Krylov Edge Spectroscopy of Symmetry-Protected Topological Phases
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 ℤN × ℤN phases, the recovered endpoint charge gives the cohomology label. The method requires no explicit ground-state wavefunction, entanglement spectrum, or guessed dressed edge or string operator. For finite-range Hamiltonians, locality eliminates full-system exact diagonalization and guarantees thermodynamic-limit convergence at fixed Krylov depth. 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.
Related
- Symmetry-permuting entanglers for non-invertible symmetry-protected topological phases
- Realization of decoherence-induced averaged symmetry-protected topological phases on quantum processors
- Agnostic and efficient detection of ℤ2 symmetry-protected topological phases in projected entangled-pair states
- Predicting Entanglement Spectra of Gapless Symmetry-Protected Topological States in One Dimension
- Fate of average symmetry-protected topological states under symmetry-preserving quantum operations
BibTeX
@misc{menzler2026krylov,
title = {{Krylov Edge Spectroscopy of Symmetry-Protected Topological Phases}},
author = {Heiko Georg Menzler and Rishabh Jha},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10676},
eprint = {2609.10676},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10676},
}
RIS
TY - UNPB
TI - Krylov Edge Spectroscopy of Symmetry-Protected Topological Phases
AU - Heiko Georg Menzler
AU - Rishabh Jha
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.10676
UR - https://arxiv.org/abs/2609.10676
AB - 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 ℤ_N × ℤ_N phases, the recovered endpoint charge gives the cohomology label. The method requires no explicit ground-state wavefunction, entanglement spectrum, or guessed dressed edge or string operator. For finite-range Hamiltonians, locality eliminates full-system exact diagonalization and guarantees thermodynamic-limit convergence at fixed Krylov depth. 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.
ER -