We present an exactly solvable model for synthetic anyons carrying non-Abelian flux. The model corresponds to a two-dimensional electron gas in a magnetic field with a specific spin interaction term, which allows only fully aligned spin states in the ground state; the ground state subspace is thus twofold degenerate. This system is perturbed with identical solenoids carrying a non-Abelian gauge potential. We explore dynamics of the ground state as these solenoids are adiabatically braided and show they behave as anyons with a non-Abelian flux. Such a system represents a middle ground between the ordinary Abelian anyons and the fully non-Abelian anyons.
Abstract Kitaev’s quantum double model is a lattice realization of Dijkgraaf–Witten topological quantum field theory. Its topologically protected ground state space has broad applications for topological quantum computation and topological quantum memory. We investigate the Z 2 symmetry enriched generalization of the model for the Abelian group in a categorical framework and present an explicit Hamiltonian construction. This model provides a lattice realization of the Z 2 symmetry of the topological phase. We discuss in detail the categorical symmetry of the phase, for which the electric-magnetic (EM) duality symmetry is a special case. The aspects of symmetry defects are investigated using the G-crossed unitary braided fusion category. By determining the corresponding anyon condensation, the gapped boundaries and boundary-bulk duality are also investigated. In the last part, an explicit lattice realization of EM duality is discussed.
Abstract We study Kitaev’s quantum double model for arbitrary finite gauge group in infinite volume, using an operator-algebraic approach. The quantum double model hosts anyonic excitations which can be identified with equivalence classes of ‘localized and transportable endomorphisms’, which produce anyonic excitations from the ground state. Following the Doplicher–Haag–Roberts (DHR) sector theory from AQFT, we organize these endomorphisms into a braided monoidal category capturing the fusion and braiding properties of the anyons. We show that this category is equivalent to
RepfD(G) Rep f D ( G ) , the representation category of the quantum double of G . This establishes for the first time the full DHR structure for a class of 2d quantum lattice models with non-abelian anyons.
Anyons exhibit a non-trivial interplay between local exclusion rules and non-local braiding and exchange phases, making a consistent commutation algebra and second-quantized formulation challenging. We develop an algebraic framework for Abelian anyons in one dimension with statistical phase
θ =
π/N that enforces a finite on-site occupancy of N-1 anyons with the exchange phase
θ between different sites. Moreover, we introduce an exact Jordan-Wigner duality between
π/3 anyons and spin-1 operators, allowing us to map a tight-binding anyon model to an XY-like spin-1 model. The model exhibits anyon-density-dependent flux, incompressible or gapless regions, and critical points with level crossings that appear as discontinuities in ground-state currents, momenta, fidelities, and correlation functions. Our second-quantization formalism establishes a novel spin anyon duality, offering a conceptually new route to realize anyons from spin Hamiltonians and to engineer corresponding device architectures.
Abstract We introduce a two‐body quantum Hamiltonian model with spins‐
21 located on the vertices of a 2D spatial lattice. The model exhibits an exact topological degeneracy in all coupling regimes. This is a remarkable non‐perturbative effect. The model has a Z2× Z2 gauge group symmetry and string‐net integrals of motion. There exists a gapped phase in which the low‐energy sector reproduces an effective topological color code model. High energy excitations fall into three families of anyonic fermions that turn out to be strongly interacting. All these, and more, are new features not present in honeycomb lattice models like Kitaev model.
We regularize compact and non-compact Abelian Chern-Simons-Maxwell theories on a spatial lattice using the Hamiltonian formulation. We consider a doubled theory with gauge fields living on a lattice and its dual lattice. The Hilbert space of the theory is a product of local Hilbert spaces, each associated with a link and the corresponding dual link. The two electric field operators associated with the link-pair do not commute. In the non-compact case with gauge group
R, each local Hilbert space is analogous to the one of a charged "particle" moving in the link-pair group space
R2 in a constant "magnetic" background field. In the compact case, the link-pair group space is a torus
U(1)2 threaded by
k units of quantized "magnetic" flux, with
k being the level of the Chern-Simons theory. The holonomies of the torus
U(1)2 give rise to two self-adjoint extension parameters, which form two non-dynamical background lattice gauge fields that explicitly break the manifest gauge symmetry from
U(1) to
Z(k). The local Hilbert space of a link-pair then decomposes into representations of a magnetic translation group. In the pure Chern-Simons limit of a large "photon" mass, this results in a
Z(k)-symmetric variant of Kitaev's toric code, self-adjointly extended by the two non-dynamical background lattice gauge fields. Electric charges on the original lattice and on the dual lattice obey mutually anyonic statistics with the statistics angle
k2π. Non-Abelian
U(k) Berry gauge fields that arise from the self-adjoint extension parameters may be interesting in the context of quantum information processing.