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We derive a $\mathbb{Z}_4$ topological invariant that extends beyond symmetryeigenvalues and Wilson loops and classifies two-dimensional insulators with a$C_4 \mathcal{T}$ symmetry. To formulate this invariant, we consider anirreducible Brillouin zone and constrain the spectrum of the open Wilson linesthat compose its boundary. We fix the gauge ambiguity of the Wilson lines byusing the Pfaffian at high symmetry momenta. As a result, we distinguish thefour $C_4 \mathcal{T}$-protected atomic insulators, each of which isadiabatically connected to a different atomic limit. We establish thecorrespondence between the invariant and the obstructed phases by constructingboth the atomic limit Hamiltonians and a $C_4 \mathcal{T}$-symmetric model thatinterpolates between them. The phase diagram shows that $C_4 \mathcal{T}$insulators allow $\pm 1$ and $2$ changes of the invariant, where the latter isoverlooked by symmetry indicators.

Pfaffian invariant identifies magnetic obstructed atomic insulators