{
  "abstract": "We study symmetric gapped ground states of one-dimensional spin $S$ chains with a continuous exponential symmetry $U(1)\\rtimes\\mathbb Z_2$ 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 $\\mathbb Z_2$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10682",
  "author_total": 2,
  "authors": [
    {
      "name": "Biao Lian"
    },
    {
      "name": "Zhen Bi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/535p9jc8vfyyc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10682",
  "fulltext_url": null,
  "id": "535p9jc8vfyyc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10682",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.mes-hall"
    ],
    [
      "arxiv",
      "cond-mat.quant-gas"
    ],
    [
      "arxiv",
      "cond-mat.str-el"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "3107|Quantum many-body systems"
    ]
  ],
  "openalex": "W7221112043",
  "pdf_url": "https://arxiv.org/pdf/2610.10682",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/535p9jc8vfyyc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:52Z",
      "native_id": "2610.10682",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10682"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.quant-gas",
      "subfield": "amo"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.mes-hall",
      "subfield": "condensed-matter"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Spin Chain with Continuous Exponential Symmetry: Topological Phase and Beyond",
  "title_text": "Spin Chain with Continuous Exponential Symmetry: Topological Phase and Beyond",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:41Z",
  "url": "https://research.draconitic.com/paper/535p9jc8vfyyc",
  "venue": "arXiv",
  "versions": []
}