Defect-free states and disclinations in toroidal nematics

We investigate the structure of the nematic director field on a toroidal surface, on the basis of two different free-energy models. The two models treat the variation of the nematic field differently; one in full-derivative form and the other in covariant-derivative form. Through solving the Euler-Lagrange equation used to minimize the energy model and conducting a simulated annealing Monte Carlo simulation, we confirm that both models produce a trivial solution as a defect-free state. In the first model, however, there exists a second-order phase transition, beyond which the energy bifurcates into a non-trivial solution as the ground state, as the torus ring radius grows. Using the simulated annealing technique on both models, we also trap the system in excited, metastable states that display nematic-field disclinations.