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Q_8 denotes the non-Abelian quaternion group with eight elements. We demonstrate that after quantum disordering these three types of tetrad orders, the systems enter fully gapped liquid phases described by Z_2, Z_4, and non-Abelian quaternion gauge field theories, respectively. The latter case realizes Kitaev's non-"},"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":"1012.5671","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.str-el","submitted_at":"2010-12-27T21:43:58Z","cross_cats_sorted":[],"title_canon_sha256":"6c641d1f77eb240ebeb26e6162aaefb418045506a4bac88aa8ad60a46197bb4f","abstract_canon_sha256":"9e623c97b83b62ce6f05bd63abb060372ecc096d3ba9a25fda7bc5ebc365ff70"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:23:03.439268Z","signature_b64":"/AKAGiCvZ3d0eVsCz3oikoykZGvwyHjyfh/1EcY1iL/cKhXe2eiVyausEXInu3n+GXMnbC7fj62NAN5iVfafBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2606554de6a96d982e5f72c60e6d5eed96ee36fa1e4c8e3848386562df6d6ca3","last_reissued_at":"2026-05-18T03:23:03.438680Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:23:03.438680Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Topological Quantum Liquids with Quaternion Non-Abelian Statistics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cond-mat.str-el","authors_text":"Andreas W.W. Ludwig, Cenke Xu","submitted_at":"2010-12-27T21:43:58Z","abstract_excerpt":"Noncollinear magnetic order is typically characterized by a \"tetrad\" ground state manifold (GSM) of three perpendicular vectors or nematic-directors. We study three types of tetrad orders in two spatial dimensions, whose GSMs are SO(3) = S^3/Z_2, S^3/Z_4, and S^3/Q_8, respectively. Q_8 denotes the non-Abelian quaternion group with eight elements. We demonstrate that after quantum disordering these three types of tetrad orders, the systems enter fully gapped liquid phases described by Z_2, Z_4, and non-Abelian quaternion gauge field theories, respectively. 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