Chiral Luttinger Liquid Research Articles

Resonant tunneling into a biased fractional quantum Hall edge.

We observe resonant tunneling into a voltage biased fractional quantum Hall effect (FQHE) edge, using atomically sharp tunnel barriers unique to cleaved-edge overgrown devices. The resonances demonstrate different tunnel couplings to the metallic lead and the FQHE edge. Weak coupling to the FQHE edge produces clear non-Fermi liquid behavior with a sixfold increase in resonance area under bias arising from the power law density of states at the FQHE edge. A simple device model uses the resonant tunneling formalism for chiral Luttinger liquids to successfully describe the data.

Plateau Behavior in the Chiral Luttinger Liquid Exponent.

The current-voltage power law exponent, alpha, for electron tunneling into chiral Luttinger liquids at the fractional quantum Hall edge is found to exhibit a plateaulike structure at alpha close to 3 as the filling factor, nu, is varied. The presence of a plateau near alpha = 3 strongly suggests a fundamental connection between alpha and the structure of the underlying quantum ground states associated with the robust incompressible nu = 1/3 Hall fluid. However, the position in the inverse filling factor where the plateau occurs can vary between samples and appears shifted to values higher than expected from theory.

Coulomb blockade in the fractional quantum Hall effect regime

We use chiral Luttinger liquid theory to study transport through a quantum dot in the fractional quantum Hall effect regime and find rich non-Fermi-liquid tunneling characteristics. In particular, we predict a remarkable Coulomb-blockade-type energy gap that is quantized in units of the noninteracting level spacing, new power-law tunneling exponents for voltages beyond threshold, and a line shape as a function of gate voltage that is dramatically different than that for a Fermi liquid. We propose experiments to use these unique spectral properties as a new probe of the fractional quantum Hall effect.

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.

Tunneling spectroscopy of localized states near the quantum Hall edge

In the paper we dscuss experimental results of M. Grayson et al. on tunneling $I$-$V$ characteristics of the quantum Hall edge. We suggest a two step tunneling mechanism involving localized electron states near the edge, which might account for discrepancy between the experimental data and the predictions of the chiral Luttinger liquid theory of the quantum Hall edge.

Bound-state instability of the chiral Luttinger liquid in one dimension

We have developed a new boot-strap method for solving a class of interacting one-dimensional chiral fermions. The conventional model for interacting right-moving electrons with spin has an SO(4) symmetry, and can be written as four interacting Majorana fermions, each with the same velocity. We have found a method for solving some cases when the velocities of these Majorana fermions are no longer equal. We demonstrate in some detail the remarkable result that corrections to the non skeleton self-energy identically vanish for these models, and this enables us to solve them exactly. For the cases where the model can be solved by bosonization, our method can be explicitly checked. However, we are also able to solve some new cases where the excitation spectrum differs qualitatively from a Luttinger liquid. Of particular interest, is the so-called SO(3) model, where a triplet of Majorana fermions moving at one velocity, interact with a single Majorana fermion moving at another velocity. We show using our method, that a sharp bound (or anti-bound) state splits off from the original Luttinger liquid continuum, cutting off the X-ray singularity to form a broad incoherent excitation with a lifetime that grows linearly with frequency.

Quantum Hall effect today

I present a brief survey of important recent developments in the quantum Hall effect. The review covers both fractional and integer regimes, from an experimentalist's perspective. The topics include direct measurements of fractional charge, composite fermion Fermi surface, spin textures, and edge state (chiral Luttinger liquid) dynamics.

From composite fermions to the Calogero-Sutherland model: Edge of the fractional quantum Hall liquid and the dimension reduction

We derive a microscopic model describing the low-lying edge excitations in the fractional quantum Hall liquid with $\nu=\frac{\nu^*} {\tilde\phi\nu^*+1}$. For $\nu^*>0$, it is found that the composite fermion model reduces to an SU$(\nu^*)$ Calogero-Sutherland model in a dimension reduction, whereas it is not exact soluble for $\nu^*<0$. However, the ground states in both cases can be found and the low-lying excitations can be shown the chiral Luttinger liquid behaviors. On the other hand, we shows that the finite temperature behavior of $G-T$ curve will deviate from the prediction of the chiral Luttinger liquid. We also point out that the suppression of the `spin' degrees of freedom agrees with very recent experiments by Chang et al. The two-boson model of Lee and Wen is described microscopically.

Shot noise and the Luttinger liquid-like properties of the FQHE

We present shot noise measurements in the Fractional Quantum Hall regime at ν= 1 3 . The two fractional edge channels which carry the current at the two opposite boundaries of a quantum Hall sample are weakly coupled by an artificial impurity, namely a Quantum Point Contact. This coupling gives rise to a current between the two edge channels and the random transfer of the charges gives rise to fluctuations of the current. For a weak coupling and for large voltages, recent experiments have established that the current noise power S I is proportional to the backscattered current I B and to the fractional charge e/3. For low voltages at low temperature, the chiral Luttinger liquid models suggest a renormalization of the coupling strength leading to strong backscattering even when the bare coupling is weak. These models predict a noise power proportional to the transmitted current I and to the electronic charge e. The new noise measurements presented here confirm the shot noise predictions at moderately low temperatures. However, at very low temperature, measurements show a surprising enhancement of the shot noise which is nearly doubled.

Breakdown of the Chiral Luttinger Liquid in One Dimension

We have developed a fermionic boot-strap method to solve a class of chiral one-dimensional fermion models. Using this scheme, we show that Luttinger liquid behavior in a gas of four interacting chiral Majorana fermions is highly sensitive to the velocity degeneracy. Upon changing the velocity of one chiral fermion, a sharp bound (or antibound) state splits off from the original Luttinger liquid continuum, cutting off the x-ray singularity to form a broad incoherent excitation with a lifetime that grows linearly with frequency.

Trial wave functions for quantum Hall edge states

Trial wave functions for Laughlin's liquids in the presence of confining potentials are considered. They lead to a description of fractional quantum Hall edge states as a prototype of the chiral Luttinger liquid. Their reliability is confirmed by Monte Carlo simulation of the equivalent classical plasma.

An apparent [formula omitted] chiral Luttinger liquid at the edge of the [formula omitted] compressible composite Fermion liquid

The conductance ( G) and current–voltage ( I− V) characteristics are investigated for electron tunneling into the edge of the ν= 1 2 gapless quantum Hall composite Fermion liquid. A non-linear, power law I– V and a power-law temperature dependence in G are observed. We deduce an I– V exponent of 1.91 (∼2) and a corresponding tunneling conductance temperature exponent of 0.77 (∼1). These values are surprisingly close to those expected for a one-dimensional chiral Luttinger liquid with an intrinsic dimensionless conductance of g= 1 2 .

The Microscopic Model of Composite Fermion-Type Excitations for ν = 1/m Edge States**The project supported in part by National Natural Science Foundation of China and the National PanDeng (Climbing-Up) Plan in China

We derive a microscopic theory of the composite fermion-type quasiparticles describing the low-lying edge excitations in the fractional quantum Hall liquid with ν = 1/m. Using the composite fermion transformation, one finds that the edge states of the system in a disc sample are described by the Calogero–Sutherland-like model (CSLM) in the one-dimensional limit. This result presents the consistency between one-dimensional and two-dimensional statistics. It is shown that the low-lying excitations, indeed, have the chiral Luttinger liquid behaviors because there is a gap between the right-and left-moving excitations of the CSLM.

Continuum of Chiral Luttinger Liquids at the Fractional Quantum Hall Edge

We study current versus voltage $(I\ensuremath{-}V)$ when tunneling into the edge of the fractional quantum Hall effect over a continuum of filling factors $(\ensuremath{\nu})$ from 1/4 to 1. Our devices manifest the power law $I\ensuremath{-}V$ behavior previously observed by Chang et al. at discrete fillings, but now with as many as six decades in current and over the whole range of filling factor suggesting the existence of a continuum of chiral Luttinger liquids. Surprisingly the exponent behaves approximately as $1/\ensuremath{\nu}$ and does not exhibit the strong plateau features predicted in recent theoretical works based on the intermixing of copropagating and counterpropagating multiple edge modes.

Tunneling into the Edge of a Compressible Quantum Hall State

We present a composite-fermion theory of tunneling into the edge of a compressible quantum Hall system. The tunneling conductance is non-Ohmic, due to slow relaxation of electromagnetic and Chern-Simons field disturbances caused by the tunneling electron. Universal results are obtained in the limit of a large number of channels involved in the relaxation. The tunneling exponent is found to be a continuous function of the Hall resistivity ${\ensuremath{\rho}}_{\mathrm{xy}}$, with a slope that is discontinuous at filling factor $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$, in the limit of vanishing bulk resistivity ${\ensuremath{\rho}}_{\mathrm{xx}}$. When $\ensuremath{\nu}$ corresponds to a principal fractional quantized Hall state, our results agree with the chiral Luttinger liquid theories of Wen and Kane, Fisher, and Polchinski.

The quantum Hall effect: Unified scaling theory and quasiparticles at the edge

We address two fundamental issues in the physics of the quantum Hall effect: a unified description of scaling behavior of conductances in the integral and fractional regimes, and a quasi-particle formulation of the chiral Luttinger Liquids that describe the dynamics of edge excitations in the fractional regime.

Aharonov-Bohm effect in the chiral Luttinger liquid

Edge states of the quantum Hall fluid provide an almost unparalled opportunity to study mesoscopic effects in a highly correlated electron system. In this paper we develop a bosonization formalism for the finite-size edge state, as described by chiral Luttinger liquid theory, and use it to study the Aharonov-Bohm effect. The problem we address may be realized experimentally by measuring the tunneling current between two edge states through a third edge state formed around an antidot in the fractional quantum Hall effect regime. A renormalization group analysis reveals the existence of a two-parameter universal scaling function G(X,Y) that describes the Aharonov-Bohm resonances. We also show that the strong renormalization of the tunneling amplitudes that couple the antidot to the incident edge states, together with the nature of the Aharonov-Bohm interference process in a chiral system, prevent the occurrence of perfect resonances as the magnetic field is varied, even at zero temperature.

The Microscopic Model of Composite Particles for ν=1/m FQHE Edge Excitations, its Bosonization and the Chiral Condition

We derive a microscopic theory of the composite particles describing the low-lying edge excitations in the fractional quantum Hall liquid with ν=1/m. Using the composite-particle transformation, one finds that the edge states of the system in a disc sample are described by the Calogero–Sutherland model with other interactions between the composite particles as perturbations. The effects of both the interactions and the confinement potential are considered, and proved unable to change the exponent g=ν=1/m. We also obtain the important chiral condition at the mean field level, and as a result justify microscopically the Chiral Luttinger Liquid model of the FQHE edge states.

Exact Ensemble Average of the Random Perturbation to the Random Fixed Point

Random scatterings among N neutral modes of the chiral Luttinger liquid can be absorbed into the random S U ( N ) rotations of local fields. Perturbation to the so-called random fixed point must be treated as the random perturbation. This random perturbation is characterized by random matrices with fixed eigenvalues. In this article the ensemble average of the random fixed point is obtained exactly. In the sequel, it is shown that simplifications which assume the two-sided S U ( N ) invariance of the random rotation lead to incorrect consequences.

Distinct universal conductances in tunneling to quantum Hall states: The role of contacts

We show that different universal values can be obtained for the two-terminal conductance of a fractional quantum Hall (FQH) state. At large voltages, or strong coupling, the conductance of a pointlike tunneling junction between an electron gas reservoir and a Laughlin FQH state at filling fraction $\ensuremath{\nu}$ saturates to a universal value $G=[2\ensuremath{\nu}/(\ensuremath{\nu}+1)]{e}^{2}/h$. We use this result to show that devices with different types of contacts between the reservoir and the FQH state lead to distinct universal values of saturation conductance that are rational multiples of ${e}^{2}/h$. The particular fraction ${e}^{2}/h$ is obtained for the case of electron tunneling in and out of a FQH liquid through two point contacts. We demonstrate that the problem of tunneling between an electron gas and a FQH state through an impurity is exactly equivalent to the problem of tunneling between a chiral Fermi liquid and a chiral Luttinger liquid. We investigate in detail the case of tunneling to a \ensuremath{\nu}=$\frac{1}{3}$ FQH state, which we show to be equivalent to the problem of tunneling between two $g$=$\frac{1}{2}$ chiral Luttinger liquids. This system provides an experimental realization of this important exactly solvable case. We use the results of the single impurity problem to consider the case of many tunneling centers coupled independently to an electron reservoir, which is relevant to recent experiments by A. Chang et al. We derive an explicit universal expression for the voltage and temperature-dependent conductance that exhibits a crossover reminiscent of a Kondo effect. This universal curve fits the experimental data over the full range of probed voltages.