Fractionalization of Coset Non-Invertible Symmetry and Exotic Hall Conductance

TL;DR

This paper explores how non-invertible symmetries in (2+1)D topological orders can fractionalize, leading to exotic Hall conductance and gapless edge states, with implications for quantum spin liquids and quantum Hall systems.

Contribution

It introduces the concept of coset non-invertible symmetry fractionalization, providing examples and a construction method for symmetry defects in topological orders.

Findings

Fractionalized non-invertible symmetries can carry fractional charges.

Systems exhibit a well-defined electric Hall conductance.

Boundary states can be gapless if symmetry is preserved.

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.