Translation invariant topological superconductors on lattice

TL;DR

This paper introduces a classification scheme for 2D topological superconductors based on four Z_2 indices, revealing 16 universal classes that persist even with interactions, and extends to 3D with 256 types.

Contribution

It defines four Z_2 topological indices for 2D superconductors with translation symmetry, classifying 16 universal classes and extending to 3D topological superconductors.

Findings

16 classes of 2D topological superconductors identified

Indices characterize states even with interactions

256 types of 3D topological superconductors

Abstract

In this paper we introduce four Z_2 topological indices zeta_k=0,1 at k=(0,0), (0,pi), (pi, 0), (pi, pi) characterizing 16 universal classes of 2D superconducting states that have translation symmetry but may break any other symmetries. The 16 classes of superconducting states are distinguished by their even/odd numbers of fermions on even-by-even, even-by-odd, odd-by-even, and odd-by-odd lattices. As a result, the 16 classes topological superconducting states exist even for interacting systems. For non-interacting systems, we find that zeta_k is the number of electrons on k=(0,0), (0,pi), (pi, 0), or (pi,pi) orbitals (mod 2) in the ground state. For 3D superconducting states with only translation symmetry, there are 256 different types of topological superconductors.