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

Dominic J. Williamson, Nick Bultinck, and Frank Verstraete

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A close connection to the theory of graded unitary fusion categories is established. AbstractWe study symmetry-enriched topological order in two-dimensional tensor network states by using graded matrix product operator algebras to represent symmetry-induced domain walls. 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.

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AbstractWe 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.Featured image: A domain wall matrix product operator in a tensor network state with symmetry-enriched topological order.Popular summaryThe 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.► BibTeX data@article{Williamson2026symmetryenriched, doi = {10.22331/q-2026-09-01-2199}, url = {https://doi.org/10.22331/q-2026-09-01-2199}, title = {Symmetry-enriched topological order in tensor networks: {D}efects, gauging and anyon condensation}, author = {Williamson, Dominic J. and Bultinck, Nick and Verstraete, Frank}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2199}, month = sep, year = {2026} }► References [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. 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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:1510.03475Cited byCould not fetch Crossref cited-by data during last attempt 2026-09-01 09:21:13: Could not fetch cited-by data for 10.22331/q-2026-09-01-2199 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-09-01 09:21:14: Cannot retrieve data from ADS due to rate limitations.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractWe 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.Featured image: A domain wall matrix product operator in a tensor network state with symmetry-enriched topological order.Popular summaryThe 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. 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Symmetry-enriched topological order in tensor networks: Defects, gauging and anyon condensation

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