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We will extend the concept of electron band Berry curvatures tosuperconducting materials. We show that this can be defined for theBogoliubov-de Gennes equation describing the superconducting state in aperiodic crystal. In addition, the concept is exploited to understand thedriving mechanism for the optical Kerr effect in time reversal symmetrybreaking superconductors. Finally, we establish a sum rule analogue to thenormal state Hall sum rule making quantitative contact between the imaginarypart of the optical conductivity and the Berry curvature. The general theorywill be applied and tested against the drosophila of the p-wave pairedmaterials Sr$_2$RuO$_4$.

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