Superconductivity and Abelian Chiral Anomalies

TL;DR

This paper explores the topological invariants of superconductors using the Bogoliubov-de Gennes framework, linking them to Abelian chiral anomalies and analyzing their implications for nodal gap structures.

Contribution

It introduces a topological classification of superconductivity based on Chern numbers and Abelian chiral anomalies, extending to nonunitary states via q-helicity analysis.

Findings

Chern numbers characterize topological invariants in superconductors.

Topological invariants relate to Dirac monopoles and doubled surfaces.

Nodal structures have topological origins explained by the invariants.

Abstract

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.