We use finite group topological lattice gauge theory, also known as the quantum double model, as a lens to explore a notion of topological order enriched by a non-invertible symmetry. For invertible symmetry enriched topological order, there is an established axiomatisation in terms of a G-crossed braided fusion category. We lay the foundations for a generalisation of this notion. By condensing an arbitrary algebra of charges in a quantum double model, we demonstrate that the category of localised excitations in the resulting theory forms a hypergroup-graded extension of the category of deconfined excitations. For every element in the hypergroup, the associated domain wall acts in a typically non-invertible way on these localised excitations. Both this action and the monoidal structure are compatible with the hypergroup grading. The actual categorical action is encoded in a Hopf monad on the category of localised excitations, and gauging the non-invertible symmetry amounts to computing the category of modules over this Hopf monad. Finally, we outline how this framework naturally extends to theories obtained by condensing algebras in a generic string-net model.
Abstract We present practical and formal methods for gauging non-invertible symmetries in (2+1)d topological quantum field theories. Along the way, we generalize various aspects of invertible 0-form gauging, including symmetry fractionalization, discrete torsion, and the fixed point theorem for symmetry action on lines. Our approach involves two complementary strands: the fusion of topological interfaces and Morita theory of fusion 2-categories. We use these methods to derive constraints on gaugeable symmetries and their duals while unifying the prescription for gauging non-invertible 0-form and 1-form symmetries and various higher structures. With a view toward recent advances in creating non-Abelian topological orders from Abelian ones, we give a simple recipe for non-invertible 0-form gauging that takes large classes of the latter to the former. We also describe conditions under which iterated gauging of invertible 0-form symmetries is equivalent to a single-step gauging of a non-invertible symmetry. We conclude with a set of concrete examples illustrating these various phenomena involving gauging symmetries of the infrared limit of the toric code.
While two-dimensional symmetry-enriched topological phases
(SETs) have been studied intensively and systematically, three-dimensional ones are still open issues. We propose an algorithmic approach of imposing global symmetry
Gs on gauge theories (denoted by
GT) with gauge group
Gg. The resulting symmetric gauge theories are dubbed ``symmetry-enriched gauge theories''
(SEG), which may be served as low-energy effective theories of three-dimensional symmetric topological quantum spin liquids. We focus on
SEGs with gauge group ${G}_{g}={\mathbb{Z}}_{{N}_{1}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{N}_{2}}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ and onsite unitary symmetry group ${G}_{s}={\mathbb{Z}}_{{K}_{1}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{K}_{2}}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ or ${G}_{s}=\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{K}_{1}}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$. Each
SEG(Gg,Gs) is described in the path-integral formalism associated with certain symmetry assignment. From the path-integral expression, we propose how to physically diagnose the ground-state properties (i.e.,
SET orders) of
SEGs in experiments of charge-loop braidings (patterns of symmetry fractionalization) and the mixed multiloop braidings among deconfined loop excitations and confined symmetry fluxes. From these symmetry-enriched properties, one can obtain the map from
SEGs to
SETs. By giving full dynamics to background gauge fields,
SEGs may be eventually promoted to a set of new gauge theories (denoted by
GT∗). Based on their gauge groups,
GT∗s may be further regrouped into different classes, each of which is labeled by a gauge group
Gg∗. Finally, a web of gauge theories involving
GT,SEG,SET, and
GT∗ is achieved. We demonstrate the above symmetry-enrichment physics and the web of gauge theories through many concrete examples.
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 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 S 3 and O(2) 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.
We examine the interplay of symmetry and topological order in
2+1 dimensional fermionic topological phases of matter. We define fermionic topological symmetries acting on the emergent topological effective theory described using braided tensor category theory. Connecting this to the
Gf fermionic symmetry of the microscopic physical system, we characterize and classify symmetry fractionalization in fermionic topological phases. We find that the physical fermion provides constraints that result in a tiered structure of obstructions and classification of fractionalization with respect to the physical fermions, the quasiparticles, and the vortices. The fractionalization of the (bosonic) symmetry
G=Gf/Z2f on the physical fermions is essentially the central extension of
G by the
Z2f fermion parity conservation that yields the fermionic symmetry
Gf. We develop an algebraic theory of
Gf symmetry defects for fermionic topological phases using
G-crossed braided tensor category theory. This formalism allows us to fully characterize and classify
2+1 dimensional fermionic symmetry enriched topological phases with on-site unitary fermionic symmetry group
Gf. We first apply this formalism to extract the minimal data specifying a general fermionic symmetry protected topological phase, and demonstrate that such phases with fixed
Gf form a group under fermionic stacking. Then we analyze general fermionic symmetry enriched topological phases and find their classification is given torsorially by the classification of the symmetry fractionalization of quasiparticles combined with the classification of fermionic symmetry protected topological phases. We illustrate our results by detailing a number of examples, including all the invertible fermionic topological phases.
Symmetry plays a crucial role in enriching topological phases of matter. Phases with intrinsic topological order that are symmetric are called symmetry-enriched topological phases (SET). In this paper, we focus on SETs in three spatial dimensions, where the intrinsic topological orders are described by Abelian gauge theory and the symmetry groups are also Abelian. As a series work of our previous research [Phys. Rev. B 94, 245120 (2016); (arXiv:1609.00985)], we study these topological phases described by twisted gauge theories with global symmetry and consider all possible topologically inequivalent charge matrices. Within each equivalence class, there is a unique pattern of symmetry fractionalization on both point-like and string-like topological excitations. In this way, we classify Abelian topological order enriched by Abelian symmetry within our field-theoretic approach. To illustrate, we concretely calculate many representative examples of SETs and discuss future directions.