{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T18:57:23Z","timestamp":1790881043070,"version":"4.1.0"},"reference-count":39,"publisher":"American Physical Society (APS)","issue":"25","license":[{"start":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T00:00:00Z","timestamp":1782172800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/link.aps.org\/licenses\/aps-default-license"}],"funder":[{"name":"Laboratoire d\u2019excellence LANEF in Grenoble","award":["ANR-10-LABX-51-01"],"award-info":[{"award-number":["ANR-10-LABX-51-01"]}]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"short-container-title":["Phys. Rev. Lett."],"accepted":{"date-parts":[[2026,5,21]]},"DOI":"10.1103\/923y-49z5","type":"journal-article","created":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T12:17:58Z","timestamp":1779365878000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["Dissipation due to Bulk Localized Low-Energy Modes in Strongly Disordered Superconductors"],"prefix":"10.1103","volume":"136","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0442-6039","authenticated-orcid":true,"given":"Anton V.","family":"Khvalyuk","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02mc6qk71","id-type":"ROR","asserted-by":"publisher"}],"name":"LPMMC"},{"id":[{"id":"https:\/\/ror.org\/02rx3b187","id-type":"ROR","asserted-by":"publisher"}],"name":"Universit\u00e9 Grenoble Alpes"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7114-5296","authenticated-orcid":true,"given":"Mikhail V.","family":"Feigel\u2019man","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02s54wa56","id-type":"ROR","asserted-by":"publisher"}],"name":"Nanocenter"},{"id":[{"id":"https:\/\/ror.org\/05060sz93","id-type":"ROR","asserted-by":"publisher"}],"name":"Jozef Stefan Institute"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"16","published-online":{"date-parts":[[2026,6,23]]},"reference":[{"key":"923y-49z5Cc1R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.111.412","type":"journal-article"},{"key":"923y-49z5Cc2R1","doi-asserted-by":"publisher","DOI":"10.1088\/1367-2630\/7\/1\/219","type":"journal-article"},{"key":"923y-49z5Cc3R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.104.027002","type":"journal-article"},{"key":"923y-49z5Cc4R1","doi-asserted-by":"publisher","DOI":"10.1088\/0034-4885\/75\/7\/072001","type":"journal-article"},{"key":"923y-49z5Cc5R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.87.052306","type":"journal-article"},{"key":"923y-49z5Cc6R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41598-018-27154-1","type":"journal-article"},{"key":"923y-49z5Cc7R1","doi-asserted-by":"publisher","DOI":"10.1088\/1367-2630\/aab7cd","type":"journal-article"},{"key":"923y-49z5Cc8R1","doi-asserted-by":"publisher","DOI":"10.1088\/0953-2048\/30\/2\/025002","type":"journal-article"},{"key":"923y-49z5Cc9R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.121.117001","type":"journal-article"},{"key":"923y-49z5Cc10R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevApplied.11.044014","type":"journal-article"},{"key":"923y-49z5Cc11R1","doi-asserted-by":"publisher","DOI":"10.1038\/nphys234","type":"journal-article"},{"key":"923y-49z5Cc12R1","doi-asserted-by":"publisher","DOI":"10.1038\/nature10930","type":"journal-article"},{"key":"923y-49z5Cc13R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-018-0097-9","type":"journal-article"},{"key":"923y-49z5Cc14R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41586-022-04947-z","type":"journal-article"},{"key":"923y-49z5Cc16R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.aop.2010.04.001","type":"journal-article"},{"key":"923y-49z5Cc17R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-020-0905-x","type":"journal-article"},{"key":"923y-49z5Cc18R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.101.157006","type":"journal-article"},{"key":"923y-49z5Cc19R1","doi-asserted-by":"publisher","DOI":"10.1038\/nphys1892","type":"journal-article"},{"key":"923y-49z5Cc20R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-018-0365-8","type":"journal-article"},{"key":"923y-49z5Cc21R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.104.224505","type":"journal-article"},{"key":"923y-49z5Cc22R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.82.184534","type":"journal-article"},{"key":"923y-49z5Cc23R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-024-02713-8","type":"journal-article"},{"key":"923y-49z5Cc24R1","unstructured":"T. Charpentier, Quantum circuits and the superconductor-insulator transition in a strongly disordered superconductor, Ph.D. thesis, Universit\u00e9 Grenoble Alpes, 2023, https:\/\/theses.hal.science\/tel-04405951.","type":"other"},{"key":"923y-49z5Cc25R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.109.144501","type":"journal-article"},{"key":"923y-49z5Cc26R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.32.5658","type":"journal-article"},{"key":"923y-49z5Cc27R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.65.014501","type":"journal-article"},{"key":"923y-49z5Cc28R1","unstructured":"In terms of original electronic operators, Sj\u2212=cj,\u2193cj,\u2191, Sj+=cj,\u2191+cj,\u2193+, Sjz=cj,\u2191+cj,\u2191+cj,\u2193+cj,\u2193\u221212.","type":"other"},{"key":"923y-49z5Cc29R1","unstructured":"A. Khvalyuk, Low-energy theory of strongly disordered superconductors, Ph.D. thesis, Universit\u00e9 Grenoble Alpes, 2025, https:\/\/theses.hal.science\/tel-05576189.","type":"other"},{"key":"923y-49z5Cc30R1","unstructured":"At finite \u03c9>0, Eq.\u00a0(7) also contains the nonlocal contributions from the phase differences on edges e other than \u27e8ij\u27e9; these terms can be naively estimated as (\u03c9\/\u03940)2d, where d is the distance between e and \u27e8ij\u27e9 on the graph. See Ref.\u00a0[25], Appendix\u00a0B.","type":"other"},{"key":"923y-49z5Cc31R1","unstructured":"The use of the 2D geometry instead of the 3D one is a technical compromise to achieve convergence of disorder averages. The qualitative results of the analysis are expected to hold both in two and three dimensions since the relevant dissipative processes are inherently microscopic.","type":"other"},{"key":"923y-49z5Cc32R1","unstructured":"The model parameters for the data in Fig.\u00a01, left, were carefully chosen to maximize mtree while still being within the range of applicability of the employed numerical methods, as detailed in Ref.\u00a0[29], Chap.\u00a05.","type":"other"},{"key":"923y-49z5Cc33R1","doi-asserted-by":"publisher","DOI":"10.1038\/srep02979","type":"journal-article"},{"key":"923y-49z5Cc34R1","doi-asserted-by":"publisher","DOI":"10.1038\/nphys2037","type":"journal-article"},{"key":"923y-49z5Cc35R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.87.184509","type":"journal-article"},{"key":"923y-49z5Cc36R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.107.L140502","type":"journal-article"},{"key":"923y-49z5Cc37R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.109.1398","type":"journal-article"},{"key":"923y-49z5Cc38R1","doi-asserted-by":"publisher","DOI":"10.21468\/SciPostPhys.17.2.066","type":"journal-article"},{"key":"923y-49z5Cc39R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.99.094506","type":"journal-article"},{"key":"923y-49z5Cc40R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.132.217002","type":"journal-article"}],"container-title":["Physical Review Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/article\/10.1103\/923y-49z5","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/923y-49z5\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T17:55:14Z","timestamp":1790877314000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/923y-49z5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,23]]},"references-count":39,"journal-issue":{"issue":"25","published-print":{"date-parts":[[2026,6]]}},"URL":"https:\/\/doi.org\/10.1103\/923y-49z5","relation":{},"ISSN":["0031-9007","1079-7114"],"issn-type":[{"value":"0031-9007","type":"print"},{"value":"1079-7114","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,23]]},"assertion":[{"value":"2025-12-13","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"\u00a9 2026 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":"2026","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":"256001"}}