Nematic phase transitions in two-dimensional systems
Simulations of nematic-isotropic transition of liquid crystals in two dimensions are performed using an O(2) vector model characterized by non linear nearest neighbour spin interaction governed by the fourth Legendre polynomial P₄. The system is studied through standard Finite-Size Scaling and c...
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irk-123456789-1210472017-06-14T03:04:50Z Nematic phase transitions in two-dimensional systems Berche, B. Paredes, R. Simulations of nematic-isotropic transition of liquid crystals in two dimensions are performed using an O(2) vector model characterized by non linear nearest neighbour spin interaction governed by the fourth Legendre polynomial P₄. The system is studied through standard Finite-Size Scaling and conformal rescaling of density profiles or correlation functions. The low temperature limit is discussed in the spin wave approximation and confirms the numerical results, while the value of the correlation function exponent at the deconfining transition seems controversial. Симуляції переходу нематик-ізотропна фаза для рідких кристалів у двох вимірах виконуються за допомогою O(2) векторної моделі, що характеризується нелінійною спіновою взаємодією найближчих сусідів, представленою четвертим поліномом Лежандра P₄ . Система досліджується стандартним скінченорозмірним скейлінгом та конформаційним перескальовуванням профілів густини чи кореляційних функцій. Границя низьких температур обговорюється у наближенні спінових хвиль і узгоджується з числовими результатами, тоді як значення експоненти кореляційної функції при переході роз’єднання виглядає контроверсійним. 2005 Article Nematic phase transitions in two-dimensional systems / B. Berche, R. Paredes // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 723–736. — Бібліогр.: 33 назв. — англ. 1607-324X PACS: 05.40.+j, 64.60.Fr, 75.10.Hk DOI:10.5488/CMP.8.4.723 http://dspace.nbuv.gov.ua/handle/123456789/121047 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Simulations of nematic-isotropic transition of liquid crystals in two dimensions
are performed using an O(2) vector model characterized by non
linear nearest neighbour spin interaction governed by the fourth Legendre
polynomial P₄. The system is studied through standard Finite-Size Scaling
and conformal rescaling of density profiles or correlation functions. The low
temperature limit is discussed in the spin wave approximation and confirms
the numerical results, while the value of the correlation function exponent
at the deconfining transition seems controversial. |
| format |
Article |
| author |
Berche, B. Paredes, R. |
| spellingShingle |
Berche, B. Paredes, R. Nematic phase transitions in two-dimensional systems Condensed Matter Physics |
| author_facet |
Berche, B. Paredes, R. |
| author_sort |
Berche, B. |
| title |
Nematic phase transitions in two-dimensional systems |
| title_short |
Nematic phase transitions in two-dimensional systems |
| title_full |
Nematic phase transitions in two-dimensional systems |
| title_fullStr |
Nematic phase transitions in two-dimensional systems |
| title_full_unstemmed |
Nematic phase transitions in two-dimensional systems |
| title_sort |
nematic phase transitions in two-dimensional systems |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121047 |
| citation_txt |
Nematic phase transitions in two-dimensional systems / B. Berche, R. Paredes // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 723–736. — Бібліогр.: 33 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT bercheb nematicphasetransitionsintwodimensionalsystems AT paredesr nematicphasetransitionsintwodimensionalsystems |
| first_indexed |
2025-07-08T19:05:46Z |
| last_indexed |
2025-07-08T19:05:46Z |
| _version_ |
1837106777213108224 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 4(44), pp. 723–736
Nematic phase transitions in
two-dimensional systems
B.Berche 1,2 ∗, R.Paredes 2 †
1 Groupe M, Laboratoire de Physique des Matériaux,
UMR CNRS 7556,
Université Henri Poincaré, Nancy 1,
F–54506 Vandœuvre les Nancy Cedex, France
2 Centro de Fı́sica,
Instituto Venezolano de Investigaciones Cientı́ficas,
Apartado 21827, Caracas 1020A, Venezuela
Received July 18, 2005
Simulations of nematic-isotropic transition of liquid crystals in two dimen-
sions are performed using an O(2) vector model characterized by non
linear nearest neighbour spin interaction governed by the fourth Legendre
polynomial P4. The system is studied through standard Finite-Size Scaling
and conformal rescaling of density profiles or correlation functions. The low
temperature limit is discussed in the spin wave approximation and confirms
the numerical results, while the value of the correlation function exponent
at the deconfining transition seems controversial.
Key words: liquid crystal, orientational transition, nematic phase,
topological transition
PACS: 05.40.+j, 64.60.Fr, 75.10.Hk
1. Ordering in two dimensions
In the context of phase transitions, two-dimensional models exhibit a very rich
variety of typical behaviours, ranging from conventional temperature-driven sec-
ond order phase transitions (e.g. Ising model) to first-order ones (e.g. q > 4-state
Potts model), with specific properties of models having continuous global symmetry
which may present defect-mediated topological phase transitions (e.g. XY model)
or even no transition at all (e.g. Heisenberg model). Models of nematic-isotropic
orientational phase transitions belong to this latter category of systems displaying a
continuous symmetry. Ordering in low dimensional systems is likely to be frustrated
by the strength of fluctuations. On qualitative grounds, let us consider for instance
∗E-mail: berche@lpm.u-nancy.fr
†E-mail: rparedes@ivic.ve
c© B.Berche, R.Paredes 723
B.Berche, R.Paredes
how fluctuations develop within the framework of Landau theory when the tempera-
ture decreases from the high temperature paramagnetic phase toward the transition
temperature. The response to a localized magnetic field applied at the origin, hδ(x),
follows from Ginzburg-Landau functional minimization of the free energy
F [m (x)] =
∫
dx
(
1
2
am2 (x) +
1
4
bm4 (x) +K|∇m (x) |2 − hδ (x)m (x)
)
.
After linearization and Fourier transform, it yields
ξ−2m̃(k) + |k|2m̃(k) =
h
2K
, (1)
where ξ−2 = a/2K is the correlation length. The response m̃(k) is here proportional
to the correlation function Fourier transform G̃(k) and the fluctuations are measured
through
kBTχ = V −1
(
〈M2〉 − 〈M〉2
)
=
∑
k
G̃ (k) . (2)
The latter sum is converted to an integral from 0 to some cutoff Λ. It is diverg-
ing with ξ in 1d (
∑
k G̃(k) ∼ ξArctan (Λξ)) and in 2d (
∑
k G̃(k) ∼ ln(Λξ)) while
it is bounded in 3d (
∑
k G̃(k) ∼ const). The divergence in 1d prevents any long
range ordering, while stable ordered state is not forbidden in three-dimensional
systems. In the intermediate case, due to the logarithmic diverging behaviour of
the intergral it is less obvious to make a final conclusion and more refined anal-
ysis is required. In his famous book on phase transitions, Cardy uses a simplified
version of Peierls argument on the existence of a phase transition at finite temper-
ature in 2d in the case of discrete symmetry and extends the argument to conti-
nuous symmetry, showing that the ordered ground state is unstable with respect
to thermal fluctuations in this latter situation. Consider a spin system with only
nearest-neighbour interactions −JSiSj and assume that the spins are represented
by classical n-component vectors. An ordered ground state may be stabilized by
symmetry breaking fields at some boundaries of the system. The variation of inter-
nal energy when a droplet of typical size l with spins progressively tilted in such a
way that at the center of the droplet the spins are pointing opposite the direction
of the field is of the order of O(l2) × J |S|2(π/l)2, where π/l is the nearest neigh-
bour spin disorientation. This result follows from the integration over the droplet
volume O(l2). Considering that the entropy is measured by the number of possible
closed loops of size O(l) in 2d, the entropy of the droplet is estimated as kB lnµl
where µ < z − 1 (z is the coordination of the lattice), so that eventually the free
energy variation is ∆F (l) = π2J |S|2 − kBT l lnµ. At any non zero temperature, the
increase of the size l of the droplet stabilizes the system and a spontaneously ordered
ground state is thus impossible. In the case of discrete symmetry, e.g. Ising model,
the energy balance would be associated with the interface only and we would get
∆F (l) = 2J |S|2l − kBT l lnµ, showing that a transition to a stable ordered ground
state is expected at the temperature around kBTc = O(2J |S|2). The two examples
are illustrated in figure 1. In the latter situation, a conventional phase transition
724
Nematic phase transitions in two-dimensional systems
� �
Figure 1. Evaluation of the free energy of a disordering droplet in two dimensions
in the case of continuous symmetry (left) and discrete symmetry (right).
toward an ordered phase is expected at finite temperature, while such an ordered
phase may only be encountered at zero temperature in the first example [1,2]. On the
other hand, an unconventional phase transition toward a quasi-long-range ordered
state may take place at finite temperature, as we discuss in the next section.
2. Two-dimensional electrodynamics and the XY model
The celebrated XY model,
H = −J
∑
(r,r′)
cos(θr − θr′) ' const +
1
2
J
∑
(r,r′)
(θr − θr′)
2, (3)
admits a phenomenological description in terms of Coulomb gas. This is a well-
known description, first given by Berezinskĭı and Kosterlitz and Thouless [3–5] and
it is worth reminding its essential steps here. Before considering this model, imagine
a point-charge q located at site r0 in a two-dimensional space, ρ(r) = qδ(r−r0). The
corresponding Coulomb potential, i.e., the solution of Poisson equation ∇
2φ(r) =
−ε−1
0 ρ(r) might be written
φ(r − r0) = −
q
2πε0
ln
|r − r0|
a
, (4)
where a is chosen such that φ(a) = 0. Consider now a spin system living in two-
dimensional space and let us call u = ∇θ the distortion field, where θ(r) is the phase
field defined by S(r) = (cos θ(r), sin θ(r)). Due to the periodicity of the phase field,
the distortion field should obey the following relation
∮
C(r0)
u dl = 2π × integer
where the contour integral is taken along a counterclockwise closed path around the
point r0. Using Stokes theorem, we may also write
∮
C(r0)
udl =
∫
d2r ẑ(∇ × u) = 2π × integer (5)
725
B.Berche, R.Paredes
which implies
∇ × u = 2π × integer× δ(r − r0)ẑ. (6)
The general solution of this equation consists of two terms, u = ∇ψ − ∇ × (ẑφ).
The curl of the first term being identically zero, we are led to a Poisson equation for
the singular term ∇
2φ = 2π×integer× δ(r−r0) where the integer ≡ n plays the
role of the total (topological) charge enclosed by the contour C. From an energetic
point of view, we may consider the kinetic energy
βH =
1
2
K
∫
d2r |u|2 (7)
which, up to an inessential constant, is the continuous approximation of equation (3)
with K = βJ . After decomposition in ψ and φ parts, the cross-term vanishes and
we get two independent contributions,
βH =
1
2
K
∫
d2r |∇ψ|2 +
1
2
K
∫
d2r |∇φ|2.
spin waves defects
(8)
This is a fundamental relation which is the basis of the factorization of the par-
tition function in a spin wave contribution and a defect (vortex) contribution,
Z = Tr exp(−βH) = ZSWZV. In the lattice version, the spin wave contribution
to the Hamiltonian βHSW = 1/2 ·K
∑
(r,r′)(ψr −ψr′)
2 is quadratic in Fourier space,
βHSW =
1
2
K
∑
q
|q|2µ2|ψq|
2, (9)
(µ is the lattice spacing and ψq the Fourier component of ψr). This expression leads
to the Gaussian model which implies [6] that
〈Sr1
· Sr2
〉 = e−
1
2
〈(ψr1
−ψr2
)2〉 ' |r1 − r2|
−1/2πK , (10)
where a temperature-dependent spin-spin critical exponent is found, ηSW = (2πK)−1.
The low temperature (LT) phase of the XY model is a quasi-long-range ordered
phase (QLRO), or a critical phase. When the temperature increases, vortices bound-
ed in pairs produce a further disordering of the system and a faster than linear
increase of the exponent η with temperature up to the temperature where the
transition takes place. The role of vortices is understood perturbatively through
the calculation of the effective screened interaction energy between the topologi-
cal charges. We consider a neutral Coulomb gas. Omitting a diverging contribution
O(lnR/a)×
∑
i ni which would occur otherwise (i.e. for a non-neutral Coulomb gas),
the vortices contribution to the Hamiltonian reads as
βHV = −
1
2
K
∫
d2r φ∇
2φ
= −2πK
∑
ij
i6=j
ninj ln
|ri − rj|
a
=
∑
ij
i6=j
βV (ri − rj), (11)
726
Nematic phase transitions in two-dimensional systems
where V (ri−rj) is the Coulomb interaction between the charges. In the perturbative
approach, we consider the effective interaction between charges ni and nj = −ni in
the presence of another screening dipole. The following result comes out [7]
e−βVeff(ri−rj) = e−βV (ri−rj)
(
1 + const × y2
0 ×
∫
dr r3e−βV (r) +O(y4
0)
)
, (12)
where y0 is the fugacity of a charge. The correction term y2
0
∫
dr r3−2πK diverges at
the deconfining transition 2πKc = 4 where the pairs of charges break. The presence
of vortices increases the disordering of the system which, below Tc, flows under
renormalization to the zero-fugacity limit where the spin-wave limit is recovered
with a renormalized temperature [5]. At the transition, the correlation function
exponent takes a universal value
ηc =
1
2πKc
=
1
4
. (13)
The Heisenberg model (O(3)) on the other hand has no transition at any finite
temperature (asymptotic freedom). An intuitive argument called escape to the third
dimension is often reported. This is the observation that vortices cannot be stable for
n > 3, since an infinitesimal amount of energy is capable of producing a spin-wave-
type excitation which eliminates the localized defect. Furthermore, a renormalization
scheme was proposed by Polyakov to treat the non linear σ-model close to the
lower critical dimension d = 2 [8–11]. The flow of the coupling constant g (the
temperature) under a change of scale s is governed by the β-function,
β(g, d) = s
∂g
∂s
∣
∣
∣
∣
0
= (d− 2)g − (n− 2)g2 +O(g3)
which shows that the model is disordered at any temperature when n > 2 in two
dimensions, since the coupling always decreases and flows to zero under renormali-
zation.
3. Nematics
3.1. Definition of the model and of the observables
We now come back to liquid crystals, the molecules of which may be idealized
as long neutral rigid rods. They are likely to interact through electrostatic inter-
actions. Therefore, Legendre polynomials appear for the description of the orien-
tational transition between a disordered isotropic high temperature phase and an
ordered nematic phase. Lattice models of nematic-isotropic transitions capture the
essentials of this extremely simplified description. The molecules are represented
by n-component unit vectors Sr (also called “spins”), located here on the sites r
of a square lattice. The interaction between molecules is restricted to the nearest
727
B.Berche, R.Paredes
neighbour pairs (r, r′), the radial dependence being kept constant and the angu-
lar dependence entering through a k-th order Legendre polynomial1, Pk(Sr · Sr′),
in terms of the scalar product between Sr and Sr′ , Sr · Sr′ . A coupling parameter
J measures the interaction intensity. One obtains the following Hamiltonian of a
lattice liquid crystal,
H = −J
∑
(r,r′)
Pk(Sr · Sr′). (14)
When the value of k in equation (14) is varied, new features may be expected, as
in the case of symmetry-breaking magnetic fields hk cos kθ added to the XY model
which change the phase diagram as investigated by José et al [7,12]. When the
polynomial order k increases, one may expect a qualitative change in the nature
of the transition, like in the case of discrete spin symmetries (Potts model) [13,14].
The value k = 2 was intensively studied. It still corresponds to the XY model for
O(2) spin symmetry, while it leads to the RP 2 or Lebwohl-Lasher model [15] for
3-component spin vectors.
The nature of the transition in this latter case is still under discussion: an early
study of Kunz and Zumbach [16] reported numerical evidence in favour of a topo-
logical transition, but more recently, several authors argued in favour of the absence
of any finite-temperature phase transition [17,18], like in the Heisenberg case. Our
own previous contributions [19,20] support the first scenario with QLRO at low
temperature like in the XY model, but one cannot exclude a finite – but extremely
large – correlation length which exceeds the maximum size available in numerical
simulations. A recent study reported extremely convincing new evidence in favour of
a topological transition [21], the transition being driven by topologically stable point
defects known as 1/2 disclination points. Eventually, in the large−n limit, there is
a proof of asymptotic freedom for values of k (in the interaction term (1 + cos θ)k)
which do not exceed a critical kc ' 4.537 . . . [18]. Above this value the transition
becomes of first order, a result which does not violate Mermin-Wagner-Hohenberg
theorem, since the correlation length is finite at the transition. For finite value of n,
the question of the nature of the transition at high k is still a challenging problem,
although there is a rigorous proof that the transition becomes of first-order for large
enough values of k for arbitrary n > 2 [22,23]. This observation suggests inspecti-
ng the effect of a higher value of k for the O(2) model as well. In the context of
orientational transitions in liquid crystals, Legendre polynomials rather than cosk θ
interactions are introduced, and we are led to the Hamiltonian of equation (14).
In this report, we consider the behaviour of an Abelian spin model (O(2) rota-
tion group) with P4-like spin interactions. We will refer to it as the P4 O(2) model
for simplicity. For 2-component vectors in a disordered phase, 〈cos2 θ〉 = 1/2 and
〈cos4 θ〉 = 3/8. In order to keep the same symmetry in the interaction as in the
P4 O(3) model already considered in the literature [24,25], but to normalize it be-
tween 0 and 1 in the limits of completely disordered and completely ordered phases
respectively, we slightly modify the Hamiltonian, considering pair interactions of
1Even order Legendre polynomials guarantee the local Z2 symmetry Sr → −Sr .
728
Nematic phase transitions in two-dimensional systems
the form Q4(x) ≡ AP4(x) + const = 8/55(35x4 − 30x2 + 15/8). The corresponding
Hamiltonian is thus defined by
HP4 O(2) = −J
∑
(r,r′)
Q4(Sr · Sr′), (15)
with Sr = (Sxr , S
y
r ), |Sr| = 1. A qualitative description of the transition is provided
by the temperature behaviour of the energy density, the specific heat, the order
parameter and the susceptibility. The internal energy is defined from the thermal
average of the Hamiltonian density, u(T ) = (dLd)−1〈H〉 and the specific heat follows
from fluctuation dissipation theorem, Cv(T ) = (LdT 2)−1(〈H2〉 − 〈H〉2). Brackets
denote the thermal average. The definition of the scalar order parameter (sometimes
called nematisation ) is deduced from the local second-rank order parameter tensor,
Mαβ(r) = Sαr S
β
r −1/2·δαβ. After space averaging, the traceless tensor L−d
∑
rM
αβ(r)
admits two opposite eigenvalues ±1/2 · η corresponding to eigenvectors n+ and n−.
The order parameter density is defined after thermal averaging by M2(T ) = 〈η〉.
The associated susceptibility is defined by the fluctuations of the order parameter
density, χM2
(T ) = 4Ld(kBT )−1(〈η2〉 − 〈η〉2).
0,2 0,4 0,6 0,8 1T
-1
-0,8
-0,6
-0,4
-0,2
u
L = 24
L = 32
L = 48
L = 64
L = 128
0,2 0,4 0,6 0,8 1T
1
C
v
0,2 0,4 0,6 0,8 1T
0
0,2
0,4
0,6
0,8
1
M
2
0,2 0,4 0,6 0,8 1T
10
-2
10
0
10
2
χ
M
2
Figure 2. Energy, specific heat, order parameter M2 and corresponding suscepti-
bility χM2
vs T for the P4 O(2) model. The full lines are only guides for the eyes.
The values of kB and J have been fixed to unity.
Simulations are performed using a standard Wolff algorithm adapted to the ex-
pression of the nearest neighbour interaction [25,26]. The spins are located on the
vertices of a simple square lattice of size L2 with periodic boundary conditions in the
two directions. Usually 106 equilibrium steps were used (measured as the number of
flipped Wolff clusters) and 106 Monte Carlo steps for the evaluation of thermal av-
erages (the autocorrelation time at kBT/J = 0.2, L = 16 is of the order of 30 MCS.
Hence the numbers of MC steps correspond roughly to 3.104 independent measure-
ments for the smallest size). A first qualitative description of the behaviour of the
system is provided by the temperature dependence of thermodynamic quantities [27].
729
B.Berche, R.Paredes
The specific heat has a maximum which does not seem to increase substantially with
the system size. This might be the sign of an essential singularity around a temper-
ature kBTc/J ' 0.70. From the order parameter variation, a smooth transition is
suspected, since there is no evolution toward a sharp jump. The susceptibility dis-
plays a non-conventional behaviour at low temperature, increasing with the system
size, which indicates a possible topological transition with a critical low temperature
phase where the susceptibility diverges at any temperature.
3.2. Characterization of the low-temperature phase
We assume the existence of a critical phase at low temperatures as suggested by
the temperature dependence results. The properties of the phase transition may be
studied using
i) Standard Finite-Size Scaling technique (FSS): in the critical low temperature
phase of a model which displays a topological transition, the physical quantities
behave like at criticality for a second-order phase transition, with power law
behaviours of the system size. The difference is that in the critical phase, the
critical exponents depend on the temperature and for any temperature below
the transition one has e.g.
M2(T ) ∼ L− 1
2
ηM2
(T ), (16)
χM2
(T ) ∼ L2−ηM2
(T ). (17)
Here ηM2
(T ) denotes the correlation function critical exponent,
〈Q2(cos(θr1
− θr2
))〉 ∼ |r1 − r2|
−ηM2
(T ). (18)
ii) Rescaling of the density profiles (or “Finite Shape Scaling”, FShS): conformally
covariant density profiles or correlation functions are expected at any tem-
perature below the transition Tc. They transform according to G(w1, w2) =
|w′(z1)|
−xσ |w′(z2)|
−xσG(z1, z2) through conformal mapping w(z) where w la-
bels the lattice sites in the transformed geometry (the one where the com-
putations are really performed), while z refers to the infinite plane where
the two-point correlations take the standard power-law expression G(z1, z2) ∼
|z1 − z2|
−ησ), and xσ = 1/2 · ησ is the scaling dimension associated to the scali-
ng field under consideration. Rather than two-point correlation functions, it
is even more convenient to work with density profiles m(w) in a finite system
with symmetry breaking fields along some surfaces in order to induce a non-
vanishing local order parameter in the bulk [28–30]. The density m(w) will be
M2(r) = 〈Q2(Sr · h∂Λ)〉. In the case of a square lattice Λ of size L × L, with
fixed boundary conditions along the four edges ∂Λ, one expects (details may
be found e.g. in [29])
m(w) ∼ [κ(w)]−
1
2
ησ ,
κ(w) = =m
[
sn
2Kw
L
]
×
∣
∣
∣
∣
(
1 − sn2 2Kw
L
) (
1 − k2sn2 2Kw
L
)∣
∣
∣
∣
− 1
2
. (19)
730
Nematic phase transitions in two-dimensional systems
Another conformal mapping which has been applied to many two-dimensional
critical systems is the logarithmic transformation w(z) = L/(2π) · ln z =
L/(2π) · ln ρ + i[Lϕ/(2π)] which maps the infinite plane onto an infinitely
long cylinder of perimeter L. The correlation functions along the axis of the
cylinder (let us say in terms of the variable u = L/(2π) · ln ρ) decay expo-
nentially at criticality, G(u1, u2) ∼ exp[−(u2 − u1)/ξ], the correlation length
amplitude on the strip being universal, ξ = L/(πη) [31]. Simulations at dif-
ferent temperatures in a system of size 10 × 10000 were performed and the
correlation function exponent thus follows from the linear behaviour
ln〈Q2[cos(θu2
− θu1
]〉 = const −
πηM2
L
(u2 − u1). (20)
0 0,2 0,4 0,6
k
B
T/J
0
0,04
0,08
0,12
0,16
η M
2
SW
L = 16 square
L = 32 square
L = 64 square
L = 128 square
10 by 10000 torus
FSS
Figure 3. Temperature variation of the correlation function exponent ηM2
(T )
deduced from conformal rescaling (open symbols) and FSS (filled symbols).
Open triangles and diamonds, which correspond to the largest systems seem
quite reliable. The dashed line shows the result of the spin-wave approximation
ηM2
(T ) = (11/64)kBT (πJ)−1.
The resulting η-exponent deduced from these different techniques is shown in
figure 3. The variation of η with the temperature is very similar to the one ob-
served in the case of the XY model and confirms the QLRO nature of the LT
phase of the model. The low temperature linear variation of this exponent is eas-
ily understood within the spin wave approximation. For O(2) model with nearest
neighbour interactions described by arbitrary polynomial in Sr · Sr′ , one is led to
an effective harmonic Hamiltonian 1/2 · J
∑
(r,r′) l(θr − θr′)
2. It yields power-law
correlations,
〈cosm(θr1
− θr2
)〉 = e−
m2
2
〈(θr1−θr2 )2〉 ∼ |r1 − r2|
−ηml (21)
with a decay exponent given by
ηml =
m2
l
ηXY =
m2
2πKl
. (22)
731
B.Berche, R.Paredes
The comparison is made visible in the figure (with m = 2 and l = 128/11 ), and as
expected, the lower the temperature the better the SW approximation.
3.3. Critical behaviour at the deconfining transition
Not only the low temperature behaviour of η is interesting, the precise value of
the η exponent at the BKT transition where some deconfining mechanism should
lead to the proliferation of unbinded topological defects is also of interest, since it
really describes the universality class of the transition. An accurate value of the
transition temperature is first needed. We performed a study of the crossing point
of U4 Binder cumulant for very large statistics (30 × 106 MCS) and large system
sizes (squares of L = 64, 80, 96 and 128 with periodic boundary conditions). The
results shown in figure 4 indicate a transition temperature of kBTBKT/J = 0.7226.
0,7 0,71 0,72 0,73
k
B
T/J
0,60
0,62
0,64
0,66
U
4
L = 64
L = 80
L = 96
L = 128
0 0,02 0,04 0,06
L
−1
0,1
0,12
0,14
0,16
0,18
0,2
η ef
f(L
)
Figure 4. Left: Crossing of the Binder parameter at the deconfining transition
at a temperature kBTBKT/J = 0.7226. Right: Size dependence of the correlati-
on function exponent at TBKT (L = 16, 32, 64, 128) and extrapolation to the
thermodynamic limit.
Then this temperature is used to perform Finite-Shape Scaling using the alge-
braic decay of density profiles inside a square with fixed boundary conditions. These
simulations are time-consuming, since the autocorrelation time increases in the low
temperature phase when T evolves towards the deconfining transition and a rather
large number of Monte Carlo steps is needed to get a satisfying number of indepen-
dent measurements. For sizes L = 16, 32, 64, we used 106 MCS for thermalization
and 30 · 106 for measurements, while “only” 20 · 106 for the largest size 128. The ex-
ponential decay of two-point correlation functions along the torus cannot be applied
at the BKT transition, since the system size being much larger than in a square
geometry, the number of MC iterations required is by far too large. In figure 4 we
plot the “effective” exponent ηeff(L) measured at TBKT for different system sizes as a
function of the inverse size. An estimate of the thermodynamic limit value (L→ ∞)
can be made using a polynomial fit (the results of quadratic and cubic fits are re-
spectively 0.118 and 0.122), but it is safer to keep the three largest sizes available,
L = 32, 64, and 128, for which a linear dependence of ηeff(L) with L−1 is observed.
732
Nematic phase transitions in two-dimensional systems
Taking into account the error bars, crossing the extreme straight lines leads to the
following value for the correlation function exponent at the deconfining transition
ηM2
(TBKT) = 0.122 ± 0.007. (23)
This value is essentially half the Kosterlitz value for the XY model.
4. Summary and open questions
The results obtained are essentially as follows:
– The P4 O(2) model displays a BKT-like transition with QLRO in the LT phase
where SWA nicely fits the nematization temperature-dependent exponents
η(T ) when T → 0.
– The value of the exponent at the deconfining transition is supposed to reach a
universal value. The numerical estimate is close to 0.125.
We believe that our conclusions are safe concerning the existence of a QLRO
phase at low temperature. The results may be understood through a naive compar-
ison with clock model in 2d. The increase of the order of the interaction polynomial
indeed increases the number of deep wells which stabilize the relative orientation of
neighbouring spins, and one is thus led to a system which is quite similar to a planar
clock model with a finite number of states, unless the fact that here we keep a conti-
nuous spin symmetry which prevents any “magnetic” long-range order at finite tem-
perature. The clock model is in the Potts universality class when q = 3, but at q > 4,
it displays a QLRO phase before conventional ordering at lower temperatures [12].
Combining this analogy with the requirements of Mermin-Wagner-Hohenberg the-
orem for continuous spin symmetry gives a natural explanation to our results. For
any type of nearest-neighbour interaction (in P1, P2 or P4) the behaviour seems to
be always described by a BKT-like transition. The similar observation that a two-
component nematic model renormalizes in two dimensions towards the XY model
was already reported in [32]. The transition is likely to be driven by a mechanism
of condensation of defects, like in the XY model, but due to the local Z2 symmetry
not only usual vortices carrying a charge 1 are stable, but also disclination points
carrying charges 1/2 should be stable. The role of these defects might be studied in
the way similar to the recent work by Dutta and Roy [21], by the comparison of the
transition in the pure model and in a modified version where a chemical potential
is artificially introduced in order to control the presence of defects.
Now, the observed value of the exponent η at the transition temperature seems
to be a bit strange. A naive application of the mechanism discussed in section 2 leads
to a value two times larger. Apart from minor modifications, the same procedure
applies herein. The Hamiltonian (15) is of the form (7) with K replaced by Kl
which does not affect the universal properties, but changes the critical temperature
by the same factor. The other modification comes from the local Z2 symmetry in
the nematic model. This changes the charges n from integers to half-integers, which
733
B.Berche, R.Paredes
modifies the factor of 2π in equation (11) to π/2. The divergence of the perturbative
term in (12) then occurs at 1/2 · πKcl = 4, where the spin wave exponent (22)
becomes ηc = m2/16 which takes the value 1/4 when m = 2. We thus have a huge
discrepancy of 100 % between the numerical determination and this prediction! A
probable source of this discrepancy may be in the mechanism involved. There is
no negative charge in the nematic model, so the pair interaction has a different
structure and a more refined analysis is therefore desirable.
Acknowledgement
We thank A.C.S. van Enter for useful correspondence. Part of the material pre-
sented here (figures 2, 3 and 4) was obtained in collaboration with Ana Fariñas and
was originally presented elsewhere.
Note added in proofs
We would like to thank S. Korshunov who draw our attention on a possible
scenario explaining the discrepancy between the numerical value of η and the pre-
diction which follows from Kosterlitz-like arguments. Twenty years ago, he studied
a similar planar XY model with mixed interactions [33]. In the parameters space
of the problem, there is a zone where, starting from the high temperature phase,
a BKT transition is first observed and then at lower temperature occurs a phase
transition governed by the presence of solitons. This latter transition belongs to the
two-dimensional Ising model (IM) universality class. According to this scenario, ours
results would thus correspond to this IM transition, but the quantity that we study
does not correspond to the order parameter of this transition (hence the value of
η is not the usual 1/4 of the IM universality class). QLRO would persist at higher
temperatures, up to a limiting value where ηM2
would indeed take its Kosterlitz-like
value m2/16.
734
Nematic phase transitions in two-dimensional systems
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735
B.Berche, R.Paredes
Нематичні фазові переходи у двовимірних
системах
Б.Берше, Р.Паредес
1 Гупа М, Лабораторія фізики матеріалів,
Університет Анрі Пуанкаре, Нансі 1,
Вандувр лє Нансі Седекс, Франція
2 Центр фізики, Інститут наукових досліджень,
Каракас, Венесуела
Отримано 18 липня 2005
Симуляції переходу нематик-ізотропна фаза для рідких кристалів
у двох вимірах виконуються за допомогою O(2) векторної моделі,
що характеризується нелінійною спіновою взаємодією найближчих
сусідів, представленою четвертим поліномом Лежандра P4 . Сис-
тема досліджується стандартним скінченорозмірним скейлінгом та
конформаційним перескальовуванням профілів густини чи коре-
ляційних функцій. Границя низьких температур обговорюється у
наближенні спінових хвиль і узгоджується з числовими резуль-
татами, тоді як значення експоненти кореляційної функції при
переході роз’єднання виглядає контроверсійним.
Ключові слова: рідкі кристали, орієнтаційний перехід, нематична
фаза, топологічний перехід
PACS: 05.40.+j, 64.60.Fr, 75.10.Hk
736
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