Order Parameter and Edge Currents of a 2D Topological Superconductor under Consecutive Symmetry Breakings*

Two-dimensional topological superconductors host topologically protected edge states guaranteed by the bulk-boundary correspondence. However, the associated edge currents are not quantized as the superconducting state breaks the electron’s U(1) symmetry. Indeed, here we show how the spin and charge edge currents can change continuously even when going through a topological phase transition.
Specifically, we investigate for a two-dimensional superconductor with Rashba spin-orbit coupling (broken mirror symmetry) and an out-of-plane Zeeman field (broken time-reversal symmetry) how the superconducting order parameter adapts under these consecutive symmetry reductions. We then examine the emerging edge currents in a strip geometry in the different topological phases of this system, namely the trivial, the helical (class DIII), and the chiral (class D) phase.
Using both a Ginzburg-Landau and a microscopic approach, we find edge currents irrespective of the topology of the system and the associated edge states. Our study underscores how edge currents in topological superconductors derive mostly from the symmetry of the system.