From Tomonaga - Luttinger to Fermi liquid in transport through a tunneling barrier

Abstract

Finite length of a one channel wire results in crossover from a TomonagaLuttinger to Fermi liquid behavior with lowering energy scale. In condition that voltage drop (V ) mostly occurs across a tunnel barrier inside the wire we found coefficients of temperature/voltage expansion of low energy conductance as a function of constant of interaction, right and left traversal times. At higher voltage the finite length contribution exhibits oscillations related to both traversal times and becomes a slowly decaying correction to the scaleinvariant V 1/g−1 dependence of the conductance. 72.10.Bg, 72.15.-v, 73.20.Dx Typeset using REVTEX 1 Quantum transport in Tomonaga-Luttinger liquids (TLL) has attracted a great deal of interest as it was suggested to be realized in a 1D constriction [1,2] and the edge state of the fractional quantum Hall (FQH) liquid [3]. Both suggestions have been supported in recent experiments on FQH [4] and on 1D [5] transport. Even more experimentalists [6] made claim on observation of the interaction effects in the 1D transport, however, without comprehensible connection with a theoretical model. Unless there is a resonant tunneling the repulsive interaction typically suppresses conductance at low energy. The experiment by Tarucha’s group [5] on 1D transport through a long wire with a weak impurity random potential inside demonstrated a crossover from TL liquid to Fermi liquid behavior at low temperature. This crossover is a finite length effect [7] and may be described in an inhomogeneous TLL model (ITLL) [8–10]. The ITLL model predicts both the conductance behavior in the Fermi liquid region up to a renormalization constant (i. e., relations between the coefficients at temperature/voltage in different integer degrees) and the interaction dependent non-analytical behavior in the TLL region. The aim of this work is to examine this crossover in the low voltage conductance of the one channel wire where its suppression is mostly determined by a high point barrier located inside the wire. Position and height of the barrier are assumed to be due to an external gate. Therefore both distances LR(L) from the barrier to the right/left reservoir and the ratio ζ = LR/(LR + LL) are assumed to be known. Then it will be shown below that just two more parameters, which are the total traversal time tL equal to sum of the right and left ones:t0 = tR + tL and the constant of the forward scattering g, are necessary for desciption of the crossover, and that the ITTL model gives quite a few ways to determine these parameters from either high voltage (V > 1/t0) or low voltage (V < 1/t0) conductance measurements. Temperature dependence of the conductance in this model was considered in [11]. Similar model turned out to be useful for experimental study of the FQH transport [4]. It was noticed recently [12] that there is a special way of connection between a ν = 1/3 FQH liquid and leads when the whole setup corresponds to the g = 1/3 ITLL model. If so, the results obtained below for spinless electrons could be directly addressed to that FQH 2 device when the FQH liquid embeds a point scatterer. Under condition that the link between two parts of the wire is weak enough it suffices to apply the tunneling Hamiltonian approach in the lowest order to describe the transport [2,13,14]. Current flowing through the weak link located, say, at x = 0 inside the wire is given by the operator J(t) = −i[Aψ R(0, t)ψL(0, t)−h.c.], (e, h̄ = 1), where A is the tunneling amplitude and ψR,L(x, t) are the electron annihilation operators in the right 0 ≤ x < LR and in the left −LL < x ≤ 0 part of the wire,respectively. The average current under voltage V applied to the left lead can be written as: < J >= 2π|A|2 ∫ dǫ[f(ǫ−V )−f(ǫ)]ρR(ǫ)ρL(ǫ−V ), where f is Fermi distribution. The problem reduces to finding of the tunneling density of states of the right (left) end of the junction ρR(L)(ǫ) which are the sum of the particle ρp(ǫ) = (1− f(ǫ))ρ(ǫ) and hole ρh(ǫ) = f(ǫ)ρ(ǫ) densities. The latters relate to the particle correlator as ρp(ǫ) = 1/(2π) ∫ dte < ψ(0, t)ψ(0, 0) > and to the hole one as ρh(ǫ) = 1/(2π) ∫ dte < ψ(0, 0)ψ(0, t) >. Tunneling density of states To calculate the tunneling density of states ρR on the right side of the weak link let us first consider spinless fermions and apply bosonization to the ψ field under condition of an elastic reflection from the boundary located at x = 0 [15,16]. ( Carrying out this calculations we will omit index ”R” below. ) Bosonic repersentation of the ψ field reads ψ(x, t) = ∑ a=r,l ψa(x, t) = (2πα) −1 ∑ ± exp{i(θ(x, t) ± φ(x, t))/2}, where ψr(l) is the right (left) going chiral component of ψ and the θ and φ fields are bosonic and mutually conjugated [θ(x, t), φ(y, t)] = 2πisgn(x − y). The elastic reflection means that ψl(0, t) = e ψr(0, t) with an appropiate phase shift δ. This results in both: φ(0, t) = δ, 1 2π ∂xθ(x, t)|x=0 = ψ r (0, t)ψr(0, t)− ψ l (0, t)ψl(0, t) = 0. (1) Then the density of particle states could be found as ρp(ǫ) = ρOEF 2π ∫ +∞ −∞ dte 1 4 [<θ(0,t)θ(0,0)>−<θ(0,0)>] (2) where the value of the free electron tuneling density was introduced as: ρO = (1+cos δ)/(πv). The problem reduces to finding the θ field correlator. It can be done for the finite length piece of the wire adiabatically connected to the lead making use of the ITTL model [8–10]. In 3 this model the Tomonaga-Luttinger interaction ( ∑ r,l ρa) 2 is switched on in the wire x < LR and switched off outside. Then the Hamiltonian takes a bosonized form H = ∫ ∞ 0 dx v 2 {u(x) (

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Cite this paper

@inproceedings{Ponomarenko2008FromT, title={From Tomonaga - Luttinger to Fermi liquid in transport through a tunneling barrier}, author={Vadim V Ponomarenko and Naoto Nagaosa}, year={2008} }