Pfaffian invariant identifies magnetic obstructed atomic insulators

TL;DR

This paper introduces a new $ ext{Z}_4$ topological invariant for 2D insulators with $C_4 ext{T}$ symmetry, enabling classification of atomic and obstructed phases beyond traditional symmetry indicators.

Contribution

The authors derive a novel $ ext{Z}_4$ invariant that extends classification methods by incorporating Wilson line spectra and Pfaffian gauge fixing, distinguishing four $C_4 ext{T}$-protected atomic insulators.

Findings

The invariant classifies four distinct $C_4 ext{T}$-protected atomic insulators.

It reveals phase transitions involving $ ext{Z}_4$ invariant changes of 1 and 2.

The invariant captures obstructed phases overlooked by symmetry indicators.

Abstract

We derive a $\mathbb{Z}_4$ topological invariant that extends beyond symmetry eigenvalues and Wilson loops and classifies two-dimensional insulators with a $C_4 \mathcal{T}$ 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 $C_4 \mathcal{T}$-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 $C_4 \mathcal{T}$-symmetric model that interpolates between them. The phase diagram shows that $C_4 \mathcal{T}$ insulators allow $\pm 1$ and $2$ changes of the invariant, where the latter is overlooked by symmetry indicators.