Euler--Chern Correspondence via Topological Superconductivity

The Fermi sea topology is characterized by the Euler characteristics ${\chi }_F$. In this paper, we examine how ${\chi }_F$ of the metallic state is inhereted 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 Euler characteristic and 3D winding number of time-reversal-invariant $p$-wave topological superconductors in three dimensions.