Pfaffian invariant identifies magnetic obstructed atomic insulators

Abstract

We derive a \mathbb{Z}_4ℤ4 topological invariant that extends beyond symmetry eigenvalues and Wilson loops and classifies two-dimensional insulators with a C_4 \mathcal{T}C4𝒯 symmetry. To formulate this invariant, we consider an irreducible Brillouin zone and constrain the spectrum of the open Wilson lines that compose its boundary. We fix the gauge ambiguity of the Wilson lines by using the Pfaffian at high symmetry momenta. As a result, we distinguish the four C_4 \mathcal{T}C4𝒯-protected atomic insulators, each of which is adiabatically connected to a different atomic limit. We establish the correspondence between the invariant and the obstructed phases by constructing both the atomic limit Hamiltonians and a C_4 \mathcal{T}C4𝒯-symmetric model that interpolates between them. The phase diagram shows that C_4 \mathcal{T}C4𝒯 insulators allow \pm 1±1 and 22 changes of the invariant, where the latter is overlooked by symmetry indicators.