## Pfaffian invariant identifies magnetic obstructed atomic insulators

Isidora Araya Day, Anastasiia Varentcova, Daniel Varjas, Anton R. Akhmerov

SciPost Phys. 15, 114 (2023) · published 25 September 2023

- doi: 10.21468/SciPostPhys.15.3.114
- Submissions/Reports

### 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.

### Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.-
^{1}Isidora Araya Day, -
^{1}Anastasiia Varentcova, -
^{1}^{2}Daniel Varjas, -
^{1}Anton R. Akhmerov

^{1}Technische Universiteit Delft / Delft University of Technology [TU Delft]^{2}Stockholm University [Univ Stockholm]

- Knut och Alice Wallenbergs Stiftelse (Knut and Alice Wallenberg Foundation) (through Organization: Knut och Alice Wallenbergs Stiftelse / Knut and Alice Wallenberg Foundation)
- Nederlandse Organisatie voor Wetenschappelijk Onderzoek / Netherlands Organisation for Scientific Research [NWO]
- Vetenskapsrådet / Swedish Research Council