Refined symmetry indicators for topological superconductors in all space groups

Abstract

Topological superconductors are exotic phases of matter featuring robust surface states that could be leveraged for topological quantum computation. A useful guiding principle for the search of topological superconductors is to relate the topological invariants with the behavior of the pairing order parameter on the normal-state Fermi surfaces. The existing formulas, however, become inadequate for the prediction of the recently proposed classes of topological crystalline superconductors. In this work, we advance the theory of symmetry indicators for topological (crystalline) superconductors to cover all space groups. Our main result is the exhaustive computation of the indicator groups for superconductors under a variety of symmetry settings. We further illustrate the power of this approach by analyzing fourfold symmetric superconductors with or without inversion symmetry and show that the indicators can diagnose topological superconductors with surface states of different dimensionalities or dictate gaplessness in the bulk excitation spectrum.

Symmetry-based indicators for topological Bogoliubov–de Gennes Hamiltonians

“…The weak-pairing limit (49) of the symmetry-based indicator z 1 agrees with Refs. [66][67][68]. The symmetry-based indicator z 1 for the full BdG Hamiltonian agrees with the forms defined in Refs.…”

“…In the weak-pairing limit, the symmetrybased indicators derived here can be expressed in terms of the normal part of the Hamiltonian only, so that in that limit our approach offers the same computational advantages as the approaches that rely on the weak-pairing limit at the outset. Very recently, a paper by Ono, Po, and Watanabe appeared [67], which bases its symmetry indicators on the full BdG Hamiltonian, be it without the Pfaffian band labels, and has a definition of an atomic limit that is consistent with ours. A recent paper by Shiozaki also reports the construction of symmetry-based indicators for Hamiltonians of BdG type and which has results very similar to ours [68].…”

“…In the weak-pairing limit, superconducting order parameter much smaller than energy scales typical for the normalstate band structure, the band labels N k s α and p k s α of the BdG Hamiltonian H (k s ) at the high-symmetry momentum k s can be expressed in terms of zero-dimensional topological invariants of the normal-state Hamiltonians h(k s ) and h(−k s ). For N k s α one has [65,67,68]…”

“…The symmetry-based indicator z 1 for the full BdG Hamiltonian agrees with the forms defined in Refs. [67,68].…”

Higher‐Order Topological Band Structures

“…[ 34,35,64 ] Symmetry‐based indicators were initially constructed for insulators, and have been recently extended to superconductors. [ 57,65–70 ] Although symmetry‐based indicators in general do not provide a full classification, their construction and subsequent application resulted in the discovery of many new topological insulator materials [ 71,72 ] . It remains to be seen if the same will be the case for topological superconductors.…”