Condensed Matter Physics, 2010, vol. 13, No. 1, p. 13601:1-17

Title: On the critical behaviour of two-dimensional liquid crystals
  A.l. Fariñas-Sánchez (Departamento de Física, Universidad Simón Bolívar, Apartado Postal 89000, Caracas 1080-A, Venezuela) ,
  R. Botet (Laboratoire de Physique des Solides, Bât. 510, CNRS UMR8502 - Université Paris-Sud, F-91405 Orsay, France) ,
  B. Berche (Statistical Physics Group, P2M Dpt, Institut Jean Lamour, Nancy Université, UMR CNRS 7198, BP 70239, F-54506 Vandœuvre les Nancy Cedex, France) ,
  R. Paredes (Instituto Venezolano de Investigaciones Científicas, Centro de Física, Laboratorio de Fisica Estadística Apdo. 20632, Caracas 1020A, Venezuela)

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.

Key words: topological transitions, critical phenomena, liquid crystals, Lebwohl-Lasher model
PACS: 64.70.M-, 64.60.Bd,, 05.70.Jk, 05.50.+q, 68.35.Rh

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