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If one included such a term, even the reciprocal part of the interaction would be invariant only under joint rotations of the lattice and the spins. This is the symmetry of all liquid crystals.","type":"other"},{"key":"2yky-45srCc23R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.6.023022","type":"journal-article"},{"key":"2yky-45srCc24R1","unstructured":"See Supplemental Material at http:\/\/link.aps.org\/supplemental\/10.1103\/2yky-45sr, which includes Refs.\u00a0[25\u201327], as well as results for the structure factor and defect dynamics for triangular lattice, along with supplementary videos.","type":"other"},{"key":"2yky-45srCc25R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.47.1525","type":"journal-article"},{"key":"2yky-45srCc26R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.54.6272","type":"journal-article"},{"key":"2yky-45srCc27R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.115.247801","type":"journal-article"},{"key":"2yky-45srCc28R1","unstructured":"In general, we must implement a coarse-graining procedure that results in a smooth nonunit vector field p, from which we can construct an equation for the director p^.","type":"other"},{"key":"2yky-45srCc30R1","doi-asserted-by":"publisher","DOI":"10.1103\/r3dx-7lrd","type":"journal-article"},{"key":"2yky-45srCc31R1","unstructured":"Superficially similar nonlinearities arise in the dynamics of polar and nematic active systems without lattice anisotropy as well; for instance, at O(\u22072), nonlinearities that remain invariant under the transformation (4) include {sin2\u03b8[(\u2202x\u03b8)2\u2212(\u2202y\u03b8)2]\u22122cos2\u03b8\u2202x\u03b8\u2202y\u03b8} and [cos2\u03b8(\u2202x2\u03b8\u2212\u2202y2\u03b8)+2sin2\u03b8\u2202x\u2202y\u03b8] with independent coefficients. Naively, it would seem that a term \u221dsin2\u03b8 may be generated from sin2\u03b8[(\u2202x\u03b8)2\u2212(\u2202y\u03b8)2] in a state with long-range order or that another such mass term may be generated from cos2\u03b8(\u2202x2\u03b8\u2212\u2202y2\u03b8). Of course, no such term is generated because unlike these nonlinearities, sin2\u03b8 or sin\u03b8 is not invariant under (4). However, such terms can be generated in the lattice-based models considered here because they are not invariant under (4).","type":"other"}],"container-title":["Physical Review Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/article\/10.1103\/2yky-45sr","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/2yky-45sr\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T21:26:36Z","timestamp":1790717196000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/2yky-45sr"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,21]]},"references-count":30,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2025,8]]}},"URL":"https:\/\/doi.org\/10.1103\/2yky-45sr","relation":{},"ISSN":["0031-9007","1079-7114"],"issn-type":[{"value":"0031-9007","type":"print"},{"value":"1079-7114","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,21]]},"assertion":[{"value":"2025-04-07","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"\u00a9 2025 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":"2025","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":"088303"}}