Selective bulk-boundary correspondence in higher-order topological insulators with anticommuting mirror and chiral symmetries

TL;DR

The paper shows that higher-order topological insulators in the BDI class protected by a chiral symmetry and two anticommuting mirror symmetries can exhibit edge-selective bulk-boundary correspondence when the mirror symmetries anticommute with the chiral operator. It analyzes 2D lattice models, including the quadrupole insulator, and introduces an edge-sensitive winding number that governs edge-selective gap closings and corner-bound states while preserving the chiral symmetry; diagonal mirrors commuting with the chiral operator do not produce such edge selectivity. The work demonstrates that breaking anticommuting mirrors yields selective hinge/edge gap closings and a topological transition described by a modified invariant, with explicit results for separable and nonseparable models and extensions to 3D. These findings provide a genuinely topological mechanism for boundary-selective phenomena in higher-order topological phases and offer guidance for experimental realization in 2D and 3D crystalline platforms.

Abstract

We investigate higher-order topological insulators protected by chiral and anticommuting mirror symmetries. Using models in the BDI class, which include the prototypical topological quadrupole insulator, we show that breaking mirror symmetries that anticommute with the chiral operator leads to edge-selective bulk-boundary correspondence, with gap closings and bound states appearing only along a subset of boundaries of the same orientation and codimension. We define a new edge-sensitive topological invariant that distinguishes this mechanism from previous reports of non-topological edge-selection effects.

Figures (7)

• Figure 1: Eigenvalue assignments $(C,M'_j)$ of bound states in the $C_4$ symmetric model \ref{['eq:ex']} with open boundaries, $f=-0.3$ and $n_1=n_2=2$. Each square shows the numerically calculated amplitude $\psi$ of one of four mutually orthogonal bound states in a $2\times2$ mesh of unit cells at the corresponding corner. The bound states at different corners are mapped to each other by $M_j$. The coordinate systems $x_1$-$x_2$ and $x'_1$-$x'_2$ are the principal axes and diagonal directions, respectively.
• Figure 2: Bulk vs. edge gaps (a) and low-energy spectra with open bounday conditions (b),(c) for the model \ref{['eq:ex']} with $f_1=-0.3$, $n_1=n_2=2$, $\Delta=0$, and variable $f_2$.
• Figure 3: Bulk vs. edge gaps (a) and low-energy spectra with open bounday conditions (b),(c) for the model \ref{['eq:ex']} with $f_1=f_2=-0.3$, $n_1=n_2=2$, and $\Delta=0$ except for variable $\Delta_1$.
• Figure 4: The spectra for $k_1$-edge (a) and $k_2$-edge (b), and corner state probability densities (c)-(e) for the model in Eq. \ref{['eq:ex']} with $f_1=f_2=-0.3$, $n_1=n_2=2$, and variable $\Delta_1$.
• Figure 5: Bulk vs. edge spectral gaps (a),(b) and energy spectra under open boundary conditions (c),(d) for the nonseparable model in Eq. \ref{['eq:nonsep']} with $\Delta(\mathbf{k}) = \Delta_1M_1$ (a),(c) and $\Delta(\mathbf{k}) = i\Delta_{11}M_1C_1\sin k_1$ (b),(d). The boxed numbers in (c) and (d) show the number of zero-energy corner-bound states. In both cases $\bar{f}_1 = \bar{f}_2 = -0.3$, $\delta f_1 = \delta f_2 = 0.2$ and $n_1 = n_2 = 1$.