Bulk-boundary correspondence in 2D chiral-symmetric higher-order topological insulators.*

Higher-order topological phases support states localized at boundaries of the system with co-dimension larger than one. Extensive work on the classification and the nature of bulk invariants characterizing such phases based on crystalline symmetries and Wilson loops has been performed. However, the question of how to find the bulk invariants of the generalized chiral-symmetric pi-flux model remains open. A classification in terms of quantized bulk quadrupole moment is insufficient since it gives a Z₂ invariant and does not account for the number of zero-energy corner states known to be a Z invariant in simple cases. These models also cannot be classified based on crystalline symmetries due to the existence of anti-commuting mirror symmetries. In this work, we study the symmetry structure of partial Wilson loops for this class of models and define a set of bulk Z invariants that fully characterize their topology. We illustrate the connection between these invariants and the number of corner states in various models with extended hopping terms, thus establishing the bulk-boundary correspondence in these models.