Symmetry indicators for topological superconductors

Abstract

The systematic diagnosis of band topology enabled by the method of "symmetry indicators" underlies the recent advances in the search for new materials realizing topological crystalline insulators. Such an efficient method has been missing for superconductors because the quasi-particle spectrum in the superconducting phase is not usually available. In this work, we establish symmetry indicators for weak-coupling superconductors that detect nontrivial topology based on the representations of the metallic band structure in the normal phase assuming a symmetry property of the gap function. We demonstrate the applications of our formulae using examples of tight-binding models and density-functional-theory band structures of realistic materials.

Symmetry-based indicators for topological Bogoliubov–de Gennes Hamiltonians

“…Furthermore, we show that there exists a consistent definition of an "atomic-limit superconductor" as an "array" of zero-dimensional superconductors [39]. Our definition is consistent with the definition in the published version of Skurativska et al [65]. 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.…”

“…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]…”

“…Our construction of symmetry-based indicators, which is based on the full classification of "zerodimensional" Hamiltonians at high-symmetry momenta [64], shows that this statement needs to be corrected to the extent that we successfully put to use a topological band label based on the Pfaffian of a BdG-type Hamiltonian H (k s )a t high-symmetry momenta k s , which has no counterpart in the normal state. A different approach to the problem of symmetry-based indicators in superconductors was taken in a later paper by these authors, together with Yanase [65], as well as by Skurativska, Neupert, and Fischer [66], who consider the "weak-pairing limit" (superconducting order parameter much smaller than energy scales typical for the normal-state band structure) and derive a classification of superconducting phases that is based on the topological classification of the normal state and the symmetry of the superconducting order parameter. An approach that uses the normal-state Hamiltonian as its sole input has the practical advantage that the symmetry-based indicators can be calculated from the vast body of band-structure knowledge available for normal phases.…”

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.…”

Magnetic Response of Majorana Kramers Pairs Protected by \(\mathbb{Z}_{2}\) Invariants

“…Recent studies have reported results useful for determining the topological invariants of 3D TCSCs from the symmetry indicators in the normal state. [44][45][46][47][48] In the present paper, we extend these studies by systematically elucidating the magnetic response of Majorana Kramers pairs with one-dimensional invariants ν[U] ∈ Z 2 protected by an order-2 symmorphic or nonsymmorphic symmetry U. In the symmorphic case, an energy gap occurs in the Majorana Kramers pair when an applied magnetic field breaks the symmetry U.…”