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

Systems with global symmetry group $O\left(2\right)$ experience topological transition in the two-dimensional space. But there is controversy about such a transition for systems with global symmetry group $O\left(3\right)$. As an example of the latter case, 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 evidence 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.

Figure 1

(Color online) $⟨m⟩∕\sigma$ is plotted vs $L^{-1}$ for the $XY$ model at $T=0.6$ (top) and for the LL model at $T=0.4$ (bottom). A linear fit is obtained for the $XY$ model $(⟨m⟩∕\sigma =31.1-16.4∕L)$. A power-law fit shows that no saturation is observed for the LL model. Both fits are shown as bold lines. The circles are the data from [6]. The number of independent realizations used to obtain the bold squares is almost two orders of magnitude larger than in [6]. The hyperscaling relation (1) is not satisfied in the thermodynamic limit by the LL model at $T=0.4$.

Figure 2

(Color online) Order parameter PDF for the $XY$ model at $T=0.6$ (top) and the LL model at $T=0.4$ (bottom) in the first-scaling form. A perfect collapse is observed for the $XY$ model. This is not the case for the LL model. The $L=768$ data are from [6], with $9\times {10}^4$ independent realizations. It is clear from this figure that the number of independent realizations used in [6] was large enough to realize the first-scaling law. No self-similarity is observed for the LL model at $T=0.4$.

Figure 3

The Binder cumulant vs the temperature for the $XY$ model $T=0.6$ (top) and the LL model (bottom). The number of independent realizations is ${10}^5$ for each $T$ and $L$. No crossing is observed for the LL model. Therefore, no evidence of any phase transition is observed for the LL model. Values of the Binder cumulant for $T=0.4$ are shown in the inset as a function of the system size. We used $6\times {10}^6$ independent realizations to obtain each point, so that the error bars are much smaller than the symbol size.

Figure 4

The 2D Binder cumulant for the Heisenberg model exhibits the same type of behavior as for the LL model (see Fig. 3).