Phase-field simulation of two-dimensional topological charges in nematic liquid crystals

The concept of topological quantum number, or topological charge, has been used extensively to describe topological defects or solitons. Nematic liquid crystals contain both integer and half-integer topological defects, making them useful models for testing the rules that govern topological defects. Here, we investigated topological defects in nematic liquid crystals using the phase-field method. If there are no defects along a loop path, the total charge number is described by an encircled loop integral. We found that the total charge number is conserved, and the conservation of defects number is determined by a boundary during the generation and annihilation of positive-negative topological defects when the loop integral is confined. These rules can be extended to other two-dimensional systems with topological defects.