The Fermi sea topology is characterized by the Euler characteristic ${\chi }_F$. In this paper, we examine how ${\chi }_F$ of the metallic state is inherited by the topological invariant of the superconducting state. We establish a correspondence between the Euler characteristic and the Chern number $C$ of $p$-wave topological superconductors without time-reversal symmetry in two dimensions. By rewriting the pairing potential ${\mathrm{\Delta }}_{\mathbf{k}}={\mathrm{\Delta }}_1-i{\mathrm{\Delta }}_2$ as a vector field $\mathbf{u}=({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2)$, we found that ${\chi }_F=C$ when $\mathbf{u}$ and fermion velocity $\mathbf{v}$ can be smoothly deformed to be parallel or antiparallel on each Fermi surface. We also discuss a similar correspondence between the Euler characteristic and three-dimensional winding number of time-reversal-invariant $p$-wave topological superconductors in three dimensions.

• Received 30 May 2023
• Accepted 22 July 2023

DOI:https://doi.org/10.1103/PhysRevResearch.5.033073

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society