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References\u00a0[31] and [68] break the invariance in the functional form (Ref.\u00a0[31] also does not include SOC) while Ref.\u00a0[29] breaks the invariance at the level of the (G)KS equations by solving the SDFT equations and by further neglecting currents in the applications.","type":"other"},{"key":"lt1f-8pz2Cc68R1","doi-asserted-by":"publisher","DOI":"10.1063\/5.0246433","type":"journal-article"},{"key":"lt1f-8pz2Cc69R1","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jpclett.4c01607","type":"journal-article"},{"key":"lt1f-8pz2Cc70R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.132.256401","type":"journal-article"},{"key":"lt1f-8pz2Cc71R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.134.106402","type":"journal-article"},{"key":"lt1f-8pz2Cc72R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.53.3764","type":"journal-article"},{"key":"lt1f-8pz2Cc73R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevMaterials.8.013802","type":"journal-article"},{"key":"lt1f-8pz2Cc74R1","unstructured":"Mathematically rigorous advancements on N representability, v representability, and related issues have been worked out in spin- and current-DFT generalizations, [75, 76, 77, 78, 79, 80, 81, 82] but not for the more general case considered here, involving both noncollinear spin density and noncollinear spin currents.","type":"other"},{"key":"lt1f-8pz2Cc75R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.65.113106","type":"journal-article"},{"key":"lt1f-8pz2Cc76R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.75.134408","type":"journal-article"},{"key":"lt1f-8pz2Cc77R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.111.153001","type":"journal-article"},{"key":"lt1f-8pz2Cc78R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.89.012515","type":"journal-article"},{"key":"lt1f-8pz2Cc79R1","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2016.v14.n4.a6","type":"journal-article"},{"key":"lt1f-8pz2Cc80R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.97.022514","type":"journal-article"},{"key":"lt1f-8pz2Cc81R1","doi-asserted-by":"publisher","DOI":"10.1021\/acsphyschemau.3c00006","type":"journal-article"},{"key":"lt1f-8pz2Cc82R1","doi-asserted-by":"publisher","DOI":"10.1063\/5.0198786","type":"journal-article"},{"key":"lt1f-8pz2Cc83R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.115.036402","type":"journal-article"},{"key":"lt1f-8pz2Cc84R1","doi-asserted-by":"publisher","DOI":"10.1038\/nchem.2535","type":"journal-article"},{"key":"lt1f-8pz2Cc85R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.133.136402","type":"journal-article"},{"key":"lt1f-8pz2Cc87R1","unstructured":"Please note that \u03c4, \u03c4\u2192 and \u03c4\u02dc are three distinct quantities.","type":"other"},{"key":"lt1f-8pz2Cc88R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.133.136401","type":"journal-article"},{"key":"lt1f-8pz2Cc89R1","unstructured":"Besides to the nc-mSCAN, the key equations derived more below also apply to other U(1)\u00d7SU(2) gauge-invariant MGGAs that can be obtained through several minimal substitutions of \u03c4 (see Ref.\u00a0[60]).","type":"other"},{"key":"lt1f-8pz2Cc90R1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1621352114","type":"journal-article"},{"key":"lt1f-8pz2Cc91R1","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198787075.001.0001","type":"book","volume-title":"Spin Current","author":"S. 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