{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T23:12:51Z","timestamp":1790723571327,"version":"4.1.0"},"reference-count":80,"publisher":"American Physical Society (APS)","issue":"8","license":[{"start":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T00:00:00Z","timestamp":1772064000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100010661","name":"Horizon 2020 Framework Programme","doi-asserted-by":"publisher","award":["101113690"],"award-info":[{"award-number":["101113690"]}],"id":[{"id":"10.13039\/100010661","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["101041435"],"award-info":[{"award-number":["101041435"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"short-container-title":["Phys. Rev. Lett."],"accepted":{"date-parts":[[2026,1,27]]},"abstract":"\n The simulation of quantum field theories, both classical and quantum, requires regularization of infinitely many degrees of freedom. However, in the context of field digitization (FD)\u2014a truncation of the local fields to\n \n \n N<\/a:mi>\n <\/a:mrow>\n <\/a:math>\n discrete values\u2014a comprehensive framework to obtain continuum results is currently missing. Here, we propose to analyze FD by interpreting the parameter\n \n \n N<\/c:mi>\n <\/c:mrow>\n <\/c:math>\n as a coupling in the renormalization group (RG) sense. As a first example, we investigate the two-dimensional (2D) classical\n \n N<\/e:mi>\n <\/e:math>\n -state clock model as a\n \n \n \n \n Z<\/g:mi>\n <\/g:mrow>\n \n N<\/g:mi>\n <\/g:mrow>\n <\/g:msub>\n <\/g:mrow>\n <\/g:math>\n FD of the\n \n \n U<\/j:mi>\n (<\/j:mo>\n 1<\/j:mn>\n )<\/j:mo>\n <\/j:mrow>\n <\/j:math>\n symmetric\n \n \n X<\/n:mi>\n Y<\/n:mi>\n <\/n:mrow>\n <\/n:math>\n model. Using effective field theory, we employ the RG to derive generalized scaling hypotheses involving the FD parameter\n \n N<\/p:mi>\n <\/p:math>\n , which allows us to relate data obtained for different\n \n N<\/r:mi>\n <\/r:math>\n -regularized models in a procedure that we term (FDS). Using numerical tensor-network calculations at finite bond dimension\n \n \u03c7<\/t:mi>\n <\/t:math>\n , we further uncover an unconventional universal crossover around a low-temperature phase transition induced by finite\n \n N<\/v:mi>\n <\/v:math>\n , demonstrating that FDS can be extended to describe the interplay of\n \n \n \u03c7<\/x:mi>\n <\/x:mrow>\n <\/x:math>\n and\n \n \n N<\/z:mi>\n <\/z:mrow>\n <\/z:math>\n . Finally, we analytically prove that our calculations for the 2D classical-statistical\n \n \n \n \n Z<\/bb:mi>\n <\/bb:mrow>\n \n N<\/bb:mi>\n <\/bb:mrow>\n <\/bb:msub>\n <\/bb:mrow>\n <\/bb:math>\n clock model are directly related to the quantum physics in the ground state of a\n \n \n (<\/eb:mo>\n 2<\/eb:mn>\n +<\/eb:mo>\n 1<\/eb:mn>\n )<\/eb:mo>\n D<\/eb:mi>\n <\/eb:mrow>\n <\/eb:math>\n \n \n \n \n Z<\/jb:mi>\n <\/jb:mrow>\n \n N<\/jb:mi>\n <\/jb:mrow>\n <\/jb:msub>\n <\/jb:mrow>\n <\/jb:math>\n lattice gauge theory, which serves as a FD of compact quantum electrodynamics. Our Letter thus paves the way for applications of FDS to quantum simulations of more complex models in higher spatial dimensions, where it could serve as a tool to analyze the continuum limit of digitized quantum field theories.\n <\/jats:p>","DOI":"10.1103\/x164-8n1w","type":"journal-article","created":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T13:03:31Z","timestamp":1769519011000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["Field Digitization Scaling in a\n \n \n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n \n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \u2282<\/mml:mo>\n U<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n Symmetric Model"],"prefix":"10.1103","volume":"136","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-9474-4155","authenticated-orcid":true,"given":"Gabriele","family":"Calliari","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/054pv6659","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Innsbruck"},{"id":[{"id":"https:\/\/ror.org\/056hzt889","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute for Quantum Optics and Quantum Information"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0411-3849","authenticated-orcid":true,"given":"Robert","family":"Ott","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/054pv6659","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Innsbruck"},{"id":[{"id":"https:\/\/ror.org\/056hzt889","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute for Quantum Optics and Quantum Information"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2144-536X","authenticated-orcid":true,"given":"Hannes","family":"Pichler","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/054pv6659","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Innsbruck"},{"id":[{"id":"https:\/\/ror.org\/056hzt889","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute for Quantum Optics and Quantum Information"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3549-7160","authenticated-orcid":true,"given":"Torsten V.","family":"Zache","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/054pv6659","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Innsbruck"},{"id":[{"id":"https:\/\/ror.org\/056hzt889","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute for Quantum Optics and Quantum Information"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"16","published-online":{"date-parts":[[2026,2,26]]},"reference":[{"key":"x164-8n1wCc1R1","doi-asserted-by":"crossref","DOI":"10.1017\/9781009212717","type":"book","volume-title":"Quantum Phases of Matter","author":"S. 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In contrast, in this Letter we focus on relating the ZN models across distinct N, interpreting them as different digitizations of the same U(1) critical properties in the vicinity of the low-temperature phase transitions.","type":"other"},{"key":"x164-8n1wCc47R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.101.060105","type":"journal-article"},{"key":"x164-8n1wCc48R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.4.023159","type":"journal-article"},{"key":"x164-8n1wCc49R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.106.024106","type":"journal-article"},{"key":"x164-8n1wCc50R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.nuclphysb.2011.09.012","type":"journal-article"},{"key":"x164-8n1wCc51R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-025-02797-w","type":"journal-article"},{"key":"x164-8n1wCc52R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.109.074501","type":"journal-article"},{"key":"x164-8n1wCc53R1","doi-asserted-by":"publisher","DOI":"10.1103\/PRXQuantum.2.030334","type":"journal-article"},{"key":"x164-8n1wCc54R1","doi-asserted-by":"publisher","DOI":"10.22331\/q-2021-02-04-393","type":"journal-article"},{"key":"x164-8n1wCc55R1","unstructured":"In this work, we are interested in the behavior N\u22655, since the detailed universal aspects of N-state clock models differ for small N<5 [47].","type":"other"},{"key":"x164-8n1wCc56R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.19.3698","type":"journal-article"},{"key":"x164-8n1wCc57R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.101.062111","type":"journal-article"},{"key":"x164-8n1wCc58R1","doi-asserted-by":"publisher","DOI":"10.1088\/1742-5468\/2014\/11\/P11022","type":"journal-article"},{"key":"x164-8n1wCc59R1","first-page":"493","type":"journal-article","volume":"32","author":"VL. Berezinskii","year":"1971","journal-title":"Sov. Phys. JETP","ISSN":"https:\/\/id.crossref.org\/issn\/0038-5646","issn-type":"print"},{"key":"x164-8n1wCc60R1","first-page":"610","type":"journal-article","volume":"34","author":"VL. Berezinskii","year":"1972","journal-title":"Sov. Phys. JETP","ISSN":"https:\/\/id.crossref.org\/issn\/0038-5646","issn-type":"print"},{"key":"x164-8n1wCc61R1","doi-asserted-by":"publisher","DOI":"10.1088\/0022-3719\/7\/6\/005","type":"journal-article"},{"key":"x164-8n1wCc62R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.98.235148","type":"journal-article"},{"key":"x164-8n1wCc63R1","doi-asserted-by":"publisher","DOI":"10.1088\/0022-3719\/11\/8\/019","type":"journal-article"},{"key":"x164-8n1wCc64R1","unstructured":"In this work, we do not investigate the second BKT transition at T=TH, whose behavior is essentially independent of N.","type":"other"},{"key":"x164-8n1wCc65R1","doi-asserted-by":"publisher","DOI":"10.1016\/S0550-3213(00)00507-1","type":"journal-article"},{"key":"x164-8n1wCc66R1","unstructured":"In this work we use the critical temperatures TL(N) numerically determined in Ref.\u00a0[48].","type":"other"},{"key":"x164-8n1wCc67R1","unstructured":"By comparing our results in Eq.\u00a0(5) to the derived \u03be behavior, we infer C\u2248|t|\/N, which can also be interpreted as an N-dependent UV initialization of gL(T,N)\u221d|t|\/N.","type":"other"},{"key":"x164-8n1wCc68R1","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/30\/1\/006","type":"journal-article"},{"key":"x164-8n1wCc69R1","unstructured":"See Supplemental Material http:\/\/link.aps.org\/supplemental\/10.1103\/x164-8n1w for a comparison between our field-digitization scaling analysis and a simple temperature rescaling.","type":"other"},{"key":"x164-8n1wCc70R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.78.024410","type":"journal-article"},{"key":"x164-8n1wCc71R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.102.255701","type":"journal-article"},{"key":"x164-8n1wCc72R1","unstructured":"Because of the truncation in \u03c7, the scaling variable x=x(\u03be) cannot be a constant in the whole critical region, but rather it quantifies the proximity to the CP.","type":"other"},{"key":"x164-8n1wCc73R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.96.220601","type":"journal-article"},{"key":"x164-8n1wCc74R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.19.3715","type":"journal-article"},{"key":"x164-8n1wCc75R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.77.054433","type":"journal-article"},{"key":"x164-8n1wCc76R1","unstructured":"For simplicity, here we state the result for an infinite plane, where the ground state is unique.","type":"other"},{"key":"x164-8n1wCc77R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.103.054418","type":"journal-article"},{"key":"x164-8n1wCc78R1","doi-asserted-by":"publisher","DOI":"10.1126\/science.aad5007","type":"journal-article"},{"key":"x164-8n1wCc79R1","doi-asserted-by":"publisher","DOI":"10.5281\/zenodo.16575343","type":"dataset"},{"key":"x164-8n1wCc80R1","doi-asserted-by":"publisher","DOI":"10.21468\/SciPostPhysLectNotes.86","type":"journal-article"},{"key":"x164-8n1wCc81R1","unstructured":"Our standard implementation of the CTMRG algorithm closely follows Ref.\u00a0[80], and our isotropic code is inspired by Ref.\u00a0[62].","type":"other"}],"container-title":["Physical Review Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/article\/10.1103\/x164-8n1w","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/x164-8n1w\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T22:52:01Z","timestamp":1790722321000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/x164-8n1w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,26]]},"references-count":80,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2026,2]]}},"URL":"https:\/\/doi.org\/10.1103\/x164-8n1w","relation":{"has-preprint":[{"id-type":"doi","id":"10.48550\/arXiv.2507.22984","asserted-by":"subject"}]},"ISSN":["0031-9007","1079-7114"],"issn-type":[{"value":"0031-9007","type":"print"},{"value":"1079-7114","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,26]]},"assertion":[{"value":"2025-08-26","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":"080403"}}