Characterizations of topological superconductors: Chern numbers, edge states and Majorana zero modes

TL;DR

This paper investigates the relationship between Chern numbers, edge states, and Majorana zero modes in topological superconductors, revealing mismatches and clarifying their distinct topological features through a checkerboard-lattice model.

Contribution

It demonstrates the non-equivalence of topological invariants and edge states in a 2D superconductor model, highlighting the need for comprehensive characterization methods.

Findings

Multiple topological phases with Chern numbers up to 4 identified.

Mismatch between Chern numbers, edge states, and Majorana modes demonstrated.

Edge states inherited from Chern insulators are not always protected.

Abstract

The topological properties in topological superconductors are usually characterized by the bulk Chern numbers, edge-state spectra, and Majorana zero modes. Whether they are equivalent or inequivalent is not well understood. Here, we investigate this issue with focus on a checkerboard-lattice model combining the Chern insulator and chiral $p$-wave superconductivity. Multiple topologically superconducting phases with Chern numbers up to $\mathcal{N}=4$ are produced. We explicitly demonstrate the mismatch between the Chern numbers, edge states and Majorana zero modes in this two-dimensional topological-superconductor model. The intrinsic reason is that some edge states in the superconducting phases inherited from the Chern-insulator phase are not protected by the particle-hole symmetry. We further check the mismatches in vortex states. Our results therefore clarify these different but complementary topological features and suggest that further considerations are required to characterize various topological superconductors.