Topological Defects and Defects-free states in toroidal nematics

We investigated the nematic ordering on a torus by means of analytic method and the method of simulated annealing, the Frank free energy, both in the standard form and covariant form, were used in the study. The defect free state was found to be the ground state in both cases. However, in the case of the standard model, there are two kinds of defective free ordering and a transition between the two occurs at a critical value of radius ratio $k=\frac{r}{R}$. The first one is $\theta=0$ in the small $k$ regime and the second one is a variable $\theta$ with position of the torus. In the case of the covariant model the ground state is confirmed to be the infinitely degenerate of $\theta$ equals to a random constant. The states with defects are the excited states, where the pairs of defects excited and, duo to the barrier between positive and negative defects, have pretty long life. The behavior of the defect state basically the same for both of the two models.