar forces into account the topological excitations would no longer be vortices,
but would be modified especially on large scales.
109
P.G.Maier, F.Schwabl
the following vortex part of the Hamiltonian [1]
Hv
t
= − 1
2t
i6=j
∑
i,j
qiqj
(
2π ln
( |xi − xj|
a
)
+ Gα|xi − xj|
)
+
µ
t
∑
i
q2
i , (21)
where the qi and xi are charge and position of the vortices, µ is the chemical po-
tential of the vortices and α is a positive constant. The most important part of the
Hamiltonian is the linear attractive force between the vortices. By means of the
methods of Kosterlitz and Thouless [3] it becomes clear that there cannot be an un-
binding transition as long as the dipolar interaction is finite. Therefore the vortices
are bound closely and do not effect the large-scale behavior. Summarizing we con-
clude that our renormalization group results describe the asymptotic behavior of the
system correctly below the transition temperature. Above the transition one expects
a high temperature phase with disordered spins and exponential correlations.
This work has been supported by the Bundesministerium für Bildung und Forschung
(BMBF) under contract No. 03–SC5–TUM 0 and by the Deutsche Forschungsge-
meinschaft (DFG) under contract No. Schw. 348/10–1/2.
References
1. Maier P.G., Schwabl F., Phys. Rev. B, 2004, 70, 207.
2. Mermin N.D., Wagner H., Phys. Rev. Lett., 1966, 17, 1133.
3. Kosterlitz J.M., Thouless D.J., J. Phys.: Condensed Matter, 1973, 6, 1181.
4. Kosterlitz J.M., J. Phys.: Condensed Matter, 1974, 7, 1046.
5. Jose J.V., Kadanoff L.P., Kirkpatrick S., Nelson D.R., Phys. Rev. B, 1977, 16, 1217.
6. Jonker B.T., Walker K.H., Kisker E., Prinz G.A., Carbone C., Phys. Rev. Lett., 1986,
57, 142.
7. Heinrich B., Urquhart K.B., Arrott A.S., Cochran J.F., Myrtle K., Purcell S.T.,
Phys. Rev. Lett., 1987, 59, 1756.
8. Koon N.C., Jonker B.T., Volkening F.A., Krebs J.J., Prinz G.A., Phys. Rev. Lett.,
1987, 59, 2463.
9. Allenspach R., J. Magn. Magn. Mat., 1994, 129, 160.
10. De’Bell K., MacIsaac A.B., Whitehead J.P., Rev. Mod. Phys., 2000, 72, 225.
11. Brézin E., Zinn-Justin J., Phys. Rev. B, 1976, 14, 3110.
12. Brézin E., Zinn-Justin J., Le Guillou J.C., Phys. Rev. D, 1976, 14, 2615.
13. Pelcovits R.A., Nelson D.R., Phys. Lett. A, 1976, 57, 23.
14. Nelson D.R., Pelcovits R.A., Phys. Rev. B, 1977, 16, 2191.
15. Pelcovits R.A., Halperin B.I., Phys. Rev. B, 1979, 19, 4614.
16. Brézin E., Zinn-Justin J., Phys. Rev. Lett., 1976, 36, 691.
17. Brezin E., Le Guillou J.C., Zinn-Justin J. Field theoretical approach to critical phe-
nomena. Phase Transitions and Critical Phenomena, vol. 6, p. 127–247. Academic
Press, London, 1976.
18. Pokrovskĭi V.L., Fĕigelman M.V., Sov. Phys. JETP, 1977, 45, 291.
19. Polyakov A.M., Phys. Lett. B, 1975, 59, 79.
20. Sak J., Phys. Rev. B, 1977, 15, 4344.
110
Two-dim xy -models with long-range interaction
Новий фазовий перехід у двовимірній xy-моделі з
далекосяжною взаємодією
П.Г.Маєр, Ф.Швабль
Інститут теоретичної фізики,
Фізичний факультет, Технічний Університет Мюнхена,
вул. Джеймса-Франка, D–85747 Ґархінґ, Німеччина
Отримано 13 січня 2004 р.
Метою цієї статті є дати огляд результатів, що стосуються впо-
рядкування і критичних властивостей двовимірних феромагнетиків,
які включають дипольні взаємодії. Ми досліджуємо двовимірну xy-
модель розширену дипольними взаємодіями. Описуючи нашу сис-
тему нелінійною σ-моделлю і використовуючи ренормалізаційно-
групові методи ми передбачаємо фазовий перехід до впорядко-
ваного стану. Цей перехід з’являється в результаті далекосяж-
ної дипольної взаємодії. В критичному режимі ми знаходимо
експоненційну поведінку для кореляційної довжини та параметра
порядку на відміну від звичайних степеневих законів. Природа
переходу виявляє разючу подібність з переходом Костерліца-
Таулеса. Ми показуємо, що існує цілий клас далекосяжних xy-
моделей, які приводять до такої нестандартної поведінки. Параме-
тризуючи розбіжності в термінах кореляційної довжини, ми здатні
обчислити критичні показники.Ці показники є точними в будь-якому
петлевому порядку.
Ключові слова: критичні явища, магнітні властивості,
низьковимірні системи
PACS: 05.70.Jk, 75.40.Cx, 75.70.Ak
111
112
|