Dirac Monopole and Spin Hall Conductance for Anisotropic Superconductivities

Abstract

Concept of the topological order is useful to characterize anisotropic superconductivities. The spin Hall conductance distinguishes superconductivities with the same symmetry. The Chern number for the spin Hall conductance is given by the covering degree of a closed surface around the Dirac monopole. It gives a clear illustration for the on-critical condition for the $d_{x^2-y^2}$ supreconductivitiy with non zero chemical potential. Non trivial topological orders on the triangular lattice are also presented and demonstrated by the topological objects.