Spin Chain with Continuous Exponential Symmetry: Topological Phase and Beyond
Abstract
We study symmetric gapped ground states of one-dimensional spin S chains with a continuous exponential symmetry U(1)rtimesℤ2 with integer base q, where the U(1) charge has a modulated unit of charge qL−j decaying exponentially with site j. The simplest model is a quantum breakdown model that can be viewed as a q-deformation of the XXZ Haldane spin chain. We show that the topology of their symmetric ground states crucially depends on S and q. For integer S, odd q allows a nontrivial ℤ2 Haldane-type symmetry-protected topological (SPT) phase with protected edge modes, while even q allows no nontrivial SPT phase. For half-odd integer S, odd q prohibits any unique symmetric gapped ground state due to a Lieb-Schultz-Mattis obstruction, whereas even q enforces a novel symmetric gapped phase which we name as the ``half-SPT" phase. Unconventionally, the half-SPT phase only has a symmetry protected edge mode on one edge, and has half of the string orders of an SPT phase. Our results uncover a new type of symmetry-protected gapped phases beyond existing classification schemes.
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BibTeX
@misc{lian2026spin,
title = {{Spin Chain with Continuous Exponential Symmetry: Topological Phase and Beyond}},
author = {Biao Lian and Zhen Bi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10682},
eprint = {2610.10682},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10682},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spin Chain with Continuous Exponential Symmetry: Topological Phase and Beyond
AU - Biao Lian
AU - Zhen Bi
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.10682
UR - https://arxiv.org/abs/2610.10682
AB - We study symmetric gapped ground states of one-dimensional spin S chains with a continuous exponential symmetry U(1)rtimesℤ_2 with integer base q, where the U(1) charge has a modulated unit of charge q^L−j decaying exponentially with site j. The simplest model is a quantum breakdown model that can be viewed as a q-deformation of the XXZ Haldane spin chain. We show that the topology of their symmetric ground states crucially depends on S and q. For integer S, odd q allows a nontrivial ℤ_2 Haldane-type symmetry-protected topological (SPT) phase with protected edge modes, while even q allows no nontrivial SPT phase. For half-odd integer S, odd q prohibits any unique symmetric gapped ground state due to a Lieb-Schultz-Mattis obstruction, whereas even q enforces a novel symmetric gapped phase which we name as the ``half-SPT" phase. Unconventionally, the half-SPT phase only has a symmetry protected edge mode on one edge, and has half of the string orders of an SPT phase. Our results uncover a new type of symmetry-protected gapped phases beyond existing classification schemes.
ER -