Symmetry-enriched topological order in tensor networks: Defects, gauging and anyon condensation
1Vienna Center for Quantum Technology, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
2Department of Physics and Astronomy, Ghent University, Krijgslaan 281 S9, B-9000 Ghent, Belgium
| Published: | 2026-09-01, volume 10, page 2199 |
| Editor: | Luca Tagliacozzo |
| Eprint: | arXiv:1711.07982v3 |
| Doi: | https://doi.org/10.22331/q-2026-09-01-2199 |
| Citation: | Quantum 10, 2199 (2026). |
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Abstract
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.
Featured image: A domain wall matrix product operator in a tensor network state with symmetry-enriched topological order.
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[1] 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).
[2] 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).
[3] Bram Vancraeynest-De Cuiper, Weronika Wiesiolek, and Frank Verstraete, "Les Houches Lecture Notes on Tensor Networks", arXiv:2512.24390, (2025).
[4] 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).
[5] 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).
[6] 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).
[7] Jong Yeon Lee, Yi-Zhuang You, and Cenke Xu, "Symmetry protected topological phases under decoherence", arXiv:2210.16323, (2022).
[8] Román Orús, "Tensor networks for complex quantum systems", Nature Reviews Physics 1 9, 538 (2019).
[9] 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).
[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).
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