Two phase transitions in the two-dimensional nematic 3-vector model with no quasi long-range order: Monte Carlo simulation of the density of states

The presence of stable topological defects in a two-dimensional (textit{d} = 2) liquid crystal model allowing molecular reorientations in three dimensions (textit{n} = 3) was largely believed to induce defect-mediated Berzenskii-Kosterlitz-Thouless (BKT) 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. We argue that in the (textit{d} = 2, textit{n} = 3) system with a biquadratic Hamiltonian, the presence of additional molecular degree 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.