Topological invariants for the Fermi surface of a time-reversal-invariant superconductor

TLDR

A time‑reversal‑invariant topological superconductor has a full bulk pairing gap and topologically protected gapless surface or edge states. We show that in the weak‑pairing limit the topological quantum number of a TRI superconductor is fully determined by Fermi‑surface properties. This determination is independent of the electronic structure away from the Fermi surface. In 3D the integer invariant equals the sign of the pairing order parameter times the first Chern number of the Berry‑phase gauge field on the Fermi surfaces, whereas in 2D and 1D the Z₂ invariant is given simply by the sign of the pairing on the Fermi surfaces, and we provide explicit expressions for the Z₂ invariant in 1D and 2D.

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.