Symmetry breaking of nematic umbilical defects through an amplitude equation

Abstract

The existence, stability properties, and bifurcation diagram of the nematic umbilical defects is studied. Close to the Fr´eedericksz transition of nematic liquid crystalswith negative anisotropic dielectric constant and homeotropic anchoring, an anisotropic Ginzburg-Landau equation for the amplitude of the tilt of the director away from the vertical axis is derived by taking the three-dimensional (3D) to 2Dlimit of the Frank-Oseen model. The anisotropic Ginzburg-Landau equation allows us to reveal themechanism of symmetry breaking of nematic umbilical defects. The positive defect is fully characterized as a function of the anisotropy, while the negative defect is characterized perturbatively. Numerical simulations show quite good agreement with the analytical results.