Fermi Surface Topological Invariants for Time Reversal Invariant Superconductors

TL;DR

This paper establishes that the topological invariants of time reversal invariant superconductors can be determined solely by Fermi surface properties, simplifying the classification of their topological phases.

Contribution

It provides a Fermi surface-based method to calculate topological invariants in TRI superconductors across different dimensions, including explicit formulas for 1D and 2D cases.

Findings

Topological quantum number in 3D depends on pairing sign and Chern number.

In 2D and 1D, the invariant depends only on the pairing sign.

Explicit expressions for Z2 invariants in 1D and 2D are derived.

Abstract

A time reversal invariant (TRI) topological superconductor has a full pairing gap in the bulk and topologically protected gapless states on the surface or at the edge. In this paper, we show that in the weak pairing limit, the topological quantum number of a TRI superconductor can be completely determined by the Fermi surface properties, and is independent of the electronic structure away from the Fermi surface. In three dimensions (3D), the integer topological quantum number in a TRI superconductor is determined by the sign of the pairing order parameter and the first Chern number of the Berry phase gauge field on the Fermi surfaces. In two (2D) and one (1D) dimension, the $Z_2$ topological quantum number of a TRI superconductor is determined simply by the sign of the pairing order parameter on the Fermi surfaces. We also obtain a generic and explicit expression of the $Z_2$ topological invariant in 1D and 2D.