Krylov Edge Spectroscopy of Symmetry-Protected Topological Phases

• URL: http://arxiv.org/abs/2609.10676v1
• Date: Wed, 09 Sep 2026 18:00:01 GMT
• Title: Krylov Edge Spectroscopy of Symmetry-Protected Topological Phases
• Abstract summary: We introduce a many-body operator-space protocol for detecting and classifying one-dimensional symmetry-protected topological phases.<n>We demonstrate the protocol in cluster, clock, Haldane, and trivial spin-1 chains.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce $Krylov$ $edge$ $spectroscopy$, a many-body operator-space protocol for detecting and classifying one-dimensional symmetry-protected topological phases from local boundary dynamics. A Hermitian boundary operator generates a semi-infinite Krylov hopping chain whose boundary weight obeys an exact zero-frequency normalizability criterion. Open-periodic, boundary-bulk, and symmetry-preserving boundary-perturbation tests identify protected boundary memory, while a finite-depth leakage residual certifies when an explicitly reconstructed operator is already near zero frequency. Classification minimizes the normalized commutator over symmetry-resolved boundary operators in fixed charge sectors. For bosonic $\mathbb{Z}_N \times \mathbb{Z}_N$ phases, the recovered endpoint charge gives the cohomology label. The method requires neither many-body exact diagonalization, an explicit ground-state wavefunction or entanglement spectrum, nor a guessed dressed edge or string operator. For finite-range Hamiltonians, locality organizes the thermodynamic limit at fixed Krylov depth before the depth limit. We demonstrate the protocol in cluster, clock, Haldane, and trivial spin-1 chains. In the exactly solvable cluster chain, the transition between the gapped topological and trivial phases manifests as a localization-delocalization transition of the Krylov edge mode on the Krylov chain. This transition occurs precisely at the bulk gap closing, and its localization-length exponent coincides with the Ising correlation-length exponent. Through operator Krylov dynamics, our work turns local boundary evolution into a direct spectroscopy of many-body topology.

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