Euler band topology in superfluids and superconductors

Real band topology often appears in systems with space-time inversion symmetry and is characterized by invariants such as the Euler and second Stiefel-Whitney classes. Here, we examine the generic band topology of Bogoliubov de-Gennes (BdG) Hamiltonians with $C_{2z}T$ symmetry, where $C_{2z}$ and $T$ are twofold rotation about the $z$ axis and time-reversal symmetries, respectively. We discuss the Euler band topology of superfluids and superconductors in the DIII and CI Altland-Zirnbauer symmetry classes, where the Euler class serves as an integer-valued topological invariant of the $4\times4$ BdG Hamiltonian. Using expressions for the Euler class under $n$-fold rotational symmetry, we derive formulas relating the Euler class to previously known topological invariants of class DIII and CI systems. We demonstrate that three-dimensional class DIII topological phases with an odd winding number, including the B phase of superfluid Helium 3, are topological superconductors or superfluids with a nontrivial Euler class. We refer to these as Euler superconductors or superfluids. Specifically, the $^3$He-B superfluid in a magnetic field is identified as an Euler superfluid. Three-dimensional class CI topological phases with twice an odd winding number are also Euler superconductors or superfluids. When spatial inversion symmetry is present, class CI superconductors with a nontrivial Euler class exhibit superconducting nodal lines with a linking structure. This phenomenon is demonstrated using a model of a three-dimensional $s_\pm$-wave superconductor. These findings provide a unified framework for exploring Euler band topology in superfluids and superconductors, connecting various phenomena associated with $T$-breaking perturbations, including Majorana Ising susceptibility and higher-order topology.