{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T21:11:31Z","timestamp":1790716291028,"version":"4.1.0"},"reference-count":50,"publisher":"American Physical Society (APS)","issue":"24","license":[{"start":{"date-parts":[[2025,6,16]],"date-time":"2025-06-16T00:00:00Z","timestamp":1750032000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/link.aps.org\/licenses\/aps-default-license"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12222406"],"award-info":[{"award-number":["12222406"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004608","name":"Natural Science Foundation of Jiangsu Province","doi-asserted-by":"publisher","award":["BK20233001"],"award-info":[{"award-number":["BK20233001"]}],"id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["2024300415"],"award-info":[{"award-number":["2024300415"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2022YFA1204701"],"award-info":[{"award-number":["2022YFA1204701"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"short-container-title":["Phys. Rev. Lett."],"accepted":{"date-parts":[[2025,5,27]]},"DOI":"10.1103\/vvrx-mljg","type":"journal-article","created":{"date-parts":[[2025,5,27]],"date-time":"2025-05-27T14:57:23Z","timestamp":1748357843000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":13,"title":["Quantum-Classical Correspondence of Non-Hermitian Symmetry Breaking"],"prefix":"10.1103","volume":"134","author":[{"given":"Zhuo-Ting","family":"Cai","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/05ryc2b20","id-type":"ROR","asserted-by":"publisher"}],"name":"National Laboratory of Solid State Microstructures"},{"id":[{"id":"https:\/\/ror.org\/01rxvg760","id-type":"ROR","asserted-by":"publisher"}],"name":"Nanjing University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9120-2761","authenticated-orcid":true,"given":"Hai-Dong","family":"Li","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/05ryc2b20","id-type":"ROR","asserted-by":"publisher"}],"name":"National Laboratory of Solid State Microstructures"},{"id":[{"id":"https:\/\/ror.org\/01rxvg760","id-type":"ROR","asserted-by":"publisher"}],"name":"Nanjing University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4807-205X","authenticated-orcid":true,"given":"Wei","family":"Chen","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/05ryc2b20","id-type":"ROR","asserted-by":"publisher"}],"name":"National Laboratory of Solid State Microstructures"},{"id":[{"id":"https:\/\/ror.org\/01rxvg760","id-type":"ROR","asserted-by":"publisher"}],"name":"Nanjing University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"16","published-online":{"date-parts":[[2025,6,16]]},"reference":[{"key":"vvrx-mljgCc1R1","doi-asserted-by":"publisher","DOI":"10.1088\/0034-4885\/70\/6\/R03","type":"journal-article"},{"key":"vvrx-mljgCc2R1","doi-asserted-by":"publisher","DOI":"10.1080\/00018732.2021.1876991","type":"journal-article"},{"key":"vvrx-mljgCc3R1","doi-asserted-by":"publisher","DOI":"10.1038\/nphys4323","type":"journal-article"},{"key":"vvrx-mljgCc4R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.80.5243","type":"journal-article"},{"key":"vvrx-mljgCc5R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.103.093902","type":"journal-article"},{"key":"vvrx-mljgCc6R1","doi-asserted-by":"publisher","DOI":"10.1038\/nphys1515","type":"journal-article"},{"key":"vvrx-mljgCc7R1","doi-asserted-by":"publisher","DOI":"10.1038\/nature11298","type":"journal-article"},{"key":"vvrx-mljgCc8R1","doi-asserted-by":"publisher","DOI":"10.1038\/nphys2927","type":"journal-article"},{"key":"vvrx-mljgCc9R1","doi-asserted-by":"publisher","DOI":"10.1126\/science.1258479","type":"journal-article"},{"key":"vvrx-mljgCc10R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41563-019-0304-9","type":"journal-article"},{"key":"vvrx-mljgCc11R1","doi-asserted-by":"publisher","DOI":"10.1038\/nmat3495","type":"journal-article"},{"key":"vvrx-mljgCc12R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41566-017-0031-1","type":"journal-article"},{"key":"vvrx-mljgCc13R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.122.237601","type":"journal-article"},{"key":"vvrx-mljgCc14R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.1.023013","type":"journal-article"},{"key":"vvrx-mljgCc15R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.113.250401","type":"journal-article"},{"key":"vvrx-mljgCc16R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.129.113601","type":"journal-article"},{"key":"vvrx-mljgCc17R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.100.054301","type":"journal-article"},{"key":"vvrx-mljgCc18R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.63.036222","type":"journal-article"},{"key":"vvrx-mljgCc19R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.104.024201","type":"journal-article"},{"key":"vvrx-mljgCc20R1","doi-asserted-by":"publisher","DOI":"10.1126\/science.aar7709","type":"journal-article"},{"key":"vvrx-mljgCc21R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.69.056216","type":"journal-article"},{"key":"vvrx-mljgCc22R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.93.015005","type":"journal-article"},{"key":"vvrx-mljgCc23R1","doi-asserted-by":"publisher","DOI":"10.1038\/nature23281","type":"journal-article"},{"key":"vvrx-mljgCc24R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.121.086803","type":"journal-article"},{"key":"vvrx-mljgCc25R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.121.026808","type":"journal-article"},{"key":"vvrx-mljgCc26R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.123.066404","type":"journal-article"},{"key":"vvrx-mljgCc27R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.124.056802","type":"journal-article"},{"key":"vvrx-mljgCc28R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.124.086801","type":"journal-article"},{"key":"vvrx-mljgCc29R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.125.126402","type":"journal-article"},{"key":"vvrx-mljgCc30R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41586-021-03375-9","type":"journal-article"},{"key":"vvrx-mljgCc31R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.112.203901","type":"journal-article"},{"key":"vvrx-mljgCc32R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-019-12599-3","type":"journal-article"},{"key":"vvrx-mljgCc33R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41928-019-0284-4","type":"journal-article"},{"key":"vvrx-mljgCc34R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.103.L241408","type":"journal-article"},{"key":"vvrx-mljgCc35R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.106.024301","type":"journal-article"},{"key":"vvrx-mljgCc36R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.132.156301","type":"journal-article"},{"key":"vvrx-mljgCc37R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.107.035306","type":"journal-article"},{"key":"vvrx-mljgCc38R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1418246","type":"journal-article"},{"key":"vvrx-mljgCc39R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1703672","type":"journal-article"},{"key":"vvrx-mljgCc40R1","unstructured":"In the pioneering work [4], Bender and Boettcher interpreted the PT transition from a semiclassical perspective. Using a specific model, they established a connection between the real or complex nature of the entire spectrum and the closure or openness of the semiclassical orbits. Our work aims to uncover a universal mechanism that applies to a broad class of systems with general pseudo-Hermitian symmetries, and can precisely describe non-Hermitian spectral properties at the level of individual states in both symmetric and symmetry-breaking phases.","type":"other"},{"key":"vvrx-mljgCc41R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1666102","type":"journal-article"},{"key":"vvrx-mljgCc42R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.106.L081402","type":"journal-article"},{"key":"vvrx-mljgCc43R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.109.045110","type":"journal-article"},{"key":"vvrx-mljgCc44R1","unstructured":"See Supplemental Material at http:\/\/link.aps.org\/supplemental\/10.1103\/vvrx-mljg for the derivations of the restriction on the propagator by \u03b7-pseudo-Hermiticity, complex path integral and classical S\u03b7 symmetry, the non-Hermitian Gutzwiller trace formula, and symmetry of semiclassical orbits and its constraint on energy spectrum, the calculation details of specific examples of continuous system, the discussion on PT transition in two-level systems and physics near the exceptional point, which includes Ref.\u00a0[45\u201349].","type":"other"},{"key":"vvrx-mljgCc45R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1665596","type":"journal-article"},{"key":"vvrx-mljgCc46R1","volume-title":"Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory","author":"R. Rajaraman","year":"1987","type":"book"},{"key":"vvrx-mljgCc47R1","doi-asserted-by":"publisher","DOI":"10.1016\/0550-3213(88)90545-7","type":"journal-article"},{"key":"vvrx-mljgCc48R1","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511789984","type":"book","volume-title":"Condensed Matter Field Theory","author":"A. Altland","year":"2010"},{"key":"vvrx-mljgCc49R1","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4916(85)90021-1","type":"journal-article"},{"key":"vvrx-mljgCc50R1","volume-title":"Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory","author":"C.\u2009M. Bender","year":"2013","type":"book"}],"container-title":["Physical Review Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/article\/10.1103\/vvrx-mljg","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/vvrx-mljg\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T20:20:59Z","timestamp":1790713259000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/vvrx-mljg"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,16]]},"references-count":50,"journal-issue":{"issue":"24","published-print":{"date-parts":[[2025,6]]}},"URL":"https:\/\/doi.org\/10.1103\/vvrx-mljg","relation":{},"ISSN":["0031-9007","1079-7114"],"issn-type":[{"value":"0031-9007","type":"print"},{"value":"1079-7114","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,16]]},"assertion":[{"value":"2024-10-13","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"\u00a9 2025 American Physical Society. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright_statement","label":"Copyright statement","group":{"name":"copyright","label":"copyright"}},{"value":"2025","name":"copyright_year","label":"Copyright year","group":{"name":"copyright","label":"copyright"}},{"value":"American Physical Society","name":"copyright_holder","label":"Copyright holder","group":{"name":"copyright","label":"copyright"}}],"article-number":"240201"}}