{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:QNFYCUC7GA7YFX4ASODADFPKIL","short_pith_number":"pith:QNFYCUC7","canonical_record":{"source":{"id":"2009.07605","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cond-mat.dis-nn","submitted_at":"2020-09-16T11:09:01Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"2c012cff331d3417a67c5a4ac56a3e268636680f6c65218070255bd507d3497e","abstract_canon_sha256":"567ab04c7af1ec958472ca4d28449ce7c2a4c9c8195118434e2728a53ca0a255"},"schema_version":"1.0"},"canonical_sha256":"834b81505f303f82df8093860195ea42dc5b3ed0b4d3057616bb87e8af6a1b6d","source":{"kind":"arxiv","id":"2009.07605","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2009.07605","created_at":"2026-07-05T02:55:32Z"},{"alias_kind":"arxiv_version","alias_value":"2009.07605v2","created_at":"2026-07-05T02:55:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.07605","created_at":"2026-07-05T02:55:32Z"},{"alias_kind":"pith_short_12","alias_value":"QNFYCUC7GA7Y","created_at":"2026-07-05T02:55:32Z"},{"alias_kind":"pith_short_16","alias_value":"QNFYCUC7GA7YFX4A","created_at":"2026-07-05T02:55:32Z"},{"alias_kind":"pith_short_8","alias_value":"QNFYCUC7","created_at":"2026-07-05T02:55:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:QNFYCUC7GA7YFX4ASODADFPKIL","target":"record","payload":{"canonical_record":{"source":{"id":"2009.07605","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cond-mat.dis-nn","submitted_at":"2020-09-16T11:09:01Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"2c012cff331d3417a67c5a4ac56a3e268636680f6c65218070255bd507d3497e","abstract_canon_sha256":"567ab04c7af1ec958472ca4d28449ce7c2a4c9c8195118434e2728a53ca0a255"},"schema_version":"1.0"},"canonical_sha256":"834b81505f303f82df8093860195ea42dc5b3ed0b4d3057616bb87e8af6a1b6d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:55:32.424421Z","signature_b64":"xihJQI5KBabuCsK1GVSn/DhcT3VHNKzc8HbF6RAkNoVIGLGQ3ZCx2Ij+ssNOPjDgBqmv48N1FIPzOXIIBXo4Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"834b81505f303f82df8093860195ea42dc5b3ed0b4d3057616bb87e8af6a1b6d","last_reissued_at":"2026-07-05T02:55:32.423948Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:55:32.423948Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2009.07605","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":"2026-07-05T02:55:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U8vqPvyG6x5LwvU4d9ejhYgtMjbi3oSaijivsv00CybzvznodmLaOYXuJiUAPQAX0uRKsdUJCTfQrGKjqBBvDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T13:10:18.682936Z"},"content_sha256":"2e97774441173770e6c7b338039382bd2faf8cf6763918eca8a971d8132e46f6","schema_version":"1.0","event_id":"sha256:2e97774441173770e6c7b338039382bd2faf8cf6763918eca8a971d8132e46f6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:QNFYCUC7GA7YFX4ASODADFPKIL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Localization transition, spectrum structure and winding numbers for one-dimensional non-Hermitian quasicrystals","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"cond-mat.dis-nn","authors_text":"Qi Zhou, Shu Chen, Yanxia Liu","submitted_at":"2020-09-16T11:09:01Z","abstract_excerpt":"By analyzing the Lyapunov exponent (LE), we develop a rigorous, fundamental scheme for the study of general non-Hermitian quasicrystals with both complex phase factor and non-reciprocal hopping. Specially, the localization-delocalization transition point, $\\mathcal{PT}$-symmetry-breaking point and the winding number transition points are determined by LEs of its dual Hermitian model. The analysis was based on Avila's global theory, and we found that winding number is directly related to the acceleration, the slope of the LE, while quantization of acceleration is the crucial ingredient of Avila"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.07605","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2009.07605/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T02:55:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+XMGEn11po53EVfOY4dA/UEurkf9rHaU8PghpZggOFt9x675eOQShv5d8GsmlWD6Rp4rTjiVx17vdolEpLD2BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T13:10:18.683434Z"},"content_sha256":"e7b1542cd3de671ecbb979999f981fc281e3074f3a26627015c02e7f287836af","schema_version":"1.0","event_id":"sha256:e7b1542cd3de671ecbb979999f981fc281e3074f3a26627015c02e7f287836af"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QNFYCUC7GA7YFX4ASODADFPKIL/bundle.json","state_url":"https://pith.science/pith/QNFYCUC7GA7YFX4ASODADFPKIL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QNFYCUC7GA7YFX4ASODADFPKIL/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":"2026-08-16T13:10:18Z","links":{"resolver":"https://pith.science/pith/QNFYCUC7GA7YFX4ASODADFPKIL","bundle":"https://pith.science/pith/QNFYCUC7GA7YFX4ASODADFPKIL/bundle.json","state":"https://pith.science/pith/QNFYCUC7GA7YFX4ASODADFPKIL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QNFYCUC7GA7YFX4ASODADFPKIL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:QNFYCUC7GA7YFX4ASODADFPKIL","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":"567ab04c7af1ec958472ca4d28449ce7c2a4c9c8195118434e2728a53ca0a255","cross_cats_sorted":["math-ph","math.MP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cond-mat.dis-nn","submitted_at":"2020-09-16T11:09:01Z","title_canon_sha256":"2c012cff331d3417a67c5a4ac56a3e268636680f6c65218070255bd507d3497e"},"schema_version":"1.0","source":{"id":"2009.07605","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2009.07605","created_at":"2026-07-05T02:55:32Z"},{"alias_kind":"arxiv_version","alias_value":"2009.07605v2","created_at":"2026-07-05T02:55:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.07605","created_at":"2026-07-05T02:55:32Z"},{"alias_kind":"pith_short_12","alias_value":"QNFYCUC7GA7Y","created_at":"2026-07-05T02:55:32Z"},{"alias_kind":"pith_short_16","alias_value":"QNFYCUC7GA7YFX4A","created_at":"2026-07-05T02:55:32Z"},{"alias_kind":"pith_short_8","alias_value":"QNFYCUC7","created_at":"2026-07-05T02:55:32Z"}],"graph_snapshots":[{"event_id":"sha256:e7b1542cd3de671ecbb979999f981fc281e3074f3a26627015c02e7f287836af","target":"graph","created_at":"2026-07-05T02:55:32Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2009.07605/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"By analyzing the Lyapunov exponent (LE), we develop a rigorous, fundamental scheme for the study of general non-Hermitian quasicrystals with both complex phase factor and non-reciprocal hopping. Specially, the localization-delocalization transition point, $\\mathcal{PT}$-symmetry-breaking point and the winding number transition points are determined by LEs of its dual Hermitian model. The analysis was based on Avila's global theory, and we found that winding number is directly related to the acceleration, the slope of the LE, while quantization of acceleration is the crucial ingredient of Avila","authors_text":"Qi Zhou, Shu Chen, Yanxia Liu","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cond-mat.dis-nn","submitted_at":"2020-09-16T11:09:01Z","title":"Localization transition, spectrum structure and winding numbers for one-dimensional non-Hermitian quasicrystals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.07605","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":"sha256:2e97774441173770e6c7b338039382bd2faf8cf6763918eca8a971d8132e46f6","target":"record","created_at":"2026-07-05T02:55:32Z","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":"567ab04c7af1ec958472ca4d28449ce7c2a4c9c8195118434e2728a53ca0a255","cross_cats_sorted":["math-ph","math.MP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cond-mat.dis-nn","submitted_at":"2020-09-16T11:09:01Z","title_canon_sha256":"2c012cff331d3417a67c5a4ac56a3e268636680f6c65218070255bd507d3497e"},"schema_version":"1.0","source":{"id":"2009.07605","kind":"arxiv","version":2}},"canonical_sha256":"834b81505f303f82df8093860195ea42dc5b3ed0b4d3057616bb87e8af6a1b6d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"834b81505f303f82df8093860195ea42dc5b3ed0b4d3057616bb87e8af6a1b6d","first_computed_at":"2026-07-05T02:55:32.423948Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:55:32.423948Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xihJQI5KBabuCsK1GVSn/DhcT3VHNKzc8HbF6RAkNoVIGLGQ3ZCx2Ij+ssNOPjDgBqmv48N1FIPzOXIIBXo4Bg==","signature_status":"signed_v1","signed_at":"2026-07-05T02:55:32.424421Z","signed_message":"canonical_sha256_bytes"},"source_id":"2009.07605","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2e97774441173770e6c7b338039382bd2faf8cf6763918eca8a971d8132e46f6","sha256:e7b1542cd3de671ecbb979999f981fc281e3074f3a26627015c02e7f287836af"],"state_sha256":"6c62fb9d50a1dd6cdc22161593312e0167f66bbab35cfd0413323c9c732f6629"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5EBS+FsymzLW+Pc+zqAXdzSf7/iyEuMfHM3N34IDZx2TwUjoWTasVxxvk2F3YlMvPUSyqaZc3h7NtfNBvXMfDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T13:10:18.686742Z","bundle_sha256":"2e3ebae4fe47afc0a9644d265cfd600c600fbfaeb75487bebf5da58a94d9693e"}}