On the critical behaviour of two-dimensional liquid crystals
The Lebwohl-Lasher (LL) model is the traditional model used to describe the nematic-isotropic transition of real liquid crystals. In this paper, we develop a numerical study of the temperature behaviour and of finite-size scaling of the two-dimensional (2D) LL-model. We discuss two possible scenario...
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| Опубліковано в: : | Condensed Matter Physics |
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| Дата: | 2010 |
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Інститут фізики конденсованих систем НАН України
2010
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| Цитувати: | On the critical behaviour of two-dimensional liquid crystals / A.l. Fariñas-Sánchez, R. Botet, B. Berche, R. Paredes // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13601: 1-17. — Бібліогр.: 49 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860270166324019200 |
|---|---|
| author | Fariñas-Sánchez, A.l. Botet, R. Berche, B. Paredes, R. |
| author_facet | Fariñas-Sánchez, A.l. Botet, R. Berche, B. Paredes, R. |
| citation_txt | On the critical behaviour of two-dimensional liquid crystals / A.l. Fariñas-Sánchez, R. Botet, B. Berche, R. Paredes // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13601: 1-17. — Бібліогр.: 49 назв. — англ. |
| collection | DSpace DC |
| container_title | Condensed Matter Physics |
| description | The Lebwohl-Lasher (LL) model is the traditional model used to describe the nematic-isotropic transition of real liquid crystals. In this paper, we develop a numerical study of the temperature behaviour and of finite-size scaling of the two-dimensional (2D) LL-model. We discuss two possible scenarios. In the first one, the 2D LL-model presents a phase transition similar to the topological transition appearing in the 2D XY-model. In the second one, the 2D LL-model does not exhibit any critical transition, but its low temperature behaviour is rather characterized by a crossover from a disordered phase to an ordered phase at zero temperature. We realize and discuss various comparisons with the 2D XY-model and the 2D Heisenberg model. Having added finite-size scaling behaviour of the order parameter and conformal mapping of order parameter profile to previous studies, we analyze the critical scaling of the probability distribution function, hyperscaling relations and stiffness order parameter and conclude that the second scenario (no critical transition) is the most plausible.
Зазначено, що модель Лебволя - Лашера (Lebwohl - Lasher) (LL) є традиційною моделлю, яка використовується для опису переходу нематик - ізотропна фаза у реальних рідких кристалах. Вивчено температурну поведінку та скінченно-мірний скейлінг двовимірної (2D) LL моделі за допомогою числових методів. Розглянуто два можливих сценарії. В одному з них, 2D LL-модель представляє фазовий перехід, подібний до топологічного переходу, який виникає в 2D XY-моделі. У другому сценарії 2D LL-модель не демонструє жодного критичного переходу, проте її низькотемпературна поведінка характеризується кросовером із невпорядкованоої у впорядковану фазу при нулі температури. Здійснено та обговорено різні порівняння із 2D XY-моделлю і 2D моделлю Гайзенберга. Доповнивши попередній розгляд скінченно-мірною поведінкою профіля параметра порядку і його конформальним відображенням, проаналізовано критичний скейлінг функції розподілу, гіперскейлінгові співвідношення і жорсткість параметра порядку та зроблено висновок про те, що другий сценарій (жодного критичного переходу) є найбільш ймовірним.
|
| first_indexed | 2025-12-07T19:05:15Z |
| format | Article |
| fulltext |
Condensed Matter Physics 2010, Vol. 13, No 1, 13601: 1–17
http://www.icmp.lviv.ua/journal
On the critical behaviour of two-dimensional liquid
crystals
A.l. Fariñas-Sánchez1, R. Botet2, B. Berche3, R. Paredes4
1 Departamento de Fı́sica, Universidad Simón Bolı́var, Apartado Postal 89000, Caracas 1080–A, Venezuela
2 Laboratoire de Physique des Solides, Bât. 510, CNRS UMR8502 – Université Paris-Sud,
F–91405 Orsay, France
3 Statistical Physics Group, P2M Dpt, Institut Jean Lamour, Nancy Université, UMR CNRS 7198, BP 70239,
F–54506 Vandœuvre les Nancy Cedex, France
4 Instituto Venezolano de Investigaciones Cientı́ficas, Centro de Fı́sica, Laboratorio de Fisica Estadı́stica
Apdo. 20632, Caracas 1020A, Venezuela
Received June 21, 2009, in final form August 31, 2009
The Lebwohl-Lasher (LL) model is the traditional model used to describe the nematic-isotropic transition of
real liquid crystals. In this paper, we develop a numerical study of the temperature behaviour and of finite-size
scaling of the two-dimensional (2D) LL-model. We discuss two possible scenarios. In the first one, the 2D
LL-model presents a phase transition similar to the topological transition appearing in the 2D XY-model. In
the second one, the 2D LL-model does not exhibit any critical transition, but its low temperature behaviour is
rather characterized by a crossover from a disordered phase to an ordered phase at zero temperature. We
realize and discuss various comparisons with the 2D XY-model and the 2D Heisenberg model. Having added
finite-size scaling behaviour of the order parameter and conformal mapping of order parameter profile to pre-
vious studies, we analyze the critical scaling of the probability distribution function, hyperscaling relations and
stiffness order parameter and conclude that the second scenario (no critical transition) is the most plausible.
Key words: topological transitions, critical phenomena, liquid crystals, Lebwohl-Lasher model
PACS: 64.70.M-, 64.60.Bd, 64.70.mf, 05.70.Jk, 05.50.+q, 68.35.Rh
1. Introduction
At high temperatures, the natural state of a liquid crystal is the isotropic phase, because all
the molecule orientations are equally probable. At low temperatures, an order along a preferred
direction may appear, it is the nematic phase. When such a material is cooled at even lower temper-
atures, other types of ordered phases are likely to appear, the description of which requires realistic
potentials (see e.g. [1]). The transition between the isotropic and the nematic phases generally oc-
curs at a critical temperature. Various experiments on three-dimensional liquid crystals, exhibited
a weak first order transition [2]. However, the corresponding two-dimensional (2D) problem is far
from being that clear. Experimentally, only 2D liquid crystals composed of long rigid interacting
rods have been studied.
First, the 2D problem must be defined through the symmetries: there is generally a continuous
global symmetry O(n), and a local Z2 symmetry. For example, the two most-studied models deal
with O(2) (the XY-model), or O(3) (the Heisenberg model) symmetries. The Hamiltonian reads
in both cases as:
H = −J
∑
〈i,j〉
Si · Sj , (1)
with Si a n-components spin, of unit length, at the lattice site i. The sum runs over all pairs of
neighbouring spins of the 2D square lattice and J > 0 is the spin-spin coupling constant.
According to the Mermin-Wagner-Hohenberg theorem [3–7], the order parameter fluctuations
prevent any spontaneous breakdown of continuous symmetries for 2D systems with short-range
interactions. However, this theorem cannot conclude in the case of transitions to a phase with no
c© A.l. Fariñas-Sánchez, R. Botet, B. Berche, R. Paredes 13601-1
http://www.icmp.lviv.ua/journal
A.l. Fariñas-Sánchez et al.
long range order. An example of such a 2D critical behaviour, which is not a continuous-symmetry
breaking, is the 2D XY-model (O(2), Abelian symmetry). This model presents a topological tran-
sition characterized by a critical low temperature phase as a plasma of pairs of vortices, at tem-
peratures under the Berezinskii-Kosterlitz-Thouless transition temperature, TBKT [8, 9]. At much
lower temperatures, the system state is dominated by spin waves. The high temperature phase
occurs when the pairs dissociate.
On the other hand, the 2D Heisenberg model (O(3), non-Abelian symmetry) has no transition
at any finite temperature (asymptotic freedom) [10–13]. The longitudinal modes of the O(3) model
are frozen for the low temperatures, and only the transverse modes are activated, leading essentially
to two Gaussian modes. The deep difference between the two models at finite temperatures is the
stability of the vortices. Indeed, the topological transition of the 2D XY-model is the result of the
stability of the topological defects, while the “third spin dimension” in the O(3) model, makes the
vortices unstable at any finite temperature.
However, the above picture has been recently questioned by the numerical evidence for a possible
transition in the 2D Heisenberg model [14–17]. Moreover, it has been shown that a quasi-long-range
order (QLRO) phase [18, 19] might appear at very low temperatures in finite systems. Indeed in
these numerical works, the thermodynamic limit remains questionable and the analytic approach
of [19] relies explicitly on the finite size of the lattice.
A model for a regular 2D liquid crystal system was proposed by Lebwohl and Lasher (LL)
[20]. Based on a lattice version of the mean field theory of Maier and Saupe [21], the molecules
are represented by n-component unit vectors and the “spin-spin” interactions are given by the
second Legendre polynomial, P2 (keeping the local Z2 symmetry). The popularity of the LL-model
comes from its ability to reproduce the weak first order phase transition observed experimentally
between the isotropic and the nematic phases [2] in the three-dimensional space. Then, the issue of
possible phase transitions in this model is attractive and has been addressed in numerous studies
[18, 22–26].
In the past twenty years, several numerical data pointed out a possible topological transition
for the 2D nematic-isotropic phase transition in liquid crystals. The situation is clear in the case of
planar rotator models (see e.g. [27, 28]), but it is still controversial for three-component models. In
[27], we have studied the nematic-isotropic transition of 2D liquid crystals using a O(2) vector model
characterized by nonlinear nearest-neighbour spin interactions governed by the fourth Legendre
polynomial P4. The system has been studied through standard finite-size scaling and conformal
rescaling of the density profiles. We also estimated the Binder cumulant [29] as a function of the
temperature for different values of system size L. This cumulant has been proved to be universal
at a critical temperature. Evidence for a topological transition at a finite temperature has been
underlined.
In the case of the three-component model, in a remarkable work [22] Kunz and Zumbach con-
cluded in favour of such a critical topological transition. Although they performed a careful and
sizable study, they were unable to decide conclusively between essential singularities or standard
power laws for the correlation length and the susceptibility, when approaching the critical temper-
ature from the high temperature phase. However, they developed a qualitative picture (pairing of
topological defects which carry most part of the system energy, sharp increase of the density of
the defects and seeming discontinuity of the rotational-rigidity modulus, finite cusp of the specific
heat, proliferation of unbounded defects at high temperature) and their conclusion clearly inclined
to the essential singularity, for a temperature TBKT. More recently, we studied the 2D LL-model
[18, 25, 26], and found signals in favour of a QLRO phase at T 6 TBKT, with a magnetization that
decays with the system size as a power law with the critical exponent η(T )/2 [26]. We obtained
good agreement with the η(T ) exponent obtained using the powerful technique of the conformal
transformation [25]. In [18], the value of η(T ) was estimated using directly the susceptibility, and
the agreement appeared to be reasonable. Possible discrepancy could come from the finite size of
the systems. This QLRO phase could be similar to the low temperature phase of the 2D XY-model.
Mondal and Roy [30], using standard finite size scaling method and Monte Carlo simulations for
the 2D LL-model, gave arguments for a continuous phase transition. The critical temperature was
13601-2
On the critical behaviour of two-dimensional liquid crystals
estimated to be Tc = 0.548 ± 0.02, a little higher than 0.513 calculated in [18, 22, 25, 26]. A still
more recent study reported evidence in favour of a topological transition [31]. Then, the transition
might be driven by topologically stable point defects known as 1
2 -disclination points.
Despite 20 years of numerical results claiming for a topological transition for the 2D LL-model,
a recent work seriously called this scenario into question. Indeed, precise estimates of the Binder
cumulant for various system sizes, have been incapable of showing any crossing point at a definite
temperature [32]. This result altogether contradicts the transition scenario at a critical point. In
the same work [32], the possibility of a QLRO phase at T = 0.4 (a value well below the “transition”
temperature estimated previously for this model) was questioned. Two strong proofs were given to
conclude that the 2D Lebwohl-Lasher model does not show any quasi-long-range ordered phase.
The proofs consisted in the violation of the hyperscaling relation between the apparent exponents of
magnetization and susceptibility, and the failure of the first-scaling collapse [33] of the probability
distribution function of the order parameter.
The main goal of the present work is to present a complementary investigation of the 2D LL-
model. Using the same tools we will compare this model with the 2D XY- and Heisenberg models,
for which the critical behaviours are not questionable. Since the 2D LL-model seems to share some
properties common with both the XY- and Heisenberg models, it appears helpful to provide the
comparisons. In section 3 we will analyze the system with and without an applied magnetic field
using the finite size scaling method (FSS). The results which follow from conformal transforma-
tion method (CT) will also be reminded [25]. We shall recover the apparent proofs for a sort of
topological transition similar to the Berezinskii-Kosterlitz-Thouless transition. However, with new
results on the shape of the tail of the probability distribution function for the magnetization, in
section 5 we will check the hyperscaling relation, study the stiffness, and see how improbable is
the reality of a critical topological transition, confirming the results of [32].
2. Definitions of the models
We will consider hereafter the 2D XY-model [8], the 2D Heisenberg model [10], and the 2D
Lebwohl-Lasher model [20]. The simulations will be performed using Monte Carlo (MC) Wolff
cluster algorithm [34] appropriately adapted to the model under consideration [22], in order to
avoid most of the critical slowing down behaviour close to transition, if any. All simulations are
realized on a square lattice of size L×L with periodic boundary conditions. The system sizes and
number of MC iterations will be specified when needed in the figure captions. The constants J and
kB are fixed to unity.
• For the 2D XY-model, the N = L2 classical spins are confined in the lattice plane x, y. The
spin Si is parameterized by: (Sx
i = cos θi, S
y
i = sin θi), with θi the angle between the direction
of the spin Si and the x-axis. According to (1), the system state is governed by the Hamil-
tonian H = −J
∑
〈i,j〉 cos(θi − θj), with the ferromagnetic coupling constant J > 0, and the
sum running over all the nearest-neighbour pairs of spins. The critical features are eventually
characterized by the power-law behaviour of the magnetization per site: m ≡ 1
N
√
(
∑
i Si)2,
which is a non-negative real number. Thermal fluctuations of the order parameter give access
to the susceptibility χ = Ld/kBT (〈m2〉 − 〈m〉2).
• For the Heisenberg model, the N = L2 classical spins can point in any direction of the
three dimensional space x, y, z, (Sx
i = sin θi cosϕi, S
y
i = sin θi sin ϕi, S
z
i = cos θi), with the
standard definitions for the local angles. Similar to the above XY-model, the Hamiltonian
has the form H = −J
∑
〈i,j〉 Si · Sj , but with three-component spins and the local order
parameter is defined by the appropriate magnetization per site.
• For the Lebwohl-Lasher model, the N = L2 classical spins can also point in any of the three
dimensional space directions x, y, z. The Hamiltonian of the system,
H = −J
∑
〈i,j〉
P2(Si · Sj), (2)
13601-3
A.l. Fariñas-Sánchez et al.
is an obvious generalization of the Heisenberg interaction which guarantees the local Z2 sym-
metry of liquid crystals. P2 is the second Legendre polynomial, that is: P2(x) = (3x2 − 1)/2.
One defines the scalar order parameter (also called nematization, or simply magnetization)
as: m ≡ 1
N
∑
i P2(cosαi), where αi is the angle between the direction of the spin Si and the
direction of symmetry breaking.
3. Qualitative analysis of the order parameter and the susceptibility
In the following we study the behaviour of the order parameter (magnetization or nematization)
m and the magnetic susceptibility (or response function) χ, as a function of the temperature for
various system sizes L.
3.1. The 2D XY-model
In figure 1, the magnetization m and the susceptibility χ (top and bottom) for the 2D XY-model
are plotted versus the temperature. Both plots have the same temperature range, wide enough to
cover the domain of temperatures where important thermodynamic changes are expected to occur
(essentially: the BKT transition, which should appear at the temperature TBKT ' 0.893).
0.6 0.7 0.8 0.9 1 1.1
T
0.1
1
10
100
1000
χ
0.2
0.4
0.6
0.8
1
m
L = 16
L = 32
L = 64
L = 96
L =128
L =256
L =512
TBKT
Figure 1. Temperature dependence of thermodynamic quantities for the 2D XY-model. Top:
magnetization (m). Bottom: susceptibility (χ). (Monte Carlo simulations with 106 MCS/spin,
after removing the first 106 MCS to achieve proper thermalization).
We can observe in figure 1 a progressive decrease of the magnetization with the system size,
at constant temperatures. Moreover, as becomes evident from figure 2, we see that these decays
follow power-laws with the system size in the whole range T 6 TBKT, according to the FSS picture
in the critical phase, m ∼ L− 1
2
η(T ), with an exponent η(T ) which depends on the value of the
temperature (to be discussed in the next section). When the value of the temperature is above
TBKT, the decay of the magnetization is faster than a power law.
On the other hand, the magnetic susceptibility at a fixed temperature (figure 1) exhibits reg-
ular increase with the system size for all temperatures at and below TBKT. It is in contrast with
an ordinary second order phase transition in which the susceptibility diverges only at the criti-
cal temperature. In particular, the susceptibility exhibits a power law behaviour with L for any
13601-4
On the critical behaviour of two-dimensional liquid crystals
10 100 1000
L
m
T = 0.60
T = 0.65
T = 0.70
T = 0.75
T = 0.80
T = 0.85
T = 0.893
T = 0.95 100
L
1
10
100
1000
10000
χ
T = 0.60
T = 0.70
T = 0.80
T = 0.893
T = 0.95
Figure 2. Behaviour of the magnetization ver-
sus the system size at different temperatures
for the 2D XY-model. Power-law behaviour
appears in the whole range T 6 TBKT. (Monte
Carlo simulations with 106 MCS/spin after re-
moving the first 106 MCS to achieve proper
thermalization).
Figure 3. Behaviour of the magnetic sus-
ceptibility versus the system size at various
temperatures for the 2D XY-model. (Monte
Carlo simulations with 106 MCS/spin after re-
moving the first 106 MCS to achieve proper
thermalization).
T 6 TBKT (figure 3). It is interesting to notice that a change in the exponent of the power-law of
the susceptibility with the system size, appears at TBKT.
Also, we see in figure 1, that the susceptibility reaches its maximum value systematically above
TBKT. The temperature at which this maximum occurs tends to TBKT when L → ∞.
The power law behaviour of the magnetization and of the magnetic susceptibility at low temper-
atures, can be interpreted as evidence of a QLRO phase for the 2D XY-model in the thermodynamic
limit, in the temperature range T < TBKT.
3.2. The 2D Heisenberg model
The 2D Heisenberg model is expected not to exhibit any phase transition [10–13]. The magnetic
susceptibility strongly diverges when the temperature vanishes [10]. However, some authors argued
in favour of a transition [14–17] and possibly an effective quasi-long-range-ordered phase [18, 19]
at very low temperatures, at least in large but finite systems.
In figure 4, one can see the behaviour of the magnetization and of the magnetic susceptibility
with the system size and temperatures for the 2D Heisenberg model. We can notice the decrease
of the values of the inflection point of m versus T , when the system size increases. In the same
way, the temperatures where the susceptibility reaches the maximum, are displaced to lower and
lower values when the system sizes are larger. As a matter of fact, if there is a positive critical
temperature analogous to the TBKT, its value should be quite small, at the thermodynamic limit.
3.3. The 2D LL-model
Now we analyze the magnetization and the magnetic susceptibility of the 2D Lebwohl-Lasher
model, using the same procedure as for the 2D XY-model. In figure 5 the magnetization, m, and
the magnetic susceptibility, χ, are shown versus the temperature for various lattice sizes L. There
is a complete qualitative agreement with the behaviour of the same quantities for the 2D XY-model
(figure 1). Indeed the ‘critical’ temperature is shifted to T ? = 0.513 [22, 25], but all the power-law
behaviours are equally recovered (figures 6 and 7), and even the change, at T ?, of the apparent
value of the exponent of the susceptibility versus the system size, or the asymptotic convergence to
T ?, of the temperatures at which the susceptibility reaches its maximum are identically noticed.
13601-5
A.l. Fariñas-Sánchez et al.
0 0.3 0.6 0.9
T
1
100
χ
0.2
0.4
0.6
0.8
1
m
L = 16
L = 32
L = 64
L = 96
L =128
L =256
L =512
0.4 0.45 0.5 0.55 0.6
T
1
10
100
1000
χ
0
0.2
0.4
0.6
0.8
m
L = 16
L = 32
L = 64
L = 96
L = 128
L = 256
L =512
T
*
Figure 4. Temperature dependence of ther-
modynamic quantities for the 2D Heisenberg
model. Top: magnetization (m). Bottom: sus-
ceptibility (χ). (Monte Carlo simulations with
106 MCS/spin after removing the first 106
MCS to achieve proper thermalization).
Figure 5. Temperature behaviour of thermo-
dynamic quantities for the 2D Lebwohl-Lasher
model. Top: magnetization (m). Bottom: mag-
netic susceptibility (χ). (Monte Carlo simula-
tion with 106 equilibrium steps (measured as
the number of flipped Wolff clusters) after re-
moving the first 106 MCS to achieve proper
thermalization).
10 100 1000
L
m T = 0.40
T = 0.42
T = 0.44
T = 0.46
T = 0.48
T = 0.50
T = 0.52
T = 0.54
10 100 1000
L
1
10
100
1000
χ
T = 0.40
T = 0.44
T = 0.48
T = 0.52
T = 0.54
Figure 6. Behaviour of the magnetization
with the system size at various temperatures
for the 2D LL-model. (Monte Carlo simula-
tion with 106 equilibrium steps (measured as
the number of flipped Wolff clusters) after re-
moving the first 106 MCS to achieve proper
thermalization).
Figure 7. Behaviour of the magnetic suscep-
tibility versus the system size at various tem-
peratures for the 2D LL-model. (Monte Carlo
simulation with 106 equilibrium steps (mea-
sured as the number of flipped Wolff clusters)
after removing the first 106 MCS to achieve
proper thermalization).
13601-6
On the critical behaviour of two-dimensional liquid crystals
3.4. Conclusion from the study of the m and χ behaviours
From this preliminary analysis, one could conclude that the 2D LL-model behaves similarly to
the 2D XY-model. Namely, one can define a BKT temperature, T ?, at which the system is critical,
and a QLRO phase seems to occur at temperatures smaller than T ?.
We shall see below that this anticipated conclusion is refuted by more refined analysis.
4. Calculations of the correlation function exponent η
In principle, very precise information can be obtained from the estimates of the fundamental
critical exponent η for the pair correlation function. This exponent may depend on the effective
temperature T in the case of a continuous line of critical points.
Several methods can be used independently in the case of the BKT transition. We shall use the
following ones below:
• Finite size scaling behaviour of the magnetization.
At the critical point of second order phase transition, the magnetization m behaves like a
power law of the system size, namely: m ∼ L−β/ν . This combination of critical exponents is
related to the exponent which describes the critical decay of an appropriate pair correlation
function, G(r) ∼ r−(d−2+η), 2β/ν = d− 2 + η, thus in two dimensions m ∼ L− 1
2
η. Here, η is
a function of T and FSS is used to estimate the critical exponents at different temperatures
(in the low temperature phase) from figure 2 or figure 6;
• FSS behaviour of the magnetic susceptibility.
The magnetic susceptibility, χ, scales as a power law involving the critical exponents γ and
ν, namely: χ ∼ Lγ/ν . Then, the hyperscaling relation1
η = 2 − γ/ν, (3)
holds, which gives an alternative estimate of the exponent η. Here too, FSS may be used to
extract an appropriate exponent from the data of figure 3 or figure 7;
• Conformal Transformation (CT) method.
Another tool can be used to estimate the value of η for very large systems: the conformal
transformation method (see Cardy [35]). It is indeed a powerful method which applies to
analyze any conformally covariant density profile (or correlation function) from the mapping
of infinite or semi-infinite geometries onto restricted geometries where simulations are actually
performed [36, 37]. We thus consider a density profile m(w) in a finite system with symmetry
breaking fields along some edges in order to induce a non-vanishing local order parameter in
the 2D bulk of the system. Here, w stands for the location in the finite system. In the case of
a square lattice of size L × L, with fixed boundary conditions along the four edges, using a
simple mathematical transformation – called the Schwarz-Christoffel transformation [38, 39]
– one expects a power law in terms of an effective locally rescaled distance
m(w) ∼ [κ(w)]−
1
2
η . (4)
This method has been extensively used in [36, 40] to estimate the pair correlation exponent
as a function of temperature for the 2D XY-model;
• Systems in an applied magnetic field.
1This scaling law is obtained using e.g. Rushbrooke equality α + 2β + γ = 2 and the hyperscaling relation in its
standard form, α = 2 − dν and the definition of the exponent η through d − 2 + η = 2β/ν. It also simply follows
from the definition of χ in terms of the correlation function, χL =
∫
Ld G(r)ddr.
13601-7
A.l. Fariñas-Sánchez et al.
According to the scaling hypothesis, the magnetization of a general d-dimensional system at
a critical temperature, must follow the scaling law:
m = H1/δG(L−1H−1/yh), (5)
where H is the applied magnetic field. The magnetic exponent is given in terms of the
magnetic field scaling dimension yh, δ = yh/(d − yh), and G is a universal function [41].
In the thermodynamic limit L−1H−1/yh � 1, the system follows the singular behaviour
m ∼ H1/δ . In the other limit L−1H−1/yh � 1, the magnetization follows m ∼ Lyh−d. As
we know that m ∼ L−η/2 for the BKT transition, one deduces in this case the hyperscaling
relation under the form:
d − yh = η/2. (6)
which again can be used to estimate the value of η.
4.1. The 2D XY-model
The FSS methods are used from figure 2 and figure 3. The results obtained by the CT method
are from the [40]. We add here a few words about the 2D XY-model in a magnetic field. The
exponent η at the BKT transition is 1/4, then yh = 15/8, and δ = 15 has the same value as for the
d = 2 Ising model. Using equation (5), an excellent collapse of Y = H/mδ versus X = HLyh at
TBKT = 0.893 has been obtained in [42]. In this work the authors obtained an additional result that
Y tends to a constant value in the limit HLyh → ∞. The saturation of Y for large magnetic fields
means a Weibull-like fieldless probability distribution function (PDF) [43] for the magnetization.
In the present work we extended this result to other temperatures.
10-2 100 102 104
HL
yh
100
H
/<
m
H
>
δ
T = 0.5
T = 0.6
T = 0.7
T = 0.8
W PDF for T = 0.5
W PDF for T = 0.6
W PDF for T = 0.7
W PDF for T = 0.8
0 0.2 0.4 0.6 0.8 1
T
0
0.1
0.2
0.3
η
spin wave solution
L = 100 C.T.
L = 200 C.T.
m ~ L-η/2
χ ~ L2-η
from yh
TBKT
Figure 8. XY-model in a magnetic field H. For
each temperature the system sizes are: L = 32,
64, 96, 128, and the magnetic field strengths
are: H = 0.1, 0.01, 0.05, 0.001, 0.005, 0.0001,
0.0005. The bold straight lines represent the
large X = HLyh limit for a Weibull-like PDF
(W) for each temperature. The number of
Wolff steps for each T , L and H is 2 × 105.
Figure 9. Dependence of the correlation func-
tion critical exponent with the temperature
for the 2D XY-model. (Monte Carlo simula-
tions with 106 MCS/spin after the removal of
the first 106 MCS to achieve proper thermali-
zation). The conformal transformation (CT)
values are from the [40]. The dashed line is
the spin-wave result η = kBT/2πJ .
Indeed, for a BKT transition there is a line of critical points for any T 6 TBKT, and the
equation (5) must be satisfied in all this range of temperatures. Then, we performed numeri-
cal simulations of the 2D XY-model in an applied magnetic field2for temperatures below TBKT.
2For the 2D XY-model or 2D LL-model in the magnetic field H, we used a Wolff algorithm similar to the one
used for the H = 0 case, with an additional spin which interacts with all the other spins, with H as the strength of
the interaction.
13601-8
On the critical behaviour of two-dimensional liquid crystals
Figure 8 shows an excellent collapse of the data, as well as the saturation of Y for several mag-
netic field strengths and several system sizes, at all the temperatures 6 TBKT investigated. The
collapse of the data gives an estimate of the critical exponent η(T ) using the hyperscaling rela-
tion (6).
This result invalidates a claim by Bramwell et al. [44], about the possible Gumbel-like distri-
butions [43] of the magnetization in the 2D XY-model at low temperatures. Indeed, the Gumbel
PDF does not lead to the saturation of the function Y in the limit HLyh → ∞.
The figure 9 shows the values of the critical exponent η for the 2D XY-model, calculated from
FSS using the magnetization or the magnetic susceptibility, with the CT method (from [40]),
and using the collapse of the data for the system in a magnetic field. All the methods lead to
determinations of η which are in excellent agreement with each other.
However, one can note that the behaviour obtained with CT is the most precise, because this
method is capable of reaching the behaviour of the infinite-size systems, with system sizes com-
putationally accessible, all shape effects being encoded (in the continuum limit) in the conformal
mapping (hence the name of Finite Shape Scaling that we tried to introduce in Ref. [27]). The
differences between the results obtained from m and from χ, are of the order of 5%. In the [32] and
[42], the authors used more than 6× 106 independent realizations for the estimate of the exponent
η. Realization of the hyperscaling relation (3) has been checked with a relative error smaller than
0.4%.
More than simple illustrations, these results allow us to compare the efficiency of different
methods for a proper definition of a quasi-long-range-order phase in systems with continuous
symmetry.
4.2. The 2D Heisenberg model
Now we compare the values measured for η for a system (the 2D Heisenberg model) which does
not exhibit any transition, at least for temperatures above T = 0.1 [10] (see figure 4). Then, we
can expect the behaviours of η to be quite different in the two cases.
0.2 0.3 0.4 0.5
T
0.00
0.10
0.20
0.30
0.40
0.50
η
L = 48 C.T.
L =100 C.T.
L =200 C.T.
L =400 C.T.
m ~ L−η/2
χ ~ L2-η
Figure 10. Dependence of the effective correlation function critical exponent with the tempera-
ture for the 2D Heisenberg model. The dashed line is the spin-wave result η = kBT/πJ [19].
The figure 10 shows the values of η calculated using the FSS and the CT methods for the 2D
Heisenberg model. The differences with the 2D XY-model are clear. It is particularly noticeable that
the estimates given by the three methods are completely different. Even the shapes of the curves
are different, and no agreement can be found. In particular, the determinations of η deduced from
m and χ, respectively, do not agree with each other. It follows that the hyperscaling relation (3)
is definitively unsatisfied.
These observations give a strong evidence for the lack of any critical behaviour in the 2D
Heisenberg model within the range of temperatures considered here.
13601-9
A.l. Fariñas-Sánchez et al.
4.3. The 2D LL-model
Now we present a similar systematic work, performed on the 2D LL-model. In particular, the
figure 11 shows results for the 2D LL-model in an applied magnetic field at the expected critical
temperature, T ?.
0.01 1 100 10000
HL
yh
0.01
0.1
1
10
H
/<
m
H
>
δ
L = 32
L = 64
L = 96
L = 128
W PDF for T* = 0.513
Figure 11. 2D Lebwohl-Lasher model in a magnetic field (H = 0.1, 0.01, 0.05, 0.001, 0.005,
0.0001, 0.0005) at T = 0.513. The bold straight line represents the large X = HLyh limit
for a Weibull-like PDF (W). Each point corresponds to an average over 100, 000 independent
realizations.
The results are consistent with the equation (5), and with the saturation of the quantity Y .
The best collapse was obtained for δ = 10.83. Then, using the hyperscaling relation (6) we found:
η = 0.338 at this temperature. This result is shown in figure 12 and the agreement with the value
of η obtained from the other methods is excellent at this temperature.
0 0.1 0.2 0.3 0.4 0.5 0.6
T
0
0.1
0.2
0.3
0.4
0.5
η
T
*
spin wave solution
L = 48 C.T.
L = 100 C.T.
L = 200 C.T.
m ~ L-η/2
χ ~ L2-η
from yh
Figure 12. Dependence of the critical exponent η with the temperature for the 2D Lebwohl-
Lasher model. (Monte Carlo simulations with 106 MCS/spin after removing the first 106 MCS
to achieve proper thermalization). The conformal transformation (CT) data are from [25]. The
dashed line is the spin-wave result η = 4kBT/3πJ .
In figure 12, we show the values of η for the 2D LL-model at various temperatures. Estimates
were done using FSS of magnetization (from [26] and figure 6) and of magnetic susceptibility (from
figure 7). They are compared to the values estimated from the CT method (from [25]), and from
the collapse of the data in a magnetic field.
Nevertheless, one should stress that the agreement between the values estimated by all the
four methods is good, but imperfect. Small discrepancies can easily be noticed. In particular, the
estimates which are based on the FSS behaviour of the magnetic susceptibility are systematically
smaller than those following from the other methods. The errors are larger than 10%, a relevant
quantity probably far out of the range of acceptable error bars. In a previous work [32], this error
13601-10
On the critical behaviour of two-dimensional liquid crystals
was only of 3%, but the authors then systematically checked the hyperscaling relation, equation (3),
and concluded that the hyperscaling relation is definitively unsatisfied at low temperature (e.g. at
T = 0.4) (see discussion in the section 5 below). This has been the first evidence for the absence
of a QLRO phase in the 2D LL-model.
5. Revisiting the behaviour of the 2D LL-model at the ‘critical’ temperature
In the previous section, we observed that the critical behaviour of the 2D XY-model and the
behaviour of the 2D LL-model were quite similar. Nevertheless, we saw small discrepancies for the
LL-model, in particular in the behaviour of the second moment of the magnetization, and on the
validity of the hyperscaling relation (3).
In [32], the authors performed intensive simulations of the 2D LL-model at the temperature
T = 0.4 well below the expected critical temperature. Clear failure of the hyperscaling relation,
failure of collapse of the PDF in the first scaling form [33], and the analysis of the Binder cumulant,
led to the conclusion that no QLRO phase exists in the 2D LL-model at this temperature.
Here, we consider the problem of the existence of a critical point at the temperature T ? =
0.513, as this temperature there was reported as a possible BKT temperature for the 2D LL-
model (see section 3 and [22, 25]). To elaborate on this question, we will report results on the
probability distribution function, the magnetization scaling and the hyperscaling relation (3) at
this temperature. We shall also compare the stiffness for the LL- and XY-models.
5.1. The first-scaling law
If the 2D Lebwohl-Lasher model is critical at the temperature T ? = 0.513, self-similarity should
be observed at this temperature because no finite characteristic length can exist at criticality.
Asymptotic self-similarity results in the so-called first-scaling law for the order parameter, here the
magnetization m, [33]:
〈m〉P (m) ≡ ΦT (z1), with z1 ≡
m
〈m〉
, (7)
with ΦT a scaling function which depends only on the actual temperature T . Equation (7) is sequel
of the standard finite-size scaling theory [45], but it is highly advantageous that equation (7) does
not require the knowledge of the values of any critical exponent. Note that, under this form, the
hyperscaling relation,
〈m〉
σ
= constant term, (8)
is automatically realized with σ the standard deviation of the order parameter. Indeed, one has:
σ2 ∝ χ/Ld, and σ scales then with the system size as: σ ∼ Lγ/2ν−1 for 2D systems. Therefore, the
ratio 〈m〉/σ should be a constant whenever the hyperscaling relation (3) is satisfied [32].
Figure 13 shows the behaviour of the PDF when the system size increases, for the 2D LL-model
at the temperature T ? = 0.513. We can observe the failure of the collapse between these curves,
especially for the largest system sizes, where the distributions tend to deviate from each other when
L increases. One must conclude that the temperature T ? = 0.513 is not critical for this model.
In two other papers published recently, one can find the same kind of curve for the 2D XY-model
at T = 0.6 < TBKT [32] and at T = TBKT [42]. The collapse of all the PDF is excellent in the two
cases, a result consistent with the known criticality of the system at TBKT and all the temperatures
below this value. In the same [32], the 2D LL-model was studied similarly at T = 0.4. Failure of
the collapse was clear in this case, leading to the conclusion that the system is not critical below
the assumed transition temperature.
In 2003, Mondal et al. [30] reported a possible transition temperature for the 2D LL-model at
the temperature T = 0.548, larger than the values reported by Kunz et al in 1992 [22]. We thus
also performed numerical simulations of the 2D LL-model at T = 0.548. In figure 14 the data are
plotted in the first-scaling form (7). The data are far from collapsing. This is actually similar to
the behaviour of a system far away from any critical point.
13601-11
A.l. Fariñas-Sánchez et al.
0.7 0.8 0.9 1 1.1 1.2
m/<m>
0
2
4
6
<
m
>
P
(m
)
L = 32
L = 64
L = 96
L = 128
L = 256
0.95 1 1.05
4
5
6
7
<
m
>
P
(m
)
L = 32
L = 64
L = 96
L = 128
L = 256
Figure 13. PDF of the magnetization for the 2D Lebwohl-Lasher model at the temperature
T ? = 0.513, plotted in the first-scaling form (7). The scaling law is not confirmed for L = 32
up to L = 256. Wolff’s single-cluster algorithm was used [34]. Each data set corresponds to the
average over 6, 000, 000 independent realizations.
0.7 0.8 0.9 1 1.1 1.2
m/<m>
0
2
4
<
m
>
P
(m
)
L = 32
L = 64
L = 96
L = 128
L = 256
Figure 14. PDF of the magnetization for the Lebwohl-Lasher model at T = 0.548, plotted in
the first-scaling form (7). Each data set corresponds to the average over 6, 000, 000 independent
realizations.
5.2. The hyperscaling relation
In this section, we numerically check the hyperscaling relation under the form (8). In figure 15,
〈m〉/σ is plotted versus 1/L for the 2D XY-model and the 2D LL-model respectively, at the BKT
temperatures reported or expected for each system, namely: TBKT = 0.893 for the XY-model and
T ? = 0.513 for the LL-model.
For the XY-model, the ratio is seen to converge to a definite value (' 14.8) at the thermody-
namic limit 1/L = 0, while for the LL-model, a power law is the best fit consistent with the data of
〈m〉/σ versus L, and this fit leads to a vanishing value of the ratio at the thermodynamic limit. We
conclude that the hyperscaling relation is very likely violated in this case, for the obvious reason
that the system is not critical at the temperature T ? = 0.513.
13601-12
On the critical behaviour of two-dimensional liquid crystals
0 0.02 0.04 0.06
13
13.5
14
14.5
15
<
m
>
/σ
0 0.02 0.04 0.06
L
-1
15
16
17
<
m
>
/σ
XY T=TBKT
Figure 15. m/σ vs. L−1 at T = TBKT for the 2D XY-model (top) and at T = T ? = 0.513 for
the 2D Lebwohl-Lasher model (bottom). For the XY-model, the data were obtained from the
[42]. For the LL-model, the blue line is a power law fit.
5.3. The stiffness modulus
The appearance of stiffness is an important consequence of the spontaneous symmetry break-
down in a system with continuous symmetry. It is closely related to correlations in the transverse
response function, any spatial variation of the order parameter, perpendicular to the direction of
the ordering, increasing the energy of the system. For this reason, the system generates a restored
strength in response to any attempt to create a new configuration: this is known as stiffness.
In the case of the 2D XY-model, this is the helicity modulus and it is defined as follows. Let
Hφ = −βJ
∑
〈i,j〉
Si · ĝφSj (9)
be the modified Hamiltonian in which ĝφ is a rotation of angle φ of the system around any axis,
say here the x-axis. The difference between the free energy F (φ) calculated with the modified
Hamiltonian, and the free energy calculated with the original Hamiltonian,
F (φ) − F (0) = −
1
β
ln
(
Tr e−βHφ
Tr e−βH0
)
, (10)
varies quadratically with the rotation angle φ, at least for φ � 1. The coefficient, usually written:
Υ, is the helicity modulus:
F (φ) − F (0) = −Υφ2 + O(φ4) (11)
and it takes the form
2Ld
Jβ
Υ =
〈
∑
〈i,j〉
Si · Ĵ
2
xSj
〉
− J
〈
(
∑
〈i,j〉
Si · ĴxSj
)2
〉
(12)
where Ĵx is the rotation operator around the x-axis.
In figure 16 the stiffness modulus for the 2D XY-model is shown versus the temperature for
various sizes from L = 16 to L = 512. The behaviour is typical of an order parameter: below
TBKT = 0.893, the plots are almost independent of the system size, and this is the sign of the rigid
phase with a finite value of the stiffness constant, while for temperatures larger than TBKT = 0.893,
13601-13
A.l. Fariñas-Sánchez et al.
the system’s response becomes weaker, leading to a decrease of the stiffness modulus. The high-
temperature phase is a disordered phase with the vanishing stiffness. The change of behaviour
in the stiffness modulus between the two phases exhibits the appearance of a topological phase
transition, with quasi-long-range order, but finite rigidity, in the low temperature phase.
0.2 0.4 0.6 0.8 1
T
0
0.2
0.4
0.6
0.8
1
Υ
L = 16
L = 32
L = 64
L = 96
L =128
L =256
L =512
TBKT
0 0.1 0.2 0.3 0.4 0.5 0.6
T
0
0.2
0.4
0.6
0.8
1
Υ
L = 25
L = 50
L = 75
L = 100
L = 150
L = 200
L = 300
L = 400
T
*
Figure 16. Temperature behaviour of the stiff-
ness modulus Υ for the 2D XY-model. (Monte
Carlo simulations with 106 MCS/spin after re-
moving the first 106 MCS to achieve proper
thermalization).
Figure 17. Temperature behaviour of the stiff-
ness modulus Υ for the 2D Lebwohl-Lasher
model. (Monte Carlo simulations with 106
MCS/spin after removing the first 106 MCS
to achieve proper thermalization).
For the 2D LL-model, the same procedure can be used with eventually the following expression
for the stiffness:
2Ld
3Jβ
Υ = −
〈
∑
〈i,j〉
(
(Si · Sj) × (Si · Ĵ
2
xSj) + (Si · ĴxSj)
2
)
〉
+ 3J
〈
(
∑
〈i,j〉
(Si · Sj) × (Si · ĴxSj)
)2
〉
.
(13)
Then, we have plotted Υ versus the temperature as it is shown in figure 17 for various sizes from
L = 25 to L = 400. Again, a behaviour similar to an order-parameter is observed in this case. At
any temperature below T ? = 0.513, one clearly identifies finite values of the stiffness independently
of the system size. Above T ?, the values of the stiffness drop to values close to 0.
However, beyond the similarities, one can remark that the stabilization to a finite value of the
stiffness in the low-temperature phase, is not similar in both models. One can see this point, in
particular analyzing the dependences with the system size. In figure 18, Υ is shown versus L for
three different temperatures of the 2D XY-model. Two of them below the BKT transition (T = 0.5
and 0.7), and the other temperature just at the BKT transition. For the lower temperatures Υ
goes fast to an asymptotic constant value. At TBKT, the stiffness modulus reaches its saturation
value algebraically. Then, it is clear that for the XY-model a sharp jump of the stiffness modulus is
observed right at the topological transition. The behaviour is completely different for the 2D LL-
model. In figure 19, the dependence of Υ with the system size L is shown for various temperatures
below T ?. The stiffness does not reach a finite value in the limit L → ∞. This behaviour is similar
to the 2D Heisenberg model [46] and to the fully frustrated Heisenberg model [47]. In figure 19,
the slopes of the curves depend on the value of the temperature. It appears to be similar to the
results by Winttel et al [47]. A detailed study of the stiffness for frustrated spin systems can be
found in [48].
13601-14
On the critical behaviour of two-dimensional liquid crystals
10 100 1000
L
0.6
0.65
0.7
0.75
0.8
0.85
0.9
Υ T = 0.893
T = 0.70
T = 0.50ΥL=Υ + ΑL-x
100
L
0.5
0.6
0.7
0.8
0.9
1
Υ
T = 0.1
T = 0.2
T = 0.3
T = 0.4
T = 0.5
Figure 18. Stiffness Υ versus system size L
for three different temperatures for the 2D
XY-model. Υ(L) goes to a finite value in the
limit L → ∞ for T 6 TBKT. (Monte Carlo
simulations with 106 MCS/spin after cance-
lation of 106 for thermalization).
Figure 19. Behaviour of stiffness modulus ver-
sus the system size for the 2D Lebwohl-Lasher
model. (Monte Carlo simulations with 106
MCS/spin after removing the first 106 MCS
to achieve proper thermalization).
6. Discussion and conclusions
Our initial goal was to give definite conclusions about the possible appearance of a critical
topological transition, for example of the Berezinskii-Kosterlitz-Thouless type, in the 2D Lebwhol-
Lasher model.
Our results exhibit a scenario more complex than expected. They can be summarized as follows:
• The behaviour of the order parameter (the magnetization) with the temperature and the
system size, pointed out a quasi-long range order phase for temperatures below the value
T ? = 0.513. One has noticed an excellent agreement of the correlation exponent η estimated
using conformal transformation, scaling of the order parameter with L, including scaling with
a magnetic field.
• If we investigate a quantity related to the second moment of the order parameter, that is the
susceptibility, a qualitative agreement with a QLRO phase behaviour is observed. However,
the value obtained for η from this quantity does not compare well with the estimates from
the order parameter. Actually, the hyperscaling relation (3) is not satisfied for T 6 T ?. This
is in clear contradiction with the possibility of a line of critical points below T ?. Then, one
has to conclude that a BKT transition is not observed for this model.
• Although not studied in the present paper, the behaviour of the Binder cumulant (which
depends on the fourth moment of the order parameter), as reported in [32], fully supports
the above conclusion: no universality is observed at any finite temperature. Signs of QLRO
just disappear if larger moments of the order parameters are used.
• As the conclusion seems to depend on the order of the moment of the magnetization, it looks
natural to use the complete probability distribution function of the order parameter to clarify
the situation. Unlike the 2D XY-model, the 2D LL-model does not show any first-scaling-law
collapse as it would be expected for a critical system. The absence of self-similarity at any
temperature discards any sort of transition in this model, which eventually appears more
similar to the Heisenberg model than to the XY-model.
• The stiffness modulus mixes a second and a fourth moment of the order parameter. For the 2D
LL-model, this quantity does not exhibit an order-parameter behaviour. At low temperatures a
13601-15
A.l. Fariñas-Sánchez et al.
logarithmic decay with L is observed, and asymptotic finite values are very unlikely, resulting
in a non-rigid phase at low temperatures.
Our conclusion, based only on numerical simulations, is that the 2D LL-model shares similari-
ties with the fully frustrated antiferromagnetic Heisenberg model (see [47, 49]). Then, the overall
behaviour could well be a sharp crossover [47], instead of a real transition to a critical phase, a
new kind of frustration preventing the topological defects of the LL-model to initiate a true phase
transition.
Acknowledgement
We gratefully acknowledge Yu. Holovatch for discussions.
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До критичної поведiнки двовимiрних рiдких кристалiв
А.I. Фарiньяс-Санчес1, Р. Боте2, Б. Берш3, Р. Паредес4
1 Унiверситет Сiмона Болiвара, Каракас 1080–A, Венесуела
2 Унiверситет Париж-Сюд, F–91405 Орсе, Францiя
3 Унiверситет Нансi, F–54506 Вандевр лє Нансi, F–54506, Францiя
4 Венесуельський iнститут наукових дослiджень, Каракас 1020A, Венесуела
Модель Лебволя-Лашера (Lebwohl-Lasher) (LL) є традицiйною моделлю, яка використовується для
опису переходу нематик-iзотропна фаза у реальних рiдких кристалах. У цiй статтi ми вивчаємо тем-
пературну поведiнку i скiнченно-мiрний скейлiнг двовимiрної (2D) ЛЛ моделi числовими методами.
Ми обговорюємо два можливих сценарiї. В одному з них, 2D LL-модель представляє фазовий пе-
рехiд, подiбний до топологiчного переходу, який виникає в 2D XY-моделi. У другому сценарiї 2D LL-
модель не демонструє жодного критичного переходу, проте її низькотемпературна поведiнка хара-
ктеризується кросовером iз невпорядкованоої у впорядковану фазу при нулi температури. Ми здiй-
снюємо та обговорюємо рiзнi порiвняння iз 2D XY-моделлю i 2D моделлю Гайзенберга. Доповнивши
попереднiй розгляд скiнченно-мiрною поведнкою профiля параметра порядку i його конформаль-
ним вiдображенням, ми аналiзуємо критичний скейлiнг функцiї розподiлу, гiперскейлiнговi спiввiд-
ношення i жорсткiсть параметра порядку та робимо висновок про те, що другий сценарiй (жодного
критичного переходу) є найбiльш ймовiрним.
Ключовi слова: топологiчнi переходи, критичнi явища, рiдкi кристали, модель Лебволя-Лашера
13601-17
Introduction
Definitions of the models
Qualitative analysis of the order parameter and the susceptibility
The 2D XY-model
The 2D Heisenberg model
The 2D LL-model
Conclusion from the study of the m and behaviours
Calculations of the correlation function exponent
The 2D XY-model
The 2D Heisenberg model
The 2D LL-model
Revisiting the behaviour of the 2D LL-model at the `critical' temperature
The first-scaling law
The hyperscaling relation
The stiffness modulus
Discussion and conclusions
|
| id | nasplib_isofts_kiev_ua-123456789-32043 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T19:05:15Z |
| publishDate | 2010 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Fariñas-Sánchez, A.l. Botet, R. Berche, B. Paredes, R. 2012-04-06T17:39:13Z 2012-04-06T17:39:13Z 2010 On the critical behaviour of two-dimensional liquid crystals / A.l. Fariñas-Sánchez, R. Botet, B. Berche, R. Paredes // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13601: 1-17. — Бібліогр.: 49 назв. — англ. 1607-324X PACS: 64.70.M-, 64.60.Bd, 64.70.mf, 05.70.Jk, 05.50.+q, 68.35.Rh https://nasplib.isofts.kiev.ua/handle/123456789/32043 The Lebwohl-Lasher (LL) model is the traditional model used to describe the nematic-isotropic transition of real liquid crystals. In this paper, we develop a numerical study of the temperature behaviour and of finite-size scaling of the two-dimensional (2D) LL-model. We discuss two possible scenarios. In the first one, the 2D LL-model presents a phase transition similar to the topological transition appearing in the 2D XY-model. In the second one, the 2D LL-model does not exhibit any critical transition, but its low temperature behaviour is rather characterized by a crossover from a disordered phase to an ordered phase at zero temperature. We realize and discuss various comparisons with the 2D XY-model and the 2D Heisenberg model. Having added finite-size scaling behaviour of the order parameter and conformal mapping of order parameter profile to previous studies, we analyze the critical scaling of the probability distribution function, hyperscaling relations and stiffness order parameter and conclude that the second scenario (no critical transition) is the most plausible. Зазначено, що модель Лебволя - Лашера (Lebwohl - Lasher) (LL) є традиційною моделлю, яка використовується для опису переходу нематик - ізотропна фаза у реальних рідких кристалах. Вивчено температурну поведінку та скінченно-мірний скейлінг двовимірної (2D) LL моделі за допомогою числових методів. Розглянуто два можливих сценарії. В одному з них, 2D LL-модель представляє фазовий перехід, подібний до топологічного переходу, який виникає в 2D XY-моделі. У другому сценарії 2D LL-модель не демонструє жодного критичного переходу, проте її низькотемпературна поведінка характеризується кросовером із невпорядкованоої у впорядковану фазу при нулі температури. Здійснено та обговорено різні порівняння із 2D XY-моделлю і 2D моделлю Гайзенберга. Доповнивши попередній розгляд скінченно-мірною поведінкою профіля параметра порядку і його конформальним відображенням, проаналізовано критичний скейлінг функції розподілу, гіперскейлінгові співвідношення і жорсткість параметра порядку та зроблено висновок про те, що другий сценарій (жодного критичного переходу) є найбільш ймовірним. We gratefully acknowledge Yu. Holovatch for discussions. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics On the critical behaviour of two-dimensional liquid crystals До критичної поведінки двовимірних рідких кристалів Article published earlier |
| spellingShingle | On the critical behaviour of two-dimensional liquid crystals Fariñas-Sánchez, A.l. Botet, R. Berche, B. Paredes, R. |
| title | On the critical behaviour of two-dimensional liquid crystals |
| title_alt | До критичної поведінки двовимірних рідких кристалів |
| title_full | On the critical behaviour of two-dimensional liquid crystals |
| title_fullStr | On the critical behaviour of two-dimensional liquid crystals |
| title_full_unstemmed | On the critical behaviour of two-dimensional liquid crystals |
| title_short | On the critical behaviour of two-dimensional liquid crystals |
| title_sort | on the critical behaviour of two-dimensional liquid crystals |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/32043 |
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