We discuss a certain class of two-dimensional quantum systems which exhibit conventional order and topological order, as well as two-dimensional quantum critical points separating these phases. All of the ground-state equal-time correlators of these theories are equal to correlation functions of a local two-dimensional classical model. The critical points therefore exhibit a time-independent form of conformal invariance. These theories characterize the universality classes of two-dimensional quantum dimer models and of quantum generalizations of the eight-vertex model, as well as Z 2 and non-abelian gauge theories. The conformal quantum critical points are relatives of the Lifshitz points of three-dimensional anisotropic classical systems such as smectic liquid crystals. In particular, the ground-state wave functional of these quantum Lifshitz points is just the statistical (Gibbs) weight of the ordinary 2D free boson, the 2D Gaussian model. The full phase diagram for the quantum eight-vertex model exhibits quantum critical lines with continuously-varying critical exponents separating phases with long-range order from a Z 2 deconfined topologically-ordered liquid phase. We show how similar ideas also apply to a well-known field theory with non-Abelian symmetry, the strong-coupling limit of 2 + 1-dimensional Yang-Mills gauge theory with a Chern-Simons term. The ground state of this theory is relevant for recent theories of topological quantum computation.

We present a new class of non-abelian spin-singlet quantum Hall states, generalizing Halperin's abelian spin-singlet states and the Read-Rezayi non-abelian quantum Hall states for spin-polarized electrons. We label the states by (k, M ) with M odd (even) for fermionic (bosonic) states, and find a filling fraction ν = 2k/(2kM + 3). The states with M = 0 are bosonic spin-singlet states characterized by an SU (3) k symmetry. We explain how an effective Landau-Ginzburg theory for the SU (3)2 state can be constructed. In general, the quasi-particles over these new quantum Hall states carry spin, fractional charge and non-abelian quantum statistics.

We study a family of fusion and modular systems realizing fusion categories Grothendieck equivalent to the representation category for so(2p + 1)2. We conjecture a classification for their monoidal equivalence classes from an analysis of their gauge invariants and define a function which gives us the number of classes.

There are many interesting parallels between systems of interacting non-Abelian anyons and quantum magnetism, occuring in ordinary SU(2) quantum magnets. Here we consider theories of so-called su(2) k anyons, well-known deformations of SU (2), in which only the first k + 1 angular momenta of SU(2) occur. In this manuscript, we discuss in particular anyonic generalizations of ordinary SU(2) spin chains with an emphasis on anyonic spin S = 1 chains. We find that the overall phase diagrams for these anyonic spin-1 chains closely mirror the phase diagram of the ordinary bilinear-biquadratic spin-1 chain including anyonic generalizations of the Haldane phase, the AKLT construction, and supersymmetric quantum critical points. A novel feature of the anyonic spin-1 chains is an additional topological symmetry that protects the gapless phases. Distinctions further arise in the form of an even/odd effect in the deformation parameter k when considering su(2) k anyonic theories with k ≥ 5, as well as for the special case of the su(2)4 theory for which the spin-1 representation plays a special role. We also address anyonic generalizations of spin-1/2 chains with a focus on the topological protection provided for their gapless ground states. Finally, we put our results into context of earlier generalizations of SU(2) quantum spin chains, in particular so-called (fused) Temperley-Lieb chains.

We investigate a class of non-Abelian spin-singlet (NASS) quantum Hall phases, proposed previously. The trial ground and quasihole excited states are exact eigenstates of certain k + 1-body interaction Hamiltonians. The k = 1 cases are the familiar Halperin Abelian spin-singlet states. We present closed-form expressions for the many-body wave functions of the ground states, which for k > 1 were previously defined only in terms of correlators in specific conformal field theories. The states contain clusters of k electrons, each cluster having either all spins up, or all spins down. The ground states are non-degenerate, while the quasihole excitations over these states show characteristic degeneracies, which give rise to non-Abelian braid statistics. Using conformal field theory methods, we derive counting rules that determine the degeneracies in a spherical geometry. The results are checked against explicit numerical diagonalization studies for small numbers of particles on the sphere.

Quantum mechanical systems, whose degrees of freedom are so-called su(2)k anyons, form a bridge between ordinary SU(2) quantum magnets (of arbitrary spin-S) and systems of interacting non-Abelian anyons. Anyonic spin-1/2 chains exhibit a topological protection mechanism that stabilizes their gapless ground states and which vanishes only in the limit (k-->infinity) of the ordinary spin-1/2 Heisenberg chain. For anyonic spin-1 chains the phase diagram closely mirrors the one of the biquadratic SU(2) spin-1 chain. Our results describe, at the same time, nucleation of different 2D topological quantum fluids within a "parent" non-Abelian quantum Hall state, arising from a macroscopic occupation with localized, interacting anyons. The edge states between the "nucleated" and the parent liquids are neutral, and correspond precisely to the gapless modes of the anyonic chains.

We show that chains of interacting Fibonacci anyons can support a wide variety of collective ground states ranging from extended critical, gapless phases to gapped phases with ground-state degeneracy and quasiparticle excitations. In particular, we generalize the Majumdar-Ghosh Hamiltonian to anyonic degrees of freedom by extending recently studied pairwise anyonic interactions to three-anyon exchanges. The energetic competition between two-and three-anyon interactions leads to a rich phase diagram that harbors multiple critical and gapped phases. For the critical phases and their higher symmetry end points we numerically establish descriptions in terms of two-dimensional conformal field theories. A topological symmetry protects the critical phases and determines the nature of gapped phases.

Abstract. Collective states of interacting non-Abelian anyons have recently been studied mostly in the context of certain fractional quantum Hall states, such as the Moore-Read state proposed to describe the physics of the quantum Hall plateau at filling fraction ν = 5/2. In this paper, we further expand this line of research and present non-unitary generalizations of interacting anyon models. In particular, we introduce the notion of Yang-Lee anyons, discuss their relation to the so-called 'Gaffnian' quantum Hall wave function and describe an elementary model for their interactions. A one-dimensional (1D) version of this modela non-unitary generalization of the original golden chain model-can be fully understood in terms of an exact algebraic solution and numerical diagonalization. We discuss the gapless theories of these chain models for general su(2) k anyonic theories and their Galois conjugates. We further introduce and solve a 1D version of the Levin-Wen model for non-unitary Yang-Lee anyons.