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We find that an odd-$S$ Haldane phase is a topologically non-trivial phase which is protected by any one of the following three global symmetries: (i) the dihedral group of $\\pi$-rotations about $x,y$ and $z$ axes; (ii) time-reversal symmetry $S^{x,y,z} \\rightarrow - S^{x,y,z}$; (iii) link inversion symmetry (reflection about a bond center), consistently with previous results [Phys. Rev. B \\textbf{81}, 064439 (2010)]. On the other hand, an even-$S$ Haldane phase"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0909.4059","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.str-el","submitted_at":"2009-09-22T19:27:18Z","cross_cats_sorted":["cond-mat.stat-mech"],"title_canon_sha256":"82596f8d96fbe3e772c95235e4c4bd524e813763965b78393a0ff57dc89205c5","abstract_canon_sha256":"eca4b38130fbb3f21a0dac22fa0583659d8406356194e5585b3d8975eb79c844"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:38.649277Z","signature_b64":"OTUbiSvrhbApXMrxHEOOCLG2skJTzBkw+j665udc+c1EwxiBqaWMG4FEHbRfw3xkh5KwaLrfKLRCgyhRlxTCAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c5312ccb519c8672ea84cce1b5907876ca1643cd95fb87f9f5b82875aa1e7df2","last_reissued_at":"2026-05-18T03:59:38.648432Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:38.648432Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Symmetry protection of topological order in one-dimensional quantum spin systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech"],"primary_cat":"cond-mat.str-el","authors_text":"Ari M. Turner, Erez Berg, Frank Pollmann, Masaki Oshikawa","submitted_at":"2009-09-22T19:27:18Z","abstract_excerpt":"We discuss the characterization and stability of the Haldane phase in integer spin chains on the basis of simple, physical arguments. We find that an odd-$S$ Haldane phase is a topologically non-trivial phase which is protected by any one of the following three global symmetries: (i) the dihedral group of $\\pi$-rotations about $x,y$ and $z$ axes; (ii) time-reversal symmetry $S^{x,y,z} \\rightarrow - S^{x,y,z}$; (iii) link inversion symmetry (reflection about a bond center), consistently with previous results [Phys. Rev. B \\textbf{81}, 064439 (2010)]. 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