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The (coskx\u2212cosky)\u03c1y\u03c30 term gaps all side Dirac cones whereas the altermagnetic term induces anisotropic momentum shifts as we show below.","type":"other"},{"key":"hz7z-m9tnCc58R1","doi-asserted-by":"publisher","DOI":"10.1126\/sciadv.aat0346","type":"journal-article"},{"key":"hz7z-m9tnCc59R1","unstructured":"See Supplemental Material at http:\/\/link.aps.org\/supplemental\/10.1103\/hz7z-m9tn for details about (Sec.\u00a0S1) Bogoliubov Fermi surfaces in superconducting topological altermagnets, (Sec.\u00a0S2) Bogoliubov Fermi surfaces in the surface spectrum of a superconducting topological altermagnet, (Sec.\u00a0S3) energy scales and experimental feasibility, (Sec.\u00a0S4) topological phase transition driven by altermagnetism at the surface, (Sec.\u00a0S5) spatial shift of Majorana zero mode in a vortex line along the x direction, and (Sec.\u00a0S6) a discussion on the possible g-wave altermagnetism topological insulator phase in EuIn2As2 and the realization of topological superconductivity, which includes Refs.\u00a0[6,60\u201363].","type":"other"},{"key":"hz7z-m9tnCc60R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.98.224509","type":"journal-article"},{"key":"hz7z-m9tnCc61R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.122.256402","type":"journal-article"},{"key":"hz7z-m9tnCc62R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.108.075150","type":"journal-article"},{"key":"hz7z-m9tnCc63R1","doi-asserted-by":"publisher","DOI":"10.1103\/k9p4-tfhd","type":"journal-article"},{"key":"hz7z-m9tnCc64R1","unstructured":"This 1D winding number is computed along the kz direction for a fixed in-plane momentum (kx,ky). More details can be found in the Ref.\u00a0[59].","type":"other"},{"key":"hz7z-m9tnCc65R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.88.035005","type":"journal-article"},{"key":"hz7z-m9tnCc66R1","unstructured":"This gapless BFS demonstrates diverse topological structures, such as a sphere and torus, when the chemical potential lies in the bulk bands and the condition tAM>tcb=12\u03942+(\u03bc\u2212|m\u2212t|)2 is satisfied (see Ref.\u00a0[59]).","type":"other"},{"key":"hz7z-m9tnCc67R1","doi-asserted-by":"publisher","DOI":"10.1103\/65q6-5wxl","type":"journal-article"},{"key":"hz7z-m9tnCc68R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.110.245417","type":"journal-article"},{"key":"hz7z-m9tnCc69R1","doi-asserted-by":"publisher","DOI":"10.1103\/dlpb-gfct","type":"journal-article"},{"key":"hz7z-m9tnCc70R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-020-14357-2","type":"journal-article"},{"key":"hz7z-m9tnCc71R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.104.094529","type":"journal-article"},{"key":"hz7z-m9tnCc72R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.127.257002","type":"journal-article"},{"key":"hz7z-m9tnCc73R1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.2119548119","type":"journal-article"},{"key":"hz7z-m9tnCc74R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.111.L140504","type":"journal-article"},{"key":"hz7z-m9tnCc75R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.96.195309","type":"journal-article"},{"key":"hz7z-m9tnCc76R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-020-20690-3","type":"journal-article"},{"key":"hz7z-m9tnCc77R1","unstructured":"The Z2 invariant can be calculated following Kitaev\u2019s formula as (\u22121)\u03bd=sgn(Pf[iH(kz=0)]Pf[iH(kz=\u03c0)]) from Pfaffian of the Hamiltonian at two points.","type":"other"},{"key":"hz7z-m9tnCc78R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.123.027003","type":"journal-article"},{"key":"hz7z-m9tnCc79R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.129.277001","type":"journal-article"},{"key":"hz7z-m9tnCc80R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-023-36347-w","type":"journal-article"},{"key":"hz7z-m9tnCc81R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.111.144501","type":"journal-article"},{"key":"hz7z-m9tnCc82R1","unstructured":"Note that the phase region (I) also appears in superconducting vortices of doped topological insulators [83], while the phase region (IV) is unique due to the presence of altermagnetic order.","type":"other"},{"key":"hz7z-m9tnCc83R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.107.097001","type":"journal-article"},{"key":"hz7z-m9tnCc84R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.124.257001","type":"journal-article"},{"key":"hz7z-m9tnCc85R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.125.037001","type":"journal-article"},{"key":"hz7z-m9tnCc86R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.100.096407","type":"journal-article"},{"key":"hz7z-m9tnCc87R1","unstructured":"Note that the eigenstates from different C2z sectors presented in Fig.\u00a04 will not hybridize. Thus the MZMs are\u00a0still protected by a superconducting energy gap as large\u00a0as \u0394.","type":"other"},{"key":"hz7z-m9tnCc88R1","unstructured":"Reference\u00a0[51] suggests to substitute the uniform magnetic field or ferromagnetic order in previous proposals, such as 1D semiconductor nanowires and 2D chiral superconductors, by altermagnetic order. References\u00a0[52, 53] utilize altermagnetic order to create higher-order topology and Majorana corner modes in 2D systems.","type":"other"},{"key":"hz7z-m9tnCc89R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-025-02822-y","type":"journal-article"},{"key":"hz7z-m9tnCc90R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-025-02864-2","type":"journal-article"},{"key":"hz7z-m9tnCc91R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-024-48493-w","type":"journal-article"},{"key":"hz7z-m9tnCc92R1","doi-asserted-by":"publisher","DOI":"10.1126\/sciadv.adn0479","type":"journal-article"},{"key":"hz7z-m9tnCc93R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.2.033342","type":"journal-article"},{"key":"hz7z-m9tnCc94R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-021-21154-y","type":"journal-article"},{"key":"hz7z-m9tnCc95R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.114.017001","type":"journal-article"},{"key":"hz7z-m9tnCc96R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-024-48610-9","type":"journal-article"}],"container-title":["Physical Review Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/article\/10.1103\/hz7z-m9tn","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/hz7z-m9tn\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T18:22:30Z","timestamp":1790878950000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/hz7z-m9tn"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,26]]},"references-count":94,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2026,8]]}},"URL":"https:\/\/doi.org\/10.1103\/hz7z-m9tn","relation":{},"ISSN":["0031-9007","1079-7114"],"issn-type":[{"value":"0031-9007","type":"print"},{"value":"1079-7114","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,26]]},"assertion":[{"value":"2025-11-26","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":"096604"}}