Simple invariants for band topology
Abstract
Despite the exhaustive understanding gathered around non-interacting topological states of matter, there is no single method capable of systematically delivering simple, numerically efficient topological invariants that is applicable to all crystalline and non-crystalline systems alike. Here we revisit the spectral localizer operator, constructed from the Hamiltonian and position operators, and show how it can be treated it as an auxiliary zero-dimensional Hamiltonian whose topology encodes the higher dimensional phases of the parent Hamiltonian. Its classification reduces every topological invariant to a matrix signature or the sign of a Pfaffian for an appropriate localizer, both of which are simple to interpret and efficient to compute in real space. We validate this approach by deriving simple real-space invariants for weak and rotationally invariant crystalline phases that were previously beyond the grasp of the spectral localizer formalism, atomic limits that escape scattering invariants, phases that evade symmetry-based indicator methods, as well as phases that had no previously known invariant. Our work provides a systematic way to construct any non-interacting topological invariant for a crystalline or non-crystalline systems, opening avenues to classify and predict the topology of previously unexplored classes of materials.
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BibTeX
@misc{chaou2026simple,
title = {{Simple invariants for band topology}},
author = {Adam Yanis Chaou and Adolfo G. Grushin and Peru d'Ornellas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07775},
eprint = {2609.07775},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07775},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Simple invariants for band topology
AU - Adam Yanis Chaou
AU - Adolfo G. Grushin
AU - Peru d'Ornellas
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07775
UR - https://arxiv.org/abs/2609.07775
AB - Despite the exhaustive understanding gathered around non-interacting topological states of matter, there is no single method capable of systematically delivering simple, numerically efficient topological invariants that is applicable to all crystalline and non-crystalline systems alike. Here we revisit the spectral localizer operator, constructed from the Hamiltonian and position operators, and show how it can be treated it as an auxiliary zero-dimensional Hamiltonian whose topology encodes the higher dimensional phases of the parent Hamiltonian. Its classification reduces every topological invariant to a matrix signature or the sign of a Pfaffian for an appropriate localizer, both of which are simple to interpret and efficient to compute in real space. We validate this approach by deriving simple real-space invariants for weak and rotationally invariant crystalline phases that were previously beyond the grasp of the spectral localizer formalism, atomic limits that escape scattering invariants, phases that evade symmetry-based indicator methods, as well as phases that had no previously known invariant. Our work provides a systematic way to construct any non-interacting topological invariant for a crystalline or non-crystalline systems, opening avenues to classify and predict the topology of previously unexplored classes of materials.
ER -