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One-particle density matrix framework for mode-shell correspondence: Characterizing topology in amorphous higher-order topological insulators

Miguel F. Martínez1, Lucien Jezequel1, Jens H. Bardarson1, Thomas Klein Kvorning1, and Julia D. Hannukainen2

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Phys. Rev. Research 8, 023320 – Published 22 June, 2026

DOI: https://doi.org/10.1103/rt48-gpfm

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 C4-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 C4-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.

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Phys. Rev. Research 8, 023320– Published 22 June, 2026
  • Received 16 October 2025
  • Accepted 22 May 2026
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