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arXiv:0809.1415 · Physical Review Letters

Nonequilibrium Transport through a Point Contact in the ν=5/2 Non-Abelian Quantum Hall State

Adrian E. Feiguin, Paul Fendley, Matthew P. A. Fisher, Chetan Nayak

Published Dec 2, 2008• 25 citations
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Abstract

We analyze charge-e/4 quasiparticle tunneling between the edges of a point contact in a non-Abelian model of the nu = 5/2 quantum Hall state in the presence of a finite voltage difference using the time-dependent density-matrix renormalization group method. We confirm that, as the voltage decreases, the system is broken into two pieces. In the limits of small and large voltage, we recover the results expected from perturbation theory about the infrared and ultraviolet fixed points. We test our methods by finding the analogous nonequilibrium current through a point contact at nu = 1/3.

DOI
10.1103/physrevlett.101.236801

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We analyze charge-e/4 quasiparticle tunneling between the edges of a point contact in a non-Abelian model of the nu = 5/2 quantum Hall state in the presence of a finite voltage difference using the time-dependent density-matrix renormalization group method. We confirm that, as the voltage decreases, the system is broken into two pieces. In the limits of small and large voltage, we recover the results expected from perturbation theory about the infrared and ultraviolet fixed points. We test our methods by finding the analogous nonequilibrium current through a point contact at nu = 1/3.
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Motivated by surprises in recent experimental findings, we study transport in a model of a quantum Hall edge system with a gate-voltage controlled constriction. A finite backscattered current at finite edge bias is explained from a Landauer-Buttiker analysis as arising from the splitting of edge current caused by the difference in the filling fractions of the bulk $({\ensuremath{\nu}}_{1})$ and constriction $({\ensuremath{\nu}}_{2})$ quantum Hall fluid regions. We develop a hydrodynamic theory for bosonic edge modes inspired by this model. The constriction region splits the incident long-wavelength chiral edge density-wave excitations among the transmitting and reflecting edge states encircling it. The competition between two interedge tunneling processes taking place inside the constriction, related by a quasiparticle-quasihole (qp-qh) symmetry, is accounted for by computing the boundary theories of the system. This competition is found to determine the strong coupling configuration of the system. A separatrix of qp-qh symmetric gapless critical states is found to lie between the relevant renormalization group (RG) flows to a metallic and an insulating configuration of the constriction system. This constitutes an interesting generalization of the Kane-Fisher quantum impurity model. The features of the RG phase diagram are also confirmed by computing various correlators and chiral linear conductances of the system. In this way, our results find excellent agreement with many recent puzzling experimental results for the cases of ${\ensuremath{\nu}}_{1}=1∕3$ and 1. We also discuss and make predictions for the case of a constriction system with ${\ensuremath{\nu}}_{2}=5∕2$.
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We compute the dependence of the tunneling current in a double point contact in the k=3 Read-Rezayi state (which is conjectured to describe an incompressible quantum hall fluid at filling fraction nu=12/5) on voltage, separation between the two contacts, and temperature. Using the tunneling hamiltonian of cond-mat/0607431, we show that the effect of quasiholes in the bulk region between the two contacts is simply an overall constant multiplying the interference term. This is the same effect as found for the differential conductivity in cond-mat/0601242; the difference is that we do an actual edge theory calculation and compute the full current-voltage curve at weak tunneling.
We compute the backscattered current in a double point-contact geometry of a quantum-Hall system at filling fraction as a function of bias voltage in the weak backscattering regime. We assume that the system is in the universality class of either the Pfaffian or anti-Pfaffian state. When the number of charge quasiparticles in the interferometer is odd, there is no interference pattern. However, the coupling between a charge quasiparticle and the edge causes it to be absorbed by the edge at low energies. Consequently, an interference pattern appears at low bias voltages and temperatures, as if there were an even number of quasiparticles in the interferometer. We relate this problem to that of a semi-infinite Ising model with a boundary magnetic field. Using the methods of perturbed boundary conformal field theory, we give an exact expression for this crossover of the interferometer as a function of bias voltage. Finally, we comment on the possible relevance of our results to recent interference experiments.