No quasi-long-range order in the two-dimensional liquid crystal

Systems with global symmetry group O(2) experience topological transition in the 2-dimensional space. But there is controversy about such a transition for systems with global symmetry group O(3). In this paper, we study the Lebwohl-Lasher model for the two-dimensional liquid crystal, using three different methods independent of the proper values of possible critical exponents. Namely, we analyze the at-equilibrium order parameter distribution function with: 1) the hyperscaling relation; 2) the first scaling collapse for the probability distribution function;and 3) the Binder's cumulant. We give strong evidences for definite lack of a line of critical points at low temperatures in the Lebwohl-Lasher model, contrary to conclusions of a number of previous numerical studies.