Topology invisible to eigenvalues in obstructed atomic insulators

TL;DR

This paper investigates how symmetry eigenvalues alone may fail to reveal the topological nature of certain atomic insulators, introducing the concept of Berry obstructed atomic limits that are distinguishable by topological invariants despite sharing the same irreducible representations.

Contribution

It generalizes the concept of topological obstructions beyond eigenvalues, identifying cases where phases with identical symmetry irreps differ by Berry phases, and provides exhaustive classification of irrep-equivalent band representations.

Findings

Identifies examples of irrep-equivalent phases distinguished by Berry phases.

Provides exhaustive lists of irrep-equivalent elementary band representations.

Proposes conditions under which irrep-equivalent bands can be topologically distinguished.

Abstract

We consider the extent to which symmetry eigenvalues reveal the topological character of bands. Specifically, we compare distinct atomic limit phases (band representations) that share the same irreducible representations (irreps) at all points in the Brillouin zone and, therefore, appear equivalent in a classification based on eigenvalues. We derive examples where such "irrep-equivalent" phases can be distinguished by a quantized Berry phase or generalization thereof. These examples constitute a generalization of the Su-Schrieffer-Heeger chain: neither phase is topological, in the sense that localized Wannier functions exist, yet there is a topological obstruction between them. We refer to two phases as "Berry obstructed atomic limits" if they have the same irreps, but differ by Berry phases. This is a distinct notion from eigenvalue obstructed atomic limits, which differ in their symmetry irreps at some point in the Brillouin zone. We compute exhaustive lists of elementary band representations that are irrep-equivalent, in all space groups, with and without time-reversal symmetry and spin-orbit coupling, and use group theory to derive a set of necessary conditions for irrep-equivalence. Finally, we conjecture, and in some cases prove, that irrep-equivalent elementary band representations that are not equivalent can be distinguished by a topological invariant.