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Rev. Lett."],"accepted":{"date-parts":[[2026,4,14]]},"abstract":"<jats:p>\n                    Superconducting altermagnets have proven to be a promising ground for emergent phenomena, but their study has involved two-dimensional systems. In this work, we investigate three-dimensional\n                    <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n                      <a:mi>d<\/a:mi>\n                    <\/a:math>\n                    - and\n                    <c:math xmlns:c=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n                      <c:mi>g<\/c:mi>\n                    <\/c:math>\n                    -wave altermagnets with spin-singlet chiral\n                    <e:math xmlns:e=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n                      <e:mi>d<\/e:mi>\n                    <\/e:math>\n                    -wave superconductivity and show the formation of crossed surface flat bands due to the interplay between superconducting and altermagnetic symmetries. We find that these crossed flat bands are topologically protected, appear at zero energy in the surface along\n                    <g:math xmlns:g=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n                      <g:mi>z<\/g:mi>\n                    <\/g:math>\n                    due to the superconducting nodal lines in the\n                    <i:math xmlns:i=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n                      <i:mi>x<\/i:mi>\n                      <i:mi>y<\/i:mi>\n                    <\/i:math>\n                    plane, and their number of corners is determined by the crystal symmetry of altermagnets. We also show that the superconducting nodal lines give rise to Bogoliubov-Fermi surfaces, which then affect the appearance of zero-energy arcs in the surface along\n                    <k:math xmlns:k=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n                      <k:mi>x<\/k:mi>\n                    <\/k:math>\n                    . Moreover, we demonstrate that the crossed flat bands or surface arcs, and Bogoliubov-Fermi surfaces give rise to the coexistence of three distinct dependences of the charge conductance on the normal transparency, hence offering a solid way for their detection and paving the way for realizing higher-dimensional topological phases using altermagnets.\n                  <\/jats:p>","DOI":"10.1103\/65q6-5wxl","type":"journal-article","created":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T16:54:42Z","timestamp":1776185682000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":7,"title":["Crossed Surface Flat Bands in Three-Dimensional Superconducting Altermagnets"],"prefix":"10.1103","volume":"136","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8535-8233","authenticated-orcid":true,"given":"Yuri","family":"Fukaya","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02pc6pc55","id-type":"ROR","asserted-by":"publisher"}],"name":"Okayama University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4753-809X","authenticated-orcid":true,"given":"Bo","family":"Lu","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/012tb2g32","id-type":"ROR","asserted-by":"publisher"}],"name":"Tianjin University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Keiji","family":"Yada","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/04chrp450","id-type":"ROR","asserted-by":"publisher"}],"name":"Nagoya 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Supplemental Material at http:\/\/link.aps.org\/supplemental\/10.1103\/65q6-5wxl for the details, which includes Refs.\u00a0[76\u201380].","type":"other"},{"key":"65q6-5wxlCc76R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.63.239","type":"journal-article"},{"key":"65q6-5wxlCc77R1","doi-asserted-by":"publisher","DOI":"10.1143\/JPSJ.64.922","type":"journal-article"},{"key":"65q6-5wxlCc78R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.62.113","type":"journal-article"},{"key":"65q6-5wxlCc79R1","volume-title":"Superconductivity","author":"J.\u2009B. Ketterson","year":"1990","type":"book"},{"key":"65q6-5wxlCc80R1","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198507550.001.0001","type":"book","volume-title":"Superconductivity, superfluids and Condensates","author":"J.\u2009F. Annett","year":"2004"},{"key":"65q6-5wxlCc81R1","unstructured":"The chosen values of the superconducting and altermagnetic amplitudes are obtained by ensuring the coexistence of altermagnetism and 3D chiral d-wave superconductivity in the self-consistent gap equation [75]. This allowed us to choose altermagnetic and superconducting amplitudes of the same order.","type":"other"},{"key":"65q6-5wxlCc82R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.110.L060508","type":"journal-article"},{"key":"65q6-5wxlCc83R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.111.224503","type":"journal-article"},{"key":"65q6-5wxlCc84R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.92.214514","type":"journal-article"},{"key":"65q6-5wxlCc85R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.83.064505","type":"journal-article"},{"key":"65q6-5wxlCc86R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.102.075430","type":"journal-article"},{"key":"65q6-5wxlCc87R1","unstructured":"This topological protection does not happen in the case when BFS exists because the winding number is ill-defined for the gapless dispersion. Thus, the surface arc state at kz\u223c0 is not expected to produce resonant states at zero energy.","type":"other"},{"key":"65q6-5wxlCc88R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.47.882","type":"journal-article"},{"key":"65q6-5wxlCc89R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.110.104515","type":"journal-article"},{"key":"65q6-5wxlCc90R1","doi-asserted-by":"publisher","DOI":"10.1007\/s10948-025-07064-w","type":"journal-article"},{"key":"65q6-5wxlCc91R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.55.5266","type":"journal-article"},{"key":"65q6-5wxlCc92R1","doi-asserted-by":"publisher","DOI":"10.1088\/1367-2630\/15\/7\/075019","type":"journal-article"},{"key":"65q6-5wxlCc93R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.101.024509","type":"journal-article"},{"key":"65q6-5wxlCc94R1","unstructured":"The characteristic energy scale that determines the conductance profile originates from the relative amplitude of the altermagnetic strengths t\u03b1 and the pair potential \u0394.","type":"other"},{"key":"65q6-5wxlCc95R1","unstructured":"The BFS in our study is similar to that in the original multiorbital model of BFS studies [96, 97, 98] from the point of view of time-reversal symmetry breaking. However, in our case, we adopt a single-orbital model, and the time-reversal symmetry breaking is entirely due to altermagnetism, which produces a zero net magnetization, hence being different from conventional magnetism needed in initial studies of BFSs [96, 97, 98].","type":"other"},{"key":"65q6-5wxlCc96R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.98.224509","type":"journal-article"},{"key":"65q6-5wxlCc97R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.118.127001","type":"journal-article"},{"key":"65q6-5wxlCc98R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.96.094526","type":"journal-article"},{"key":"65q6-5wxlCc99R1","unstructured":"We note that \u03c3\u00af(k\u02dc,eV=0) given by 2\u03c3\u00afN0 and \u03c3\u00afN2 originates due to perfect resonance of Andreev reflections [55, 60, 101] and Andreev reflection without resonance [60, 100].","type":"other"},{"key":"65q6-5wxlCc100R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.69.731","type":"journal-article"},{"key":"65q6-5wxlCc101R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.74.3451","type":"journal-article"},{"key":"65q6-5wxlCc102R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.107.077003","type":"journal-article"},{"key":"65q6-5wxlCc103R1","doi-asserted-by":"publisher","DOI":"10.1143\/JPSJ.71.2102","type":"journal-article"},{"key":"65q6-5wxlCc104R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.66.094507","type":"journal-article"},{"key":"65q6-5wxlCc105R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.66.174502","type":"journal-article"},{"key":"65q6-5wxlCc106R1","doi-asserted-by":"publisher","DOI":"10.1143\/JPSJ.72.2157","type":"journal-article"},{"key":"65q6-5wxlCc107R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.75.473","type":"journal-article"},{"key":"65q6-5wxlCc108R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.93.025006","type":"journal-article"},{"key":"65q6-5wxlCc109R1","doi-asserted-by":"publisher","DOI":"10.1038\/s43586-022-00133-7","type":"journal-article"},{"key":"65q6-5wxlCc110R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.96.015003","type":"journal-article"},{"key":"65q6-5wxlCc111R1","doi-asserted-by":"publisher","DOI":"10.1126\/science.1218726","type":"journal-article"},{"key":"65q6-5wxlCc112R1","doi-asserted-by":"publisher","DOI":"10.1038\/nphys2671","type":"journal-article"},{"key":"65q6-5wxlCc113R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-021-27082-1","type":"journal-article"},{"key":"65q6-5wxlCc114R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-025-03000-w","type":"journal-article"}],"container-title":["Physical Review Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/article\/10.1103\/65q6-5wxl","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/65q6-5wxl\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T12:35:42Z","timestamp":1780576542000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/65q6-5wxl"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,4]]},"references-count":103,"journal-issue":{"issue":"22","published-print":{"date-parts":[[2026,6]]}},"URL":"https:\/\/doi.org\/10.1103\/65q6-5wxl","relation":{"has-preprint":[{"id-type":"doi","id":"10.48550\/arXiv.2510.14724","asserted-by":"subject"}]},"ISSN":["0031-9007","1079-7114"],"issn-type":[{"value":"0031-9007","type":"print"},{"value":"1079-7114","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,4]]},"assertion":[{"value":"2025-10-25","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"Published by the 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":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/","name":"creative_commons_license","label":"Creative Commons license","group":{"name":"licenses","label":"Licenses"}},{"value":"\u00a92026 The authors","name":"copyright_holder","label":"Copyright holder","group":{"name":"copyright","label":"copyright"}}],"article-number":"226001"}}