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However, we stress that such coupling preserves the same symmetry properties and thus merely renormalizes the NMEE magnitude without introducing additional tensor components.","type":"other"},{"key":"r3x2-kb93Cc43R1","unstructured":"See Supplemental Material at http:\/\/link.aps.org\/supplemental\/10.1103\/r3x2-kb93 for derivation details based on the response theory, Keldysh Green\u2019s function method, and semiclassical theory; for the definition of the spin-space quantum geometric tensors; for the symmetry analysis with respect to the chiral C, time-reversal T, and inversion P symmetries of the quantum geometry tensors; as well as for the geometry origins corresponding effects summarized in Table\u00a0I, which also includes Refs.\u00a0[27,30,41,44\u201347].","type":"other"},{"key":"r3x2-kb93Cc44R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.134.116301","type":"journal-article"},{"key":"r3x2-kb93Cc46R1","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781139023979","type":"book","volume-title":"Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction","author":"G. 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Considering the response from all states must vanish, we have \u0393\u03b1\u03b3,\u03b2=0 when \u03bc lies in the gap.","type":"other"},{"key":"r3x2-kb93Cc60R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.101.094429","type":"journal-article"},{"key":"r3x2-kb93Cc61R1","doi-asserted-by":"publisher","DOI":"10.1126\/sciadv.aar3566","type":"journal-article"},{"key":"r3x2-kb93Cc62R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.96.094406","type":"journal-article"},{"key":"r3x2-kb93Cc63R1","unstructured":"In our effectively 2D model, \u03b4s is defined per unit area (\u03bcB\/nm2). To facilitate comparison with experimentally reported three-dimensional quantities, we convert it to an effective volume density using a characteristic thickness h\u22480.63\u2009\u2009nm for CuMnAs [62], i.e., \u03b4s3D=\u03b4s2D\/h.","type":"other"},{"key":"r3x2-kb93Cc64R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.97.126603","type":"journal-article"},{"key":"r3x2-kb93Cc65R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41586-023-06101-9","type":"journal-article"},{"key":"r3x2-kb93Cc66R1","unstructured":"Experimentally, one can apply an ac current and an ac magnetic field to modulate the electric and magnetic fields at frequencies \u03c9E and \u03c9B, respectively. Under such dual modulation, the NMEE-induced spin magnetization appears at the sum and difference frequencies, \u03c9E\u00b1\u03c9B, while the conventional magnetization induced solely by the magnetic field remains at \u03c9B.","type":"other"},{"key":"r3x2-kb93Cc67R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.124.067203","type":"journal-article"},{"key":"r3x2-kb93Cc68R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.109.235413","type":"journal-article"}],"container-title":["Physical Review Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/article\/10.1103\/r3x2-kb93","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/r3x2-kb93\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T17:44:55Z","timestamp":1790876695000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/r3x2-kb93"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,4]]},"references-count":65,"journal-issue":{"issue":"22","published-print":{"date-parts":[[2026,6]]}},"URL":"https:\/\/doi.org\/10.1103\/r3x2-kb93","relation":{},"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":"2025-11-14","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":"226302"}}