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We study the multi-gap topology of the periodic spectra of Wilson loopoperators (WLOs) in mirror symmetric insulators. We develop two topologicalinvariants each associated with a mirror-invariant gap in the Wilson loopspectrum. We propose that both topological invariants in combination determinethe general higher-order bulk-boundary correspondence in 2D mirror symmetric,boundary-obstructed topological insulators. Finally, we demonstrate that thesenew multi-gap topological invariants apply to anomalous cases beyond thosecaptured by the nested Wilson loop, and we subsequently develop anunderstanding of the correlation between WLOs along two orthogonal directions.

Multi-gap topology of the Wilson loop operator in mirror symmetric insulators