Topological transitions in two-dimensional lattice models of liquid crystals

The phase diagram of hard-core nematogenic models in three-dimensional space can be studied by means of Onsager’s theory, and, on the other hand, the critical properties of continuous interaction potentials can be investigated using the molecular field approach pioneered by Maier and Saupe. Comparison between these treatments shows a certain formal similarity, reflecting their common variational root; on this basis, hard-core potential models can be mapped onto continuous ones, via their excluded volume. Some years ago, this line of reasoning had been applied to hard spherocylinders, hence the continuous potential $G\left(\tau \right)=a+b\sqrt{1-{\tau }^2},$ $b>0$ had been used to define a mesogenic model on a three-dimensional lattice [S. Romano, Int. J. Mod. Phys. B 9, 85 (1995)]; in the formula, $\tau$ denotes the scalar product between the two unit vectors defining particle orientations. Here we went on by addressing the same interaction potential on a two-dimensional lattice. Our analysis based on extensive Monte Carlo simulations found evidence of a topological transition, and the critical behavior in its vicinity was studied in detail. Results obtained for the present model were compared with those already obtained in the literature for interaction potentials defined by Legendre polynomials of second and fourth orders in the scalar product $\tau$.

Figure 1

Simulation results for the scaled potential energy $U_N=U/U_0$, as well as short-range order parameters ${\sigma }_2$ and ${\sigma }_4$ obtained with the largest sample size. Unless otherwise stated or shown, here and in the following figures, statistical errors fall within symbol sizes.

Figure 2

Simulation results for the configurational specific heat, obtained with different sample sizes. The statistical errors, not shown here, range between 1 and 5 %.

Figure 3

Simulation results for the second-rank order parameter ${\overline{P}}_2$ obtained with different sample sizes.

Figure 4

Simulation results for the fourth-rank order parameter ${\overline{P}}_4$ obtained with different sample sizes

Figure 5

Simulation results for the logarithm of the nematic susceptibility $\chi$, obtained with different sample sizes; the statistical errors on $\chi$, not shown, range up to 3%. The solid curve is obtained by fitting to Eq. (10) with data corresponding to $L=160$.

Figure 6

Simulation results for the logarithm of the nematic susceptibility $\chi$ for the $-P_4$ model; the statistical errors on $\chi$, not shown, range up to 3%.