{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:G7NAM3GK3HNMQKV5O642T3ZGCR","short_pith_number":"pith:G7NAM3GK","canonical_record":{"source":{"id":"1307.0716","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-07-02T14:45:58Z","cross_cats_sorted":["cond-mat.stat-mech","math.MP"],"title_canon_sha256":"bf0e8e5658d71ece73c39ce9826837d3b6773d5748b7aad90f6ba38784adf1e2","abstract_canon_sha256":"5af383fe47f923f58a0119833030bfc272a42703eb8d593538304e0959c27738"},"schema_version":"1.0"},"canonical_sha256":"37da066ccad9dac82abd77b9a9ef26145efb17b4162d51cff1886db9e45cd4d0","source":{"kind":"arxiv","id":"1307.0716","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.0716","created_at":"2026-05-18T02:55:36Z"},{"alias_kind":"arxiv_version","alias_value":"1307.0716v2","created_at":"2026-05-18T02:55:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.0716","created_at":"2026-05-18T02:55:36Z"},{"alias_kind":"pith_short_12","alias_value":"G7NAM3GK3HNM","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"G7NAM3GK3HNMQKV5","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"G7NAM3GK","created_at":"2026-05-18T12:27:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:G7NAM3GK3HNMQKV5O642T3ZGCR","target":"record","payload":{"canonical_record":{"source":{"id":"1307.0716","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-07-02T14:45:58Z","cross_cats_sorted":["cond-mat.stat-mech","math.MP"],"title_canon_sha256":"bf0e8e5658d71ece73c39ce9826837d3b6773d5748b7aad90f6ba38784adf1e2","abstract_canon_sha256":"5af383fe47f923f58a0119833030bfc272a42703eb8d593538304e0959c27738"},"schema_version":"1.0"},"canonical_sha256":"37da066ccad9dac82abd77b9a9ef26145efb17b4162d51cff1886db9e45cd4d0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:36.331564Z","signature_b64":"9e16SZwnACWnaY6qIJBsO9G22d3V83yFQpzO5zmFCICK+rAVoKwJwBjvw31XOSXEbaZdnAdk6Ie/v/xgLbncDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"37da066ccad9dac82abd77b9a9ef26145efb17b4162d51cff1886db9e45cd4d0","last_reissued_at":"2026-05-18T02:55:36.330831Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:36.330831Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.0716","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:55:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4sjW070sQ+p1pCJFhT7GjCGmwzxeogAUxlgHdd0ZuEOCKvGsGuKbiPXkhyn3li2EZsF1mIsTg1tHxQDBRs5qDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T23:34:27.702935Z"},"content_sha256":"eed37a503eb3092ba66503b5500709e86dadf3084f7b3406aca36fdc87c565da","schema_version":"1.0","event_id":"sha256:eed37a503eb3092ba66503b5500709e86dadf3084f7b3406aca36fdc87c565da"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:G7NAM3GK3HNMQKV5O642T3ZGCR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On gapped phases with a continuous symmetry and boundary operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","math.MP"],"primary_cat":"math-ph","authors_text":"Bruno Nachtergaele, Sven Bachmann","submitted_at":"2013-07-02T14:45:58Z","abstract_excerpt":"We discuss the role of compact symmetry groups, G, in the classification of gapped ground state phases of quantum spin systems. We consider two representations of G on infinite subsystems. First, in arbitrary dimensions, we show that the ground state spaces of models within the same G-symmetric phase carry equivalent representations of the group for each finite or infinite sublattice on which they can be defined and on which they remain gapped. This includes infinite systems with boundaries or with non-trivial topologies. Second, for two classes of one-dimensional models, by two different meth"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0716","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:55:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1HrwHdNTMX8+Oj8dvzwvnZ8kC01Dt0r7V3/zhU89FzGe3WBinlyFcu6qjjF+rHDcMf4tCC01dCnJ1VLY51plDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T23:34:27.703356Z"},"content_sha256":"466c77a25c5eaaee07bf89397c63efa8e7cafb6ff0c21a325dfb8e87acbaf111","schema_version":"1.0","event_id":"sha256:466c77a25c5eaaee07bf89397c63efa8e7cafb6ff0c21a325dfb8e87acbaf111"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G7NAM3GK3HNMQKV5O642T3ZGCR/bundle.json","state_url":"https://pith.science/pith/G7NAM3GK3HNMQKV5O642T3ZGCR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G7NAM3GK3HNMQKV5O642T3ZGCR/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-05T23:34:27Z","links":{"resolver":"https://pith.science/pith/G7NAM3GK3HNMQKV5O642T3ZGCR","bundle":"https://pith.science/pith/G7NAM3GK3HNMQKV5O642T3ZGCR/bundle.json","state":"https://pith.science/pith/G7NAM3GK3HNMQKV5O642T3ZGCR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G7NAM3GK3HNMQKV5O642T3ZGCR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:G7NAM3GK3HNMQKV5O642T3ZGCR","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"5af383fe47f923f58a0119833030bfc272a42703eb8d593538304e0959c27738","cross_cats_sorted":["cond-mat.stat-mech","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-07-02T14:45:58Z","title_canon_sha256":"bf0e8e5658d71ece73c39ce9826837d3b6773d5748b7aad90f6ba38784adf1e2"},"schema_version":"1.0","source":{"id":"1307.0716","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.0716","created_at":"2026-05-18T02:55:36Z"},{"alias_kind":"arxiv_version","alias_value":"1307.0716v2","created_at":"2026-05-18T02:55:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.0716","created_at":"2026-05-18T02:55:36Z"},{"alias_kind":"pith_short_12","alias_value":"G7NAM3GK3HNM","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"G7NAM3GK3HNMQKV5","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"G7NAM3GK","created_at":"2026-05-18T12:27:45Z"}],"graph_snapshots":[{"event_id":"sha256:466c77a25c5eaaee07bf89397c63efa8e7cafb6ff0c21a325dfb8e87acbaf111","target":"graph","created_at":"2026-05-18T02:55:36Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We discuss the role of compact symmetry groups, G, in the classification of gapped ground state phases of quantum spin systems. We consider two representations of G on infinite subsystems. First, in arbitrary dimensions, we show that the ground state spaces of models within the same G-symmetric phase carry equivalent representations of the group for each finite or infinite sublattice on which they can be defined and on which they remain gapped. This includes infinite systems with boundaries or with non-trivial topologies. Second, for two classes of one-dimensional models, by two different meth","authors_text":"Bruno Nachtergaele, Sven Bachmann","cross_cats":["cond-mat.stat-mech","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-07-02T14:45:58Z","title":"On gapped phases with a continuous symmetry and boundary operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0716","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:eed37a503eb3092ba66503b5500709e86dadf3084f7b3406aca36fdc87c565da","target":"record","created_at":"2026-05-18T02:55:36Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"5af383fe47f923f58a0119833030bfc272a42703eb8d593538304e0959c27738","cross_cats_sorted":["cond-mat.stat-mech","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-07-02T14:45:58Z","title_canon_sha256":"bf0e8e5658d71ece73c39ce9826837d3b6773d5748b7aad90f6ba38784adf1e2"},"schema_version":"1.0","source":{"id":"1307.0716","kind":"arxiv","version":2}},"canonical_sha256":"37da066ccad9dac82abd77b9a9ef26145efb17b4162d51cff1886db9e45cd4d0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"37da066ccad9dac82abd77b9a9ef26145efb17b4162d51cff1886db9e45cd4d0","first_computed_at":"2026-05-18T02:55:36.330831Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:55:36.330831Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9e16SZwnACWnaY6qIJBsO9G22d3V83yFQpzO5zmFCICK+rAVoKwJwBjvw31XOSXEbaZdnAdk6Ie/v/xgLbncDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:55:36.331564Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.0716","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:eed37a503eb3092ba66503b5500709e86dadf3084f7b3406aca36fdc87c565da","sha256:466c77a25c5eaaee07bf89397c63efa8e7cafb6ff0c21a325dfb8e87acbaf111"],"state_sha256":"2068ac3e6433a4c44e98987dca22ec77897e2c6d68c86d1fadb3e9f56e9f1afd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Bzo+OTQ7cL9QUrExzIar4QflzN1ckj6lqnaRJ/SGlM/sK8w8+srVj5WYVxWAt8ZbwYFVtG85kwfMDatXDwqPAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T23:34:27.705537Z","bundle_sha256":"917d7076ba703d7f06ee270e3d3a22ecd38c60597b4776590cb94658a45977dc"}}