Superconductivity and Abelian chiral anomalies

Motivated by the geometric character of spin Hall conductance, the topological invariants of generic superconductivity are discussed based on the Bogoliuvov-de Gennes equation on lattices. They are given by the Chern numbers of degenerate condensate bands for unitary order, which are realizations of Abelian chiral anomalies for non-Abelian connections. The three types of Chern numbers for the $x$, $y$, and $z$ directions are given by covering degrees of some doubled surfaces around the Dirac monopoles. For nonunitary states, several topological invariants are defined by analyzing the so-called $q$ helicity. Topological origins of the nodal structures of superconducting gaps are also discussed.

Figure 1

(Color online) Examples of closed surfaces $R_1\left(T_{xy}^2\right)$ which are cut by the $R_Y-R_Z$ plane. The monopole is at the origin, $t=d_z=1$, $\mu =-5$: (a)$k_z=0$ and (b)$k_z=-2\pi ∕5$.