Symmetries of Vortex Lattice Solutions of the Bogoliubov–de Gennes Equation in a Two-Dimensional Square Lattice.

This report describes symmetry properties of tetragonal vortex lattice solutions of the Bogoliubov–de Gennes equation in a two-dimensional square lattice in a uniform magnetic field. The invariance group of a tetragonal vortex lattice solution is expressed in a form of G_{( l)} = ( e + tC_{2 x}) $$\widetilde C_4^l L$$ ( l = 0, 2, ± 1), where tC_{2 x} is a space π rotation around the x-axis accompanied with time reversal, $$\widetilde C_4^l$$ is a kind of fourfold rotation group, and L is the magnetic translational group of the vortex lattice state. We give a new, refined definition of local symmetric order parameters (OPs) ( s-wave, d-wave, and p-wave), which have a well-defined nature such that the OP (e.g., s-wave OP) at the translated site by a lattice vector (of the vortex lattice) from a site ( m, n) is expressed by the OP (e.g., s-wave) at the site ( m, n) times a phase factor. Winding numbers around the origin of s-wave and d-wave OPs are obtained for four types of solutions G_{( l)} ( l = 0, 2, ± 1). It is shown that all energy bands of quasiparticles of a vortex lattice state are doubly degenerate. [ABSTRACT FROM AUTHOR]