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

Suman Aich, Babak Seradjeh

TL;DR

The paper tackles higher-order topological phases in two-dimensional chiral-symmetric Bloch systems with two anticommuting mirrors. It develops momentum-space diagnostics—mirror-filtered winding numbers for Wannier Hamiltonians and periodicized Wilson-line invariants—that capture Wannier-band topology and provide a bulk–boundary framework for Wilson loops. While these invariants robustly diagnose Wannier gaps and edge-related features, their direct mapping to corner-bound states is not universal, revealing subtle distinctions between Wannier-sector topology and physical boundary modes. The work points to open challenges in formulating momentum-space invariants for general nonseparable models and motivates extending these constructions to other symmetry classes and disordered settings.

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 $π$-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.

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

TL;DR

The paper tackles higher-order topological phases in two-dimensional chiral-symmetric Bloch systems with two anticommuting mirrors. It develops momentum-space diagnostics—mirror-filtered winding numbers for Wannier Hamiltonians and periodicized Wilson-line invariants—that capture Wannier-band topology and provide a bulk–boundary framework for Wilson loops. While these invariants robustly diagnose Wannier gaps and edge-related features, their direct mapping to corner-bound states is not universal, revealing subtle distinctions between Wannier-sector topology and physical boundary modes. The work points to open challenges in formulating momentum-space invariants for general nonseparable models and motivates extending these constructions to other symmetry classes and disordered settings.

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 -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.

Paper Structure

This paper contains 11 sections, 10 theorems, 44 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Model Hamiltonians
  3. Wilson loops and edge spectra
  4. Topological invariants and bulk-boundary correspondence
  5. Chiral winding numbers
  6. Mirror-filtered invariants of the Wannier Hamiltonian
  7. Invariants of the Wannier spectrum
  8. Summary and outlook
  9. Chiral winding numbers
  10. Reformulations of the winding number
  11. Symmetry properties of the winding number

Key Result

Theorem 1

The winding number can be expressed as where $q$ is the unitary obtained by flattening $d$, i.e. $q(k) = u(k)v^\dagger(k)$ given the singular-value decomposition $d(k) = u(k)D(k)v^\dagger(k)$.

Figures (6)

  • Figure 1: Bulk and edge gaps (a), (b), and Wannier gaps (c), (d) for the separable model in Eq. \ref{['eq:ff']} ($\delta f_1 = \delta f_2=0$) with $\bar{f}_1=-0.3$ and variable $\bar{f}_2$ (left panels), and $C_4$ symmetric model with variable $\bar{f}$ (right panels). In both cases $n_1=n_2=2$, $\Delta=0$.
  • Figure 2: Bulk and edge gaps (a), (b), and Wannier gaps (c), (d) for the nonseparable model in Eq. \ref{['eq:ff']} with $\delta f_1 = -0.1$ and variable $\delta f_2$ (left panels), and for the $C_4$ symmetric model with variable $\delta f$ (right panels). In both cases $n_1=n_2=1$, $\bar{f}_1=\bar{f}_2=-0.3$, and $\Delta=0$. The vertical blue and green lines show bulk and edge gap closings, respectively. The shaded region in (d) marks the vanishingly small value of the Wannier gap.
  • Figure 3: Bulk and edge gaps (a), (b), Wannier gaps (c), (d), for the $C_4$ symmetric model \ref{['eq:bc']} with variable $\bar{g}$ and $\bar{f} =0.11$, $\delta f=1.11$ (left panels), and $\bar{f} =-0.3$, $\delta f=0.6$ (right panels). The vertical blue and green lines show bulk and edge gap closings, respectively. The vertical dashed lines in (d) show the kinks observed in the Wannier gap. The shaded region in (c) marks the vanishingly small value of the Wannier gap.
  • Figure 4: Bulk-boundary correspondence in the separable model in Eq. \ref{['eq:ff']} ($\delta f_1 = \delta f_2=0$) showing the bulk invariants (a)-(d), and the low-energy spectrum of the open system (e), (f). The parameters correspond to those in Fig. \ref{['fig:BEWsep']}. In (a)-(d), some values are offset slightly to distinguish the coinciding values. In (e) and (f), the boxed numbers show the number of zero-energy corner-bound states.
  • Figure 5: Bulk-boundary correspondence in the non-separable model in Eq. \ref{['eq:ff']} showing the bulk invariants (a)-(d), and the low-energy spectrum of the open system (e), (f). The parameters correspond to those in Fig. \ref{['fig:BEWnonsep']}. In (e) and (f), the boxed numbers show the number of zero-energy corner-bound states.
  • ...and 1 more figures

Theorems & Definitions (20)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 3.1
  • proof
  • Corollary 3.2: Change of basis
  • proof
  • ...and 10 more