## Fractionalization of coset non-invertible symmetry and exotic Hall conductance

Po-Shen Hsin, Ryohei Kobayashi, Carolyn Zhang

SciPost Phys. 17, 095 (2024) · published 27 September 2024

- doi: 10.21468/SciPostPhys.17.3.095
- Submissions/Reports

### 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.

### Cited by 1

### Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.-
^{1}^{2}Po-Shen Hsin, -
^{3}Ryohei Kobayashi, -
^{4}Carolyn Zhang

^{1}King's College London [KCL]^{2}Mani L. Bhaumik Institute for Theoretical Physics^{3}Institute for Advanced Study [IAS]^{4}Harvard University

- Kavli Institute for Theoretical Physics, University of California, Santa Barbara (through Organization: Kavli Institute for Theoretical Physics [KITP])
- King's College London
- National Science Foundation [NSF]
- Simons Foundation
- Society of Fellows, Harvard University
- University of Maryland