{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T19:01:12Z","timestamp":1790881272146,"version":"4.1.0"},"reference-count":84,"publisher":"American Physical Society (APS)","issue":"1","license":[{"start":{"date-parts":[[2026,7,21]],"date-time":"2026-07-21T00:00:00Z","timestamp":1784592000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/link.aps.org\/licenses\/aps-default-license"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["26K06945"],"award-info":[{"award-number":["26K06945"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"short-container-title":["Phys. Rev. E"],"accepted":{"date-parts":[[2026,7,1]]},"DOI":"10.1103\/8jxd-l7rp","type":"journal-article","created":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T15:37:05Z","timestamp":1782920225000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Stability and interactions of hopfions in confined chiral systems: Role of the background phase"],"prefix":"10.1103","volume":"114","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7100-7144","authenticated-orcid":true,"given":"Andrey O.","family":"Leonov","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/03t78wx29","id-type":"ROR","asserted-by":"publisher"}],"name":"Hiroshima University"},{"id":[{"id":"https:\/\/ror.org\/03t78wx29","id-type":"ROR","asserted-by":"publisher"}],"name":"Hiroshima University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"16","published-online":{"date-parts":[[2026,7,21]]},"reference":[{"key":"8jxd-l7rpCc1R1","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511617034","type":"book","volume-title":"Topological Solitons","author":"N. Manton","year":"2004"},{"key":"8jxd-l7rpCc2R1","doi-asserted-by":"crossref","DOI":"10.1017\/9781108555623","type":"book","volume-title":"Topological and Non-Topological Solitons in Scalar Field Theories","author":"Y.\u00a0M. Shnir","year":"2018"},{"key":"8jxd-l7rpCc3R1","first-page":"767","type":"journal-article","volume":"73","author":"G. Volovik","year":"1977","journal-title":"Zh. Eksp. Teor. Fiz."},{"key":"8jxd-l7rpCc3R2","first-page":"401","type":"journal-article","volume":"46","author":"G. Volovik","year":"1977","journal-title":"Sov. Phys. JETP"},{"key":"8jxd-l7rpCc4R1","volume-title":"Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory","author":"R. Rajaraman","year":"1982","type":"book"},{"key":"8jxd-l7rpCc5R1","doi-asserted-by":"publisher","DOI":"10.1103\/s8xh-m9qb","type":"journal-article"},{"key":"8jxd-l7rpCc6R1","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8121\/aad521","type":"journal-article"},{"key":"8jxd-l7rpCc7R1","doi-asserted-by":"publisher","DOI":"10.1038\/387058a0","type":"journal-article"},{"key":"8jxd-l7rpCc8R1","volume-title":"Differential Forms in Algebraic Topology","author":"R. Bott","year":"2013","type":"book"},{"key":"8jxd-l7rpCc9R1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.33.5.117","type":"journal-article"},{"key":"8jxd-l7rpCc10R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.56.5194","type":"journal-article"},{"key":"8jxd-l7rpCc11R1","doi-asserted-by":"publisher","DOI":"10.1126\/sciadv.aao3820","type":"journal-article"},{"key":"8jxd-l7rpCc12R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.88.177002","type":"journal-article"},{"key":"8jxd-l7rpCc13R1","doi-asserted-by":"publisher","DOI":"10.1126\/science.1148694","type":"journal-article"},{"key":"8jxd-l7rpCc14R1","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8113\/48\/20\/205202","type":"journal-article"},{"key":"8jxd-l7rpCc15R1","doi-asserted-by":"publisher","DOI":"10.1038\/nphys2560","type":"journal-article"},{"key":"8jxd-l7rpCc16R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-021-26171-5","type":"journal-article"},{"key":"8jxd-l7rpCc17R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.65.100512","type":"journal-article"},{"key":"8jxd-l7rpCc18R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.5099991","type":"journal-article"},{"key":"8jxd-l7rpCc19R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.107.104404","type":"journal-article"},{"key":"8jxd-l7rpCc20R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-020-16258-w","type":"journal-article"},{"key":"8jxd-l7rpCc21R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevX.7.011006","type":"journal-article"},{"key":"8jxd-l7rpCc22R1","doi-asserted-by":"publisher","DOI":"10.1038\/nmat4826","type":"journal-article"},{"key":"8jxd-l7rpCc23R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.92.053603","type":"journal-article"},{"key":"8jxd-l7rpCc24R1","doi-asserted-by":"publisher","DOI":"10.1088\/1361-648X\/ab5e3c","type":"journal-article"},{"key":"8jxd-l7rpCc26R1","doi-asserted-by":"publisher","DOI":"10.1088\/0370-1328\/82\/2\/306","type":"journal-article"},{"key":"8jxd-l7rpCc27R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1704233","type":"journal-article"},{"key":"8jxd-l7rpCc28R1","volume-title":"Proceedings of the NATO Advanced Study Institute","author":"P. De Gennes","year":"1975","type":"conference-proceedings"},{"key":"8jxd-l7rpCc29R1","doi-asserted-by":"publisher","DOI":"10.1103\/7nly-96st","type":"journal-article"},{"key":"8jxd-l7rpCc30R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.121.187201","type":"journal-article"},{"key":"8jxd-l7rpCc31R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-026-03236-0","type":"journal-article"},{"key":"8jxd-l7rpCc32R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.90.042502","type":"journal-article"},{"key":"8jxd-l7rpCc33R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.104.044701","type":"journal-article"},{"key":"8jxd-l7rpCc34R1","first-page":"960","type":"journal-article","volume":"19","author":"I. Dzyaloshinskii","year":"1964","journal-title":"Sov. Phys. JETP"},{"key":"8jxd-l7rpCc35R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.120.91","type":"journal-article"},{"key":"8jxd-l7rpCc36R1","doi-asserted-by":"crossref","DOI":"10.1201\/9780203023013","type":"book","volume-title":"Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments","author":"P. Oswald","year":"2005"},{"key":"8jxd-l7rpCc37R1","doi-asserted-by":"crossref","DOI":"10.1007\/b97416","type":"book","volume-title":"Soft Matter Physics: An Introduction","author":"M. Kleman","year":"2003"},{"key":"8jxd-l7rpCc38R1","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-031-70200-6","type":"book","volume-title":"Introduction to Topological Defects and Solitons: In Liquid Crystals, Magnets, and Related Materials","author":"J.\u00a0V. Selinger","year":"2024"},{"key":"8jxd-l7rpCc39R1","volume-title":"Defects in Liquid Crystals: Computer Simulations, Theory and Experiments","author":"S. Shiyanovskii","year":"2001","type":"book"},{"key":"8jxd-l7rpCc40R1","doi-asserted-by":"publisher","DOI":"10.1051\/jphyscol1969413","type":"journal-article"},{"key":"8jxd-l7rpCc41R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.85.094429","type":"journal-article"},{"key":"8jxd-l7rpCc42R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.108.267201","type":"journal-article"},{"key":"8jxd-l7rpCc43R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.89.025008","type":"journal-article"},{"key":"8jxd-l7rpCc44R1","doi-asserted-by":"publisher","DOI":"10.1088\/0034-4885\/59\/11\/002","type":"journal-article"},{"key":"8jxd-l7rpCc45R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.93.064416","type":"journal-article"},{"key":"8jxd-l7rpCc46R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.92.144402","type":"journal-article"},{"key":"8jxd-l7rpCc47R1","doi-asserted-by":"publisher","DOI":"10.3390\/magnetism3040023","type":"journal-article"},{"key":"8jxd-l7rpCc48R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.111.174450","type":"journal-article"},{"key":"8jxd-l7rpCc49R1","doi-asserted-by":"publisher","DOI":"10.1080\/10587250108028283","type":"journal-article"},{"key":"8jxd-l7rpCc50R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.4899186","type":"journal-article"},{"key":"8jxd-l7rpCc51R1","doi-asserted-by":"publisher","DOI":"10.1103\/645q-nz7t","type":"journal-article"},{"key":"8jxd-l7rpCc52R1","doi-asserted-by":"publisher","DOI":"10.1126\/science.1166767","type":"journal-article"},{"key":"8jxd-l7rpCc53R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.5.033033","type":"journal-article"},{"key":"8jxd-l7rpCc54R1","doi-asserted-by":"publisher","DOI":"10.1209\/epl\/i2001-00513-x","type":"journal-article"},{"key":"8jxd-l7rpCc55R1","doi-asserted-by":"publisher","DOI":"10.1038\/ncomms3635","type":"journal-article"},{"key":"8jxd-l7rpCc56R1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1314654110","type":"journal-article"},{"key":"8jxd-l7rpCc57R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1652780","type":"journal-article"},{"key":"8jxd-l7rpCc58R1","doi-asserted-by":"publisher","DOI":"10.1016\/0038-1098(68)90024-0","type":"journal-article"},{"key":"8jxd-l7rpCc59R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.112.217801","type":"journal-article"},{"key":"8jxd-l7rpCc60R1","doi-asserted-by":"publisher","DOI":"10.1364\/OL.43.001850","type":"journal-article"},{"key":"8jxd-l7rpCc61R1","doi-asserted-by":"publisher","DOI":"10.1126\/science.adn6812","type":"journal-article"},{"key":"8jxd-l7rpCc62R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.103.024431","type":"journal-article"},{"key":"8jxd-l7rpCc63R1","doi-asserted-by":"publisher","DOI":"10.1038\/ncomms9193","type":"journal-article"},{"key":"8jxd-l7rpCc64R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.95.094423","type":"journal-article"},{"key":"8jxd-l7rpCc65R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.91.224407","type":"journal-article"},{"key":"8jxd-l7rpCc66R1","doi-asserted-by":"publisher","DOI":"10.1021\/acs.nanolett.2c03106","type":"journal-article"},{"key":"8jxd-l7rpCc68R1","doi-asserted-by":"publisher","DOI":"10.1016\/S0370-1573(00)00056-9","type":"journal-article"},{"key":"8jxd-l7rpCc69R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.59.5562","type":"journal-article"},{"key":"8jxd-l7rpCc70R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.72.061707","type":"journal-article"},{"key":"8jxd-l7rpCc71R1","volume-title":"Les Houches Summer School in Theoretical Physics, 1973: Molecular Fluids","author":"R.\u00a0B. Meyer","year":"1976","type":"book"},{"key":"8jxd-l7rpCc72R1","volume-title":"The Physics of Liquid Crystals","author":"P.-G. De Gennes","year":"1995","type":"book"},{"key":"8jxd-l7rpCc73R1","doi-asserted-by":"publisher","DOI":"10.1080\/15421407608084348","type":"journal-article"},{"key":"8jxd-l7rpCc74R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.80.5742","type":"journal-article"},{"key":"8jxd-l7rpCc75R1","doi-asserted-by":"publisher","DOI":"10.1080\/02678299408029147","type":"journal-article"},{"key":"8jxd-l7rpCc76R1","doi-asserted-by":"publisher","DOI":"10.3390\/nano14060504","type":"journal-article"},{"key":"8jxd-l7rpCc77R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.98.054404","type":"journal-article"},{"key":"8jxd-l7rpCc78R1","doi-asserted-by":"publisher","DOI":"10.1088\/0953-8984\/28\/35\/35LT01","type":"journal-article"},{"key":"8jxd-l7rpCc79R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.97.134403","type":"journal-article"},{"key":"8jxd-l7rpCc80R1","doi-asserted-by":"publisher","DOI":"10.1088\/1361-648X\/ab9bc8","type":"journal-article"},{"key":"8jxd-l7rpCc81R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.105.094404","type":"journal-article"},{"key":"8jxd-l7rpCc82R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cpc.2015.07.001","type":"journal-article"},{"key":"8jxd-l7rpCc83R1","first-page":"14","type":"journal-article","volume":"29","author":"H. Harborth","year":"1974","journal-title":"Elem. Math."},{"key":"8jxd-l7rpCc84R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.106.224428","type":"journal-article"},{"key":"8jxd-l7rpCc85R1","doi-asserted-by":"publisher","DOI":"10.1088\/1367-2630\/18\/6\/065003","type":"journal-article"}],"container-title":["Physical Review E"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/article\/10.1103\/8jxd-l7rp","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/8jxd-l7rp\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T18:07:43Z","timestamp":1790878063000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/8jxd-l7rp"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,21]]},"references-count":84,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,7]]}},"URL":"https:\/\/doi.org\/10.1103\/8jxd-l7rp","relation":{},"ISSN":["2470-0045","2470-0053"],"issn-type":[{"value":"2470-0045","type":"print"},{"value":"2470-0053","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,21]]},"assertion":[{"value":"2026-04-06","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"\u00a92026 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":"015416"}}