Eddy Ardonne

Topological Order and Conformal Quantum Critical Points

Abstract We discuss a certain class of two-dimensional quantum systems which exhibit conventional order and topological order, as well as 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 two-dimensional free boson, the two-dimensional 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.

A New Class of Non-Abelian Spin-Singlet Quantum Hall States

We present a new class of non-abelian spin-singlet quantum Hall states, generalizing Halperins 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

Fusion products of Kirillov–Reshetikhin modules and fermionic multiplicity formulas

\nu=2k/(2kM+3)

Fermionic Characters and Arbitrary Highest-Weight Integrable -Modules

. 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.

Collective States of Interacting Fibonacci Anyons

Abstract We give a complete description of the graded multiplicity space which appears in the Feigin–Loktev fusion product of graded Kirillov–Reshetikhin modules for all simple Lie algebras. This construction is used to obtain an upper bound formula for the fusion coefficients in these cases. The formula generalizes the case of g = A r of our previous paper, where the multiplicities are generalized Kostka polynomials. In the case of other Lie algebras, the formula is the fermionic side of the X = M conjecture. In the cases where the Kirillov–Reshetikhin conjecture, regarding the decomposition formula for tensor products of KR-modules, has been proven in its original, restricted form, our result provides a proof of the conjectures of Feigin and Loktev regarding the fusion product multiplicities.

Non-Abelian spin-singlet quantum Hall states: wave functions and quasihole state counting

This paper contains the generalization of the Feigin-Stoyanovsky construction to all integrable -modules. We give formulas for the q-characters of any highest-weight integrable module of as a linear combination of the fermionic q-characters of the fusion products of a special set of integrable modules. The coefficients in the sum are the entries of the inverse matrix of generalized Kostka polynomials in q−1. We prove the conjecture of Feigin and Loktev regarding the q-multiplicities of irreducible modules in the graded tensor product of rectangular highest weight-modules in the case of . We also give the fermionic formulas for the q-characters of the (non-level-restricted) fusion products of rectangular highest-weight integrable -modules.

Non-abelian quantum Hall states -- exclusion statistics, K-matrices and duality

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.

Non-abelian statistics in the interference noise of the Moore–Read quantum Hall state

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.  2001 Elsevier Science B.V. All rights reserved.

Separation of spin and charge in paired spin-singlet quantum Hall states

We study excitations in edge theories for non-abelian quantum Hall states, focussing on the spin polarized states proposed by Read and Rezayi and on the spin singlet states proposed by two of the authors. By studying the exclusion statistics properties of edge-electrons and edge-quasiholes, we arrive at a novel K-matrix structure. Interestingly, the duality between the electron and quasihole sectors links the pseudoparticles that are characteristic for non-abelian statistics with composite particles that are associated to the “pairing physics” of the non-abelian quantum Hall states.

Degeneracy of non-Abelian quantum Hall states on the torus: domain walls and conformal field theory

We propose noise oscillation measurements in a double point contact, accessible with current technology, to seek for a signature of the non-abelian nature of the ν = 5/2 quantum Hall state. Calculating the voltage and temperature dependence of the current and noise oscillations, we predict the non-abelian nature to materialize through a multiplicity of the possible outcomes: two qualitatively different frequency dependences of the nonzero interference noise. Comparison between our predictions for the Moore–Read state with experiments on ν = 5/2 will serve as a much needed test for the nature of the ν = 5/2 quantum Hall state.