Wilson loop invariants and bulk-boundary correspondence in higher-order topological insulators with two anticommuting mirror symmetries

TL;DR

This paper explores the bulk-boundary correspondence in higher-order topological insulators with anticommuting mirror symmetries, introducing new momentum-space invariants to characterize Wannier band topology, especially in nonseparable models.

Contribution

It develops gauge-independent mirror-filtered winding numbers and adapted Wilson line invariants to better understand higher-order topological phases with complex symmetries.

Findings

Separable models' invariants predict corner states accurately.

Nonseparable models require new momentum-space diagnostics.

New invariants clarify the role of symmetries in topological phases.

Abstract

We investigate the higher-order bulk-boundary correspondence in a family of chiral-symmetric Bloch Hamiltonians with anticommuting mirror symmetries. These models generalize the $\pi$-flux square lattice, the prototypical topological quadrupole insulator, and include both separable and nonseparable models with extended and diagonal hopping. For separable systems, the product of subsystem chiral winding numbers correctly predicts the number of zero-energy corner states. However, this invariant fails in nonseparable models, motivating the development of new momentum-space diagnostics. We introduce gauge-independent mirror-filtered winding numbers for Wannier Hamiltonians, constructed by projecting mirror eigenstates onto the occupied subspace. Furthermore, by adapting periodicized Wilson lines from chiral Floquet theory to the case with momentum-dependent chiral operator, we define new invariants associated directly with Wannier gaps. These invariants provide a detailed characterization of Wannier band topology. Our results clarify the interplay between chiral symmetry, mirror symmetries, and Wilson loops in higher-order topological phases and point to open challenges in formulating momentum-space invariants for general nonseparable models.