Topological transition in a two-dimensional three-vector model: A non-Boltzmann Monte Carlo study

Two-dimensional three-vector ($d=2,n=3$) lattice model of a liquid crystal (LC) system with order parameter space ($\mathcal{R}$) described by the fundamental group ${\mathrm{\Pi }}_1\left(\mathcal{R}\right)=Z_2$ was recently investigated based on non-Boltzmann Monte Carlo simulations. Its results indicated that the system did not undergo a topological transition condensing to a low temperature critical state as was reported earlier. Instead, a crossover to a nematic phase was observed, induced by the onset of a competing relevant length scale. This mechanism is further probed here by assigning a more restrictive $\mathcal{R}$ symmetry with ${\mathrm{\Pi }}_1\left(\mathcal{R}\right)=\mathbb{Q}$ (the discrete and non-Abelian group of quaternions), thus engaging the three spin degrees in the formation of point topological defects (disclinations). The results reported here indicate that such a choice of symmetry of the Hamiltonian with suitable model parameters leads to a defect-mediated transition to a low-temperature phase with topological order. It is characterized by a line of critical points with quasi-long-range order of its three spin degrees. The associated temperature-dependent power-law exponent decreases progressively and vanishes linearly as temperature tends to zero. The high-temperature disordered phase shows exponential spin correlations and their temperature-dependent lengths exhibit an essential singular divergence as the system approaches the topological transition point. Biaxial LC models have the required $\mathcal{R}$ symmetry owing to their tensor orientational orders and are suggested to serve as prototype examples to exhibit topological transition in ($d=2,n=3$) lattice models.

Figure 1

Temperature variation of specific heat (per site) at lattice sizes $L=60$, 80, 100, 120, and 150. Insets show the temperature variation of (a) size independent energy per site, and (b) topological density at $L=150$.

Figure 2

Temperature variation of orientational order parameters at lattice sizes $L=60$, 80, 100, 120, and 150. Insets show the temperature variation of (a) uniaxial susceptibility, (b) finite-size scaling plot of the peak value of susceptibility ${\chi }_0\left(L\right)$. The dashed arrows are indicative of the variation of the parameter with an increase in system size.

Figure 3

Temperature variation of topological parameter ${\delta }_z$ at lattice sizes $L=60$, 80, 100, 120, and 150. Inset (a) shows inflexion point of ${\delta }_z$ at $L=80$ (dashed vertical line) indicating the unbinding transition temperature $T_{U\delta }$; (b) finite-size scaling plot of $T_{U\delta }\left(L\right)$.

Figure 4

Spatial variation of correlation functions $G\left(r\right)$ at representative temperatures bracketing the transition ($L=150$). $G\left(r\right)$ fits well to exponential decays above $T=0.75$, whereas it exhibits a power-law variation below $T=0.73$. These bounding decays are indicated in the figure as dashed lines. The fit curves both above and below the transition temperature are superimposed on the corresponding data points as solid lines at a few representative temperatures ($T=0.17$, 0.35, 0.53, 0.7, 0.78, and 1.05).

Figure 5

Temperature variation of the exponent $\eta \left(T\right)$ and correlation length $\xi \left(T\right)$ at $L=150$. The curved dashed line (left) is the power-law fit of $\eta \left(T\right)$, and the dashed-dotted line (right) is a divergence fit of $\xi \left(T\right)$ as indicated in the text. The short dashed straight line from the origin is a linear fit of $\eta \left(T\right)$ data at low temperatures. The inset shows the critical contribution to the $C_v$ (short dashes) superposed on the $C_v$ profile (long dashes) depicted over the entire temperature range. The critical contribution vanishes at the transition temperature $T\approx 0.727$.