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(2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
by Kansei Inamura, Sheng-Jie Huang, Apoorv Tiwari, Sakura Schafer-Nameki
Submission summary
| Authors (as registered SciPost users): | Kansei Inamura |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2506.09177v2 (pdf) |
| Date submitted: | Dec. 16, 2025, 10:41 a.m. |
| Submitted by: | Kansei Inamura |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
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| Approach: | Theoretical |
Abstract
Gapped phases in 2+1 dimensional quantum field theories with fusion 2-categorical symmetries were recently classified and characterized using the Symmetry Topological Field Theory (SymTFT) approach arXiv:2408.05266, arXiv:2502.20440. In this paper, we provide a systematic lattice model construction for all such gapped phases. Specifically, we consider "all-boson type" fusion 2-category symmetries, all of which are obtainable from 0-form symmetry groups $G$ (possibly with an 't Hooft anomaly) via generalized gauging--that is, by stacking with an $H$-symmetric TFT and gauging a subgroup $H$. The continuum classification directly informs the lattice data, such as the generalized gauging that determines the symmetry category, and the data that specifies the gapped phase. We construct commuting projector Hamiltonians and ground states applicable to any non-chiral gapped phase with such symmetries. We also describe the ground states in terms of tensor networks. In light of the length of the paper, we include a self-contained summary section presenting the main results and examples.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
We thank the referees for their in-depth reading and comments on our paper. Let us reply to each referee in turn and explain what changes we have implemented in the revised version.
Referee 1:
We have taken the suggestion on board and have added Table 1 in Section 2.3 to summarize the key phases that we discussed.
Referee 2:
No changes requested
Referee 3:
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Below (4.1), we have added explanations regarding the conventions.
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Regarding the topological excitations, we believe that it should be possible to write down the string operators as in the Levin-Wen model. However, computing the symmetry action (including the symmetry fractionalization) on the quasi-particle excitations may involve a technical difficulty because such a computation technique has not been well-established for general non-invertible symmetries as far as we are aware. As such, developing the excitation theory for our models may not be straightforward in practice. It would indeed be an interesting direction to explore in the future.
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(a) The minimal boundaries correspond to $2\text{Vec}_G^{\omega}$-module 2-categories of the form ${}_{\text{Vec}_H^{\nu}} (2\text{Vec}_G^{\omega})$, i.e., the 2-category of left $\text{Vec}_H^{\nu}$-modules in $2\text{Vec}_G^{\omega}$. The non-minimal boundaries correspond to $2\text{Vec}_G^{\omega}$-module 2-categories ${}_A (2\text{Vec}_G^{\omega})$ for more general $G$-graded $\omega$-twisted fusion categories $A$. In both cases, the module 2-categories are indecomposable. The relation between topological boundaries and module 2-categories is briefly mentioned in Section 2.5 (on page 58) and discussed in more detail in Appendix D.
(b) Indeed, all topological boundary conditions can be obtained by generalized gauging (meaning stacking with an $H$-symmetric TFT and gauging $H$). In fact, Lagrangian algebras of $\mathcal{Z}(2\text{Vec}_G^{\omega})$ are classified rigorously in arXiv:2411.13367, and the classification data are consistent with the data of the generalized gauging. A more physical proof can be found in Section 5 of arXiv:2408.05266 (see also the sequel arXiv:2502.20440). For topological boundaries labeled by module 2-categories (i.e., non-chiral topological boundaries), we have provided a detailed physical argument supporting this fact in Appendix D.
(c) We have described a systematic way to construct lattice models directly using the data of a module 2-category in Section 4.2 (for a general fusion 2-category), Section 5.2 (for $2\text{Vec}_G$), and Section 6.2 (for $2\text{Vec}_G^{\omega}$). A module 2-category provides a mathematical way of defining generalized gauging on the lattice. The generalized gauging defined via module 2-categories is equivalent to the one defined in terms of stacking and gauging, as argued in Section 4.2.
We thank the referee for finding all these typos. We have corrected them.
List of changes
- We added Table 1 in Section 2.3.
- We expanded the explanations below (4.1).
- We fixed minor typos and added references.