{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,15]],"date-time":"2026-08-15T01:14:25Z","timestamp":1786756465761,"version":"build-2736575974"},"reference-count":87,"publisher":"American Physical Society (APS)","issue":"24","license":[{"start":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T00:00:00Z","timestamp":1765152000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Hightech Agenda Bayern Plus"}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"short-container-title":["Phys. Rev. B"],"accepted":{"date-parts":[[2025,11,12]]},"abstract":"<jats:p>\n                    Topological phase transitions beyond anyon condensation remain poorly understood. A notable example is the transition between the toric code (TC) and double semion (DS) phases, which has two distinct\n                    <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <a:msub>\n                        <a:mi mathvariant=\"double-struck\">Z<\/a:mi>\n                        <a:mn>2<\/a:mn>\n                      <\/a:msub>\n                    <\/a:math>\n                    topological orders in (2+1) dimensions [\n                    <c:math xmlns:c=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <c:mrow>\n                        <c:mo>(<\/c:mo>\n                        <c:mn>2<\/c:mn>\n                        <c:mo>+<\/c:mo>\n                        <c:mn>1<\/c:mn>\n                        <c:mo>)<\/c:mo>\n                        <c:mi mathvariant=\"normal\">D<\/c:mi>\n                      <\/c:mrow>\n                    <\/c:math>\n                    ]. Previous studies reveal that the transition between them can be either first order or via an intermediate phase, thus the existence of a direct continuous transition between them remains a long-standing problem. Motivated by the fact that both phases can arise from condensing distinct anyons in the\n                    <e:math xmlns:e=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <e:msub>\n                        <e:mi mathvariant=\"double-struck\">Z<\/e:mi>\n                        <e:mn>4<\/e:mn>\n                      <\/e:msub>\n                    <\/e:math>\n                    topological order, we introduce a perturbed\n                    <g:math xmlns:g=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <g:msub>\n                        <g:mi mathvariant=\"double-struck\">Z<\/g:mi>\n                        <g:mn>4<\/g:mn>\n                      <\/g:msub>\n                    <\/g:math>\n                    quantum double (QD) model to study the TC-DS transition. We confirm the existence of a continuous\n                    <i:math xmlns:i=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <i:mrow>\n                        <i:mo>(<\/i:mo>\n                        <i:mn>2<\/i:mn>\n                        <i:mo>+<\/i:mo>\n                        <i:mn>1<\/i:mn>\n                        <i:mo>)<\/i:mo>\n                        <i:mi mathvariant=\"normal\">D<\/i:mi>\n                      <\/i:mrow>\n                    <\/i:math>\n                    XY* transition between the TC and DS phases by mapping it to a two-coupled quantum Ising model. Importantly, using the condensation order parameters and the area law coefficients of the Wilson loops, we further reveal that\n                    <k:math xmlns:k=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <k:msub>\n                        <k:mi mathvariant=\"double-struck\">Z<\/k:mi>\n                        <k:mn>4<\/k:mn>\n                      <\/k:msub>\n                    <\/k:math>\n                    anyons, fractionalized from the\n                    <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <m:msub>\n                        <m:mi mathvariant=\"double-struck\">Z<\/m:mi>\n                        <m:mn>2<\/m:mn>\n                      <\/m:msub>\n                    <\/m:math>\n                    topological orders, become deconfined at the transition between\n                    <o:math xmlns:o=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <o:msub>\n                        <o:mi mathvariant=\"double-struck\">Z<\/o:mi>\n                        <o:mn>2<\/o:mn>\n                      <\/o:msub>\n                    <\/o:math>\n                    topologically ordered phases. Our results open a path toward developing a theoretical framework for topological phase transitions beyond anyon condensation.\n                  <\/jats:p>","DOI":"10.1103\/1bjs-mf2l","type":"journal-article","created":{"date-parts":[[2025,11,12]],"date-time":"2025-11-12T16:30:07Z","timestamp":1762965007000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":3,"title":["Continuous topological phase transition between\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:msub>\n                        <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:msub>\n                    <\/mml:math>\n                    topologically ordered phases"],"prefix":"10.1103","volume":"112","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1243-344X","authenticated-orcid":true,"given":"Qi","family":"Zhang","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/05cvf7v30","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute of Physics, Chinese Academy of Sciences"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7723-4855","authenticated-orcid":true,"given":"Wen-Tao","family":"Xu","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02kkvpp62","id-type":"ROR","asserted-by":"publisher"}],"name":"Technical University of Munich"},{"id":[{"id":"https:\/\/ror.org\/04xrcta15","id-type":"ROR","asserted-by":"publisher"}],"name":"Munich Center for Quantum Science and Technology (MCQST)"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"16","published-online":{"date-parts":[[2025,12,8]]},"reference":[{"key":"1bjs-mf2lCc1R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.48.1559","type":"journal-article"},{"key":"1bjs-mf2lCc2R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.50.1395","type":"journal-article"},{"key":"1bjs-mf2lCc3R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1499754","type":"journal-article"},{"key":"1bjs-mf2lCc4R1","doi-asserted-by":"publisher","DOI":"10.1016\/S0003-4916(02)00018-0","type":"journal-article"},{"key":"1bjs-mf2lCc5R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.aop.2005.10.005","type":"journal-article"},{"key":"1bjs-mf2lCc6R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.71.045110","type":"journal-article"},{"key":"1bjs-mf2lCc7R1","doi-asserted-by":"publisher","DOI":"10.1126\/science.abi8378","type":"journal-article"},{"key":"1bjs-mf2lCc8R1","doi-asserted-by":"publisher","DOI":"10.1126\/science.abi8794","type":"journal-article"},{"key":"1bjs-mf2lCc9R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41586-023-06934-4","type":"journal-article"},{"key":"1bjs-mf2lCc10R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-024-02529-6","type":"journal-article"},{"key":"1bjs-mf2lCc11R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.98.070602","type":"journal-article"},{"key":"1bjs-mf2lCc12R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.122.176401","type":"journal-article"},{"key":"1bjs-mf2lCc13R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevX.11.041008","type":"journal-article"},{"key":"1bjs-mf2lCc14R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.104.155119","type":"journal-article"},{"key":"1bjs-mf2lCc15R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.106.205139","type":"journal-article"},{"key":"1bjs-mf2lCc16R1","doi-asserted-by":"publisher","DOI":"10.21468\/SciPostPhys.17.2.053","type":"journal-article"},{"key":"1bjs-mf2lCc17R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.82.085114","type":"journal-article"},{"key":"1bjs-mf2lCc18R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.85.195104","type":"journal-article"},{"key":"1bjs-mf2lCc19R1","doi-asserted-by":"publisher","DOI":"10.1038\/ncomms9284","type":"journal-article"},{"key":"1bjs-mf2lCc20R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.95.235119","type":"journal-article"},{"key":"1bjs-mf2lCc21R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.96.155127","type":"journal-article"},{"key":"1bjs-mf2lCc22R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.97.195124","type":"journal-article"},{"key":"1bjs-mf2lCc23R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.124.130603","type":"journal-article"},{"key":"1bjs-mf2lCc24R1","doi-asserted-by":"publisher","DOI":"10.1038\/s42005-020-00479-y","type":"journal-article"},{"key":"1bjs-mf2lCc25R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.104.045131","type":"journal-article"},{"key":"1bjs-mf2lCc26R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevX.11.041014","type":"journal-article"},{"key":"1bjs-mf2lCc27R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.109.165103","type":"journal-article"},{"key":"1bjs-mf2lCc29R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.90.035117","type":"journal-article"},{"key":"1bjs-mf2lCc30R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.103.L140412","type":"journal-article"},{"key":"1bjs-mf2lCc31R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.103.144437","type":"journal-article"},{"key":"1bjs-mf2lCc32R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.98.165115","type":"journal-article"},{"key":"1bjs-mf2lCc33R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.6.043256","type":"journal-article"},{"key":"1bjs-mf2lCc35R1","doi-asserted-by":"publisher","DOI":"10.1103\/PRXQuantum.3.010353","type":"journal-article"},{"key":"1bjs-mf2lCc36R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.94.035133","type":"journal-article"},{"key":"1bjs-mf2lCc37R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.94.155123","type":"journal-article"},{"key":"1bjs-mf2lCc38R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevX.9.031041","type":"journal-article"},{"key":"1bjs-mf2lCc39R1","doi-asserted-by":"publisher","DOI":"10.1126\/science.1091806","type":"journal-article"},{"key":"1bjs-mf2lCc40R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.70.144407","type":"journal-article"},{"key":"1bjs-mf2lCc41R1","unstructured":"The Z4 QD model is also known called the Z4 toric code model. In this work, \u201cTC\u201d refers exclusively to the Z2 toric code model, and we avoid calling HQD the Z4 TC model to prevent confusion.","type":"other"},{"key":"1bjs-mf2lCc42R1","unstructured":"See Supplemental Material at http:\/\/link.aps.org\/supplemental\/10.1103\/1bjs-mf2l for a review on the anyon condensation theory of the Z4 QD model, details of the duality transformation, additional numerical results, and an analysis of the critical exponents at the MCP based on the O(2) \u03d54 field theory, which includes Refs.   [75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92].","type":"other"},{"key":"1bjs-mf2lCc43R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.89.181601","type":"journal-article"},{"key":"1bjs-mf2lCc44R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.79.045316","type":"journal-article"},{"key":"1bjs-mf2lCc45R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.nuclphysb.2014.07.003","type":"journal-article"},{"key":"1bjs-mf2lCc46R1","doi-asserted-by":"publisher","DOI":"10.1146\/annurev-conmatphys-033117-054154","type":"journal-article"},{"key":"1bjs-mf2lCc47R1","unstructured":"Alternative, the phase transitions to the TC phase by condensing the flux m2 by considering the Hamiltonian HQD\u2212hx\u2211eXe2 with hx&gt;0.657.","type":"other"},{"key":"1bjs-mf2lCc48R1","unstructured":"The phase transition point can be determined by mapping the Hamiltonian HQD\u2212hz\u2211eZe2 to the transverse-field Ising model HIsing=\u2212\u2211\u2329i,j\u232a\u03c3iz\u03c3jz\u2212h\u2211i\u03c3ix. This yields the critical point at hz=2\/hcTFIM, where hcTFIM=3.044330(6) is the known critical value of the transverse-field Ising model   [58].","type":"other"},{"key":"1bjs-mf2lCc49R1","unstructured":"Note that AvBpAvBp\u2020=Av2 and AvBpAv\u2020Bp=Bp2 are conserved quantities. Although the Hamiltonian used here differs from HDS\u2032=\u2212\u2211\u2329vp\u232a\u2032(AvBp+Bp2+H.c.)\u2212\u2211eWe in Ref.\u00a0  [35], they share the same ground state.","type":"other"},{"key":"1bjs-mf2lCc50R1","doi-asserted-by":"publisher","DOI":"10.1007\/JHEP02(2015)172","type":"journal-article"},{"key":"1bjs-mf2lCc51R1","doi-asserted-by":"publisher","DOI":"10.1146\/annurev-conmatphys-040721-021029","type":"journal-article"},{"key":"1bjs-mf2lCc52R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physrep.2023.11.002","type":"journal-article"},{"key":"1bjs-mf2lCc53R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.99.205139","type":"journal-article"},{"key":"1bjs-mf2lCc54R1","doi-asserted-by":"publisher","DOI":"10.1016\/0370-2693(83)91171-1","type":"journal-article"},{"key":"1bjs-mf2lCc55R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41534-025-01030-z","type":"journal-article"},{"key":"1bjs-mf2lCc56R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1665530","type":"journal-article"},{"key":"1bjs-mf2lCc57R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.51.659","type":"journal-article"},{"key":"1bjs-mf2lCc58R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.102.094101","type":"journal-article"},{"key":"1bjs-mf2lCc59R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.3.3918","type":"journal-article"},{"key":"1bjs-mf2lCc60R1","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-017-1737-7","type":"journal-article"},{"key":"1bjs-mf2lCc61R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.134.146503","type":"journal-article"},{"key":"1bjs-mf2lCc62R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.117.210401","type":"journal-article"},{"key":"1bjs-mf2lCc63R1","doi-asserted-by":"publisher","DOI":"10.1143\/JPSJ.65.891","type":"journal-article"},{"key":"1bjs-mf2lCc64R1","doi-asserted-by":"publisher","DOI":"10.1143\/JPSJ.66.3040","type":"journal-article"},{"key":"1bjs-mf2lCc65R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.98.235148","type":"journal-article"},{"key":"1bjs-mf2lCc66R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.28.240","type":"journal-article"},{"key":"1bjs-mf2lCc67R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physrep.2022.12.002","type":"journal-article"},{"key":"1bjs-mf2lCc68R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.86.115109","type":"journal-article"},{"key":"1bjs-mf2lCc69R1","doi-asserted-by":"publisher","DOI":"10.21468\/SciPostPhys.14.2.013","type":"journal-article"},{"key":"1bjs-mf2lCc70R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01206315","type":"journal-article"},{"key":"1bjs-mf2lCc71R1","type":"book","volume-title":"Lattice Gauge Theory: A Challenge in Large-Scale Computing","author":"M. 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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":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/","name":"creative_commons_license","label":"Creative Commons license","group":{"name":"licenses","label":"Licenses"}},{"value":"\u00a92025 The authors","name":"copyright_holder","label":"Copyright holder","group":{"name":"copyright","label":"copyright"}}],"article-number":"L241107"}}