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 (R) described by the fundamental group Pi(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 Pi(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.