Fractionalization of itinerant anyons in one-dimensional chains.

We construct models of interacting itinerant non-Abelian anyons moving along one-dimensional chains, focusing, in particular, on itinerant Ising anyon chains, and derive effective anyonic t-J models for the low-energy sectors. Solving these models by exact diagonalization, we find a fractionalization of the anyons into charge and (non-Abelian) anyonic degrees of freedom--a generalization of spin-charge separation of electrons which occurs in Luttinger liquids. A detailed description of the excitation spectrum by combining spectra for charge and anyonic sectors requires a subtle coupling between charge and anyonic excitations at the microscopic level (which we also find to be present in Luttinger liquids), despite the macroscopic fractionalization.

Competing topological orders in the ν=12/5 quantum Hall state.

We provide numerical evidence that a p(x)-ip(y) paired Bonderson-Slingerland (BS) non-Abelian hierarchy state is a strong candidate for the observed ν=12/5 quantum Hall plateau. We confirm the existence of a gapped incompressible ν=12/5 quantum Hall state with shift S=2 on the sphere, matching that of the BS state. The exact ground state of the Coulomb interaction at S=2 is shown to have a large overlap with the BS trial wave function. Larger overlaps are obtained with BS-type wave functions that are hierarchical descendants of general p(x)-ip(y) weakly paired states at ν=5/2. We perform a finite-size scaling analysis of the ground-state energies for ν=12/5 states at shifts corresponding to the BS (S=2) and 3-clustered Read-Rezayi (S=-2) universality classes. This analysis reveals very tight competition between these two non-Abelian topological orders.

Hierarchical nature of the quantum Hall effects.

I demonstrate that the wave function for a ν=n+ν[over ˜] quantum Hall state with Landau levels 0,1,…,n-1 filled and a filling fraction ν[over ˜] quantum Hall state with 0<ν[over ˜]≤1 in the nth Landau level can be obtained hierarchically from the ν=n state by introducing quasielectrons which are then projected into the (conjugate of the) ν[over ˜] state. In particular, the ν[over ˜]=1 case produces the filled Landau level wave functions hierarchically, thus establishing the hierarchical nature of the integer quantum Hall states. It follows that the composite fermion description of fractional quantum Hall states fits within the hierarchy theory of the fractional quantum Hall effect. I also demonstrate this directly by generating the composite fermion ground-state wave functions via application of the hierarchy construction to fractional quantum Hall states, starting from the ν=1/m Laughlin states.

Numerical calculation of the neutral fermion gap at the ν=5/2 fractional quantum Hall state.

We present the first numerical computation of the neutral fermion gap, Δ(F), in the ν=5/2 quantum Hall state, which is analogous to the energy gap for a Bogoliubov-de Gennes quasiparticle in a superconductor. We find Δ(F)≈0.027e(2)/εℓ(0), comparable to the charge gap. We also deduce an effective Fermi velocity v(F) for neutral fermions from the low-energy spectra for odd numbers of electrons, and thereby obtain a correlation length ξ(F)=v(F)/Δ(F)≈1.3ℓ(0). We comment on implications for experiments, topological quantum information processing, and electronic mechanisms of superconductivity.

Topological quantum buses: coherent quantum information transfer between topological and conventional qubits.

We propose computing bus devices that enable quantum information to be coherently transferred between topological and conventional qubits. We describe a concrete realization of such a topological quantum bus acting between a topological qubit in a Majorana wire network and a conventional semiconductor double quantum dot qubit. Specifically, this device measures the joint (fermion) parity of these two different qubits by using the Aharonov-Casher effect in conjunction with an ancilliary superconducting flux qubit that facilitates the measurement. Such a parity measurement, together with the ability to apply Hadamard gates to the two qubits, allows one to produce states in which the topological and conventional qubits are maximally entangled and to teleport quantum states between the topological and conventional quantum systems.

Implementing arbitrary phase gates with Ising anyons.

Ising-type non-Abelian anyons are likely to occur in a number of physical systems, including quantum Hall systems, where recent experiments support their existence. In general, non-Abelian anyons may be utilized to provide a topologically error-protected medium for quantum information processing. However, the topologically protected operations that may be obtained by braiding and measuring topological charge of Ising anyons are precisely the Clifford gates, which are not computationally universal. The Clifford gate set can be made universal by supplementing it with single-qubit pi/8-phase gates. We propose a method of implementing arbitrary single-qubit phase gates for Ising anyons by running a current of anyons with interfering paths around computational anyons.

Splitting the topological degeneracy of non-Abelian anyons.

We examine tunneling of topological charge between non-Abelian anyons as a perturbation of the long-range effective theory of a topologically ordered system. We obtain energy corrections in terms of the anyons' universal algebraic structure and nonuniversal tunneling amplitudes. We find that generic tunneling completely lifts the topological degeneracy of non-Abelian anyons. This degeneracy splitting is exponentially suppressed for long distances between anyons, but leaves no topological protection at shorter distances. We also show that general interactions of anyons can be expressed in terms of the transfer of topological charge, and thus can be treated effectively as tunneling interactions.

Measurement-only topological quantum computation.

We remove the need to physically transport computational anyons around each other from the implementation of computational gates in topological quantum computing. By using an anyonic analog of quantum state teleportation, we show how the braiding transformations used to generate computational gates may be produced through a series of topological charge measurements.

Decoherence of anyonic charge in interferometry measurements.

We examine interferometric measurements of the topological charge of (non-Abelian) anyons. The target's topological charge is measured from its effect on the interference of probe particles sent through the interferometer. We find that superpositions of distinct anyonic charges a and a' in the target decohere (exponentially in the number of probes particles used) when the probes have nontrivial monodromy with the charges that may be fused with a to give a'.

Probing non-Abelian statistics with quasiparticle interferometry.

We examine interferometric experiments in systems that exhibit non-Abelian braiding statistics, expressing outcomes in terms of the modular S-matrix. In particular, this result applies to fractional quantum Hall interferometry, and we give a detailed treatment of the Read-Rezayi states, providing explicit predictions for the recently observed nu = 12/5 plateau.

Detecting non-Abelian statistics in the nu = 5/2 fractional quantum hall state.

In this Letter we propose an interferometric experiment to detect non-Abelian quasiparticle statistics--one of the hallmark characteristics of the Moore-Read state expected to describe the observed fractional quantum Hall effect plateau at nu = 5/2. The implications for using this state for constructing a topologically protected qubit as has been recently proposed by Das Sarma et al. are also addressed.