# Atomic limit and inversion-symmetry indicators for topological superconductors

## TL;DR

This paper develops a new symmetry indicator based on atomic limits for inversion-symmetric topological superconductors, enabling refined classification and bulk-boundary analysis using only normal-state data.

## Contribution

It introduces a $ ext{Z}_{2^d}$-valued symmetry indicator for superconductors, extending symmetry indicator methods to superconducting systems and higher-order phases.

## Key findings

- Defines atomic limit for Bogoliubov-de-Gennes Hamiltonians
- Derives a $ ext{Z}_{2^d}$ symmetry indicator for superconductors
- Connects the indicator to bulk-boundary correspondence and higher-order phases

## Abstract

Symmetry indicators have proven to be extremely helpful in identifying topologically non-trivial crystalline insulators using symmetry-group representations of their Bloch states. An extension of this approach to superconducting systems requires defining an appropriate atomic limit for Bogoliubov-de-Gennes Hamiltonians. Here, we introduce such a notion of atomic limit and derive a $\mathbb{Z}_{2^d}$-valued symmetry indicator for inversion-symmetric superconductors of $d$ dimensions. This indicator allows for a refined topological classification including higher-order phases for systems in the superconducting symmetry classes D and DIII. We further elucidate the bulk-boundary correspondence of these phases using Dirac surface theories. Requiring only the normal-state band structure and the superconducting order-parameter symmetry as an input, this indicator is well suited for a search of topological superconductors using first-principles calculations.