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Zhenghan Wang
open-in-new
<h2 id="education-and-career">Education and career</h2> <p>Wang graduated with a <a href="/facts/Bachelor_of_Science/9n3gvDf7">B.S.</a> and <a href="/facts/Master_of_Science/5xKwvXjm">M.S.</a> from the <a href="/facts/University_of_Science_and_Technology_of_China/1LfcA52q">University of Science and Technology of China</a> in 1989 and obtained his <a href="/facts/Doctor_of_Philosophy/paJumdN8">Ph.D.</a> in 1993 from <a href="/facts/University_of_California%2c_San_Diego/srMClpNv">UC San Diego</a> under the supervision of <a href="/facts/Michael_Freedman/FxESCuA7">Michael Freedman</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> From 1993 to 1996 Wang taught as an assistant professor at the <a href="/facts/University_of_Michigan/6e1Ea5xh">University of Michigan</a> and from 1996 to 2007 Wang taught at <a href="/facts/Indiana_University_Bloomington/44gyhspU">Indiana University Bloomington</a>. For the majority of this time, Wang specialized in the topology of <a href="/facts/4-manifold/XwVxwqnr">4-manifolds</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p>In 2005, Wang moved to Santa Barbara to serve as a lead scientist in the newly founded research institute Microsoft Station Q. At Station Q, Wang worked with Michael Freedman (the station's director and his former Ph.D. advisor) on the foundations of <a href="/facts/Topological_quantum_computing/paJBK88q">topological quantum computing</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> Since 2012 Wang has served as a full professor at <a href="/facts/University_of_California%2c_Santa_Barbara/G7UvBWlK">UC Santa Barbara</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> From 2013 to 2020 Wang served as a distinguished visiting research chair at the <a href="/facts/Perimeter_Institute_for_Theoretical_Physics/QM6XFQve">Perimeter Institute for Theoretical Physics</a> as well.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> Wang was included in the 2019 class of fellows of the <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <h2 id="research">Research</h2> <h3>Topological Quantum Computing</h3> <p>Zhenghan Wang's most notable contributions are in the field of <a href="/facts/Topological_quantum_computer/paJBK88q">topological quantum computation</a>. In a series of early papers with Michael Freedman, <a href="/facts/Michael_J._Larsen/UqqdS6JE">Michael J. Larsen</a>, and <a href="/facts/Alexei_Kitaev/yF6fPcey">Alexei Kitaev</a>, Wang established the abstract equivalence of topological quantum computation with the <a href="/facts/Quantum_circuit/yuFDFFPa">quantum circuit model</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a><a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> The implication of these works for <a href="/facts/Topological_phase/EfEnKRSw">topological phases</a> is that the Fibonacci <a href="/facts/Anyon/hfgpCFjf">anyon model</a> can be used to make a universal quantum computer, and the implication of these works for quantum circuits is the <a href="/facts/Aharonov%25E2%2580%2593Jones%25E2%2580%2593Landau_algorithm/F3V57tWd">Aharonov–Jones–Landau algorithm</a>.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> Wang has also introduced several other schemes for universal topological quantum computation using <a href="/facts/Anyon/hfgpCFjf">anyons</a> which are more likely to be experimentally realizable.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a><a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a><a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p> <h3>The Algebraic Theory of Topological Phases</h3> <p>Outside of direct applications to topological quantum computing, Wang has made many contributions to the formal algebraic theory of two dimensional <a href="/facts/Topological_order/EfEnKRSw">topological quantum phases of matter</a>. This includes work on the structure and classification of <a href="/facts/Boson/XtPFcjb5">bosonic</a> topological order (modular tensor categories),<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a><a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a><a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a><a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> <a href="/facts/Fermion/LEckdDsR">fermionic</a> topological order (super-modular tensor categories),<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a><a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a><a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> and <a href="/facts/Symmetry-protected_topological_order/h8CVbg1M">symmetry-enriched</a> topological order (G-crossed modular tensor categories).<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a><a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a><a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a> Wang has also worked more specifically on the theory of the <a href="/facts/Fractional_quantum_Hall_effect/bkRtdmxX">fractional quantum Hall effect</a><a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a><a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a><a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a> and anyonic chains.<a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a><a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a><a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a><a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a> </p> <h3>Higher Dimensional Topological Quantum Field Theory</h3> <p>In addition to his work on two dimensional <a href="/facts/Topological_order/EfEnKRSw">topological order</a>, Wang has also worked in the theory of three dimensional <a href="/facts/Topological_quantum_field_theory/zHjrp3b8">topological quantum field theory</a>. Here he is most well known introducing the Walker-Wang model along with his coauthor Kevin Walker.<a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a><a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a><a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a> This theory has been used to describe the boundaries of <a href="/facts/Topological_insulator/pbrbxG9X">topological insulators</a><a class="footnote-ref" id="fnref:37" href="#fn:37"><sup>37</sup></a> and to construct nontrivial <a href="/facts/Quantum_cellular_automaton/Wwp9FU8N">quantum cellular automata</a>.<a class="footnote-ref" id="fnref:38" href="#fn:38"><sup>38</sup></a> Wang has also made contributions to the theory of three dimensional <a href="/facts/Fracton_(subdimensional_particle)/7HIDPMwc">fracton phases</a><a class="footnote-ref" id="fnref:39" href="#fn:39"><sup>39</sup></a><a class="footnote-ref" id="fnref:40" href="#fn:40"><sup>40</sup></a> </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"Engineering New Physical Reality: Quantum Computing with Topological Materials". Institute for Optical Science. 12 March 2021. Retrieved 30 March 2024. <a href="https://opticalscience.osu.edu/events/zhenghan-wang-engineering-new-physical-reality-quantum-computing-topological-materials" target="_blank">https://opticalscience.osu.edu/events/zhenghan-wang-engineering-new-physical-reality-quantum-computing-topological-materials</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Lin, Xiao-Song; Wang, Zhenghan (1996-04-26), On Ohtsuki's invariants of integral homology 3-spheres, I, arXiv:q-alg/9509009 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Wang, Zhenghan (1995). "Classification of closed nonorientable 4-manifolds with infinite cyclic fundamental group" (PDF). Mathematical Research Letters. 2 (3): 339–344. doi:10.4310/MRL.1995.v2.n3.a11 – via intlpress. <a href="https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1995/0002/0003/MRL-1995-0002-0003-a011.pdf" target="_blank">https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1995/0002/0003/MRL-1995-0002-0003-a011.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Wang, Zhenghan; Freedman, Michael (1994). "CP2-STABLE THEORY" (PDF). Mathematical Research Letters. 1 (1): 4 – via intlpress. <a href="https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1994/0001/0001/MRL-1994-0001-0001-a005.pdf" target="_blank">https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1994/0001/0001/MRL-1994-0001-0001-a005.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>"Zhenghan Wang at Microsoft Research". Microsoft Research. Retrieved 2024-03-30. <a href="https://www.microsoft.com/en-us/research/people/zhenghwa/" target="_blank">https://www.microsoft.com/en-us/research/people/zhenghwa/</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>"Engineering New Physical Reality: Quantum Computing with Topological Materials". Institute for Optical Science. 12 March 2021. Retrieved 30 March 2024. <a href="https://opticalscience.osu.edu/events/zhenghan-wang-engineering-new-physical-reality-quantum-computing-topological-materials" target="_blank">https://opticalscience.osu.edu/events/zhenghan-wang-engineering-new-physical-reality-quantum-computing-topological-materials</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>"Engineering New Physical Reality: Quantum Computing with Topological Materials". Institute for Optical Science. 12 March 2021. Retrieved 30 March 2024. <a href="https://opticalscience.osu.edu/events/zhenghan-wang-engineering-new-physical-reality-quantum-computing-topological-materials" target="_blank">https://opticalscience.osu.edu/events/zhenghan-wang-engineering-new-physical-reality-quantum-computing-topological-materials</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>"Fellows of the American Mathematical Society". American Mathematical Society. Retrieved 2024-03-30. <a href="https://www.ams.org/fellows_by_year.cgi?year=2019" target="_blank">https://www.ams.org/fellows_by_year.cgi?year=2019</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Freedman, Michael H.; Kitaev, Alexei; Larsen, Michael J.; Wang, Zhenghan (2002-09-20), Topological Quantum Computation, arXiv:quant-ph/0101025 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Freedman, Michael; Larsen, Michael; Wang, Zhenghan (2000-02-01), A modular functor which is universal for quantum computation, arXiv:quant-ph/0001108, Bibcode:2000quant.ph..1108F <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Freedman, Michael H.; Kitaev, Alexei; Wang, Zhenghan (2002-06-01). "Simulation of topological field theories by quantum computers". Communications in Mathematical Physics. 227 (3): 587–603. arXiv:quant-ph/0001071. Bibcode:2002CMaPh.227..587F. doi:10.1007/s002200200635. ISSN 0010-3616. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Freedman, Michael H.; Larsen, Michael J.; Wang, Zhenghan (2002-06-01). "The two-eigenvalue problem and density of Jones representation of braid groups". Communications in Mathematical Physics. 228 (1): 177–199. arXiv:math/0103200. Bibcode:2002CMaPh.228..177F. doi:10.1007/s002200200636. ISSN 0010-3616. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Aharonov, Dorit; Jones, Vaughan; Landau, Zeph (2006-04-10), A Polynomial Quantum Algorithm for Approximating the Jones Polynomial, arXiv:quant-ph/0511096 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Cui, Shawn X.; Hong, Seung-Moon; Wang, Zhenghan (August 2015). "Universal quantum computation with weakly integral anyons". Quantum Information Processing. 14 (8): 2687–2727. arXiv:1401.7096. Bibcode:2015QuIP...14.2687C. doi:10.1007/s11128-015-1016-y. ISSN 1570-0755. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Levaillant, Claire; Bauer, Bela; Freedman, Michael; Wang, Zhenghan; Bonderson, Parsa (2015-07-01). "Universal Gates via Fusion and Measurement Operations on SU$(2)_4$ Anyons". Physical Review A. 92 (1): 012301. arXiv:1504.02098. doi:10.1103/PhysRevA.92.012301. ISSN 1050-2947. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Cong, Iris; Cheng, Meng; Wang, Zhenghan (2017-10-25). "Universal Quantum Computation with Gapped Boundaries". Physical Review Letters. 119 (17): 170504. arXiv:1707.05490. Bibcode:2017PhRvL.119q0504C. doi:10.1103/PhysRevLett.119.170504. ISSN 0031-9007. PMID 29219455. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Rowell, Eric; Stong, Richard; Wang, Zhenghan (2009-11-09), On classification of modular tensor categories, arXiv:0712.1377 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Bruillard, Paul; Ng, Siu-Hung; Rowell, Eric C.; Wang, Zhenghan (2015-07-21). "Rank-finiteness for modular categories". Journal of the American Mathematical Society. 29 (3): 857–881. arXiv:1310.7050. doi:10.1090/jams/842. ISSN 0894-0347. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Hong, Seung-moon; Rowell, Eric; Wang, Zhenghan (2008-06-10), On exotic modular tensor categories, arXiv:0710.5761 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Ng, Siu-Hung; Rowell, Eric C.; Wang, Zhenghan; Wen, Xiao-Gang (September 2023). "Reconstruction of modular data from $SL_2(\mathbb{Z})$ representations". Communications in Mathematical Physics. 402 (3): 2465–2545. arXiv:2203.14829. doi:10.1007/s00220-023-04775-w. ISSN 0010-3616. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Bonderson, Parsa; Rowell, Eric C.; Zhang, Qing; Wang, Zhenghan (2018-07-16). "Congruence Subgroups and Super-Modular Categories". Pacific Journal of Mathematics. 296 (2): 257–270. arXiv:1704.02041. doi:10.2140/pjm.2018.296.257. ISSN 0030-8730. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Bruillard, Paul; Galindo, Cesar; Hagge, Tobias; Ng, Siu-Hung; Plavnik, Julia Yael; Rowell, Eric C.; Wang, Zhenghan (2017-04-01). "Fermionic Modular Categories and the 16-fold Way". Journal of Mathematical Physics. 58 (4): 041704. arXiv:1603.09294. Bibcode:2017JMP....58d1704B. doi:10.1063/1.4982048. ISSN 0022-2488. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>Bruillard, Paul; Galindo, César; Ng, Siu-Hung; Plavnik, Julia Yael; Rowell, Eric C.; Wang, Zhenghan (2017-05-23), Classification of super-modular categories by rank, arXiv:1705.05293 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>Barkeshli, Maissam; Bonderson, Parsa; Cheng, Meng; Wang, Zhenghan (2019-09-20). "Symmetry Fractionalization, Defects, and Gauging of Topological Phases". Physical Review B. 100 (11): 115147. arXiv:1410.4540. Bibcode:2019PhRvB.100k5147B. doi:10.1103/PhysRevB.100.115147. ISSN 2469-9950. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>Cui, Shawn X.; Galindo, César; Plavnik, Julia Yael; Wang, Zhenghan (December 2016). "On Gauging Symmetry of Modular Categories". Communications in Mathematical Physics. 348 (3): 1043–1064. arXiv:1510.03475. Bibcode:2016CMaPh.348.1043C. doi:10.1007/s00220-016-2633-8. ISSN 0010-3616. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> <li id="fn:26"><p>Cui, Shawn X.; Zini, Modjtaba Shokrian; Wang, Zhenghan (2018-09-01), On Generalized Symmetries and Structure of Modular Categories, arXiv:1809.00245 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li> <li id="fn:27"><p>Wen, Xiao-Gang; Wang, Zhenghan (2008-10-09). "Topological properties of Abelian and non-Abelian quantum Hall states from the pattern of zeros". Physical Review B. 78 (15): 155109. arXiv:0803.1016. doi:10.1103/PhysRevB.78.155109. ISSN 1098-0121. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li> <li id="fn:28"><p>Peterson, Michael R.; Wu, Yang-Le; Cheng, Meng; Barkeshli, Maissam; Wang, Zhenghan; Sarma, Sankar Das (2015-07-02). "Abelian and Non-Abelian States in $\nu=2/3$ Bilayer Fractional Quantum Hall Systems". Physical Review B. 92 (3): 035103. arXiv:1502.02671. Bibcode:2015PhRvB..92c5103P. doi:10.1103/PhysRevB.92.035103. ISSN 1098-0121. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></p></li> <li id="fn:29"><p>Lu, Yuan-Ming; Wen, Xiao-Gang; Wang, Zhenghan; Wang, Ziqiang (2010-03-17). "Non-Abelian Quantum Hall States and their Quasiparticles: from the Pattern of Zeros to Vertex Algebra". Physical Review B. 81 (11): 115124. arXiv:0910.3988. Bibcode:2010PhRvB..81k5124L. doi:10.1103/PhysRevB.81.115124. ISSN 1098-0121. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></p></li> <li id="fn:30"><p>Gils, Charlotte; Ardonne, Eddy; Trebst, Simon; Huse, David A.; Ludwig, Andreas W. W.; Troyer, Matthias; Wang, Zhenghan (2013-06-17). "Anyonic quantum spin chains: Spin-1 generalizations and topological stability". Physical Review B. 87 (23): 235120. arXiv:1303.4290. Bibcode:2013PhRvB..87w5120G. doi:10.1103/PhysRevB.87.235120. ISSN 1098-0121. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></p></li> <li id="fn:31"><p>Zini, Modjtaba Shokrian; Wang, Zhenghan (2018-08-06), Conformal Field Theories as Scaling Limit of Anyonic Chains, arXiv:1706.08497 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></p></li> <li id="fn:32"><p>Jiang, Hong-Chen; Rachel, Stephan; Weng, Zheng-Yu; Zhang, Shou-Cheng; Wang, Zhenghan (2010-12-14). "Critical theory of the topological quantum phase transition in a spin-2 chain". Physical Review B. 82 (22): 220403. arXiv:1010.4273. doi:10.1103/PhysRevB.82.220403. ISSN 1098-0121. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></p></li> <li id="fn:33"><p>Feiguin, Adrian; Trebst, Simon; Ludwig, Andreas W. W.; Troyer, Matthias; Kitaev, Alexei; Wang, Zhenghan; Freedman, Michael H. (2007-04-20). "Interacting anyons in topological quantum liquids: The golden chain". Physical Review Letters. 98 (16): 160409. arXiv:cond-mat/0612341. Bibcode:2007PhRvL..98p0409F. doi:10.1103/PhysRevLett.98.160409. ISSN 0031-9007. PMID 17501404. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></p></li> <li id="fn:34"><p>"Walker-Wang model in nLab". ncatlab.org. Retrieved 2024-03-23. <a href="https://ncatlab.org/nlab/show/Walker-Wang+model" target="_blank">https://ncatlab.org/nlab/show/Walker-Wang+model</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></p></li> <li id="fn:35"><p>Walker, Kevin; Wang, Zhenghan (2011-04-27), (3+1)-TQFTs and Topological Insulators, arXiv:1104.2632 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></p></li> <li id="fn:36"><p>Williamson, Dominic J.; Wang, Zhenghan (February 2017). "Hamiltonian models for topological phases of matter in three spatial dimensions". Annals of Physics. 377: 311–344. arXiv:1606.07144. Bibcode:2017AnPhy.377..311W. doi:10.1016/j.aop.2016.12.018. ISSN 0003-4916. <a href="https://dx.doi.org/10.1016/j.aop.2016.12.018" target="_blank">https://dx.doi.org/10.1016/j.aop.2016.12.018</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></p></li> <li id="fn:37"><p>Burnell, F. 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