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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 |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2307.01266v3 (pdf) |
| Date accepted: | 2025-02-25 |
| Date submitted: | 2025-01-23 09:50 |
| Submitted by: | Tiwari, Apoorv |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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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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- 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
Referee 1:
-- Statements emphasizing that the Pontryagin group of an Abelian group is isomorphic to the group itself appear at several places in the draft. See for example above eq. 3.2. However we would like to emphasize that the symmetry category upon gauging an Abelian group in d+1 dimensions is not isomorphic to the Abelian group, but instead is the higher representation category dRep(G). This statement can be found in the last paragraph in Section 2.1.
-- That is correct. Gauging a symmetry can indeed relate short-range and long range entangled phases as the referee points out. These are dualities in the sense that they are spectrum and correlation function preserving isomorphisms between two distinct quantum systems. In certain special cases, the quantum system before and after gauging are isomorphic. In such cases, the codimension-1 gauging interface can be treated as a (non-invertible) 0-form symmetry operator.
-- We remind the referee that the operator Z_e^{p} is not within the Z_n 1-form symmetric bond algebra, i.e., it takes us out of the physical constrained state space with 1-form symmetry. If the referee insists on working with an unconstrained state space, they may add the 1-form symmetry generators associated to each vertex of the lattice and take the corresponding coupling constant to infinity, then one reproduces the claimed Z_{n/p} topological order.
Referee 2:
-- We thank the referee for pointing these typos out. We now have corrected them.
-- This is indeed a very interesting question. Gauging non-Abelian (sub) symmetries lead to more interesting symmetry categories which are necessarily non-invertible. In general, in d+1 spatial dimensions, gauging a non-Abelian G 0-form symmetry, produces a dual quantum system with a dRep(G) symmetry. This contains a non-invertible 1-form subsymmetry Rep(G) corresponding to topological Wilson lines obtained after gauging G. All other symmetry generators in dRep(G) are obtainable as condensation defects of the Rep(G) lines. We have now added these comments as well as corresponding references at the end of Sec. 2.1.
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