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Fractionalization of Coset Non-Invertible Symmetry and Exotic Hall Conductance
by Po-Shen Hsin, Ryohei Kobayashi, Carolyn Zhang
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Submission summary
| Authors (as registered SciPost users): | Po-Shen Hsin · Ryohei Kobayashi |
| Submission information | |
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| Preprint Link: | scipost_202408_00025v1 (pdf) |
| Date accepted: | 2024-09-13 |
| Date submitted: | 2024-08-22 02:35 |
| Submitted by: | Kobayashi, Ryohei |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We investigate fractionalization of non-invertible symmetry in (2+1)D topological orders. We focus on coset non-invertible symmetries obtained by gauging non-normal subgroups of invertible $0$-form symmetries. These symmetries can arise as global symmetries in quantum spin liquids, given by the quotient of the projective symmetry group by a non-normal subgroup as invariant gauge group. We point out that such coset non-invertible symmetries in topological orders can exhibit symmetry fractionalization: each anyon can carry a "fractional charge" under the coset non-invertible symmetry given by a gauge invariant superposition of fractional quantum numbers. We present various examples using field theories and quantum double lattice models, such as fractional quantum Hall systems with charge conjugation symmetry gauged and finite group gauge theory from gauging a non-normal subgroup. They include symmetry enriched $S_3$ and $O(2)$ gauge theories. We show that such systems have a fractionalized continuous non-invertible coset symmetry and a well-defined electric Hall conductance. The coset symmetry enforces a gapless edge state if the boundary preserves the continuous non-invertible symmetry. We propose a general approach for constructing coset symmetry defects using a "sandwich" construction: non-invertible symmetry defects can generally be constructed from an invertible defect sandwiched by condensation defects. The anomaly free condition for finite coset symmetry is also identified.
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Author comments upon resubmission
List of changes
1. In response to Report 1. 1, below (3.1) we added the comment that the orbit [g] has the length |K|/|K_g| for the stabilizer group K_g of the group element g.
2. In response to Report 1. 2, above Section 3.3.2, we added a few sentences to connect the coset symmetry to the non-invertible domain wall in U(1)8 from condensing the charge 4 Wilson line. We also cited 1012.0911.
3. In response to Report 1. 3, at the beginning of Section 6 we added a description of the parton construction for the spin liquids.
4. In response to Report 2, we omitted the word "continuous" since the construction is valid for either G is continuous or discrete.
Published as SciPost Phys. 17, 095 (2024)