Role of Topology and Symmetry for the Edge Currents of a 2D Superconductor

TL;DR

This paper investigates how symmetry and topology influence edge currents in 2D topological superconductors, revealing that symmetry plays a crucial role in non-quantized edge currents regardless of topological state.

Contribution

It systematically analyzes the impact of symmetry reductions on edge properties in 2D superconductors using both Ginzburg-Landau and microscopic models.

Findings

Edge (spin) currents can exist independently of bulk topology.

Topological edge states evolve continuously across phase transitions.

Symmetry, rather than topology, governs non-quantized edge currents.

Abstract

The bulk-boundary correspondence guarantees topologically protected edge states in a two-dimensional topological superconductor. Unlike in topological insulators, these edge states are, however, not connected to a quantized (spin) current as the electron number is not conserved in a Bogolyubov-de Gennes Hamiltonian. Still, edge currents are in general present. Here, we use the two-dimensional Rashba system as an example to systematically analyze the effect symmetry reductions have on the order-parameter mixing and the edge properties in a superconductor of Altland-Zirnbauer class DIII (time-reversal-symmetry preserving) and D (time-reversal-symmetry breaking). In particular, we employ both Ginzburg-Landau and microscopic modeling to analyze the bulk superconducting properties and edge currents appearing in a strip geometry. We find edge (spin) currents independent of bulk topology and associated topological edge states which evolve continuously even when going through a phase transition into a topological state. Our findings emphasize the importance of symmetry over topology for the understanding of the non-quantized edge currents.