Bulk-boundary correspondence in 2D mirror-symmetric higher-order topological insulators

We study the higher-order bulk-boundary correspondence in a family of generalized models with extended hopping terms and anti-commuting mirror symmetries. Specifically, we study the symmetry structure of Wilson loops for this class of models and define a set of bulk Z invariants that characterize their topology. We illustrate the bulk-boundary correspondence by showing these invariants match the number of corner states in an open geometry. We also show that corner bound states are robust and their number is correctly captured by our bulk Z invariants in the presence of chiral symmetry breaking terms in the Hamiltonian.