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Euler band topology in superfluids and superconductors

Shingo Kobayashi, Manabu Sato, Akira Furusaki

Abstract

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.

Euler band topology in superfluids and superconductors

Abstract

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 symmetry, where and are twofold rotation about the 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 BdG Hamiltonian. Using expressions for the Euler class under -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 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 -wave superconductor. These findings provide a unified framework for exploring Euler band topology in superfluids and superconductors, connecting various phenomena associated with -breaking perturbations, including Majorana Ising susceptibility and higher-order topology.

Paper Structure

This paper contains 29 equations, 3 figures, 1 table.

Figures (3)

  • Figure 1: (Color online) (a) Schematic illustration of gapless surface states (red) in topological phases with a nonzero 3D winding number [Eq. (\ref{['eq:w3d']})]. (b) and (c) Gapless surface states under application of a $T$-breaking but $C_{2z}T$-preserving perturbation [e.g., magnetic field in the [110] direction] to the Euler superconductor (superfluid) shown in (a), as implied by the relation between the 3D winding number and Euler class in Eqs. (\ref{['eq:Euler-DIII_3D']}) and (\ref{['eq:Euler-CI_3D']}). We infer the existence of hinge states from the relation between the second SW class and the Chern-Simons invariant Ahn2019, where $\bar{e}_{2z}$ is an odd integer.
  • Figure 2: (Color online) (a) TRIMs in the 3D BZ of orthorhombic systems. (b) Deformation of the $k_z=0,\pi$ planes to a sphere.
  • Figure 3: (Color online) The inner Fermi surface (gray), superconducting gap nodes (red), and the lines of band degeneracies of the normal-state Hamiltonian (blue) for the BdG Hamiltonian in Eqs. (\ref{['eq:normal_CI']}) and (\ref{['eq:gap_CI']}) are shown. We change the parameter from (a) $v_4=0$ to (b) $v_4 = 0.0065$ to demonstrate the splitting of the nodal loops. The other parameters are fixed as $(t,\mu,v_1,v_2,v_3,\Delta_0,\Delta_1)=(1,2,0.3,0.3,0.3,0.2,-0.9998)$.