{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T03:02:50Z","timestamp":1785898970422,"version":"3.56.0"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"11","license":[{"start":{"date-parts":[[2009,9,20]],"date-time":"2009-09-20T00:00:00Z","timestamp":1253404800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Nature Phys"],"published-print":{"date-parts":[[2009,11]]},"DOI":"10.1038\/nphys1396","type":"journal-article","created":{"date-parts":[[2009,9,20]],"date-time":"2009-09-20T18:29:08Z","timestamp":1253471348000},"page":"834-839","source":"Crossref","is-referenced-by-count":65,"title":["Topology-driven quantum phase transitions in time-reversal-invariant anyonic quantum liquids"],"prefix":"10.1038","volume":"5","author":[{"given":"Charlotte","family":"Gils","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Simon","family":"Trebst","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alexei","family":"Kitaev","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andreas W. W.","family":"Ludwig","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Matthias","family":"Troyer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhenghan","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2009,9,20]]},"reference":[{"key":"BFnphys1396_CR1","doi-asserted-by":"publisher","first-page":"7387","DOI":"10.1103\/PhysRevB.40.7387","volume":"40","author":"X-G Wen","year":"1989","unstructured":"Wen, X.-G. Vacuum degeneracy of chiral spin states in compactified space. Phys. Rev. B 40, 7387\u20137390 (1989).","journal-title":"Phys. Rev. B"},{"key":"BFnphys1396_CR2","doi-asserted-by":"publisher","first-page":"1395","DOI":"10.1103\/PhysRevLett.50.1395","volume":"50","author":"RB Laughlin","year":"1983","unstructured":"Laughlin, R. B. Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395\u20131398 (1983).","journal-title":"Phys. Rev. Lett."},{"key":"BFnphys1396_CR3","doi-asserted-by":"publisher","first-page":"9377","DOI":"10.1103\/PhysRevB.41.9377","volume":"41","author":"X-G Wen","year":"1990","unstructured":"Wen, X.-G. & Niu, Q. Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces. Phys. Rev. B 41, 9377\u20139396 (1990).","journal-title":"Phys. Rev. B"},{"key":"BFnphys1396_CR4","doi-asserted-by":"publisher","first-page":"1881","DOI":"10.1103\/PhysRevLett.86.1881","volume":"86","author":"R Moessner","year":"2001","unstructured":"Moessner, R. & Sondhi, S. L. Resonating valence bond phase in the triangular lattice quantum dimer model. Phys. Rev. Lett. 86, 1881\u20131884 (2001).","journal-title":"Phys. Rev. Lett."},{"key":"BFnphys1396_CR5","doi-asserted-by":"publisher","first-page":"224412","DOI":"10.1103\/PhysRevB.65.224412","volume":"65","author":"L Balents","year":"2002","unstructured":"Balents, L., Fisher, M. P. A. & Girvin, S. M. Fractionalization in an easy-axis Kagome antiferromagnet. Phys. Rev. B 65, 224412 (2002).","journal-title":"Phys. Rev. B"},{"key":"BFnphys1396_CR6","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1038\/415503a","volume":"415","author":"LB Ioffe","year":"2002","unstructured":"Ioffe, L. B. et al. Topologically protected quantum bits using Josephson junction arrays. Nature 415, 503\u2013506 (2002).","journal-title":"Nature"},{"key":"BFnphys1396_CR7","doi-asserted-by":"publisher","first-page":"045110","DOI":"10.1103\/PhysRevB.71.045110","volume":"71","author":"M Levin","year":"2005","unstructured":"Levin, M. & Wen, X.-G. String-net condensation: A physical mechanism for topological phases. Phys. Rev. B 71, 045110 (2005).","journal-title":"Phys. Rev. B"},{"key":"BFnphys1396_CR8","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1016\/S0003-4916(02)00018-0","volume":"303","author":"A Kitaev","year":"2003","unstructured":"Kitaev, A. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2\u201330 (2003).","journal-title":"Ann. Phys."},{"key":"BFnphys1396_CR9","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1016\/j.aop.2005.10.005","volume":"321","author":"A Kitaev","year":"2006","unstructured":"Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2\u2013111 (2006).","journal-title":"Ann. Phys."},{"key":"BFnphys1396_CR10","doi-asserted-by":"publisher","first-page":"017205","DOI":"10.1103\/PhysRevLett.102.017205","volume":"102","author":"G Jackeli","year":"2009","unstructured":"Jackeli, G. & Khaliullin, G. Mott insulators in the strong spin\u2013orbit coupling limit: From Heisenberg to a quantum compass and Kitaev models. Phys.\u00a0Rev.\u00a0Lett. 102, 017205 (2009).","journal-title":"Phys.\u00a0Rev.\u00a0Lett."},{"key":"BFnphys1396_CR11","doi-asserted-by":"publisher","first-page":"1329","DOI":"10.1126\/science.1167106","volume":"323","author":"BJ Kim","year":"2009","unstructured":"Kim, B. J. et al. Phase-sensitive observation of a spin-orbital Mott state in Sr2IrO4 . Science 323, 1329\u20131332 (2009).","journal-title":"Science"},{"key":"BFnphys1396_CR12","first-page":"1","volume":"37","author":"JM Leinaas","year":"1977","unstructured":"Leinaas, J. M. & Myrheim, J. On the theory of identical particles. Il\u00a0Nuovo\u00a0Cimento 37, 1\u201323 (1977).","journal-title":"Il\u00a0Nuovo\u00a0Cimento"},{"key":"BFnphys1396_CR13","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1016\/S0550-3213(99)00095-4","volume":"547","author":"P Bouwknegt","year":"1999","unstructured":"Bouwknegt, P. & Schoutens, K. Exclusion statistics in conformal field theory\u2014generalized fermions and spinons for level-1 WZW theories. Nucl.\u00a0Phys.\u00a0B 547, 501\u2013537 (1999).","journal-title":"Nucl.\u00a0Phys.\u00a0B"},{"key":"BFnphys1396_CR14","doi-asserted-by":"publisher","first-page":"8084","DOI":"10.1103\/PhysRevB.59.8084","volume":"59","author":"N Read","year":"1999","unstructured":"Read, N. & Rezayi, E. Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited Landau level. Phys. Rev. B 59, 8084\u20138092 (1999).","journal-title":"Phys. Rev. B"},{"key":"BFnphys1396_CR15","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1016\/S0550-3213(01)00308-X","volume":"612","author":"JK Slingerland","year":"2001","unstructured":"Slingerland, J. K. & Bais, F. A. Quantum groups and non-Abelian braiding in quantum Hall systems. Nucl. Phys. B 612, 229\u2013290 (2001).","journal-title":"Nucl. Phys. B"},{"key":"BFnphys1396_CR16","doi-asserted-by":"publisher","first-page":"726","DOI":"10.1038\/nphys678","volume":"3","author":"HP B\u00fcchler","year":"2007","unstructured":"B\u00fcchler, H. P., Micheli, A. & Zoller, P. Three-body interactions with cold polar molecules. Nature Phys. 3, 726\u2013731 (2007).","journal-title":"Nature Phys."},{"key":"BFnphys1396_CR17","doi-asserted-by":"publisher","first-page":"3113","DOI":"10.1016\/j.aop.2008.04.011","volume":"323","author":"P Fendley","year":"2008","unstructured":"Fendley, P. Topological order from quantum loops and nets. Ann. Phys. 323, 3113\u20133136 (2008).","journal-title":"Ann. Phys."},{"key":"BFnphys1396_CR18","doi-asserted-by":"publisher","first-page":"511","DOI":"10.1103\/PhysRev.97.511","volume":"97","author":"JA Wheeler","year":"1955","unstructured":"Wheeler, J. A. Geons. Phys. Rev. 97, 511\u2013536 (1955).","journal-title":"Phys. Rev."},{"key":"BFnphys1396_CR19","doi-asserted-by":"publisher","first-page":"604","DOI":"10.1016\/0003-4916(57)90050-7","volume":"2","author":"JA Wheeler","year":"1957","unstructured":"Wheeler, J. A. On the nature of quantum geometrodynamics. Ann. Phys. 2, 604\u2013614 (1957).","journal-title":"Ann. Phys."},{"key":"BFnphys1396_CR20","doi-asserted-by":"publisher","first-page":"1575","DOI":"10.1103\/PhysRevLett.52.1575","volume":"52","author":"D Friedan","year":"1985","unstructured":"Friedan, D., Qiu, Z. & Shenker, S. Conformal invariance, unitarity, and critical exponents in two dimensions. Phys. Rev. Lett. 52, 1575\u20131578 (1985).","journal-title":"Phys. Rev. Lett."},{"key":"BFnphys1396_CR21","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1016\/0550-3213(87)90155-6","volume":"280","author":"A Cappelli","year":"1987","unstructured":"Cappelli, A., Itzykson, C. & Zuber, J. B. Modular invariant partition functions in two dimensions. Nucl. Phys. B 280, 445\u2013465 (1987).","journal-title":"Nucl. Phys. B"},{"key":"BFnphys1396_CR22","doi-asserted-by":"publisher","first-page":"L1229","DOI":"10.1088\/0305-4470\/20\/18\/003","volume":"20","author":"V Pasquier","year":"1987","unstructured":"Pasquier, V. Lattice derivation of modular invariant partition functions on the torus. J. Phys. A 20, L1229\u2013L1238 (1987).","journal-title":"J. Phys. A"},{"key":"BFnphys1396_CR23","doi-asserted-by":"publisher","first-page":"162","DOI":"10.1016\/0550-3213(87)90332-4","volume":"285","author":"V Pasquier","year":"1987","unstructured":"Pasquier, V. Two-dimensional critical systems labelled by Dynkin diagrams. Nucl. Phys. B 285, 162\u2013172 (1987).","journal-title":"Nucl. Phys. B"},{"key":"BFnphys1396_CR24","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1098\/rspa.1971.0067","volume":"322","author":"N Temperley","year":"1971","unstructured":"Temperley, N. & Lieb, E. Relations between percolation and colouring problem and other graph-theoretical problems associated with regular planar lattices\u2014some exact results for percolation problem. Proc. R. Soc. Lond. A 322, 251\u2013280 (1971).","journal-title":"Proc. R. Soc. Lond. A"},{"key":"BFnphys1396_CR25","doi-asserted-by":"publisher","first-page":"160409","DOI":"10.1103\/PhysRevLett.98.160409","volume":"98","author":"A Feiguin","year":"2007","unstructured":"Feiguin, A. et al. Interacting anyons in topological quantum liquids: The golden chain. Phys. Rev. Lett. 98, 160409 (2007).","journal-title":"Phys. Rev. Lett."},{"key":"BFnphys1396_CR26","doi-asserted-by":"publisher","first-page":"070401","DOI":"10.1103\/PhysRevLett.103.070401","volume":"103","author":"C Gils","year":"2009","unstructured":"Gils, C. et al. Collective states of interacting anyons, edge states, and the nucleation of topological liquids. Phys. Rev. Lett. 103, 070401 (2009).","journal-title":"Phys. Rev. Lett."},{"key":"BFnphys1396_CR27","doi-asserted-by":"publisher","first-page":"937","DOI":"10.1103\/PhysRevLett.67.937","volume":"67","author":"FDM Haldane","year":"1991","unstructured":"Haldane, F. D. M. \u2018Fractional statistics\u2019 in arbitary dimensions: A generalization of the Pauli principle. Phys. Rev. Lett. 67, 937\u2013940 (1991).","journal-title":"Phys. Rev. Lett."},{"key":"BFnphys1396_CR28","doi-asserted-by":"publisher","first-page":"066401","DOI":"10.1103\/PhysRevLett.94.066401","volume":"94","author":"M Freedman","year":"2005","unstructured":"Freedman, M., Nayak, C. & Shtengel, K. Extended Hubbard model with ring exchange: A route to a non-Abelian topological phase. Phys. Rev. Lett. 94, 066401 (2005).","journal-title":"Phys. Rev. Lett."},{"key":"BFnphys1396_CR29","doi-asserted-by":"publisher","first-page":"3682","DOI":"10.1103\/PhysRevD.19.3682","volume":"19","author":"E Fradkin","year":"1979","unstructured":"Fradkin, E. & Shenker, S. H. Phase diagrams of lattice gauge theories with Higgs fields. Phys. Rev. D 19, 3682\u20133697 (1979).","journal-title":"Phys. Rev. D"},{"key":"BFnphys1396_CR30","doi-asserted-by":"publisher","first-page":"070602","DOI":"10.1103\/PhysRevLett.98.070602","volume":"98","author":"S Trebst","year":"2007","unstructured":"Trebst, S., Werner, P., Troyer, M., Shtengel, K. & Nayak, C. Breakdown of a topological phase: Quantum phase transition in a loop gas with tension. Phys. Rev. Lett. 98, 070602 (2007).","journal-title":"Phys. Rev. Lett."},{"key":"BFnphys1396_CR31","doi-asserted-by":"publisher","first-page":"024412","DOI":"10.1103\/PhysRevB.72.024412","volume":"72","author":"P Fendley","year":"2005","unstructured":"Fendley, P. & Fradkin, E. Realizing non-Abelian statistics in time-reversal-invariant systems. Phys. Rev. B 72, 024412 (2005).","journal-title":"Phys. Rev. B"},{"key":"BFnphys1396_CR32","doi-asserted-by":"publisher","first-page":"428","DOI":"10.1016\/j.aop.2004.01.006","volume":"310","author":"M Freedman","year":"2004","unstructured":"Freedman, M., Nayak, C., Shtengel, K., Walker, K. & Wang, Z. A class of P,T-invariant topological phases of interacting electrons. Ann. Phys. 310, 428\u2013492 (2004).","journal-title":"Ann. Phys."},{"key":"BFnphys1396_CR33","doi-asserted-by":"crossref","unstructured":"Greenblatt, R. L., Aizenman, M. & Lebowitz, J. L. Rounding of first order transitions in low-dimensional quantum systems with quenched disorder. Preprint at &lt;\n                    http:\/\/arxiv.org\/abs\/0907.2419v1\n                    \n                  &gt; (2009).","DOI":"10.1103\/PhysRevLett.103.197201"},{"key":"BFnphys1396_CR34","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/BF01238857","volume":"123","author":"G Moore","year":"1989","unstructured":"Moore, G. & Seiberg, N. Classical and quantum conformal field theory. Commun. Math. Phys. 123, 177\u2013254 (1989).","journal-title":"Commun. Math. Phys."},{"key":"BFnphys1396_CR35","volume-title":"Infinite Dimensional Lie Algebras and Groups","author":"AN Kirillov","year":"1988","unstructured":"Kirillov, A. N. & Reshetikhin, N. Yu. Infinite Dimensional Lie Algebras and Groups (ed. Kac, V. G., World Scientific, 1988)."},{"key":"BFnphys1396_CR36","doi-asserted-by":"publisher","first-page":"1187","DOI":"10.1016\/j.jmmm.2006.10.304","volume":"310","author":"AF Albuquerque","year":"2007","unstructured":"Albuquerque, A. F. et al. The ALPS project release 1.3: Open-source software for strongly correlated systems. J. Magn. Magn. Mater. 310, 1187\u20131193 (2007).","journal-title":"J. Magn. Magn. Mater."}],"container-title":["Nature Physics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.nature.com\/articles\/nphys1396.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/www.nature.com\/articles\/nphys1396","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/www.nature.com\/articles\/nphys1396.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T23:41:42Z","timestamp":1684453302000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/nphys1396"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,9,20]]},"references-count":36,"journal-issue":{"issue":"11","published-print":{"date-parts":[[2009,11]]}},"alternative-id":["BFnphys1396"],"URL":"https:\/\/doi.org\/10.1038\/nphys1396","relation":{},"ISSN":["1745-2473","1745-2481"],"issn-type":[{"value":"1745-2473","type":"print"},{"value":"1745-2481","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,9,20]]}}}