34 research outputs found
Braiding and fusion of non-Abelian vortex anyons
We demonstrate that certain vortices in spinor Bose-Einstein condensates are
non-Abelian anyons and may be useful for topological quantum computation. We
perform numerical experiments of controllable braiding and fusion of such
vortices, implementing the actions required for manipulating topological
qubits. Our results suggest that a new platform for topological quantum
information processing could potentially be developed by harnessing non-Abelian
vortex anyons in spinor Bose-Einstein condensates.Comment: 16 pages, 3 figures, 6 supplementary figures; added details of the
H-charge, J. K. Slingerland added to author lis
An investigation of pre-crystalline order, ruling out Pauli crystals and introducing Pauli anti-crystals
Fluid states of matter can locally exhibit characteristics of the onset of
crystalline order. Traditionally this has been theoretically investigated using
multipoint correlation functions. However new measurement techniques now allow
multiparticle configurations of cold atomic systems to be observed directly.
This has led to a search for new techniques to characterize the configurations
that are likely to be observed. One of these techniques is the configuration
density (CD), which has been used to argue for the formation of "Pauli
crystals" by non-interacting electrons in e.g. a harmonic trap. We show here
that such Pauli crystals do not exist, but that other other interesting spatial
structures can occur in the form of an "anti-Crystal", where the fermions
preferentially avoid a lattice of positions surrounding any given fermion.
Further, we show that configuration densities must be treated with great care
as naive application can lead to the identification of crystalline structures
which are artifacts of the method and of no physical significance. We analyze
the failure of the CD and suggest methods that might be more suitable for
characterizing multiparticle correlations which may signal the onset of
crystalline order. In particular, we introduce neighbour counting statistics
(NCS), which is the full counting statistics of the particle number in a
neighborhood of a given particle. We test this on two dimensional systems with
emerging triangular and square crystal structures.Comment: 17 pages, 8 figures; v2 Title chang
Condensate-induced transitions between topologically ordered phases
We investigate transitions between topologically ordered phases in two spatial dimensions induced by the
condensation of a bosonic quasiparticle. To this end, we formulate an extension of the theory of symmetrybreaking
phase transitions which applies to phases with topological excitations described by quantum groups or
modular tensor categories. This enables us to deal with phases whose quasiparticles have noninteger quantum
dimensions and obey braid statistics. Many examples of such phases can be constructed from two-dimensional
rational conformal field theories, and we find that there is a beautiful connection between quantum group
symmetry breaking and certain well-known constructions in conformal field theory, notably the coset construction,
the construction of orbifold models, and more general conformal extensions. Besides the general framework,
many representative examples are worked out in detail
Theory of Topological Edges and Domain Walls
We investigate domain walls between topologically ordered phases in two spatial dimensions. We
present a method which allows for the determination of the superselection sectors of excitations of such
walls and which leads to a unified description of the kinematics of a wall and the two phases to either side
of it. This incorporates a description of scattering processes at domain walls which can be applied to
questions of transport through walls. In addition to the general formalism, we give representative
examples including domain walls between the Abelian and non-Abelian topological phases of Kitaev’s
honeycomb lattice model in a magnetic field, as well as recently proposed domain walls between spin
polarized and unpolarized non-Abelian fractional quantum Hall states at different filling fractions
Towards a non-abelian electric-magnetic symmetry: the skeleton group
We propose an electric-magnetic symmetry group in non-abelian gauge theory, which we call the skeleton group. We work in the context of non-abelian unbroken gauge symmetry, and provide evidence for our proposal by relating the representation theory of the skeleton group to the labelling and fusion rules of charge sectors. We show that the labels of electric, magnetic and dyonic sectors in non-abelian Yang-Mills theory can be interpreted in terms of irreducible representations of the skeleton group. Decomposing tensor products of these representations thus gives a set of fusion rules which contain information about the full fusion rules of these charge sectors. We demonstrate consistency of the skeleton's fusion rules with the known fusion rules of the purely electric and purely magnetic magnetic sectors, and extract new predictions for the fusion rules of dyonic sectors in particular cases. We also implement S-duality and show that the fusion rules obtained from the skeleton group commute with S-duality
Clebsch–Gordan and 6 j -coefficients for rank 2 quantum groups
We calculate (
q
-deformed) Clebsch–Gordan and 6
j
-coefficients for rank 2
quantum groups. We explain in detail how such calculations are done,
which should allow the reader to perform similar calculations in other cases.
Moreover, we tabulate the
q
-Clebsch–Gordan and 6
j
-coefficients explicitly, as
well as some other topological data associated with theories corresponding to
rank 2 quantum groups. Finally, we collect some useful properties of the fusion
rules of particular conformal field theories
Dynamics and level statistics of interacting fermions in the lowest Landau level
We consider the unitary dynamics of interacting fermions in the lowest Landau level, on spherical and toroidal geometries. The dynamics are driven by the interaction Hamiltonian which, viewed in the basis of single-particle Landau orbitals, contains correlated pair hopping terms in addition to static repulsion. This setting and this type of Hamiltonian has a significant history in numerical studies of fractional quantum Hall (FQH) physics, but the many-body quantum dynamics generated by such correlated hopping has not been explored in detail. We focus on initial states containing all the fermions in one block of orbitals. We characterize in detail how the fermionic liquid spreads out starting from such a state. We identify and explain differences with regular (single-particle) hopping Hamiltonians. Such differences are seen, e.g. in the entanglement dynamics, in that some initial block states are frozen or near-frozen, and in density gradients persisting in long-time equilibrated states. Examining the level spacing statistics, we show that the most common Hamiltonians used in FQH physics are not integrable, and explain that GOE statistics (level statistics corresponding to the Gaussian orthogonal ensemble) can appear in many cases despite the lack of time-reversal symmetry
Clebsch-Gordan and 6j-coefficients for rank two quantum groups
We calculate (q-deformed) Clebsch-Gordan and 6j-coefficients for rank two
quantum groups. We explain in detail how such calculations are done, which
should allow the reader to perform similar calculations in other cases.
Moreover, we tabulate the q-Clebsch-Gordan and 6j-coefficients explicitly, as
well as some other topological data associated with theories corresponding to
rank-two quantum groups. Finally, we collect some useful properties of the
fusion rules of particular conformal field theories.Comment: 43 pages. v2: minor changes and added references. For mathematica
notebooks containing the various q-CG and 6j symbols, see
http://arxiv.org/src/1004.5456/an
Numerical simulation of non-Abelian anyons
Two-dimensional systems such as quantum spin liquids or fractional quantum Hall systems exhibit anyonic excitations that possess more general statistics than bosons or fermions. This exotic statistics makes it challenging to solve even a many-body system of non-interacting anyons. We introduce an algorithm that allows to simulate anyonic tight-binding Hamiltonians on two-dimensional lattices. The algorithm is directly derived from the low energy topological quantum field theory and is suited for general Abelian and non-Abelian anyon models. As concrete examples, we apply the algorithm to study the energy level spacing statistics, which reveals level repulsion for free semions, Fibonacci anyons, and Ising anyons. Additionally, we simulate nonequilibrium quench dynamics, where we observe that the density distribution becomes homogeneous for large times - indicating thermalization