Nematic phase transitions in two-dimensional systems

Berche Berche,Paredes Paredes

doi:10.5488/cmp.8.4.723

Abstract

Simulations of nematic-isotropic transition of liquid crystals in two dimensions are performed using an $O(2)$ vector model characterised by non linear nearest neighbour spin interaction governed by the fourth Legendre polynomial $P_4$. 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.

Highlights

In the context of phase transitions, two-dimensional models exhibit a very rich variety of typical behaviours, ranging from conventional temperature-driven second 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)
Due to the logarithmic diverging behaviour of the intergral it is less obvious to make a final conclusion and more refined analysis 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 temperature in 2d in the case of discrete symmetry and extends the argument to continuous symmetry, showing that the ordered ground state is unstable with respect to thermal fluctuations in this latter situation
– The P4 O(2) model displays a BKT-like transition with quasi-long-range ordered phase (QLRO) in the low temperature (LT) phase where SWA nicely fits the nematization temperature-dependent exponents η(T ) when T → 0

Summary

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 second 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). Due to the logarithmic diverging behaviour of the intergral it is less obvious to make a final conclusion and more refined analysis 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 temperature in 2d in the case of discrete symmetry and extends the argument to continuous symmetry, showing that the ordered ground state is unstable with respect to thermal fluctuations in this latter situation. The variation of internal 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), where π/l is the nearest neighbour spin disorientation This result follows from the integration over the droplet volume O(l2). An unconventional phase transition toward a quasi-long-range ordered state may take place at finite temperature, as we discuss

Two-dimensional electrodynamics and the XY model

Definition of the model and of the observables

Characterization of the low-temperature phase

Critical behaviour at the deconfining transition

Summary and open questions