Using the Renormalized Energy from Ginzburg-Landau Theory to Compute Solution Landscapes in the Reduced Landau-de Gennes Theory

Abstract

We study nematic liquid crystal phases confined to bounded, two-dimensional (2D) domains, in the reduced Landau-de Gennes (rLdG) theoretical framework. The rLdG framework is equivalent to the celebrated Ginzburg-Landau (GL) theoretical setting, under certain assumptions. In the asymptotic limit of large domains (or low temperatures), we recall the concept of the finite-dimensional renormalized energy in the GL theory [Bethuel et al., GL Vortices, Prog. Nonlinear Differ. Equ. Appl. 13, Birkhäuser, 1994.] and generalize it to polygons with sharp corners. The renormalized energy is fully defined by the triplet, $(\mathbf{a},\mathbf{c}, \mathbf{d}$), of topological degrees of the interior defects ($\mathbf{d}$) and the winding number contribution of polygon vertices ($\mathbf{c}$) and interior defect locations ($\mathbf{a}$), subject to sum rules dictated by the imposed Dirichlet boundary condition. We use the critical points of the renormalized energy to construct initial conditions for the rLdG solver, recover known energy-minimizing and non-energy minimizing saddle points of the rLdG energy on a disc, regular hexagon and rectangle along with the discovery of previously unreported rLdG solution branches. The renormalized energy approach is computationally substantially more efficient than a direct PDE-solver in the rLdG setting but misses solution branches with line defects and finite-size effects.