{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:FPX4ZDBVAOGUDQWTMMXPHGBGFD","short_pith_number":"pith:FPX4ZDBV","schema_version":"1.0","canonical_sha256":"2befcc8c35038d41c2d3632ef3982628eaad10ab4552c21148687c0f8bb3dbff","source":{"kind":"arxiv","id":"1107.0405","version":1},"attestation_state":"computed","paper":{"title":"The BCS gap equation for spin-polarized fermions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.quant-gas","cond-mat.supr-con","math.MP"],"primary_cat":"math-ph","authors_text":"Abraham Freiji, Christian Hainzl, Robert Seiringer","submitted_at":"2011-07-02T19:06:21Z","abstract_excerpt":"We study the BCS gap equation for a Fermi gas with unequal population of spin-up and spin-down states. For $\\cosh(\\delta_\\mu/T) \\leq 2$, with $T$ the temperature and $\\delta_\\mu$ the chemical potential difference, the question of existence of non-trivial solutions can be reduced to spectral properties of a linear operator, similar to the unpolarized case studied previously in \\cite{FHNS,HHSS,HS}. For $\\cosh(\\delta_\\mu/T) > 2$ the phase diagram is more complicated, however. We derive upper and lower bounds for the critical temperature, and study their behavior in the small coupling limit."},"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":"1107.0405","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-07-02T19:06:21Z","cross_cats_sorted":["cond-mat.quant-gas","cond-mat.supr-con","math.MP"],"title_canon_sha256":"3ba139627a3fe0525ae94ad29afdce93b7ce224e10ee5f5c03ba4ab6e9520917","abstract_canon_sha256":"043270a274f2f45d175b19aef8ab7f02be37b2543a715c9ae04f4eaeacff5d02"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:01:25.573692Z","signature_b64":"D+ZbEi3hxp9cfHGkYqFZETyK7uJJdTMpMPllTpzc0FxNm1sAMUmmRKb3SfypvqJIiIjjVq5MqfPrNT28bnwyBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2befcc8c35038d41c2d3632ef3982628eaad10ab4552c21148687c0f8bb3dbff","last_reissued_at":"2026-05-18T02:01:25.573187Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:01:25.573187Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The BCS gap equation for spin-polarized fermions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.quant-gas","cond-mat.supr-con","math.MP"],"primary_cat":"math-ph","authors_text":"Abraham Freiji, Christian Hainzl, Robert Seiringer","submitted_at":"2011-07-02T19:06:21Z","abstract_excerpt":"We study the BCS gap equation for a Fermi gas with unequal population of spin-up and spin-down states. For $\\cosh(\\delta_\\mu/T) \\leq 2$, with $T$ the temperature and $\\delta_\\mu$ the chemical potential difference, the question of existence of non-trivial solutions can be reduced to spectral properties of a linear operator, similar to the unpolarized case studied previously in \\cite{FHNS,HHSS,HS}. For $\\cosh(\\delta_\\mu/T) > 2$ the phase diagram is more complicated, however. We derive upper and lower bounds for the critical temperature, and study their behavior in the small coupling limit."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.0405","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1107.0405","created_at":"2026-05-18T02:01:25.573261+00:00"},{"alias_kind":"arxiv_version","alias_value":"1107.0405v1","created_at":"2026-05-18T02:01:25.573261+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.0405","created_at":"2026-05-18T02:01:25.573261+00:00"},{"alias_kind":"pith_short_12","alias_value":"FPX4ZDBVAOGU","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_16","alias_value":"FPX4ZDBVAOGUDQWT","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_8","alias_value":"FPX4ZDBV","created_at":"2026-05-18T12:26:28.662955+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FPX4ZDBVAOGUDQWTMMXPHGBGFD","json":"https://pith.science/pith/FPX4ZDBVAOGUDQWTMMXPHGBGFD.json","graph_json":"https://pith.science/api/pith-number/FPX4ZDBVAOGUDQWTMMXPHGBGFD/graph.json","events_json":"https://pith.science/api/pith-number/FPX4ZDBVAOGUDQWTMMXPHGBGFD/events.json","paper":"https://pith.science/paper/FPX4ZDBV"},"agent_actions":{"view_html":"https://pith.science/pith/FPX4ZDBVAOGUDQWTMMXPHGBGFD","download_json":"https://pith.science/pith/FPX4ZDBVAOGUDQWTMMXPHGBGFD.json","view_paper":"https://pith.science/paper/FPX4ZDBV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1107.0405&json=true","fetch_graph":"https://pith.science/api/pith-number/FPX4ZDBVAOGUDQWTMMXPHGBGFD/graph.json","fetch_events":"https://pith.science/api/pith-number/FPX4ZDBVAOGUDQWTMMXPHGBGFD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FPX4ZDBVAOGUDQWTMMXPHGBGFD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FPX4ZDBVAOGUDQWTMMXPHGBGFD/action/storage_attestation","attest_author":"https://pith.science/pith/FPX4ZDBVAOGUDQWTMMXPHGBGFD/action/author_attestation","sign_citation":"https://pith.science/pith/FPX4ZDBVAOGUDQWTMMXPHGBGFD/action/citation_signature","submit_replication":"https://pith.science/pith/FPX4ZDBVAOGUDQWTMMXPHGBGFD/action/replication_record"}},"created_at":"2026-05-18T02:01:25.573261+00:00","updated_at":"2026-05-18T02:01:25.573261+00:00"}