A One-Particle Density Matrix Framework for Mode-Shell Correspondence: Characterizing Topology in Amorphous Higher-Order Topological Insulators

TL;DR

This work reframes higher_order topology as a property of the quantum state by formulating a mode_shell correspondence directly from the one_particle density matrix for Gaussian states with a chiral constraint. It defines a mode index $\mathcal{I}_{\rm mode}$ and a shell index $\mathcal{I}_{\rm shell}$, showing that in 1D the shell index reduces to the local chiral marker and that in 2D the indices diagnose intrinsic higher_order topology, demonstrated on the BBH model with $C_4$ symmetry and on an amorphous counterpart. The approach remains valid without translation invariance and extends to interacting states with a gapped bulk via band_flattening, providing a practical, state_based bulk_boundary diagnostic for higher_order topology in crystalline and amorphous materials. Overall, the framework offers a robust, real_space method to characterize topology directly from the quantum state, with potential generalizations to other symmetries and higher dimensions.

Abstract

We present a framework for characterizing higher-order topological phases directly from the one-particle density matrix, without any reference to an underlying Hamiltonian. Our approach extends the mode-shell correspondence, originally formulated for single-particle Hamiltonians, to Gaussian states subject to chiral constraints. In this correspondence, the mode index counts topological boundary modes, while the shell index quantifies the bulk topology in a region surrounding the modes, providing a bulk-boundary diagnostic. In one-dimensional topological insulators, the shell index reduces to the local chiral marker, recovering the winding number in the translation-invariant limit. We apply the mode-shell correspondence to a $C_4$-symmetric higher-order topological insulator with a chiral constraint and show that a fractional shell index implies that the higher-order phase is intrinsic. The one-particle density matrix is formulated in real space, so the mode-shell correspondence also applies to models without translation invariance. By introducing structural disorder into the $C_4$-symmetric higher-order insulator, we show that the mode-shell correspondence remains a meaningful diagnostic in amorphous structures. The mode-shell correspondence generalizes to interacting states with a gapped bulk spectrum in the one-particle density matrix, providing a practical and diverse route to characterize higher-order topology from the quantum state itself.

Figures (3)

• Figure 1: Example of the choice of regions $\mathcal{A}$, its complement $\mathcal{A}^c$, and $\mathcal{A_{\rm shell}}\subset \mathcal{A}$ in a one-dimensional chain with the topological zero modes, depicted in blue, at the two ends of the chain. The solid line represents the boundary between $\mathcal{A}$ and $\mathcal{A}^c$, and the dotted line represents the shell, the boundary of $\mathcal{A_{\rm shell}}$.
• Figure 2: (a) Spectrum of the restricted one-particle density matrix $\rho_\mathcal{A}$ for the ground state of the BBH model at half filling, with parameters $\gamma=0.5, \eta=1$ and defined on a square lattice of linear size $L=20$. $\lambda$ denotes the eigenvalues and $N_\lambda$ the eigenvector number. (b) Combined density of states of the four zero modes in the energy spectrum of the BBH Hamiltonian. The region $\mathcal{A}$ contains the lattice sites with $x<L/2$ and $y<L/2$, enclosed by the blue box, and the region $\mathcal{A}_{\rm shell}$ is shaded in light blue. (c) Mode index density, defined in Eq. (\ref{['eq:mode_density']}), within the region $\mathcal{A}$. The box in the upper right corner indicates the total mode index, computed by summing the mode index density over real space. (d) Shell index density, defined in Eq. (\ref{['eq:shell_density']}), within the region $\mathcal{A}$. The box in the upper right corner indicates the total shell index, computed by summing the shell index density over real space. The region $\mathcal{A}_{\rm shell}$ is shaded in light blue in (c) and (d).
• Figure 3: (a) Spectrum of the restricted one-particle density matrix $\rho_\mathcal{A}$ for the ground state of the BBH model at half filling, with parameters $\gamma=0.5, \eta=1$ and defined on an amorphous structure of linear size $L=20$, and amorphicity $w=0.1$. $\lambda$ denotes the eigenvalues and $N_\lambda$ the eigenvector number. (b) Combined density of states of the four zero modes in the energy spectrum of the BBH Hamiltonian. The region $\mathcal{A}$ contains the lattice sites with $x<L/2$ and $y<L/2$, enclosed by the blue box, and the region $\mathcal{A}_{\rm shell}$ is shaded in light blue. (c) Mode index density, defined in Eq. (\ref{['eq:mode_density']}), within the region $\mathcal{A}$. The box in the upper right corner indicates the total mode index, computed by summing the mode density over real space. (d) Shell index density, defined in Eq. (\ref{['eq:shell_density']}), within the region $\mathcal{A}$. The box in the upper right corner indicates the total shell index, computed by summing the shell index density over real space. The region $\mathcal{A}_{\rm shell}$ is shaded in light blue in (c) and (d).