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arXiv:cond-mat/0604064 · Physical Review Letters

Dynamical Disentanglement across a Point Contact in a Non-Abelian Quantum Hall State

Paul Fendley, Matthew P. A. Fisher, Chetan Nayak

Published Jul 17, 2006• 48 citations
arXivHTMLPDF

Abstract

We analyze the tunneling of non-Abelian quasiparticles between the edges of a quantum Hall droplet at the Landau level filling fraction nu=5/2, assuming that the electrons in the first excited Landau level organize themselves in the non-Abelian Moore-Read Pfaffian state. By bosonizing the edge theory, we show that an effective spin-1/2 degree of freedom emerges in the description of a point contact. We show how the crossover from the high-temperature regime of weak quasiparticle tunneling between the edges of the droplet, with the 4-terminal Rxx approximately T(-3/2), to the low-temperature limit, with Rxx(-1/10)(h/e2) approximately-T4, is closely related to the two-channel Kondo effect. We give a physical interpretation for the entropy loss of ln(2[square root of 2) in this crossover.

DOI
10.1103/physrevlett.97.036801

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We analyze the tunneling of non-Abelian quasiparticles between the edges of a quantum Hall droplet at the Landau level filling fraction nu=5/2, assuming that the electrons in the first excited Landau level organize themselves in the non-Abelian Moore-Read Pfaffian state. By bosonizing the edge theory, we show that an effective spin-1/2 degree of freedom emerges in the description of a point contact. We show how the crossover from the high-temperature regime of weak quasiparticle tunneling between the edges of the droplet, with the 4-terminal Rxx approximately T(-3/2), to the low-temperature limit, with Rxx(-1/10)(h/e2) approximately-T4, is closely related to the two-channel Kondo effect. We give a physical interpretation for the entropy loss of ln(2[square root of 2) in this crossover.
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We study a quantum point contact in the fractional quantum Hall regime at Landau level filling factors $\ensuremath{\nu}=1/3$ and 5/25/25/2. By using exact diagonalizations in the cylinder geometry, we identify the edge modes in the presence of a parabolic confining potential. By changing the sign of the potential, we can access both the tunneling through the bulk of the fluid and the tunneling between spatially separated droplets. This geometry is realized in the quantum point contact geometry for two-dimensional electron gases. In the case of the model Moore-Read Pfaffian state at filling factor $\ensuremath{\nu}=5/2$, we identify the conformal towers of many-body eigenstates including the non-Abelian sector. By a Monte-Carlo technique, we compute the various scaling exponents that characterize the edge modes. In the case of hard-core interactions whose ground states are exact model wave functions, we find equality of neutral and charged velocities, both bosonic and fermionic, for the Pfaffian state.
In this paper, we study the coupling between a quantum dot and the edge of a non-Abelian fractional quantum Hall state. We assume the dot is small enough that its level spacing is large compared to both the temperature and the coupling to the spatially proximate bulk non-Abelian fractional quantum Hall state. We focus on the physics of level degeneracy with electron number on the dot. The physics of such a resonant level is governed by a kkk-channel Kondo model when the quantum Hall state is a Read-Rezayi state at filling fraction $\ensuremath{\nu}=2+k/(k+2)$ or its particle-hole conjugate at $\ensuremath{\nu}=2+2/(k+2)$. The kkk-channel Kondo model is channel symmetric even without fine tuning any couplings in the former state; in the latter, it is generically channel asymmetric. The two limits exhibit non-Fermi-liquid and Fermi-liquid properties, respectively, and therefore may be distinguished. By exploiting the mapping between the resonant level model and the multichannel Kondo model, we discuss the thermodynamic and transport properties of the system. In the special case of k=2k=2k=2, our results provide a distinct venue to distinguish between the Pfaffian and anti-Pfaffian states at filling fraction $\ensuremath{\nu}=5/2$. We present numerical estimates for realizing this scenario in experiment.
Identifying and understanding interacting systems that can host non-Abelian topological phases with fractionalized quasiparticles have attracted intense attentions in the past twenty years. Theoretically, it is possible to realize a rich variety of such states by coupling two Abelian fractional quantum Hall (FQH) states together through gapping out part of the low energy degrees of freedom. So far, there are some indications, but no robust example has been established in bilayer systems for realizing the non-Abelian state in the past. Here, we present a phase diagram of a double-layer bosonic FQH system based on the exact diagonalization and density-matrix renormalization group (DMRG) calculations, which demonstrate a potential regime with the emergence of the non-Abelian bosonic Moore-Read state. We start from the Abelian phase with fourfold topological degeneracies on torus geometry when the two layers are weakly coupled. With the increase of interlayer tunneling, we find an intermediate regime with a threefold groundstate degeneracy and a finite fractional drag Hall conductance. We find the different topological sectors in consistent with Moore-Read state by inserting different fluxes in adiabatic DMRG study. We also extract the modular S−\mathcal{S}-S− matrix, which supports the emergence of the non-Abelian Ising anyon quasiparticle in this system.