Hall conductivity as bulk signature of topological transitions in superconductors

TL;DR

This paper demonstrates that the Hall conductivity's derivatives reveal topological phase transitions in superconductors, providing a bulk signature where charge quantization fails due to non-conservation.

Contribution

It introduces a method to detect topological transitions in superconductors through the behavior of Hall conductivity derivatives, extending understanding beyond charge quantization limitations.

Findings

Hall conductivity evolves continuously across topological phases.

Derivatives of Hall conductivity exhibit sharp features at transitions.

Analysis includes p-wave superconductors with spin-orbit coupling and Zeeman splitting.

Abstract

Topological superconductors may undergo transitions between phases with different topological numbers which, like the case of topological insulators, are related to the presence of gapless (Majorana) edge states. In $\mathbb{Z}$ topological insulators the charge Hall conductivity is quantized, being proportional to the number of gapless states running at the edge. In a superconductor, however, charge is not conserved and, therefore, $\sigma_{xy}$ is not quantized, even in the case of a $\mathbb{Z}$ topological superconductor. Here it is shown that while the $\sigma_{xy}$ evolves continuously between different topological phases of a $\mathbb{Z}$ topological superconductor, its derivatives display sharp features signaling the topological transitions. We consider in detail the case of a triplet superconductor with p-wave symmetry in the presence of Rashba spin-orbit (SO) coupling and externally applied Zeeman spin splitting. Generalization to the cases where the pairing vector is not aligned with that of the SO coupling is given. We generalize also to the cases where the normal system is already topologically non-trivial.