[Paper Review] Fractionalization of Coset Non-Invertible Symmetry and Exotic Hall Conductance

Po-Shen Hsin, Ryohei Kobayashi+1 more|arXiv (Cornell University)|May 30, 2024

Advanced Physical and Chemical Molecular InteractionsChemistry3 citations

TL;DR

This paper introduces a framework for fractionalizing coset non-invertible symmetries $G/K$ in (2+1)D topological phases, where $K$ is a non-normal subgroup of $G$. It demonstrates that such symmetries can lead to exotic Hall conductance even when the symmetry is non-invertible, via a 'sandwich' construction of defects and explicit lattice models, with key examples in $O(2)$ and $S_3$ gauge theories showing fractionalized anyons and mixed anomalies.

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.

Motivation & Objective

• To extend symmetry fractionalization from invertible to non-invertible symmetries, specifically coset symmetries $G/K$ where $K$ is non-normal in $G$.
• To understand how such non-invertible symmetries, arising in quantum spin liquids and gauged FQH states, can fractionalize in topological order.
• To establish a systematic method for constructing and classifying fractionalization patterns of coset non-invertible symmetries in (2+1)D.
• To demonstrate that these symmetries can support a well-defined electric Hall conductance even when the symmetry is non-invertible.
• To explore the role of anomalies and deconfined critical points in systems with fractionalized non-invertible symmetries.

Proposed method

• Constructs non-invertible defects via a 'sandwich' mechanism: combining invertible symmetry defects with condensation defects that condense a subgroup $K$.
• Uses Chern-Simons field theory to describe bulk topological order, with $O(2)_{2k}$ and $S_3$ quantum double models as key examples.
• Analyzes symmetry fractionalization by decomposing representations of $G$ into those of $K$, showing how coset symmetry acts on Wilson lines.
• Derives fractionalization data on the lattice by examining how anyons braid with symmetry defects and how Wilson lines transform under coset symmetry.
• Applies anomaly inflow and consistency conditions to classify possible fractionalization patterns for finite coset symmetries.
• Studies deconfined critical points by gauging a discrete subgroup $K$ of $G$, leading to mixed anomalies between 1-form and coset symmetries.

Figure 1: The continuous non-invertible symmetry defect of the gauged theory $\mathcal{T}/K$ is understood as an invertible defect sandwiched by a pair of half gauging defects.

Experimental results

Research questions

• RQ1 How can non-invertible symmetries of the form $G/K$—where $K$ is non-normal in $G$—be fractionalized in (2+1)D topological phases?
• RQ2 What is the physical meaning and measurable consequence of fractionalization in such non-invertible symmetries, particularly in terms of transport response?
• RQ3 Can a well-defined Hall conductance be defined in systems with non-invertible symmetries, and how does it differ from the standard $U(1)$ Hall conductance?
• RQ4 How do the braiding and fusion properties of anyons transform under the action of coset non-invertible symmetry?
• RQ5 What are the implications of mixed anomalies between 1-form and coset symmetries for deconfined quantum criticality?

Key findings

• The $O(2)_{2k}$ Chern-Simons theory, obtained by gauging the $Z_2$ charge conjugation symmetry of a $U(1)_{2k}$ FQH state, exhibits a continuous non-invertible symmetry $(U(1) timesbZ_2)/bZ_2$, known as cosine symmetry.
• This cosine symmetry is fractionalized: anyons in the $O(2)_{2k}$ theory carry fractional charge under the symmetry, forming superpositions of opposite fractional charges related by charge conjugation.
• Despite the non-invertible nature of the symmetry, the electric Hall conductance remains well-defined and is even under charge conjugation, allowing a consistent response function.
• In the $S_3$ quantum double model, the coset symmetry $S_3/bZ_2$ acts non-trivially on $bZ_2$ Wilson lines, mapping them to the vacuum line, a transformation impossible for invertible symmetries.
• The fractionalization of $S_3/bZ_2$ symmetry leads to a mixed anomaly between the $bZ_2$ 1-form symmetry and the coset symmetry, preventing confinement at critical points.
• The defect sandwich construction provides a general mechanism to build non-invertible defects from invertible symmetries and condensation defects, enabling systematic classification of fractionalization patterns.

Figure 2: (a): The junction of the cosine symmetry defects that corresponds to the fusion channel $\tilde{U}_{[g]}\times\tilde{U}_{[g^{\prime}]}\to\tilde{U}_{[gkg^{\prime}k^{-1}]}$ . (b): Conjugating the invertible defects by $k\in K$ in the network of coset symmetry defects leads to an another expr