Symmetry-enriched topological order in tensor networks: Defects, gauging and anyon condensation

We study symmetry-enriched topological order in two-dimensional tensor network states by using graded matrix product operator algebras to represent symmetry-induced domain walls. A close connection to the theory of graded unitary fusion categories is established. Tensor network representations of the topological defect superselection sectors are constructed for all domain walls. The emergent symmetry-enriched topological order is extracted from these representations, including the symmetry action on the underlying anyons. Dual phase transitions, induced by gauging a global symmetry, and condensation of a bosonic subtheory, are analyzed and the relationship between topological orders on either side of the transition is derived. Several examples are worked through explicitly.

The interplay of symmetry and topology in entangled states of quantum matter produce physical phenomena such as the fractionalization of charge on anyonic quasiparticle excitations. Tensor networks provide a tractable classical framework for the simulation of complex quantum states. In this work we extend that framework to include general symmetry actions on topological states of quantum matter in two dimensions.

[1] Lev Davidovich Landau and Evgenii Mikhailovich Lifshitz. ``Course of theoretical physics''. Pergamon Press. (1965).

[2] G. B. Segal. ``The definition of conformal field theory''. Pages 165–171. Springer Netherlands. Dordrecht (1988). https:/​/​doi.org/​10.1007/​978-94-015-7809-7_9

[3] Edward Witten. ``Topological quantum field theory''. Commun. Math. Phys. 117, 353–386 (1988). https:/​/​doi.org/​10.1007/​BF01223371

[4] Michael Atiyah. ``Topological quantum field theories''. Inst. Hautes Études Sci. Publ. Math. 68, 175–186 (1988). https:/​/​doi.org/​10.1007/​BF02698547

[5] Franz J Wegner. ``Duality in generalized Ising models and phase transitions without local order parameters''. J. Math. Phys. 12, 2259–2272 (1971). https:/​/​doi.org/​10.1063/​1.1665530

[6] John Michael Kosterlitz and David James Thouless. ``Ordering, metastability and phase transitions in two-dimensional systems''. J. Phys. C: Solid State Phys. 6, 1181 (1973). https:/​/​doi.org/​10.1088/​0022-3719/​6/​7/​010

[7] X.-G. Wen. ``Vacuum degeneracy of chiral spin states in compactified space''. Phys. Rev. B 40, 7387–7390 (1989). https:/​/​doi.org/​10.1103/​PhysRevB.40.7387

[8] Torbjörn Einarsson. ``Fractional statistics on a torus''. Phys. Rev. Lett. 64, 1995–1998 (1990). https:/​/​doi.org/​10.1103/​physrevlett.64.1995

[9] X.-G. Wen. ``Topological Orders in Rigid States''. Int. J. Mod. Phys. B 04, 239–271 (1990). https:/​/​doi.org/​10.1142/​S0217979290000139

[10] F. D. M. Haldane. ``Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State''. Phys. Rev. Lett. 50, 1153–1156 (1983). https:/​/​doi.org/​10.1103/​PhysRevLett.50.1153

[11] Zheng-Cheng Gu and Xiao-Gang Wen. ``Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order''. Phys. Rev. B 80, 155131 (2009). arXiv:0903.1069. https:/​/​doi.org/​10.1103/​PhysRevB.80.155131 arXiv:0903.1069

[12] Frank Pollmann, Ari M Turner, Erez Berg, and Masaki Oshikawa. ``Entanglement spectrum of a topological phase in one dimension''. Phys. Rev. B 81, 064439 (2010). arXiv:0910.1811. https:/​/​doi.org/​10.1103/​PhysRevB.81.064439 arXiv:0910.1811

[13] Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, and Xiao-Gang Wen. ``Symmetry protected topological orders and the group cohomology of their symmetry group''. Phys. Rev. B 87, 155114 (2013). arXiv:1106.4772. https:/​/​doi.org/​10.1103/​PhysRevB.87.155114 arXiv:1106.4772

[14] Vladimir Turaev. ``Homotopy field theory in dimension 3 and crossed group-categories''. preprint, (2000). arXiv:math/​0005291. arXiv:math/0005291

[15] Alexander Kirillov Jr. ``On $g$–equivariant modular categories''. preprint, (2004). arXiv:math/​0401119. arXiv:math/0401119

[16] Alexei Kitaev. ``Anyons in an exactly solved model and beyond''. Ann. Phys. 321, 2 – 111 (2006). arXiv:cond-mat/​0506438. https:/​/​doi.org/​10.1016/​j.aop.2005.10.005 arXiv:cond-mat/0506438

[17] Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik. ``On braided fusion categories I''. Selecta Mathematica 16, 1–119 (2010). arXiv:0906.0620. https:/​/​doi.org/​10.1007/​s00029-010-0017-z arXiv:0906.0620

[18] Pavel Etingof, Dmitri Nikshych, and Victor Ostrik. ``Fusion categories and homotopy theory''. preprint, (2009). arXiv:0909.3140. arXiv:0909.3140

[19] H. Bombin. ``Topological order with a twist: Ising anyons from an abelian model''. Phys. Rev. Lett. 105, 030403 (2010). arXiv:1004.1838. https:/​/​doi.org/​10.1103/​PhysRevLett.105.030403 arXiv:1004.1838

[20] Ling-Yan Hung and Xiao-Gang Wen. ``Quantized topological terms in weak-coupling gauge theories with a global symmetry and their connection to symmetry-enriched topological phases''. Phys. Rev. B 87, 165107 (2013). arXiv:1212.1827. https:/​/​doi.org/​10.1103/​PhysRevB.87.165107 arXiv:1212.1827

[21] Andrej Mesaros and Ying Ran. ``Classification of symmetry enriched topological phases with exactly solvable models''. Phys. Rev. B 87, 155115 (2013). arXiv:1212.0835. https:/​/​doi.org/​10.1103/​PhysRevB.87.155115 arXiv:1212.0835

[22] Maissam Barkeshli, Chao Ming Jian, and Xiao Liang Qi. ``Theory of defects in abelian topological states''. Physical Review B - Condensed Matter and Materials Physics 88 (2013). https:/​/​doi.org/​10.1103/​PhysRevB.88.235103

[23] Maissam Barkeshli, Parsa Bonderson, Meng Cheng, and Zhenghan Wang. ``Symmetry, defects, and gauging of topological phases''. preprint, (2014). arXiv:1410.4540. https:/​/​doi.org/​10.1103/​PhysRevB.100.115147 arXiv:1410.4540

[24] Nicolas Tarantino, Netanel H Lindner, and Lukasz Fidkowski. ``Symmetry fractionalization and twist defects''. New J. Phys. 18, 035006 (2016). arXiv:1506.06754. https:/​/​doi.org/​10.1088/​1367-2630/​18/​3/​035006 arXiv:1506.06754

[25] Jeffrey CY Teo, Taylor L Hughes, and Eduardo Fradkin. ``Theory of twist liquids: gauging an anyonic symmetry''. Ann. Phys. 360, 349–445 (2015). arXiv:1503.06812. https:/​/​doi.org/​10.1016/​j.aop.2015.05.012 arXiv:1503.06812

[26] Chris Heinrich, Fiona Burnell, Lukasz Fidkowski, and Michael Levin. ``Symmetry-enriched string nets: Exactly solvable models for set phases''. Phys. Rev. B 94, 235136 (2016). arXiv:1606.07816. https:/​/​doi.org/​10.1103/​PhysRevB.94.235136 arXiv:1606.07816

[27] Meng Cheng, Zheng-Cheng Gu, Shenghan Jiang, and Yang Qi. ``Exactly solvable models for symmetry-enriched topological phases''. Phys. Rev. B 96, 115107 (2017). arXiv:1606.08482. https:/​/​doi.org/​10.1103/​PhysRevB.96.115107 arXiv:1606.08482

[28] Salman Beigi, Peter W. Shor, and Daniel Whalen. ``The quantum double model with boundary: Condensations and symmetries''. Commun. Math. Phys. 306, 663–694 (2011). arXiv:1006.5479. https:/​/​doi.org/​10.1007/​s00220-011-1294-x arXiv:1006.5479

[29] Benjamin J. Brown, Stephen D. Bartlett, Andrew C. Doherty, and Sean D. Barrett. ``Topological entanglement entropy with a twist''. Phys. Rev. Lett. 111, 220402 (2013). arXiv:1303.4455. https:/​/​doi.org/​10.1103/​PhysRevLett.111.220402 arXiv:1303.4455

[30] Beni Yoshida. ``Gapped boundaries, group cohomology and fault-tolerant logical gates''. Ann. Phys. 377, 387 – 413 (2017). arXiv:1509.03626. https:/​/​doi.org/​10.1016/​j.aop.2016.12.014 arXiv:1509.03626

[31] Fernando Pastawski and Beni Yoshida. ``Fault-tolerant logical gates in quantum error-correcting codes''. Phys. Rev. A 91, 012305 (2015). arXiv:1408.1720. https:/​/​doi.org/​10.1103/​PhysRevA.91.012305 arXiv:1408.1720

[32] Sergey Bravyi and Robert König. ``Classification of topologically protected gates for local stabilizer codes''. Phys. Rev. Lett. 110, 170503 (2013). arXiv:1206.1609. https:/​/​doi.org/​10.1103/​PhysRevLett.110.170503 arXiv:1206.1609

[33] Michael E. Beverland, Oliver Buerschaper, Robert König, Fernando Pastawski, John Preskill, and Sumit Sijher. ``Protected gates for topological quantum field theories''. J. Math. Phys. 57, 022201 (2016). arXiv:1409.3898. https:/​/​doi.org/​10.1063/​1.4939783 arXiv:1409.3898

[34] Beni Yoshida. ``Topological color code and symmetry-protected topological phases''. Phys. Rev. B 91, 245131 (2015). arXiv:1503.07208. https:/​/​doi.org/​10.1103/​PhysRevB.91.245131 arXiv:1503.07208

[35] Jacob C Bridgeman, Stephen D Bartlett, and Andrew C Doherty. ``Tensor networks with a twist: Anyon-permuting domain walls and defects in peps''. preprint, (2017). arXiv:1704.04221. https:/​/​doi.org/​10.1103/​PhysRevX.8.021074 arXiv:1704.04221

[36] Paul Webster and Stephen D Bartlett. ``Locality-preserving logical operators in topological stabiliser codes''. preprint (2017). arXiv:1709.00020. https:/​/​doi.org/​10.1103/​PhysRevA.97.012330 arXiv:1709.00020

[37] F Verstraete, J. I. Cirac, and V Murg. ``Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems''. Adv. Phys. 57, 143–224 (2009). arXiv:0907.2796. https:/​/​doi.org/​10.1080/​14789940801912366 arXiv:0907.2796

[38] Román Orús. ``A practical introduction to tensor networks: Matrix product states and projected entangled pair states''. Ann. Phys. 349, 117–158 (2014). arXiv:1306.2164. https:/​/​doi.org/​10.1016/​j.aop.2014.06.013 arXiv:1306.2164

[39] Jacob C Bridgeman and Christopher T Chubb. ``Hand-waving and interpretive dance: an introductory course on tensor networks''. J. Phys. A 50, 223001 (2017). arXiv:1603.03039. https:/​/​doi.org/​10.1088/​1751-8121/​aa6dc3 arXiv:1603.03039

[40] Steven R. White. ``Density matrix formulation for quantum renormalization groups''. Phys. Rev. Lett. 69, 2863–2866 (1992). https:/​/​doi.org/​10.1103/​PhysRevLett.69.2863

[41] Stellan Östlund and Stefan Rommer. ``Thermodynamic Limit of Density Matrix Renormalization''. Phys. Rev. Lett. 75, 3537–3540 (1995). arXiv:cond-mat/​9503107. https:/​/​doi.org/​10.1103/​PhysRevLett.75.3537 arXiv:cond-mat/9503107

[42] J. Dukelsky, M. A. Martín-Delgado, T. Nishino, and G. Sierra. ``Equivalence of the variational matrix product method and the density matrix renormalization group applied to spin chains''. EPL 43, 457 (1998). arXiv:cond-mat/​9710310. https:/​/​doi.org/​10.1209/​epl/​i1998-00381-x arXiv:cond-mat/9710310

[43] Ian Affleck, Tom Kennedy, Elliott H. Lieb, and Hal Tasaki. ``Rigorous results on valence-bond ground states in antiferromagnets''. Phys. Rev. Lett. 59, 799–802 (1987). https:/​/​doi.org/​10.1103/​PhysRevLett.59.799

[44] M. Fannes, B. Nachtergaele, and R. F. Werner. ``Finitely correlated states on quantum spin chains''. Commun. Math. Phys. 144, 443–490 (1992). https:/​/​doi.org/​10.1007/​BF02099178

[45] A. Klümper, A. Schadschneider, and J. Zittartz. ``Matrix Product Ground States for One-Dimensional Spin-1 Quantum Antiferromagnets''. EPL 24, 293 (1993). arXiv:cond-mat/​9307028. https:/​/​doi.org/​10.1209/​0295-5075/​24/​4/​010 arXiv:cond-mat/9307028

[46] D. Pérez-García, F. Verstraete, M. M. Wolf, and J. I. Cirac. ``Matrix product state representations''. Quantum Info. Comput. 7, 401–430 (2007). arXiv:quant-ph/​0608197. arXiv:quant-ph/0608197

[47] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen. ``Classification of gapped symmetric phases in one-dimensional spin systems''. Phys. Rev. B 83, 035107 (2011). arXiv:1008.3745. https:/​/​doi.org/​10.1103/​PhysRevB.83.035107 arXiv:1008.3745

[48] Norbert Schuch, David Pérez-García, and Ignacio Cirac. ``Classifying quantum phases using matrix product states and projected entangled pair states''. Phys. Rev. B 84, 165139 (2011). arXiv:1010.3732. https:/​/​doi.org/​10.1103/​PhysRevB.84.165139 arXiv:1010.3732

[49] J.I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete. ``Matrix product density operators: Renormalization fixed points and boundary theories''. Ann. Phys. 378, 100 – 149 (2017). arXiv:1606.00608. https:/​/​doi.org/​10.1016/​j.aop.2016.12.030 arXiv:1606.00608

[50] Tomotoshi Nishino, Yasuhiro Hieida, Kouichi Okunishi, Nobuya Maeshima, Yasuhiro Akutsu, and Andrej Gendiar. ``Two-Dimensional Tensor Product Variational Formulation''. Progr. Theor. Phys. 105, 409 (2001). arXiv:cond-mat/​0011103. https:/​/​doi.org/​10.1143/​PTP.105.409 arXiv:cond-mat/0011103

[51] F. Verstraete and J. I. Cirac. ``Renormalization algorithms for quantum-many body systems in two and higher dimensions'' (2004). arXiv:cond-mat/​0407066. arXiv:cond-mat/0407066

[52] N. Schuch, I. Cirac, and D. Perez-Garcia. ``Peps as ground states: Degeneracy and topology''. Ann. Phys. 325, 2153 – 2192 (2010). arXiv:1001.3807. https:/​/​doi.org/​10.1016/​j.aop.2010.05.008 arXiv:1001.3807

[53] Oliver Buerschaper. ``Twisted injectivity in projected entangled pair states and the classification of quantum phases''. Ann. Phys. 351, 447 – 476 (2014). arXiv:1307.7763. https:/​/​doi.org/​10.1016/​j.aop.2014.09.007 arXiv:1307.7763

[54] Mehmet B. Şahinoğlu, D. Williamson, N. Bultinck, M. Mariën, J. Haegeman, N. Schuch, and F. Verstraete. ``Characterizing topological order with matrix product operators''. preprint, (2014). arXiv:1409.2150. https:/​/​doi.org/​10.1007/​s00023-020-00992-4 arXiv:1409.2150

[55] N. Bultinck, M. Mariën, D.J. Williamson, Mehmet B. Şahinoğlu, J. Haegeman, and F. Verstraete. ``Anyons and matrix product operator algebras''. Ann. Phys. 378, 183 – 233 (2017). arXiv:1511.08090. https:/​/​doi.org/​10.1016/​j.aop.2017.01.004 arXiv:1511.08090

[56] J.C. Bridgeman, S. T. Flammia, and D. Poulin. ``Detecting Topological Order with Ribbon Operators''. Phys. Rev. B 94, 205123 (2016). arXiv:1603.02275. https:/​/​doi.org/​10.1103/​PhysRevB.94.205123 arXiv:1603.02275

[57] Dominic J Williamson, Nick Bultinck, Jutho Haegeman, and Frank Verstraete. ``Fermionic Matrix Product Operators and Topological Phases of Matter''. preprint, (2016). arXiv:1609.02897. arXiv:1609.02897

[58] Nick Bultinck, Dominic J. Williamson, Jutho Haegeman, and Frank Verstraete. ``Fermionic projected entangled-pair states and topological phases''. J. Phys. A (2017). arXiv:1707.00470. https:/​/​doi.org/​10.1088/​1751-8121/​aa99cc arXiv:1707.00470

[59] Michael Müger. ``From subfactors to categories and topology I: Frobenius algebras in and Morita equivalence of tensor categories''. J. of Pure Appl. Algebr. 180, 81 – 157 (2003). arXiv:math/​0111204. https:/​/​doi.org/​10.1016/​S0022-4049(02)00247-5 arXiv:math/0111204

[60] Michael A. Levin and Xiao-Gang Wen. ``String-net condensation: A physical mechanism for topological phases''. Phys. Rev. B 71, 045110 (2005). arXiv:cond-mat/​0404617. https:/​/​doi.org/​10.1103/​PhysRevB.71.045110 arXiv:cond-mat/0404617

[61] Oliver Buerschaper, Miguel Aguado, and Guifré Vidal. ``Explicit tensor network representation for the ground states of string-net models''. Phys. Rev. B 79, 085119 (2009). arXiv:0809.2393. https:/​/​doi.org/​10.1103/​PhysRevB.79.085119 arXiv:0809.2393

[62] Zheng-Cheng Gu, Michael Levin, Brian Swingle, and Xiao-Gang Wen. ``Tensor-product representations for string-net condensed states''. Phys. Rev. B 79, 085118 (2009). arXiv:0809.2821. https:/​/​doi.org/​10.1103/​PhysRevB.79.085118 arXiv:0809.2821

[63] Shuo Yang, Thorsten B. Wahl, Hong-Hao Tu, Norbert Schuch, and J. Ignacio Cirac. ``Chiral projected entangled-pair state with topological order''. Phys. Rev. Lett. 114, 106803 (2015). arXiv:1411.6618. https:/​/​doi.org/​10.1103/​PhysRevLett.114.106803 arXiv:1411.6618

[64] J. Dubail and N. Read. ``Tensor network trial states for chiral topological phases in two dimensions and a no-go theorem in any dimension''. Phys. Rev. B 92, 205307 (2015). arXiv:1307.7726. https:/​/​doi.org/​10.1103/​PhysRevB.92.205307 arXiv:1307.7726

[65] T. B. Wahl, H.-H. Tu, N. Schuch, and J. I. Cirac. ``Projected Entangled-Pair States Can Describe Chiral Topological States''. Phys. Rev. Lett. 111, 236805 (2013). arXiv:1308.0316. https:/​/​doi.org/​10.1103/​PhysRevLett.111.236805 arXiv:1308.0316

[66] Dominic J. Williamson, Nick Bultinck, Michael Mariën, Mehmet B. Şahinoğlu, Jutho Haegeman, and Frank Verstraete. ``Matrix product operators for symmetry-protected topological phases: Gauging and edge theories''. Phys. Rev. B 94, 205150 (2016). arXiv:1412.5604. https:/​/​doi.org/​10.1103/​PhysRevB.94.205150 arXiv:1412.5604

[67] Shlomo Gelaki, Deepak Naidu, and Dmitri Nikshych. ``Centers of graded fusion categories''. Algebra & Number Theory 3, 959–990 (2009). arXiv:0905.3117. https:/​/​doi.org/​10.2140/​ant.2009.3.959 arXiv:0905.3117

[68] Liang Chang, Meng Cheng, Shawn X Cui, Yuting Hu, Wei Jin, Ramis Movassagh, Pieter Naaijkens, Zhenghan Wang, and Amanda Young. ``On enriching the Levin-Wen model with symmetry''. J. Phys. A 48, 12FT01 (2015). arXiv:1412.6589. https:/​/​doi.org/​10.1088/​1751-8113/​48/​12/​12FT01 arXiv:1412.6589

[69] Adrian Ocneanu. ``Chirality for operator algebras''. Subfactors (Kyuzeso, 1993)Pages 39–63 (1994).

[70] David Evans and Yasuyuki Kawahigashi. ``On Ocneanu's theory of asymptotic inclusions for subfactors, topological quantum field theories and quantum doubles''. Int. J. Math. 6, 205–228 (1995). https:/​/​doi.org/​10.1142/​s0129167x95000468

[71] David E Evans and Yasuyuki Kawahigashi. ``Quantum symmetries on operator algebras''. Volume 147. Clarendon Press Oxford. (1998).

[72] Kenneth G. Wilson. ``Confinement of quarks''. Phys. Rev. D 10, 2445–2459 (1974). https:/​/​doi.org/​10.1103/​PhysRevD.10.2445

[73] John Kogut and Leonard Susskind. ``Hamiltonian formulation of wilson's lattice gauge theories''. Phys. Rev. D 11, 395–408 (1975). https:/​/​doi.org/​10.1103/​PhysRevD.11.395

[74] Michael Levin and Zheng-Cheng Gu. ``Braiding statistics approach to symmetry-protected topological phases''. Phys. Rev. B 86, 115109 (2012). arXiv:1202.3120. https:/​/​doi.org/​10.1103/​PhysRevB.86.115109 arXiv:1202.3120

[75] Jutho Haegeman, Karel Van Acoleyen, Norbert Schuch, J Ignacio Cirac, and Frank Verstraete. ``Gauging quantum states: from global to local symmetries in many-body systems''. Phys. Rev. X 5, 011024 (2015). arXiv:1407.1025. https:/​/​doi.org/​10.1103/​PhysRevX.5.011024 arXiv:1407.1025

[76] F. A. Bais and J. K. Slingerland. ``Condensate-induced transitions between topologically ordered phases''. Phys. Rev. B 79, 045316 (2009). arXiv:0808.0627. https:/​/​doi.org/​10.1103/​PhysRevB.79.045316 arXiv:0808.0627

[77] A.Yu. Kitaev. ``Fault-tolerant quantum computation by anyons''. Ann. Phys. 303, 2 – 30 (2003). arXiv:quant-ph/​9707021. https:/​/​doi.org/​10.1016/​S0003-4916(02)00018-0 arXiv:quant-ph/9707021

[78] Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik. ``On fusion categories''. Ann. Math.Pages 581–642 (2005). arXiv:math/​0203060. https:/​/​doi.org/​10.4007/​annals.2005.162.581 arXiv:math/0203060

[79] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik. ``Tensor categories''. Volume 205. American Mathematical Soc. (2015). https:/​/​doi.org/​10.1090/​surv/​205

[80] Bojko Bakalov and Alexander A Kirillov. ``Lectures on tensor categories and modular functors''. Volume 21. American Mathematical Soc. (2001). https:/​/​doi.org/​10.1090/​ulect/​021

[81] Zohar Nussinov and Gerardo Ortiz. ``Sufficient symmetry conditions for topological quantum order''. Proceedings of the National Academy of Sciences 106, 16944–16949 (2009). arXiv:cond-mat/​0605316. https:/​/​doi.org/​10.1073/​pnas.0803726105 arXiv:cond-mat/0605316

[82] Zohar Nussinov and Gerardo Ortiz. ``A symmetry principle for topological quantum order''. Annals of Physics 324, 977 – 1057 (2009). arXiv:cond-mat/​0702377. https:/​/​doi.org/​10.1016/​j.aop.2008.11.002 arXiv:cond-mat/0702377

[83] Emilio Cobanera, Gerardo Ortiz, and Zohar Nussinov. ``The bond-algebraic approach to dualities''. Advances in Physics 60, 679–798 (2011). arXiv:1103.2776. https:/​/​doi.org/​10.1080/​00018732.2011.619814 arXiv:1103.2776

[84] David Aasen, Roger S K Mong, and Paul Fendley. ``Topological Defects on the Lattice I: The Ising model''. J. Phys. A 49, 354001 (2016). arXiv:1601.07185. https:/​/​doi.org/​10.1088/​1751-8113/​49/​35/​354001 arXiv:1601.07185

[85] André Joyal and Ross Street. ``The geometry of tensor calculus, I''. Advances in Mathematics 88, 55 – 112 (1991). https:/​/​doi.org/​10.1016/​0001-8708(91)90003-P

[86] A. Joyal and R. Street. ``Braided tensor categories''. Advances in Mathematics 102, 20 – 78 (1993). https:/​/​doi.org/​10.1006/​aima.1993.1055

[87] V.G. Turaev and O.Y. Viro. ``State sum invariants of 3-manifolds and quantum 6j-symbols''. Topology 31, 865 – 902 (1992). https:/​/​doi.org/​10.1016/​0040-9383(92)90015-A

[88] John Barrett and Bruce Westbury. ``Invariants of piecewise-linear 3-manifolds''. Trans. Am. Math. Soc. 348, 3997–4022 (1996). arXiv:hep-th/​9311155. https:/​/​doi.org/​10.1090/​S0002-9947-96-01660-1 arXiv:hep-th/9311155

[89] Kevin Walker. ``On Witten’s 3-manifold invariants & TQFTs''. preprints, available at http:/​/​canyon23.net/​math (1991,2006). url: http:/​/​canyon23.net/​math. http:/​/​canyon23.net/​math

[90] Michael Freedman, Chetan Nayak, Kevin Walker, and Zhenghan Wang. ``On picture $(2+1)$-TQFTs''. Pages 19–106. WORLD SCIENTIFIC. (2011). arXiv:0806.1926. https:/​/​doi.org/​10.1142/​9789812819116_0002 arXiv:0806.1926

[91] Robert Koenig, Greg Kuperberg, and Ben W. Reichardt. ``Quantum computation with Turaev-Viro codes''. Ann. Phys. 325, 2707 – 2749 (2010). arXiv:1002.2816. https:/​/​doi.org/​10.1016/​j.aop.2010.08.001 arXiv:1002.2816

[92] D. Pérez-García, F. Verstraete, M. M. Wolf, and J. I. Cirac. ``Peps as unique ground states of local hamiltonians''. Quantum Info. Comput. 8, 650–663 (2008). arXiv:0707.2260. arXiv:0707.2260

[93] Mehmet Burak Sahinoglu, Michael Walter, and Dominic J Williamson. ``A tensor network framework for topological order in any dimension''. preprint, available at https:/​/​sites.google.com/​site/​dominicjw/​papers (2015). url: https:/​/​sites.google.com/​site/​dominicjw/​papers. https:/​/​sites.google.com/​site/​dominicjw/​papers

[94] Michaël Mignard and Peter Schauenburg. ``Modular categories are not determined by their modular data''. preprint, (2017). arXiv:1704.04221. https:/​/​doi.org/​10.1103/​PhysRevX.8.021074 arXiv:1704.04221

[95] Michael Muger. ``From subfactors to categories and topology II: The quantum double of tensor categories and subfactors''. J. Pure Appl. Algebr. 180, 159 – 219 (2003). arXiv:math/​0111205. https:/​/​doi.org/​10.1016/​S0022-4049(02)00248-7 arXiv:math/0111205

[96] Maissam Barkeshli, Parsa Bonderson, Chao-Ming Jian, Meng Cheng, and Kevin Walker. ``Reflection and time reversal symmetry enriched topological phases of matter: path integrals, non-orientable manifolds, and anomalies''. preprint, (2016). arXiv:1612.07792. https:/​/​doi.org/​10.1007/​s00220-019-03475-8 arXiv:1612.07792

[97] Shawn X Cui and Zhenghan Wang. ``State sum invariants of three manifolds from spherical multi-fusion categories''. preprint, (2017). arXiv:1702.07113. https:/​/​doi.org/​10.1142/​S0218216517501048 arXiv:1702.07113

[98] Dominic J. Williamson. ``Fractal symmetries: Ungauging the cubic code''. Phys. Rev. B 94, 155128 (2016). arXiv:1603.05182. https:/​/​doi.org/​10.1103/​PhysRevB.94.155128 arXiv:1603.05182

[99] R. Walter Ogburn and John Preskill. ``Topological quantum computation''. Pages 341–356. Springer Berlin Heidelberg. Berlin, Heidelberg (1999). https:/​/​doi.org/​10.1007/​3-540-49208-9_31

[100] Michael Freedman, Alexei Kitaev, Michael Larsen, and Zhenghan Wang. ``Topological quantum computation''. Bull. Am. Math. Soc. 40, 31–38 (2003). arXiv:quant-ph/​0101025. https:/​/​doi.org/​10.1090/​S0273-0979-02-00964-3 arXiv:quant-ph/0101025

[101] Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma. ``Non-abelian anyons and topological quantum computation''. Rev. Mod. Phys. 80, 1083–1159 (2008). arXiv:0707.1889. https:/​/​doi.org/​10.1103/​RevModPhys.80.1083 arXiv:0707.1889

[102] Zhenghan Wang. ``Topological quantum computation''. Volume 112. American Mathematical Soc. (2010). https:/​/​doi.org/​10.1090/​cbms/​112

[103] Gregory Moore and Nathan Seiberg. ``Polynomial equations for rational conformal field theories''. Phys. Lett. B 212, 451 – 460 (1988). https:/​/​doi.org/​10.1016/​0370-2693(88)91796-0

[104] Gregory Moore and Nathan Seiberg. ``Classical and quantum conformal field theory''. Commun. Math. Phys. 123, 177–254 (1989). https:/​/​doi.org/​10.1007/​BF01238857

[105] Saunders Mac Lane. ``Categories for the working mathematician''. Volume 5. Springer. (2013). https:/​/​doi.org/​10.1007/​978-1-4612-9839-7

[106] Zhu-Xi Luo, Ethan Lake, and Yong-Shi Wu. ``The structure of fixed-point tensor network states characterizes the patterns of long-range entanglement''. Phys. Rev. B 96, 035101 (2017). arXiv:1611.01140. https:/​/​doi.org/​10.1103/​PhysRevB.96.035101 arXiv:1611.01140

[107] Carlos Fernández-González, Roger S. K. Mong, Olivier Landon-Cardinal, David Pérez-García, and Norbert Schuch. ``Constructing topological models by symmetrization: A projected entangled pair states study''. Phys. Rev. B 94, 155106 (2016). arXiv:1608.00594. https:/​/​doi.org/​10.1103/​PhysRevB.94.155106 arXiv:1608.00594

[108] Tian Lan and Xiao-Gang Wen. ``Topological quasiparticles and the holographic bulk-edge relation in $(2+1)$-dimensional string-net models''. Phys. Rev. B 90, 115119 (2014). arXiv:1311.1784. https:/​/​doi.org/​10.1103/​PhysRevB.90.115119 arXiv:1311.1784

[109] J. Haah. ``An invariant of topologically ordered states under local unitary transformations''. Commun. Math. Phys. 342, 771–801 (2016). arXiv:1407.2926. https:/​/​doi.org/​10.1007/​s00220-016-2594-y arXiv:1407.2926

[110] David Aasen, Ethan Lake, and Kevin Walker. ``Fermion condensation and super pivotal categories''. preprint (2017). arXiv:1709.01941. https:/​/​doi.org/​10.1063/​1.5045669 arXiv:1709.01941

[111] David Aasen, Paul Fendley, and Roger S. K. Mong. ``Topological Defects on the Lattice: Dualities and Degeneracies'' (2020). arXiv:2008.08598. arXiv:2008.08598

[112] Adrian Ocneanu. ``Operator algebras, topology and subgroups of quantum symmetry— construction of subgroups of quantum groups''. Taniguchi Conference on Mathematics Nara 1998, Adv. Stud. Pure Math., 31, 235–263 (2001).

[113] David Aasen. ``Lessons from the tube master''. private communication (2017).

[114] Yuanjie Ren, Nathanan Tantivasadakarn, and Dominic J. Williamson. ``Efficient preparation of solvable anyons with adaptive quantum circuits''. Physical Review X 15 (2025). https:/​/​doi.org/​10.1103/​b9hf-gx4f

[115] Yi Zhang, Tarun Grover, Ari Turner, Masaki Oshikawa, and Ashvin Vishwanath. ``Quasiparticle statistics and braiding from ground-state entanglement''. Phys. Rev. B 85, 235151 (2012). arXiv:1111.2342. https:/​/​doi.org/​10.1103/​PhysRevB.85.235151 arXiv:1111.2342

[116] Alexei Kitaev and John Preskill. ``Topological entanglement entropy''. Phys. Rev. Lett. 96, 110404 (2006). arXiv:hep-th/​0510092. https:/​/​doi.org/​10.1103/​PhysRevLett.96.110404 arXiv:hep-th/0510092

[117] Michael Levin and Xiao-Gang Wen. ``Detecting Topological Order in a Ground State Wave Function''. Phys. Rev. Lett. 96, 110405 (2006). arXiv:cond-mat/​0510613. https:/​/​doi.org/​10.1103/​PhysRevLett.96.110405 arXiv:cond-mat/0510613

[118] Xiao-Liang Qi, Hosho Katsura, and Andreas W. W. Ludwig. ``General relationship between the entanglement spectrum and the edge state spectrum of topological quantum states''. Phys. Rev. Lett. 108, 196402 (2012). arXiv:1103.5437. https:/​/​doi.org/​10.1103/​PhysRevLett.108.196402 arXiv:1103.5437

[119] S. Flammia, A. Hamma, T. Hughes, and X.-G. Wen. ``Topological entanglement Renyi entropy and reduced density matrix structure''. Phys. Rev. Lett. 103, 261601 (2009). arXiv:0909.3305. https:/​/​doi.org/​10.1103/​PhysRevLett.103.261601 arXiv:0909.3305

[120] V Pasquier. ``Operator content of the ade lattice models''. Journal of Physics A: Mathematical and General 20, 5707 (1987). https:/​/​doi.org/​10.1088/​0305-4470/​20/​16/​043

[121] Erik Verlinde. ``Fusion rules and modular transformations in 2d conformal field theory''. Nuclear Physics B 300, 360 – 376 (1988). https:/​/​doi.org/​10.1016/​0550-3213(88)90603-7

[122] Xie Chen, Zheng-Xin Liu, and Xiao-Gang Wen. ``Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations''. Phys. Rev. B 84, 235141 (2011). arXiv:1106.4752. https:/​/​doi.org/​10.1103/​PhysRevB.84.235141 arXiv:1106.4752

[123] Xiao-Gang Wen. ``Quantum orders and symmetric spin liquids''. Phys. Rev. B 65, 165113 (2002). arXiv:cond-mat/​0107071. https:/​/​doi.org/​10.1103/​PhysRevB.65.165113 arXiv:cond-mat/0107071

[124] Shenghan Jiang and Ying Ran. ``Anyon condensation and a generic tensor-network construction for symmetry-protected topological phases''. Phys. Rev. B 95, 125107 (2017). arXiv:1611.07652. https:/​/​doi.org/​10.1103/​PhysRevB.95.125107 arXiv:1611.07652

[125] José Garre-Rubio, Sofyan Iblisdir, and David Pérez-García. ``Symmetry reduction induced by anyon condensation: a tensor network approach''. preprint, (2017). arXiv:1702.08759. https:/​/​doi.org/​10.1103/​PhysRevB.96.155123 arXiv:1702.08759

[126] Kasper Duivenvoorden, Mohsin Iqbal, Jutho Haegeman, Frank Verstraete, and Norbert Schuch. ``Entanglement phases as holographic duals of anyon condensates''. Phys. Rev. B 95, 235119 (2017). arXiv:1702.08469. https:/​/​doi.org/​10.1103/​PhysRevB.95.235119 arXiv:1702.08469

[127] Juven C. Wang, Luiz H. Santos, and Xiao-Gang Wen. ``Bosonic anomalies, induced fractional quantum numbers, and degenerate zero modes: The anomalous edge physics of symmetry-protected topological states''. Phys. Rev. B 91, 195134 (2015). arXiv:1403.5256. https:/​/​doi.org/​10.1103/​PhysRevB.91.195134 arXiv:1403.5256

[128] Ryan Thorngren and Curt von Keyserlingk. ``Higher SPT's and a generalization of anomaly in-flow''. preprint, (2015). arXiv:1511.02929. arXiv:1511.02929

[129] Lukasz Fidkowski and Ashvin Vishwanath. ``Realizing anomalous anyonic symmetries at the surfaces of 3d gauge theories''. preprint, (2015). arXiv:1511.01502. https:/​/​doi.org/​10.1103/​PhysRevB.96.045131 arXiv:1511.01502

[130] Lukasz Fidkowski, Xie Chen, and Ashvin Vishwanath. ``Non-abelian topological order on the surface of a 3d topological superconductor from an exactly solved model''. Phys. Rev. X 3, 041016 (2013). arXiv:1305.5851. https:/​/​doi.org/​10.1103/​PhysRevX.3.041016 arXiv:1305.5851

[131] F. J. Burnell, Xie Chen, Lukasz Fidkowski, and Ashvin Vishwanath. ``Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order''. Phys. Rev. B 90, 245122 (2014). arXiv:1302.7072. https:/​/​doi.org/​10.1103/​PhysRevB.90.245122 arXiv:1302.7072

[132] Xie Chen, F. J. Burnell, Ashvin Vishwanath, and Lukasz Fidkowski. ``Anomalous symmetry fractionalization and surface topological order''. Phys. Rev. X 5, 041013 (2015). arXiv:1403.6491. https:/​/​doi.org/​10.1103/​PhysRevX.5.041013 arXiv:1403.6491

[133] Cain Edie-Michell. ``Equivalences of graded categories''. preprint (2017). arXiv:1711.00645. arXiv:1711.00645

[134] Alexei Davydov, Dmitri Nikshych, and Victor Ostrik. ``On the structure of the witt group of braided fusion categories''. Selecta Mathematica 19, 237–269 (2013). arXiv:1109.5558. https:/​/​doi.org/​10.1007/​s00029-012-0093-3 arXiv:1109.5558

[135] Alexei Davydov, Michael Müger, Dmitri Nikshych, and Victor Ostrik. ``The witt group of non-degenerate braided fusion categories''. Journal für die reine und angewandte Mathematik 2013, 135–177 (2013). arXiv:1009.2117. https:/​/​doi.org/​10.1515/​crelle.2012.014 arXiv:1009.2117

[136] Alexei Kitaev and Liang Kong. ``Models for gapped boundaries and domain walls''. Commun. Math. Phys. 313, 351–373 (2012). arXiv:1104.5047. https:/​/​doi.org/​10.1007/​s00220-012-1500-5 arXiv:1104.5047

[137] Ching-Yu Huang, Xie Chen, and Frank Pollmann. ``Detection of symmetry-enriched topological phases''. Phys. Rev. B 90, 045142 (2014). arXiv:1312.3093. https:/​/​doi.org/​10.1103/​PhysRevB.90.045142 arXiv:1312.3093

[138] Robbert Dijkgraaf and Edward Witten. ``Topological gauge theories and group cohomology''. Commun. Math. Phys. 129, 393–429 (1990). https:/​/​doi.org/​10.1007/​BF02096988

[139] Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, and Xiao-Gang Wen. ``Symmetry-protected topological orders in interacting bosonic systems''. Science 338, 1604–1606 (2012). arXiv:1301.0861. https:/​/​doi.org/​10.1126/​science.1227224 arXiv:1301.0861

[140] Dominic V. Else and Chetan Nayak. ``Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge''. Phys. Rev. B 90, 235137 (2014). arXiv:1409.5436. https:/​/​doi.org/​10.1103/​PhysRevB.90.235137 arXiv:1409.5436

[141] Xie Chen, Bei Zeng, Zheng-Cheng Gu, Isaac L. Chuang, and Xiao-Gang Wen. ``Tensor product representation of a topological ordered phase: Necessary symmetry conditions''. Phys. Rev. B 82, 165119 (2010). arXiv:1003.1774. https:/​/​doi.org/​10.1103/​PhysRevB.82.165119 arXiv:1003.1774

[142] Jutho Haegeman, V Zauner, N Schuch, and Frank Verstraete. ``Shadows of anyons and the entanglement structure of topological phases''. Nat. Commun. 6 (2015). arXiv:1410.5443. https:/​/​doi.org/​10.1038/​ncomms9284 arXiv:1410.5443

[143] Sujeet K Shukla, M Burak Şahinoğlu, Frank Pollmann, and Xie Chen. ``Boson condensation and instability in the tensor network representation of string-net states''. preprint, (2016). arXiv:1610.00608. https:/​/​doi.org/​10.1103/​PhysRevB.98.125112 arXiv:1610.00608

[144] Michaël Mariën, Jutho Haegeman, Paul Fendley, and Frank Verstraete. ``Condensation-driven phase transitions in perturbed string nets''. preprint, (2016). arXiv:1607.05296. https:/​/​doi.org/​10.1103/​PhysRevB.96.155127 arXiv:1607.05296

[145] Davide Gaiotto, Anton Kapustin, Nathan Seiberg, and Brian Willett. ``Generalized global symmetries''. J. High Energy Phys. 2015, 172 (2015). arXiv:1412.5148. https:/​/​doi.org/​10.1007/​JHEP02(2015)172 arXiv:1412.5148

[146] Anton Kapustin and Ryan Thorngren. ``Higher symmetry and gapped phases of gauge theories''. preprint, (2013). arXiv:1309.4721. arXiv:1309.4721

[147] Dominic J. Williamson et al. ``Matrix product operator symmetry-breaking and anyon condensation''. in preparation (2017).

[148] M. de Wild Propitius. ``Topological interactions in broken gauge theories''. PhD thesis. University of Amsterdam. (1995). arXiv:hep-th/​9511195. arXiv:hep-th/9511195

[149] Mark de Wild Propitius. ``(Spontaneously broken) Abelian Chern-Simons theories''. Nucl. Phys. B 489, 297 – 359 (1997). arXiv:hep-th/​9606029. https:/​/​doi.org/​10.1016/​S0550-3213(97)00005-9 arXiv:hep-th/9606029

[150] Jacob C. Bridgeman and Dominic J. Williamson. ``Anomalies and entanglement renormalization''. Phys. Rev. B 96, 125104 (2017). arXiv:1703.07782. https:/​/​doi.org/​10.1103/​PhysRevB.96.125104 arXiv:1703.07782

[151] Xie Chen and Ashvin Vishwanath. ``Towards gauging time-reversal symmetry: A tensor network approach''. Phys. Rev. X 5, 041034 (2015). arXiv:1401.3736. https:/​/​doi.org/​10.1103/​PhysRevX.5.041034 arXiv:1401.3736

[152] Ryan Thorngren and Dominic V Else. ``Gauging spatial symmetries and the classification of topological crystalline phases''. preprint, (2016). arXiv:1612.00846. https:/​/​doi.org/​10.1103/​PhysRevX.8.011040 arXiv:1612.00846

[153] Kevin Walker. ``Codimension-1 defects, categorified group actions, and condensing fermions''. talk at IPAM workshop (Jan. 26-30, 2015). url: http:/​/​www.ipam.ucla.edu/​abstract/​?tid=12400&pcode=STQ2015. http:/​/​www.ipam.ucla.edu/​abstract/​?tid=12400&pcode=STQ2015

[154] Lakshya Bhardwaj, Davide Gaiotto, and Anton Kapustin. ``State sum constructions of spin-tfts and string net constructions of fermionic phases of matter''. J. High Energy Phys. 2017, 96 (2017). arXiv:1605.01640. https:/​/​doi.org/​10.1007/​JHEP04(2017)096 arXiv:1605.01640

[155] Anton Kapustin and Ryan Thorngren. ``Fermionic SPT phases in higher dimensions and bosonization''. preprint, (2017). arXiv:1701.08264. https:/​/​doi.org/​10.1007/​JHEP10(2017)080 arXiv:1701.08264

[156] Yidun Wan and Chenjie Wang. ``Fermion condensation and gapped domain walls in topological orders''. J. High Energy Phys. 2017, 172 (2017). arXiv:1607.01388. https:/​/​doi.org/​10.1007/​JHEP03(2017)172 arXiv:1607.01388

[157] Davide Gaiotto and Anton Kapustin. ``Spin TQFTs and fermionic phases of matter''. Int. J. Mod. Phys. A 31, 1645044 (2016). arXiv:1505.05856. https:/​/​doi.org/​10.1142/​S0217751X16450445 arXiv:1505.05856

[158] Nicolas Tarantino and Lukasz Fidkowski. ``Discrete spin structures and commuting projector models for two-dimensional fermionic symmetry-protected topological phases''. Phys. Rev. B 94, 115115 (2016). arXiv:1604.02145. https:/​/​doi.org/​10.1103/​PhysRevB.94.115115 arXiv:1604.02145

[159] Brayden Ware, Jun Ho Son, Meng Cheng, Ryan V. Mishmash, Jason Alicea, and Bela Bauer. ``Ising anyons in frustration-free majorana-dimer models''. Phys. Rev. B 94, 115127 (2016). arXiv:1605.06125. https:/​/​doi.org/​10.1103/​PhysRevB.94.115127 arXiv:1605.06125

[160] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen. ``Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order''. Phys. Rev. B 82, 155138 (2010). arXiv:1004.3835. https:/​/​doi.org/​10.1103/​PhysRevB.82.155138 arXiv:1004.3835

[161] Dominic J Williamson and Zhenghan Wang. ``Hamiltonian models for topological phases of matter in three spatial dimensions''. Ann. Phys. 377, 311 – 344 (2017). arXiv:1606.07144. https:/​/​doi.org/​10.1016/​j.aop.2016.12.018 arXiv:1606.07144

[162] Dominic J. Williamson et al. ``Symmetry-enriched topological order and fault-tolerant logic gates''. in preparation (2017).

[163] Cain Edie-Michell. ``The Brauer-Picard groups of the $ADE$ fusion categories''. preprint (2017). arXiv:1709.04721. arXiv:1709.04721

[164] Cain Edie-Michell and Scott Morrison. ``A field guide to categories with $A_n$ fusion rules''. preprint (2017). arXiv:1710.07362. arXiv:1710.07362

[165] Jürgen Fuchs, Ingo Runkel, and Christoph Schweigert. ``TFT construction of RCFT correlators I: Partition functions''. Nucl. Phys. B 646, 353–497 (2002). arXiv:hep-th/​0204148. https:/​/​doi.org/​10.1016/​S0550-3213(02)00744-7 arXiv:hep-th/0204148

[166] Jürg Fröhlich, Jürgen Fuchs, Ingo Runkel, and Christoph Schweigert. ``Kramers-wannier duality from conformal defects''. Phys. Rev. Lett. 93, 070601 (2004). arXiv:cond-mat/​0404051. https:/​/​doi.org/​10.1103/​PhysRevLett.93.070601 arXiv:cond-mat/0404051

[167] Jürg Fröhlich, Jürgen Fuchs, Ingo Runkel, and Christoph Schweigert. ``Duality and defects in rational conformal field theory''. Nucl. Phys. B 763, 354 – 430 (2007). arXiv:hep-th/​0607247. https:/​/​doi.org/​10.1016/​j.nuclphysb.2006.11.017 arXiv:hep-th/0607247

[168] Roger S K Mong, David J Clarke, Jason Alicea, Netanel H Lindner, and Paul Fendley. ``Parafermionic conformal field theory on the lattice''. J. Phys. A 47, 452001 (2014). arXiv:1406.0846. https:/​/​doi.org/​10.1088/​1751-8113/​47/​45/​452001 arXiv:1406.0846

[169] Yi-Zhuang You, Zhen Bi, Alex Rasmussen, Kevin Slagle, and Cenke Xu. ``Wave function and strange correlator of short-range entangled states''. Phys. Rev. Lett. 112, 247202 (2014). arXiv:1312.0626. https:/​/​doi.org/​10.1103/​PhysRevLett.112.247202 arXiv:1312.0626

[170] Robijn Vanhove, Matthias Bal, Dominic J. Williamson, Nick Bultinck, Jutho Haegeman, and Frank Verstraete. ``Mapping Topological to Conformal Field Theories through strange Correlators''. Phys. Rev. Lett. 121, 177203 (2018). arXiv:1801.05959. https:/​/​doi.org/​10.1103/​PhysRevLett.121.177203 arXiv:1801.05959

[171] Robijn Vanhove, Laurens Lootens, Hong-Hao Tu, and Frank Verstraete. ``Topological aspects of the critical three-state potts model''. Journal of Physics A: Mathematical and Theoretical 55 (2022). https:/​/​doi.org/​10.1088/​1751-8121/​ac68b1

[172] Andras Molnar, Alberto Ruiz de Alarcón, José Garre-Rubio, Norbert Schuch, J. Ignacio Cirac, and David Pérez-García. ``Matrix product operator algebras i: representations of weak hopf algebras and projected entangled pair states'' (2022). url: http:/​/​arxiv.org/​abs/​2204.05940. arXiv:2204.05940

[173] Alberto Ruiz de Alarcón, José Garre-Rubio, András Molnár, and David Pérez-García. ``Matrix product operator algebras ii: Phases of matter for 1d mixed states'' (2022).

[174] Yuhan Liu, Andras Molnar, Xiao-Qi Sun, Frank Verstraete, Kohtaro Kato, and Laurens Lootens. ``Trading mathematical for physical simplicity: Bialgebraic structures in matrix product operator symmetries'' (2025). url: http:/​/​arxiv.org/​abs/​2509.03600. arXiv:2509.03600

[175] Ryan Thorngren and Yifan Wang. ``Fusion category symmetry i: Anomaly in-flow and gapped phases'' (2019). url: http:/​/​arxiv.org/​abs/​1912.02817. arXiv:1912.02817

[176] David Aasen, Paul Fendley, and Roger S. K. Mong. ``Topological defects on the lattice: Dualities and degeneracies'' (2020). url: http:/​/​arxiv.org/​abs/​2008.08598. arXiv:2008.08598

[177] Laurens Lootens, Clement Delcamp, Gerardo Ortiz, and Frank Verstraete. ``Dualities in one-dimensional quantum lattice models: symmetric hamiltonians and matrix product operator intertwiners''. PRX Quantum 4, 020357 (2021). https:/​/​doi.org/​10.1103/​PRXQuantum.4.020357

[178] Laurens Lootens, Clement Delcamp, and Frank Verstraete. ``Dualities in one-dimensional quantum lattice models: topological sectors''. PRX Quantum 5, 010338 (2022). https:/​/​doi.org/​10.1103/​PRXQuantum.5.010338

[179] Jacob C. Bridgeman, Laurens Lootens, and Frank Verstraete. ``Invertible bimodule categories and generalized schur orthogonality'' (2022).

[180] Lukas Haller, Wen Tao Xu, Yu Jie Liu, and Frank Pollmann. ``Quantum phase transition between symmetry enriched topological phases in tensor-network states''. Physical Review Research 5, 043078 (2023). https:/​/​doi.org/​10.1103/​PhysRevResearch.5.043078

[181] Bram Vancraeynest-De Cuiper, José Garre-Rubio, Frank Verstraete, Kevin Vervoort, Dominic J. Williamson, and Laurens Lootens. ``From gauging to duality in one-dimensional quantum lattice models'' (2025). url: http:/​/​doi.org/​10.21468/​SciPostPhys.20.4.113. https:/​/​doi.org/​10.21468/​SciPostPhys.20.4.113

[182] Corey Jones, Kylan Schatz, and Dominic J. Williamson. ``Quantum cellular automata and categorical dualities of spin chains''. Communications in Mathematical Physics 2026 407:4 407, 66– (2026). https:/​/​doi.org/​10.1007/​s00220-026-05571-y

[183] Laurens Lootens, Jürgen Fuchs, Jutho Haegeman, Christoph Schweigert, and Frank Verstraete. ``Matrix product operator symmetries and intertwiners in string-nets with domain walls''. SciPost Physics 10 (2021). https:/​/​doi.org/​10.21468/​SCIPOSTPHYS.10.3.053

[184] Laurens Lootens, Bram Vancraeynest-De Cuiper, Norbert Schuch, and Frank Verstraete. ``Mapping between morita-equivalent string-net states with a constant depth quantum circuit''. Physical Review B 105 (2022). https:/​/​doi.org/​10.1103/​PhysRevB.105.085130

[185] Kylan Schatz. ``Boundary symmetries of (2+1)d topological orders'' (2024). url: https:/​/​arxiv.org/​abs/​2408.10832v1. arXiv:2408.10832v1

[186] Chien Hung Lin and Fiona J. Burnell. ``Anyon condensation in string-net models''. Physical Review B 110, 115127 (2024). https:/​/​doi.org/​10.1103/​PhysRevB.110.115127

[187] Daniel Barter, Jacob C. Bridgeman, and Corey Jones. ``Domain Walls in Topological Phases and the Brauer–Picard Ring for Vec(Z/​pZ)''. Communications in Mathematical Physics 2019 369:3 369, 1167–1185 (2019). https:/​/​doi.org/​10.1007/​s00220-019-03338-2

[188] Tian Lan. ``Tube category, tensor renormalization and topological holography''. Communications in Mathematical Physics 2025 406:8 406, 196– (2025). https:/​/​doi.org/​10.1007/​s00220-025-05383-6

[189] Nathanan Tantivasadakarn, Ryan Thorngren, Ashvin Vishwanath, and Ruben Verresen. ``Long-range entanglement from measuring symmetry-protected topological phases''. Physical Review X 14 (2021). https:/​/​doi.org/​10.1103/​PhysRevX.14.021040

[190] Sergey Bravyi, Isaac Kim, Alexander Kliesch, and Robert Koenig. ``Adaptive constant-depth circuits for manipulating non-abelian anyons'' (2022). url: http:/​/​arxiv.org/​abs/​2205.01933. arXiv:2205.01933

[191] Laurens Lootens, Clement Delcamp, Dominic Williamson, and Frank Verstraete. ``Low-depth unitary quantum circuits for dualities in one-dimensional quantum lattice models''. Physical Review Letters 134 (2023). https:/​/​doi.org/​10.1103/​PhysRevLett.134.130403

[192] Nathanan Tantivasadakarn, Xinyu Liu, and Xie Chen. ``Sequential circuit as generalized symmetry on lattice'' (2025). url: http:/​/​arxiv.org/​abs/​2507.22394. arXiv:2507.22394

[193] Shawn X. Cui, César Galindo, Julia Yael Plavnik, and Zhenghan Wang. ``On gauging symmetry of modular categories''. Communications in Mathematical Physics 348, 1043–1064 (2016). arXiv:1510.03475. https:/​/​doi.org/​10.1007/​s00220-016-2633-8 arXiv:1510.03475

[1] J. Ignacio Cirac, David Pérez-García, Norbert Schuch, and Frank Verstraete, "Matrix product states and projected entangled pair states: Concepts, symmetries, theorems", Reviews of Modern Physics 93 4, 045003 (2021).

[2] Román Orús, "Tensor networks for complex quantum systems", Nature Reviews Physics 1 9, 538 (2019).

[3] Jong Yeon Lee, Chao-Ming Jian, and Cenke Xu, "Quantum Criticality Under Decoherence or Weak Measurement", PRX Quantum 4 3, 030317 (2023).

[4] Nathan Seiberg, Sahand Seifnashri, and Shu-Heng Shao, "Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space", SciPost Physics 16 6, 154 (2024).

[5] Ying-Hsuan Lin, Masaki Okada, Sahand Seifnashri, and Yuji Tachikawa, "Asymptotic density of states in 2d CFTs with non-invertible symmetries", Journal of High Energy Physics 2023 3, 94 (2023).

[6] Laurens Lootens, Clement Delcamp, Gerardo Ortiz, and Frank Verstraete, "Dualities in One-Dimensional Quantum Lattice Models: Symmetric Hamiltonians and Matrix Product Operator Intertwiners", PRX Quantum 4 2, 020357 (2023).

[7] Anasuya Lyons, Chiu Fan Bowen Lo, Nathanan Tantivasadakarn, Ashvin Vishwanath, and Ruben Verresen, "Protocols for Creating Anyons and Defects via Gauging", Physical Review Letters 135 20, 200405 (2025).

[8] David T. Stephen, Hendrik Poulsen Nautrup, Juani Bermejo-Vega, Jens Eisert, and Robert Raussendorf, "Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter", Quantum 3, 142 (2019).

[9] Clement Delcamp and Apoorv Tiwari, "Higher categorical symmetries and gauging in two-dimensional spin systems", SciPost Physics 16 4, 110 (2024).

[10] Christopher Fechisin, Nathanan Tantivasadakarn, and Victor V. Albert, "Noninvertible Symmetry-Protected Topological Order in a Group-Based Cluster State", Physical Review X 15 1, 011058 (2025).

[11] Pranay Gorantla, Shu-Heng Shao, and Nathanan Tantivasadakarn, "Tensor Networks for Noninvertible Symmetries in <inline-formula><mml:math><mml:mrow><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mi>D</mml:mi></mml:mrow></mml:math></inline-formula> and Beyond", Physical Review X 15 4, 041006 (2025).

[12] Jong Yeon Lee, Yi-Zhuang You, and Cenke Xu, "Symmetry protected topological phases under decoherence", arXiv:2210.16323, (2022).

[13] Ignacio Cirac, David Perez-Garcia, Norbert Schuch, and Frank Verstraete, "Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems", arXiv:2011.12127, (2020).

[14] Robijn Vanhove, Laurens Lootens, Maarten Van Damme, Ramona Wolf, Tobias J. Osborne, Jutho Haegeman, and Frank Verstraete, "Critical Lattice Model for a Haagerup Conformal Field Theory", Physical Review Letters 128 23, 231602 (2022).

[15] Bram Vancraeynest-De Cuiper, José Garre-Rubio, Frank Verstraete, Kevin Vervoort, Dominic J. Williamson, and Laurens Lootens, "From gauging to duality in one-dimensional quantum lattice models", SciPost Physics 20 4, 113 (2026).

[16] Dominic J. Williamson, Zhen Bi, and Meng Cheng, "Fractonic matter in symmetry-enriched U (1 ) gauge theory", Physical Review B 100 12, 125150 (2019).

[17] Ying-Hsuan Lin and Shu-Heng Shao, "Bootstrapping noninvertible symmetries", Physical Review D 107 12, 125025 (2023).

[18] Abhijat Sarma, Yimu Bao, Nayan Myerson-Jain, Thomas Kiely, and Cenke Xu, "Effective conformal field theory generated from pure and dephased Chern insulator", Physical Review B 112 8, 085130 (2025).

[19] Bram Vancraeynest-De Cuiper and Clement Delcamp, "Twisted gauging and topological sectors in (2+1)d Abelian lattice gauge theories", SciPost Physics 19 2, 054 (2025).

[20] Lin Chen, Kaixin Ji, Haochen Zhang, Ce Shen, Ruoshui Wang, Xiangdong Zeng, and Ling-Yan Hung, "<inline-formula><mml:math display="inline"><mml:msub><mml:mrow><mml:mi>CFT</mml:mi></mml:mrow><mml:mi>D</mml:mi></mml:msub></mml:math></inline-formula> from <inline-formula><mml:math display="inline"><mml:msub><mml:mrow><mml:mi>TQFT</mml:mi></mml:mrow><mml:mrow><mml:mi>D</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula> via Holographic Tensor Network, and Precision Discretization of <inline-formula><mml:math display="inline"><mml:msub><mml:mi>CFT</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula>", Physical Review X 14 4, 041033 (2024).

[21] Robijn Vanhove, Laurens Lootens, Hong-Hao Tu, and Frank Verstraete, "Topological aspects of the critical three-state Potts model", Journal of Physics A Mathematical General 55 23, 235002 (2022).

[22] Abhinav Prem and Dominic Williamson, "Gauging permutation symmetries as a route to non-Abelian fractons", SciPost Physics 7 5, 068 (2019).

[23] Clement Delcamp and Apoorv Tiwari, "From gauge to higher gauge models of topological phases", Journal of High Energy Physics 2018 10, 49 (2018).

[24] Markus S. Kesselring, Fernando Pastawski, Jens Eisert, and Benjamin J. Brown, "The boundaries and twist defects of the color code and their applications to topological quantum computation", Quantum 2, 101 (2018).

[25] Jong Yeon Lee, Yi-Zhuang You, and Cenke Xu, "Symmetry protected topological phases under decoherence", Quantum 9, 1607 (2025).

[26] Bram Vancraeynest-De Cuiper, Weronika Wiesiolek, and Frank Verstraete, "Les Houches Lecture Notes on Tensor Networks", arXiv:2512.24390, (2025).

[27] Robijn Vanhove, Matthias Bal, Dominic J. Williamson, Nick Bultinck, Jutho Haegeman, and Frank Verstraete, "Mapping Topological to Conformal Field Theories through strange Correlators", Physical Review Letters 121 17, 177203 (2018).

[28] Lukas Haller, Wen-Tao Xu, Yu-Jie Liu, and Frank Pollmann, "Quantum phase transition between symmetry enriched topological phases in tensor-network states", Physical Review Research 5 4, 043078 (2023).

[29] Alex Bullivant and Clement Delcamp, "Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases", Journal of High Energy Physics 2021 7, 25 (2021).

[30] Robijn Vanhove, Vibhu Ravindran, David T. Stephen, Xiao-Gang Wen, and Xie Chen, "Duality via sequential quantum circuit in the topological holography formalism", Physical Review B 112 3, 035173 (2025).

[31] Shuhei Ohyama and Shinsei Ryu, "Higher Berry phase from projected entangled pair states in (<inline-formula><mml:math><mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula>) dimensions", Physical Review B 111 4, 045112 (2025).

[32] Kansei Inamura, Sheng-Jie Huang, Apoorv Tiwari, and Sakura Schafer-Nameki, "(2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry", arXiv:2506.09177, (2025).

[33] Laurens Lootens, Clement Delcamp, Dominic Williamson, and Frank Verstraete, "Low-Depth Unitary Quantum Circuits for Dualities in One-Dimensional Quantum Lattice Models", Physical Review Letters 134 13, 130403 (2025).

[34] Zhian Jia, Dagomir Kaszlikowski, and Sheng Tan, "Boundary and domain wall theories of 2d generalized quantum double model", Journal of High Energy Physics 2023 7, 160 (2023).

[35] Meng Cheng and Dominic J. Williamson, "Relative anomaly in (1 +1 )d rational conformal field theory", Physical Review Research 2 4, 043044 (2020).

[36] Jacob C. Bridgeman, Laurens Lootens, and Frank Verstraete, "Invertible Bimodule Categories and Generalized Schur Orthogonality", Communications in Mathematical Physics 402 3, 2691 (2023).

[37] David T. Stephen, José Garre-Rubio, Arpit Dua, and Dominic J. Williamson, "Subsystem symmetry enriched topological order in three dimensions", Physical Review Research 2 3, 033331 (2020).

[38] Yuanjie Ren, Nathanan Tantivasadakarn, and Dominic J. Williamson, "Efficient Preparation of Solvable Anyons with Adaptive Quantum Circuits", Physical Review X 15 3, 031060 (2025).

[39] Nathanan Tantivasadakarn, Wenjie Ji, and Sagar Vijay, "Non-Abelian hybrid fracton orders", Physical Review B 104 11, 115117 (2021).

[40] Laurens Lootens, Bram Vancraeynest-De Cuiper, Norbert Schuch, and Frank Verstraete, "Mapping between Morita-equivalent string-net states with a constant depth quantum circuit", Physical Review B 105 8, 085130 (2022).

[41] Chien-Hung Lin and Fiona J. Burnell, "Anyon condensation in string-net models", Physical Review B 110 11, 115127 (2024).

[42] Clement Delcamp, "Tensor network approach to electromagnetic duality in (3+1)d topological gauge models", Journal of High Energy Physics 2022 8, 149 (2022).

[43] Po-Shen Hsin, David Stephen, Arpit Dua, and Dominic Williamson, "Subsystem symmetry fractionalization and foliated field theory", SciPost Physics 18 5, 147 (2025).

[44] Clement Delcamp and Norbert Schuch, "On tensor network representations of the (3+1)d toric code", Quantum 5, 604 (2021).

[45] Sam Roberts and Stephen D. Bartlett, "Symmetry-Protected Self-Correcting Quantum Memories", Physical Review X 10 3, 031041 (2020).

[46] Qing-Rui Wang and Meng Cheng, "Exactly solvable models for U(1) symmetry-enriched topological phases", Physical Review B 106 11, 115104 (2022).

[47] Daniel Barter, Jacob C. Bridgeman, and Corey Jones, "Domain Walls in Topological Phases and the Brauer-Picard Ring for Vec(Z/pZ)", Communications in Mathematical Physics 369 3, 1167 (2019).

[48] Dominic J. Williamson and Meng Cheng, "Designer non-Abelian fractons from topological layers", Physical Review B 107 3, 035103 (2023).

[49] Laurens Lootens, Robijn Vanhove, Jutho Haegeman, and Frank Verstraete, "Galois Conjugated Tensor Fusion Categories and Nonunitary Conformal Field Theory", Physical Review Letters 124 12, 120601 (2020).

[50] Jacob C. Bridgeman and Daniel Barter, "Computing data for Levin-Wen with defects", Quantum 4, 277 (2020).

[51] Yu-Jie Liu, Kirill Shtengel, and Frank Pollmann, "Simulating two-dimensional topological quantum phase transitions on a digital quantum computer", Physical Review Research 6 4, 043256 (2024).

[52] Pranay Gorantla, Abhinav Prem, Nathanan Tantivasadakarn, and Dominic J. Williamson, "String membrane nets from higher-form gauging: An alternate route to <inline-formula><mml:math><mml:mi>p</mml:mi></mml:math></inline-formula>-string condensation", Physical Review B 112 12, 125124 (2025).

[53] Jacob C. Bridgeman, Stephen D. Bartlett, and Andrew C. Doherty, "Tensor networks with a twist: Anyon-permuting domain walls and defects in projected entangled pair states", Physical Review B 96 24, 245122 (2017).

[54] David T. Stephen, Arpit Dua, José Garre-Rubio, Dominic J. Williamson, and Michael Hermele, "Fractionalization of subsystem symmetries in two dimensions", Physical Review B 106 8, 085104 (2022).

[55] Yang Liu, Z. Y. Xie, Hong-Gang Luo, and Jize Zhao, "Long-range spin-orbital order in the spin-orbital SU (2 )×SU (2 )×U (1 ) model", Physical Review B 107 4, L041106 (2023).

[56] Finn Gagliano, Andrea Grigoletto, and Kantaro Ohmori, "Higher Representations and Quark Confinement", arXiv:2501.09069, (2025).

[57] Joseph Sullivan, Thomas Iadecola, and Dominic J. Williamson, "Planar p-string condensation: Chiral fracton phases from fractional quantum Hall layers and beyond", Physical Review B 103 20, 205301 (2021).

[58] Laurens Lootens, Jürgen Fuchs, Jutho Haegeman, Christoph Schweigert, and Frank Verstraete, "Matrix product operator symmetries and intertwiners in string-nets with domain walls", arXiv:2008.11187, (2020).

[59] Wen-Tao Xu, Jose Garre-Rubio, and Norbert Schuch, "Complete characterization of non-Abelian topological phase transitions and detection of anyon splitting with projected entangled pair states", Physical Review B 106 20, 205139 (2022).

[60] Dominic J. Williamson, Clement Delcamp, Frank Verstraete, and Norbert Schuch, "On the stability of topological order in tensor network states", Physical Review B 104 23, 235151 (2021).

[61] Zhian Jia, Dagomir Kaszlikowski, and Sheng Tan, "Electric-magnetic duality and Z2 symmetry enriched Abelian lattice gauge theory", Journal of Physics A Mathematical General 57 25, 255203 (2024).

[62] Laurens Lootens, Robijn Vanhove, and Frank Verstraete, "Cardy states, defect lines and chiral operators of coset CFTs on the lattice", arXiv:1907.02520, (2019).

[63] Clement Delcamp and Nafiz Ishtiaque, "Symmetry Topological Field Theory and Non-Abelian Kramers─Wannier Dualities of Generalised Ising Models", Annales Henri Poincaré 27 6, 2273 (2026).

[64] Tian Lan, "Tube Category, Tensor Renormalization and Topological Holography", Communications in Mathematical Physics 406 8, 196 (2025).

[65] Pavel Putrov and Rajath Radhakrishnan, "Braidings on topological operators, anomaly of higher-form symmetries and the SymTFT", arXiv:2503.13633, (2025).

[66] Julian Boesl, Yu-Jie Liu, Frank Pollmann, and Michael Knap, "Skeleton of Isometric Tensor Network States for Abelian String-Net Models", Physical Review Letters 137 2, 020405 (2026).

[67] Chao Xu, Yixin Ma, and Shenghan Jiang, "Unveiling correlated two-dimensional topological insulators through fermionic tensor network states—classification, edge theories and variational wavefunctions", Reports on Progress in Physics 87 10, 108001 (2024).

[68] Jacob C. Bridgeman, Daniel Barter, and Corey Jones, "Fusing Binary Interface Defects in Topological Phases: The $\operatorname{Vec}(\mathbb{Z}/p\mathbb{Z})$ case", arXiv:1810.09469, (2018).

[69] Jacob C. Bridgeman and Daniel Barter, "Computing defects associated to bounded domain wall structures: the <inline-formula><mml:math><mml:mi>Z</mml:mi><mml:mo>/</mml:mo><mml:mi>p</mml:mi><mml:mi>Z</mml:mi></mml:math></inline-formula> case", Journal of Physics A Mathematical General 53 23, 235206 (2020).

[70] Matthias Bal, Dominic J. Williamson, Robijn Vanhove, Nick Bultinck, Jutho Haegeman, and Frank Verstraete, "Mapping topological to conformal field theories through strange correlators", arXiv:1801.05959, (2018).

[71] José Garre-Rubio, Mohsin Iqbal, and David T. Stephen, "String order parameters for symmetry fractionalization in an enriched toric code", Physical Review B 103 12, 125104 (2021).

[72] Kfir Dolev, Vladimir Calvera, Samuel S. Cree, and Dominic J. Williamson, "Gauging the bulk: generalized gauging maps and holographic codes", Journal of High Energy Physics 2022 5, 158 (2022).

[73] Sujeet K. Shukla, Tyler D. Ellison, and Lukasz Fidkowski, "Tensor network approach to two-dimensional bosonization", Physical Review B 101 15, 155105 (2020).

[74] Chao-Yi Chen, Bing-Xin Lao, Xin-Yang Yu, and Ling-Yan Hung, "Galois conjugates of string-net model", Physical Review D 105 6, 065009 (2022).

[75] Jacob C. Bridgeman, Alexander Hahn, Tobias J. Osborne, and Ramona Wolf, "Gauging defects in quantum spin systems: A case study", Physical Review B 101 13, 134111 (2020).

[76] Qi Zhang and Wen-Tao Xu, "Continuous topological phase transition between <inline-formula><mml:math><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula> topologically ordered phases", Physical Review B 112 24, L241107 (2025).

[77] Kylan Schatz, "Boundary symmetries of (2+1)D topological orders", Letters in Mathematical Physics 115 3, 69 (2025).

[78] Nick Bultinck and Meng Cheng, "Filling constraints on fermionic topological order in zero magnetic field", Physical Review B 98 16, 161119 (2018).

[79] Jun Ho Son and Jason Alicea, "Commuting-projector Hamiltonians for chiral topological phases built from parafermions", Physical Review B 97 24, 245144 (2018).

[80] Li-Ping Yang, Y. F. Fu, Z. Y. Xie, and T. Xiang, "Efficient calculation of three-dimensional tensor networks", Physical Review B 107 16, 165127 (2023).

[81] Hongyu Wang, Yuting Hu, and Yidun Wan, "Extend the Levin-Wen model to two-dimensional topological orders with gapped boundary junctions", Journal of High Energy Physics 2022 7, 88 (2022).

[82] José Garre-Rubio and Sofyan Iblisdir, "Local order parameters for symmetry fractionalization", arXiv:1905.00602, (2019).

[83] Kevin Slagle, "Quantum Gauge Networks: A New Kind of Tensor Network", Quantum 7, 1113 (2023).

[84] Xiangdong Zeng, Ruoshui Wang, Ce Shen, and Ling-Yan Hung, "Virasoro generators in the Fibonacci model tensor network: Tackling finite-size effects", Physical Review B 107 24, 245146 (2023).

[85] Sam Roberts and Stephen D. Bartlett, "Symmetry-protected self-correcting quantum memories", arXiv:1805.01474, (2018).

[86] Yun-Tak Oh and Hyun-Yong Lee, "Nested Shadows of Anyons:A Framework for Identifying Topological Phases", arXiv:2502.18811, (2025).

[87] Anne E. B. Nielsen, Benedikt Herwerth, J. Ignacio Cirac, and Germán Sierra, "Field tensor network states", Physical Review B 103 15, 155130 (2021).

[88] Jacob C. Bridgeman, Daniel Barter, and Corey Jones, "Fusing binary interface defects in topological phases: The Z /p Z case", Journal of Mathematical Physics 60 12, 121701 (2019).

[89] Nianrui Fu, Yu Zhao, and Yidun Wan, "Symmetry-enriched topological phases and their gauging: a string-net model realization", Journal of High Energy Physics 2026 5, 16 (2026).

[90] Anna Francuz, Laurens Lootens, Frank Verstraete, and Jacek Dziarmaga, "Variational methods for characterizing matrix product operator symmetries", Physical Review B 104 19, 195152 (2021).

[91] Nianrui Fu, Siyuan Wang, Yu Zhao, and Yidun Wan, "Nonlinear symmetry-fragmentation of nonabelian anyons in symmetry-enriched topological phases: a string-net model realization", Journal of High Energy Physics 2026 3, 91 (2026).

[92] D.-S. Wang, "A comparative study of universal quantum computing models: towards a physical unification", arXiv:2108.07909, (2021).

[93] Kansei Inamura, Sheng-Jie Huang, Apoorv Tiwari, and Sakura Schäfer-Nameki, "(2+1)d lattice models and tensor networks for gapped phases with categorical symmetry", SciPost Physics 20 2, 043 (2026).

[94] Avi Vadali, Robijn Vanhove, Ruben Verresen, Jason Alicea, and Pablo Sala, "Statistical Mechanics and Symmetries of Non-Abelian Anyon Proliferation: From Deformation to Decoherence", arXiv:2606.12527, (2026).

[95] Lea E. Bottini, Clement Delcamp, Edmund Heng, Campbell K. McLauchlan, and Dominic J. Williamson, "Topological lattice gauge theory enriched by non-invertible symmetry", arXiv:2605.28688, (2026).

[96] Yacine Ikhlef, "Non-invertible symmetries and modular invariance in lattice models", arXiv:2606.31313, (2026).

[97] Luisa Eck, Peter Huston, Kyle Kawagoe, and David Penneys, "Non-invertible symmetry enriched string net topological orders", arXiv:2605.28794, (2026).

[98] Nianrui Fu, Siyuan Wang, Yu Zhao, and Yidun Wan, "Anyon Condensation In Symmetry-Enriched Topological Phases: $G$-Grading of Multifusion Categories", arXiv:2607.24740, (2026).

[99] Yun-Tak Oh and Hyun-Yong Lee, "Identifying topological phases through global and local symmetries of the quantum transfer matrix", Scientific Reports 16, 1586 (2026).

The above citations are from SAO/NASA ADS (last updated successfully 2026-09-15 21:40:18). The list may be incomplete as not all publishers provide suitable and complete citation data.

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