The equivalence of topological indices for U(1)rtimesℤ2-symmetric quantum spin chains
Abstract
Symmetry-protected topological (SPT) phases in quantum spin chains are distinguished by topological indices. Ogata's operator-algebraic indices apply to a broad range of symmetry classes without requiring continuous symmetry, whereas the elementary twist index of Tasaki requires U(1) symmetry. We prove their equivalence in the common setting of integer-spin chains with on-site U(1)rtimesℤ2 symmetry and a symmetric locally-unique gapped ground state, without assuming translation invariance or a matrix product representation. A charge-fluctuation estimate yields a strong-limit construction of half-chain rotations with their phases fixed. An exact group commutator identity then identifies the Ogata index with the limit of local twist expectations and gives an explicit error bound. This provides a computable formula for the index when a lower bound on the gap and the relevant ground-state expectations are available. Combining the equivalence with a recent boundary-field gap theorem of OpenAI and our earlier twist-index calculation, we obtain a spin-rotation-invariant locally-unique gapped ground state of the infinite S=1 antiferromagnetic Heisenberg chain with both indices equal to −1, without assuming infinite-volume uniqueness. The author formulated the problem and proposed the basic strategy, while ChatGPT Astra made substantial contributions to the development of the proof.
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BibTeX
@misc{tasaki2026equivalence,
title = {{The equivalence of topological indices for $\mathrmU(1)\rtimes\mathbbZ_2$-symmetric quantum spin chains}},
author = {Hal Tasaki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.34119},
eprint = {2609.34119},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34119},
}
RIS
TY - UNPB
TI - The equivalence of topological indices for U(1)rtimesℤ_2-symmetric quantum spin chains
AU - Hal Tasaki
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2609.34119
UR - https://arxiv.org/abs/2609.34119
AB - Symmetry-protected topological (SPT) phases in quantum spin chains are distinguished by topological indices. Ogata's operator-algebraic indices apply to a broad range of symmetry classes without requiring continuous symmetry, whereas the elementary twist index of Tasaki requires U(1) symmetry. We prove their equivalence in the common setting of integer-spin chains with on-site U(1)rtimesℤ_2 symmetry and a symmetric locally-unique gapped ground state, without assuming translation invariance or a matrix product representation. A charge-fluctuation estimate yields a strong-limit construction of half-chain rotations with their phases fixed. An exact group commutator identity then identifies the Ogata index with the limit of local twist expectations and gives an explicit error bound. This provides a computable formula for the index when a lower bound on the gap and the relevant ground-state expectations are available. Combining the equivalence with a recent boundary-field gap theorem of OpenAI and our earlier twist-index calculation, we obtain a spin-rotation-invariant locally-unique gapped ground state of the infinite S=1 antiferromagnetic Heisenberg chain with both indices equal to −1, without assuming infinite-volume uniqueness. The author formulated the problem and proposed the basic strategy, while ChatGPT Astra made substantial contributions to the development of the proof.
ER -