The Berry curvature of the Bogoliubov quasiparticle Bloch states in the unconventional superconductor Sr$_2$RuO$_4$

TL;DR

This paper extends Berry curvature concepts to superconductors, specifically analyzing the Bogoliubov quasiparticle states in Sr$_2$RuO$_4$, to understand optical Kerr effects and establish sum rules linking conductivity and Berry curvature.

Contribution

It introduces a framework for Berry curvature in superconductors and applies it to Sr$_2$RuO$_4$, connecting optical properties with topological aspects.

Findings

Berry curvature defined for superconducting Bogoliubov states

Sum rule linking optical conductivity and Berry curvature

Application to Sr$_2$RuO$_4$ revealing topological effects

Abstract

We will extend the concept of electron band Berry curvatures to superconducting materials. We show that this can be defined for the Bogoliubov-de Gennes equation describing the superconducting state in a periodic crystal. In addition, the concept is exploited to understand the driving mechanism for the optical Kerr effect in time reversal symmetry breaking superconductors. Finally, we establish a sum rule analogue to the normal state Hall sum rule making quantitative contact between the imaginary part of the optical conductivity and the Berry curvature. The general theory will be applied and tested against the drosophila of the p-wave paired materials Sr$_2$RuO$_4$.