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Two-dimensional liquid crystal (LC) models of interacting V-shaped bent-core molecules with two rigid rodlike identical segments connected at a fixed angle (θ=112^{∘}) are investigated. The model assigns equal biquadratic tensor coupling among constituents of the interacting neighboring molecules on a square lattice, allowing for reorientations in three dimensions (d=2, n=3). We find evidence of two temperature-driven topological transitions mediated by point disclinations associated with the three ordering directors, condensing the LC medium successively into uniaxial and biaxial phases. With Monte Carlo simulations, temperature dependencies of the system energy, specific heat, orientational order parameters, topological order parameters, and densities of unbound topological defects of the different ordering directors are computed. The high-temperature transition results in topological ordering of disclinations of the primary director, imparting uniaxial symmetry to the phase. The low-temperature transition precipitates simultaneous topological ordering of defects of the remaining directors, resulting in biaxial symmetry. The correlation functions, quantifying spatial variations of the orientational correlations of the molecular axes show exponential decays in the high-temperature (disordered) phase, and power-law decays in the low-temperature (biaxial) phase. Differing temperature dependencies of the topological parameters point to a significant degree of cross coupling among the uniaxial and biaxial tensors of interacting molecules. This simplified Hamiltonian leaves θ as the only free model parameter, and the system traces a θ-dependent trajectory in a plane of the phenomenological parameter space.

Two-dimensional three-vector (d=2,n=3) lattice model of a liquid crystal (LC) system with order parameter space (R) described by the fundamental group Π_{1}(R)=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 R symmetry with Π_{1}(R)=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 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.

The presence of stable topological defects in a two-dimensional (d=2) liquid crystal model allowing molecular reorientations in three dimensions (n=3) was largely believed to induce a defect-mediated Berzenskii-Kosterlitz-Thouless-type transition to a low temperature phase with quasi-long-range order. However, earlier Monte Carlo (MC) simulations could not establish certain essential signatures of the transition, suggesting further investigations. We study this model by computing its equilibrium properties through MC simulations, based on the determination of the density of states of the system. Our results show that, on cooling, the high temperature disordered phase deviates from its initial progression towards the topological transition, crossing over to a new fixed point, condensing into a nematic phase with exponential correlations of its director fluctuations. The thermally induced topological kinetic processes continue, however, limited to the length scales set by the nematic director fluctuations, and lead to a second topological transition at a lower temperature. It is argued that in the (d=2, n=3) system with an attractive biquadratic Hamiltonian, the presence of additional molecular degrees of freedom and local Z_{2} symmetry associated with lattice sites together promote the onset of an additional relevant scaling field at matching length scales in the high temperature region, leading to a crossover.

General quadratic Hamiltonian models, describing the interaction between liquid-crystal molecules (typically with D_{2h} symmetry), take into account couplings between their uniaxial and biaxial tensors. While the attractive contributions arising from interactions between similar tensors of the participating molecules provide for eventual condensation of the respective orders at suitably low temperatures, the role of cross coupling between unlike tensors is not fully appreciated. Our recent study with an advanced Monte Carlo technique (entropic sampling) showed clearly the increasing relevance of this cross term in determining the phase diagram (contravening in some regions of model parameter space), the predictions of mean-field theory, and standard Monte Carlo simulation results. In this context, we investigated the phase diagrams and the nature of the phases therein on two trajectories in the parameter space: one is a line in the interior region of biaxial stability believed to be representative of the real systems, and the second is the extensively investigated parabolic path resulting from the London dispersion approximation. In both cases, we find the destabilizing effect of increased cross-coupling interactions, which invariably result in the formation of local biaxial organizations inhomogeneously distributed. This manifests as a small, but unmistakable, contribution of biaxial order in the uniaxial phase. The free-energy profiles computed in the present study as a function of the two dominant order parameters indicate complex landscapes. On the one hand, these profiles account for the unusual thermal behavior of the biaxial order parameter under significant destabilizing influence from the cross terms. On the other, they also allude to the possibility that in real systems, these complexities might indeed be inhibiting the formation of a low-temperature biaxial order itself-perhaps reflecting the difficulties in their ready realization in the laboratory.