Topological order in two-dimensions can be described in terms of deconfined quasiparticle excitations—anyons—and their braiding statistics. However, it has recently been realized that this data does not completely describe the situation in the presence of an unbroken global symmetry. In this case, there can be multiple distinct quantum phases with the same anyons and statistics, but with different patterns of symmetry fractionalization—termed symmetry enriched topological order. When the global symmetry group G , which we take to be discrete, does not change topological superselection sectors—i.e. does not change one type of anyon into a different type of anyon—one can imagine a local version of the action of G around each anyon. This leads to projective representations and a group cohomology description of symmetry fractionalization, with the second cohomology group H 2 ( G , A a b e l i a n ) being the relevant group. In this paper, we treat the general case of a symmetry group G possibly permuting anyon types. We show that despite the lack of a local action of G , one can still make sense of a so-called twisted group cohomology description of symmetry fractionalization, and show how this data is encoded in the associativity of fusion rules of the extrinsic ‘twist’ defects of the symmetry. Furthermore, building on work of Hermele (2014 Phys. Rev. B 90 184418 ), we construct a wide class of exactly-solvable models which exhibit this twisted symmetry fractionalization, and connect them to our formal framework.
Topological phases in two dimensions support anyonic quasiparticle excitations that obey neither bosonic nor fermionic statistics. These anyon structures often carry global symmetries that relate distinct anyons with similar fusion and statistical properties. Anyonic symmetries associate topological defects or fluxes in topological phases. As the symmetries are global and static, these extrinsic defects are semiclassical objects that behave disparately from conventional quantum anyons. Remarkably, even when the topological states supporting them are Abelian, they are generically non-Abelian and powerful enough for topological quantum computation. In this article, I review the most recent theoretical developments on symmetries and defects in topological phases.
Abstract In this work, we explore a microscopic realization of three types of anyonic symmetries in a Z 2 × Z 2 topological order, corresponding to a double toric code. These symmetries act as nontrivial permutations on the anyon labels of the parent state. We consider a setup consisting of two decoupled Wen plaquette models stacked on top of each other and introduce dislocations that modify the Hamiltonian, giving rise to localized twist defects, eventually inducing interactions between the layers. In this context, branch cuts act as sources of anyon permutations when they cross it. We characterize the defects by calculating their quantum dimensions, and we also consider double loop operators around them that allow us to determine the non-Abelian fusion rules between the defects, including when they carry different anyon permutations.
We describe the braiding statistics of topological twist defects in fractional quantum Hall (FQH) states. In particular, we study Abelian bosonic
(mmn) FQH states with bilayer symmetries. Twist defects are staircases that lift orbiting quasiparticles from one layer to another. These point defects exhibit semiclassical non-Abelian characteristics. We develop a procedure to evaluate consistent basis transformations (
F symbols) of the defect fusion theory. Unlike quantum anyonic excitations of a topological phase, the exchange and braiding statistics of twist defects is not characterized by modular transformations, but instead a subcollection known as congruent transformations. We also characterize the projective nature of unitary braiding operations and relate it to the global
Z2 bilayer symmetry.
We investigate whether the braid group representation on
N anyons uniquely determines the modular data (
S and
T matrices) of a non-Abelian topological order, and whether this framework classifies all fusion rules for Majorana zero modes (MZMs) in 2D topological superconductors. We show that the local
N-anyon braid group representation constrains the modular data up to finite gauge ambiguity but does not uniquely fix the global
S-matrix without specifying the underlying spin structure. For MZMs, the braid generators satisfy the Ising anyon
R-matrix with the asymptotic Hecke relation
σi2=1 (up to phase), which rigidly fixes the local
F and
R symbols of the Ising category. Consequently, the fusion rules are strictly of Ising type (
σ×σ=1+ψ), precluding alternative non-Abelian fusion rules. The
T-matrix is determined by topological spins, but the full
S-matrix depends on the spin structure of the underlying manifold, reflecting the super-modular structure of the theory. ## 1. Introduction Topological quantum computation exploits the non-Abelian statistics of anyons—quasiparticle excitations in two-dimensional systems whose braiding operations are represented by non-commuting unitary matrices on a degenerate ground-state manifold [1]. A central question is whether the local braiding data of
N anyons suffices to reconstruct the full modular data of the underlying topological order (TO), or whether additional global information is required. This question has direct physical consequences for 2D topological superconductors (TSCs) hosting Majorana zero modes (MZMs), where the anyonic excitations are vortices binding MZMs and the fermionic quasiparticle is a transparent (local) fermion [2,3]. The modular data of a TO consists of the
S-matrix, which encodes the mutual braiding statistics of anyons, and the
T-matrix, which encodes the topological spins. Together, these matrices form a projective representation of t
We describe the mathematical theory of topological quantum computing with symmetry defects in the language of fusion categories and unitary representations. Symmetry defects together with anyons are modeled by G-crossed braided extensions of unitary modular tensor categories. The algebraic data of these categories afford projective unitary representations of the braid group. Elements in the image of such representations correspond to quantum gates arising from exchanging anyons and symmetry defects in topological phases of matter with symmetry. We provide some small examples that highlight features of practical interest for quantum computing. In particular, symmetry defects show the potential to generate non-abelian statistics from abelian topological phases and to be used as as a tool to enlarge the set of quantum gates accessible to an anyonic device.