Monte Carlo evaluation of non-Abelian statistics

We develop a general framework to (numerically) study adiabatic braiding of quasiholes in fractional quantum Hall systems. Specifically, we investigate the Moore-Read (MR) state at nu=1/2 filling factor, a known candidate for non-Abelian statistics, which appears to actually occur in nature. The non-Abelian statistics of MR quasiholes is demonstrated explicitly for the first time, confirming the results predicted by conformal field theories.     

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.     

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 with an electronic Mach-Zehnder interferometer

Fractionally charged quasiparticles in the quantum Hall state with a filling factor nu=5/2 are expected to obey non-Abelian statistics. We demonstrate that their statistics can be probed by transport measurements in an electronic Mach-Zehnder interferometer. The tunneling current through the interferometer exhibits a characteristic dependence on the magnetic flux and a nonanalytic dependence on the tunneling amplitudes which can be controlled by gate voltages.     

Fractional statistics in the fractional quantum Hall effect

A microscopic confirmation of the fractional statistics of the quasiparticles in the fractional quantum Hall effect has so far been lacking. We calculate the statistics of the composite-fermion quasiparticles at nu=1/3 and nu=2/5 by evaluating the Berry phase for a closed loop encircling another composite-fermion quasiparticle. A careful consideration of subtle perturbations in the trajectory due to the presence of an additional quasiparticle is crucial for obtaining the correct value of the statistics. The conditions for the applicability of the fractional statistics concept are discussed.     

Minimal setup for non-Abelian braiding of Majorana zero modes

Braiding Majorana zero modes(MZMs) is the key procedure toward topological quantum computation. However, the complexity of the braiding manipulation hinders its experimental realization. Here we propose an experimental setup consisting of MZMs and a quantum dot state which can substantially simplify the braiding protocol of MZMs. Such braiding scheme, corresponding to a specific closed loop in the parameter space, is quite universal and can be realized in various platforms. Moreover, the braiding results can be measured and manifested through electric current, providing a simple and novel way to detect the non-Abelian statistics of MZMs.     

Non-Abelian braiding of lattice bosons

We report on a numerical experiment in which we use time-dependent potentials to braid non-Abelian quasiparticles. We consider lattice bosons in a uniform magnetic field within the fractional quantum Hall regime, where ν, the ratio of particles to flux quanta, is near 1/2, 1, or 3/2. We introduce time-dependent potentials which move quasiparticle excitations around one another, explicitly simulating a braiding operation which could implement part of a gate in a quantum computation. We find that different braids do not commute for ν near 1 and 3/2, with Berry matrices, respectively, consistent with Ising and Fibonacci anyons. Near ν=1/2, the braids commute.     

Multichannel Kondo models in non-Abelian quantum Hall droplets

We study the coupling between a quantum dot and the edge of a non-Abelian fractional quantum Hall state which is spatially separated from it by an integer quantum Hall state. Near a resonance, the physics at energy scales below the level spacing of the edge states of the dot is governed by a k-channel Kondo model when the quantum Hall state is a Read-Rezayi state at filling fraction nu=2+k/(k+2) or its particle-hole conjugate at nu=2+2/(k+2). The k-channel Kondo model is channel isotropic even without fine-tuning in the former state; in the latter, it is generically channel anisotropic. In the special case of k=2, our results provide a new venue, realized in a mesoscopic context, to distinguish between the Pfaffian and anti-Pfaffian states at filling fraction nu=5/2.     

Boundary conformal field theory and tunneling of edge quasiparticles in non-Abelian topological states

We explain how (perturbed) boundary conformal field theory allows us to understand the tunneling of edge quasiparticles in non-Abelian topological states. The coupling between a bulk non-Abelian quasiparticle and the edge is due to resonant tunneling to a zero mode on the quasiparticle, which causes the zero mode to hybridize with the edge. This can be reformulated as the flow from one conformally invariant boundary condition to another in an associated critical statistical mechanical model. Tunneling from one edge to another at a point contact can split the system in two, either partially or completely. This can be reformulated in the critical statistical mechanical model as the flow from one type of defect line to another. We illustrate these two phenomena in detail in the context of the ν=5/2 quantum Hall state and the critical Ising model. We briefly discuss the case of Fibonacci anyons and conclude by explaining the general formulation and its physical interpretation.     

Effect of Landau level mixing on braiding statistics

We examine the effect of Landau level mixing on the braiding statistics of quasiparticles of Abelian and non-Abelian quantum Hall states. While path dependent geometric phases can perturb the Abelian part of the statistics, we find that the non-Abelian properties remain unchanged to an accuracy that is exponentially small in the distance between quasiparticles.     

Evolution of quasiparticle charge in the fractional quantum hall regime

The charge of quasiparticles in a fractional quantum Hall (FQH) liquid, tunneling through a partly reflecting constriction with transmission t, was determined via shot noise measurements. In the nu = 1/3 FQH state, a charge smoothly evolving from e(*) = e/3 for t(1/3) congruent with 1 to e(*) = e for t(1/3)<1 was determined, agreeing with chiral Luttinger liquid theory. In the nu = 2/5 FQH state the quasiparticle charge evolves smoothly from e(*) = e/5 at t(2/5) congruent with 1 to a maximum charge less than e(*) = e/3 at t(2/5)<1. Thus it appears that quasiparticles with an approximate charge e/5 pass a barrier they see as almost opaque.     

Switching noise as a probe of statistics in the fractional quantum Hall effect

We propose an experiment to probe the unconventional quantum statistics of quasiparticles in fractional quantum Hall states by measurement of current noise. The geometry we consider is that of a Hall bar where two quantum point contacts introduce two interfering amplitudes for backscattering. Thermal fluctuations of the number of quasiparticles enclosed between the two point contacts introduce current noise, which reflects the statistics of the quasiparticles. We analyze Abelian nu=1/q states and the non-Abelian nu=5/2 state.     

e/3 Laughlin quasiparticle primary-filling nu=1/3 interferometer

We report experimental realization of a quasiparticle interferometer where the entire system is in 1/3 primary fractional quantum Hall state. The interferometer consists of chiral edge channels coupled by quantum-coherent tunneling in two constrictions, thus enclosing an Aharonov-Bohm area. We observe magnetic flux and charge periods h/e and e/3, equivalent to the creation of one quasielectron in the island. Quantum theory predicts a 3h/e flux period for charge e/3, integer statistics particles. Thus, the observed periods demonstrate the anyonic braiding statistics of Laughlin quasiparticles.     

Characterizing neutral modes of fractional states in the second Landau level

Fractionally charged quasiparticles, which obey non-abelian statistics, were predicted to exist in the ν=8/3, ν=5/2, and ν=7/3 fractional quantum Hall states (in the second Landau level). Here we present measurements of charge and neutral modes in these states. For both ν=7/3 and ν=8/3 states, we found a quasiparticle charge e=1/3 and an upstream neutral mode only in ν=8/3-excluding the possibility of non-abelian Read-Rezayi states and supporting Laughlin-like states. The absence of an upstream neutral mode in the ν=7/3 state also proves that edge reconstruction was not present in the ν=7/3 state, suggesting its absence also in ν=5/2 state, and thus may provide further support for the non-abelian anti-pfaffian nature of the ν=5/2 state.     

The Haldane-Rezayi quantum Hall state and conformal field theory

We propose field theories for the bulk and edge of a quantum Hall state in the universality class of the Haldane-Rezayi wavefunction. The bulk theory is associated with the c = −2 conformal field theory. The topological properties of the state, such as the quasiparticle braiding statistics and ground state degeneracy on a torus, may be deduced from this conformal field theory. The 10-fold degeneracy on a torus is explained by the existence of a logarithmic operator in the c = −2 theory; this operator corresponds to a novel bulk excitation in the quantum Hall state. We argue that the edge theory is the c = 1 chiral Dirac fermion, which is related in a simple way to the c = −2 theory of the bulk. This theory is reformulated as a truncated version of a doublet of Dirac fermions in which the SU(2) symmetry - which corresponds to the spin-rotational symmetry of the quantum Hall system - is manifest and non-local. We make predictions for the current-voltage characteristics for transport through point contacts.     

Nonlinear quasiparticle tunneling between fractional quantum hall edges

Remarkable nonlinearities in the differential tunneling conductance between fractional quantum Hall edge states at a constriction are observed in the weak-backscattering regime. In the nu=1/3 state a peak develops as temperature is increased and its width is determined by the fractional charge. In the range 2/3相似文献   

Edge excitations and non-Abelian statistics in the moore-read state: A numerical study in the presence of coulomb interaction and edge confinement

We study the ground state and low-energy excitations of fractional quantum Hall systems on a disk at a filling fraction nu = 5/2, with Coulomb interaction and a background confining potential. We find the Moore-Read ground state is stable within a finite but narrow window in parameter space. The corresponding low-energy excitations contain a fermionic branch and a bosonic branch, with widely different velocities. A short-range repulsive potential can stabilize a charge +e/4 quasihole at the center, leading to a different edge excitation spectrum due to the change of boundary conditions for Majorana fermions, clearly indicating the non-Abelian nature of the quasihole.     

Proposed experiments to probe the non-Abelian nu = 5/2 quantum hall state

We propose several experiments to test the non-Abelian nature of quasiparticles in the fractional quantum Hall state at nu = 5/2. In a simplified version of the experiment suggested by [S. Das Sarma, M. Freedman, and C. Nayak, Phys. Rev. Lett. 94, 166802 (2005).], interference is turned on and off when the number of localized quasiparticles between the interfering paths varies between even and odd. We find analogous effects in the thermodynamic properties of closed systems.     

Bulk-edge coupling in the non-Abelian nu=5/2 quantum hall interferometer

Recent schemes for probing non-Abelian statistics in the quantum Hall effect are based on geometries where current-carrying quasiparticles flow along edges that encircle bulk quasiparticles, which are localized. Here we consider one such scheme, the Fabry-Perot interferometer, and analyze how its interference patterns are affected by a coupling that allows tunneling of neutral Majorana fermions between the bulk and edge. While at weak coupling this tunneling degrades the interference signal, we find that at strong coupling, the bulk quasiparticle becomes essentially absorbed by the edge and the intereference signal is fully restored. Furthermore, we find that the strength of the coupling can be tuned by the source-drain voltage.