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Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
by Heidar Moradi, Ömer M. Aksoy, Jens H. Bardarson, Apoorv Tiwari
Submission summary
| Authors (as registered SciPost users): | Ömer M. Aksoy · Jens H Bardarson · Apoorv Tiwari |
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| Preprint Link: | https://arxiv.org/abs/2307.01266v2 (pdf) |
| Date submitted: | 2023-09-21 12:16 |
| Submitted by: | Tiwari, Apoorv |
| Submitted to: | SciPost Physics |
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| Academic field: | Physics |
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| Approach: | Theoretical |
Abstract
We investigate the gauging of higher-form finite Abelian symmetries and their sub-groups in quantum spin models in spatial dimensions $d=2$ and 3. Doing so, we naturally uncover gauged models with dual higher-group symmetries and potential mixed 't Hooft anomalies. We demonstrate that the mixed anomalies manifest as the symmetry fractionalization of higher-form symmetries participating in the mixed anomaly. Gauging is realized as an isomorphism or duality between the bond algebras that generate the space of quantum spin models with the dual generalized symmetry structures. We explore the mapping of gapped phases under such gauging related dualities for 0-form and 1-form symmetries in spatial dimension $d=2$ and 3. In $d=2$, these include several non-trivial dualities between short-range entangled gapped phases with 0-form symmetries and 0-form symmetry enriched Higgs and (twisted) deconfined phases of the gauged theory with possible symmetry fractionalizations. Such dualities also imply strong constraints on several unconventional, i.e., deconfined or topological transitions. In $d=3$, among others, we find, dualities between topological orders via gauging of 1-form symmetries. Hamiltonians self-dual under gauging of 1-form symmetries host emergent non-invertible symmetries, realizing higher-categorical generalizations of the Tambara-Yamagami fusion category.
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The manuscript discusses
(1) gauging a finite Abelian subgroup higher-form symmetry in lattice models, exam the dual higher-form symmetry and demonstrate the anomalies from group extension as discussed in e.g. [18],[3] from field theory perspective.
(2) Using the gauging procedure, the authors models that are self-dual under gauging and thus enjoy Kramers-Wannier type non-invertible duality symmetry as in [24],[25].
The discussion is systematic and the I recommend publication provided the following comments are addressed:
- For finite Abelian groups $G= prod Z_{N_i}$ with integers $N_i$, maybe add that the dual is isomorphic to itself $G^vee\cong G$
- The manuscript used a terminology that equates "gauging" with "duality". However, gauging a symmetry in general leads to a different theory, e.g. SPT v.s. topological order. The former is short range entangled while the later has long range entanglement, but they can be related by gauging a symmetry. In general, gauging a symmetry corresponds to a topological interface between different theories, and only when the theory is "self-dual" under gauging the topological interface becomes a topological domain wall within the same theory such as the Kramers-Wannier type duality symmetry.
- In (6.27), the local Z_e^p on each edge e and the usual vertex term A_v= prod XX^dagX X^dag X X^dag both commute with the Hamiltonian, but they are not Hamiltonian terms and also do not commute. Therefore, the model has local logical degrees of freedom. For instance, the states |0> and Z_e^p|0> for each edge e must be different due to different eigenvalues of the vertex term A_v. So I don't think the GSD is just (n/p)^b3 and the ground state subspace is not just Z_{n/p} topological gauge theory.
Recommendation
Ask for minor revision