Monte Carlo investigation of critical properties of the Landau point of a biaxial liquid-crystal system

Extensive Monte Carlo simulations are performed to investigate the critical properties of a special singular point usually known as the Landau point. The singular behavior is studied in the case when the order parameter is a tensor of rank 2. Such an order parameter is associated with a nematic-liquid-crystal phase. A three-dimensional lattice dispersion model that exhibits a direct biaxial nematic-to-isotropic phase transition at the Landau point is thus chosen for the present study. Finite-size scaling and cumulant methods are used to obtain precise values of the critical exponent $\nu =0.713\left(4\right)$, the ratio $\gamma /\nu =1.85\left(1\right)$, and the fourth-order critical Binder cumulant $U^{*}=0.6360\left(1\right)$. Estimated values of the exponents are in good agreement with renormalization-group predictions.

Figure 1

Schematic diagram of a symmetric V-shaped molecule. The three molecular principal (symmetry) axes are $x, y$, and $z$ with the $x$ axis normal to the plane of the page.

Figure 2

The susceptibility of biaxial order parameter $\chi (T^{*},L)$ generated by multiple histogram reweighting plotted against dimensionless temperature, $T^{*}$, for five lattice sizes.

Figure 3

The Binder cumulant $U_4(T^{*},L)$ generated by the multiple histogram reweighting plotted against dimensionless temperature, $T^{*}$, for five lattice sizes.

Figure 4

The derivative of the Binder cumulant $d{U}_4/dK$ is plotted as a function of the dimensionless temperature, $T^{*}$, for five lattice sizes.

Figure 5

Plot of $U_4(T^{*},L)$ vs $T^{*}$ for several lattice sizes. For increasing lattice size, a shift of the intersection toward higher values of $U_4$ becomes visible as a larger scale is chosen. The intersections with the cumulant for $L=28$ are shown by triangles.

Figure 6

Extrapolation of the temperature values of the crossings of the $L=28$ curve of $U_4$ with all other curves against the inverse logarithm of $b=L^{\prime }/L$. The critical temperature $T_C=1.1298\pm 0.0001$.

Figure 7

Extrapolation of the fourth-order Binder cumulant values of the crossings of the $L=28$ curve of $U_4$ with all other curves against the inverse logarithm of $b=L^{\prime }/L$. The critical Binder cumulant $U^{*}=0.6360\pm 0.0001$.

Figure 8

Variation of $ln{\chi }_{\text{max}}$ with $lnL$ (slope, $\gamma /\nu =1.85\pm 0.01$).

Figure 9

$\text{ln}{(d{U}_4/dK)}_{\text{max}}$ vs $ln\left(L\right)$. The exponent $\nu =0.713\pm 0.004$.