# Topologically stable gapless phases of time-reversal invariant superconductors

## TL;DR

This paper demonstrates that time-reversal invariant superconductors in two and three dimensions can host topologically stable Fermi points and lines, with implications for surface states and phase diagrams.

## Contribution

It establishes the existence of stable Fermi points and lines in such superconductors and proves a fermion doubling theorem analog, linking bulk invariants to surface states.

## Key findings

- Fermi points in 2D are multiples of four
- Fermi lines in 3D come in pairs
- Topologically stable Fermi lines lead to gapless regions

## Abstract

We show that time-reversal invariant superconductors in d=2 (d=3) dimensions can support topologically stable Fermi points (lines), characterized by an integer topological charge. Combining this with the momentum space symmetries present, we prove analogs of the fermion doubling theorem: for d=2 lattice models admitting a spin X electron-hole structure, the number of Fermi points is a multiple of four, while for d=3, Fermi lines come in pairs. We show two implications of our findings for topological superconductors in d=3: first, we relate the bulk topological invariant to a topological number for the surface Fermi points in the form of an index theorem. Second, we show that the existence of topologically stable Fermi lines results in extended gapless regions in a generic topological superconductor phase diagram.