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Berezinskii","year":"1971","journal-title":"Sov. Phys. JETP","ISSN":"https:\/\/id.crossref.org\/issn\/0038-5646","issn-type":"print"},{"key":"6jpk-yj56Cc44R1","doi-asserted-by":"publisher","DOI":"10.1088\/0022-3719\/6\/7\/010","type":"journal-article"},{"key":"6jpk-yj56Cc45R1","doi-asserted-by":"publisher","DOI":"10.1088\/0022-3719\/7\/6\/005","type":"journal-article"},{"key":"6jpk-yj56Cc46R1","unstructured":"We note that the universal nature of these two transitions for 4&lt;q\u22648 has long been a subject of debate [18, 20, 21, 22, 27, 28]. However, a series of recent studies has provided compelling evidence that both transitions are of BKT type for all q&gt;4 [24, 25, 28, 29, 31, 34, 35, 37, 40, 41].","type":"other"},{"key":"6jpk-yj56Cc47R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.48.3519","type":"journal-article"},{"key":"6jpk-yj56Cc48R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.84.174407","type":"journal-article"},{"key":"6jpk-yj56Cc49R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.86.134410","type":"journal-article"},{"key":"6jpk-yj56Cc50R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.108.045702","type":"journal-article"},{"key":"6jpk-yj56Cc51R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.87.144406","type":"journal-article"},{"key":"6jpk-yj56Cc52R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.91.032145","type":"journal-article"},{"key":"6jpk-yj56Cc53R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.104.024118","type":"journal-article"},{"key":"6jpk-yj56Cc54R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.107.034124","type":"journal-article"},{"key":"6jpk-yj56Cc55R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.108.054124","type":"journal-article"},{"key":"6jpk-yj56Cc56R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.109.104419","type":"journal-article"},{"key":"6jpk-yj56Cc57R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.109.064422","type":"journal-article"},{"key":"6jpk-yj56Cc58R1","doi-asserted-by":"publisher","DOI":"10.1093\/ptep\/ptae061","type":"journal-article"},{"key":"6jpk-yj56Cc59R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.338570","type":"journal-article"},{"key":"6jpk-yj56Cc60R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.62.2056","type":"journal-article"},{"key":"6jpk-yj56Cc61R1","doi-asserted-by":"publisher","DOI":"10.1051\/jphyscol:19888543","type":"journal-article"},{"key":"6jpk-yj56Cc62R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.44.10057","type":"journal-article"},{"key":"6jpk-yj56Cc63R1","doi-asserted-by":"publisher","DOI":"10.1209\/epl\/i1997-00325-6","type":"journal-article"},{"key":"6jpk-yj56Cc64R1","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/30\/8\/013","type":"journal-article"},{"key":"6jpk-yj56Cc65R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.61.6114","type":"journal-article"},{"key":"6jpk-yj56Cc66R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.111.104428","type":"journal-article"},{"key":"6jpk-yj56Cc67R1","unstructured":"For q=4, the J1\u2212J2 clock model is exactly equivalent to two decoupled J1\u2212J2 Ising models. This correspondence arises because the clock angles \u03b8i\u2208{0,\u03c0\/2,\u03c0,3\u03c0\/2} can be mapped onto two Ising variables \u03c3i,\u03c4i\u2208{\u00b11} as 0\u2192(+1,+1), \u03c0\/2\u2192(+1,\u22121), \u03c0\u2192(\u22121,\u22121), and 3\u03c0\/2\u2192(\u22121,+1). Consequently, the spin-spin interaction term transforms as cos(\u03b8i\u2212\u03b8j)\u219212(\u03c3i\u03c3j+\u03c4i\u03c4j).","type":"other"},{"key":"6jpk-yj56Cc68R1","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198517962.001.0001","type":"book","volume-title":"Monte\u00a0Carlo Methods in Statistical Physics","author":"M.\u2009E.\u2009J. Newman","year":"1999"},{"key":"6jpk-yj56Cc69R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1699114","type":"journal-article"},{"key":"6jpk-yj56Cc70R1","unstructured":"See Supplemental Material at http:\/\/link.aps.org\/supplemental\/10.1103\/6jpk-yj56 for technical details of the CMC and CTMRG methods used in this study, as well as additional results from CMC and CTMRG simulations, including the characterization of first-order phase transitions, that further support and complement the findings presented in the main text.","type":"other"},{"key":"6jpk-yj56Cc71R1","doi-asserted-by":"publisher","DOI":"10.1143\/JPSJ.65.891","type":"journal-article"},{"key":"6jpk-yj56Cc72R1","doi-asserted-by":"publisher","DOI":"10.1143\/JPSJ.66.3040","type":"journal-article"},{"key":"6jpk-yj56Cc73R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.113.046402","type":"journal-article"},{"key":"6jpk-yj56Cc74R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.98.235148","type":"journal-article"},{"key":"6jpk-yj56Cc75R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.99.120601","type":"journal-article"},{"key":"6jpk-yj56Cc76R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.78.205116","type":"journal-article"},{"key":"6jpk-yj56Cc77R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.115.180405","type":"journal-article"},{"key":"6jpk-yj56Cc78R1","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-030-34489-4","type":"book","volume-title":"Tensor Network Contractions: Methods and Applications to Quantum Many-Body Systems","author":"S.-J. Ran","year":"2020"},{"key":"6jpk-yj56Cc79R1","doi-asserted-by":"publisher","DOI":"10.7566\/JPSJ.91.062001","type":"journal-article"},{"key":"6jpk-yj56Cc80R1","doi-asserted-by":"crossref","DOI":"10.1017\/9781009398671","type":"book","volume-title":"Density Matrix and Tensor Network Renormalization","author":"T. Xiang","year":"2023"},{"key":"6jpk-yj56Cc81R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.17.1133","type":"journal-article"},{"key":"6jpk-yj56Cc82R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01646487","type":"journal-article"},{"key":"6jpk-yj56Cc83R1","unstructured":"Here, Zq\u2212 signifies that the effective spins in the infrared limit take odd multiples of \u03c0\/q.","type":"other"},{"key":"6jpk-yj56Cc84R1","doi-asserted-by":"publisher","DOI":"10.1016\/0370-2693(94)91007-3","type":"journal-article"},{"key":"6jpk-yj56Cc85R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.90.227902","type":"journal-article"},{"key":"6jpk-yj56Cc86R1","doi-asserted-by":"publisher","DOI":"10.1088\/1742-5468\/2004\/06\/P06002","type":"journal-article"},{"key":"6jpk-yj56Cc87R1","doi-asserted-by":"publisher","DOI":"10.1126\/science.1091806","type":"journal-article"},{"key":"6jpk-yj56Cc88R1","volume-title":"50 Years of the Renormalization Group","author":"T. Senthil","year":"2024","type":"book"},{"key":"6jpk-yj56Cc89R1","unstructured":"The QLRO\u2295Z2 NM phase itself, on the other hand, is not within the Landau paradigm.","type":"other"},{"key":"6jpk-yj56Cc90R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.87.235106","type":"journal-article"},{"key":"6jpk-yj56Cc91R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.105.155159","type":"journal-article"},{"key":"6jpk-yj56Cc92R1","unstructured":"Recall that an operator is relevant if its scaling dimension is smaller than the Euclidean dimensions (2 in our case). 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