Abstract
The generalized quantum double lattice realization of 2d topological orders based on Hopf algebras is discussed in this work. Both left-module and right-module constructions are investigated. The ribbon operators and the classification of topological excitations based on the representations of the quantum double of Hopf algebras are discussed. To generalize the model to a 2d surface with boundaries and surface defects, we present a systematic construction of the boundary Hamiltonian and domain wall Hamiltonian. The algebraic data behind the gapped boundary and domain wall are comodule algebras and bicomodule algebras. The topological excitations in the boundary and domain wall are classified by bimodules over these algebras. The ribbon operator realization of boundary-bulk duality is also discussed. Finally, via the Hopf tensor network representation of the quantum many-body states, we solve the ground state of the model in the presence of the boundary and domain wall.
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Acknowledgments
Z.J. acknowledges Liang Kong and Zhenghan Wang for many helpful discussions on topological order, boundary-bulk duality, and the mathematical theory of various topological defects. He also acknowledges Yuting Hu for his help. S.T. would like to thank Uli Walther for his constant encouragement and stimulating conversations. He also appreciates Shawn X. Cui and Bowen Yan for many helpful discussions. We also acknowledge Juven C. Wang and Eric Samperton for bringing our attention to several related references. All authors are grateful for the referee’s valuable suggestions. Z.J. and D.K. are supported by National Research Foundation, Singapore, and A*Star under the CQT Bridging Grant. S.T. was in part supported by NSF grant DMS-2100288 and by Simons Foundation Collaboration Grant for Mathematicians #580839.
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Jia, Z., Kaszlikowski, D. & Tan, S. Boundary and domain wall theories of 2d generalized quantum double model. J. High Energ. Phys. 2023, 160 (2023). https://doi.org/10.1007/JHEP07(2023)160
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DOI: https://doi.org/10.1007/JHEP07(2023)160