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arXiv:cond-mat/9501134 · Physical review. B, Condensed matter

NonlinearI(V) characteristics of Luttinger liquids and gated Hall bars

Scot R. Renn, Daniel P. Arovas

Published Jun 15, 1995• 25 citations
arXivHTMLPDF

Abstract

Nonlinear current voltage characteristics of a disordered Luttinger liquid are calculated using a perturbative formalism. One finds the nonuniversal power-law characteristics of the form I(V)\ensuremath{\sim}${\mathit{V}}^{1/(2\mathit{g}\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}\mathrm{\ensuremath{-}}1)}$, which is valid both in the superfluid phase when I is small and also in the insulator phase when I is large. Mesoscopic voltage fluctuations are also calculated. One finds Var(\ensuremath{\Delta}V)\ensuremath{\sim}${\mathit{I}}^{4\mathit{g}\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}\mathrm{\ensuremath{-}}3}$. Both the I(V) characteristic and the voltage fluctuations exhibit universal power-law behavior at the superfluid insulator transition where g\ifmmode \tilde{}\else \~{}\fi{}=3/2. The possible application of these results to the nonlinear transport properties of gated Hall bars is discussed.

DOI
10.1103/physrevb.51.16832

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Nonlinear current voltage characteristics of a disordered Luttinger liquid are calculated using a perturbative formalism. One finds the nonuniversal power-law characteristics of the form I(V)\ensuremath{\sim}V1/(2g\ifmmode~\else˜\fi\ensuremath−1){\mathit{V}}^{1/(2\mathit{g}\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}\mathrm{\ensuremath{-}}1)}V1/(2g\ifmmode~\else˜\fi\ensuremath−1), which is valid both in the superfluid phase when I is small and also in the insulator phase when I is large. Mesoscopic voltage fluctuations are also calculated. One finds Var(\ensuremath{\Delta}V)\ensuremath{\sim}I4g\ifmmode~\else˜\fi\ensuremath−3{\mathit{I}}^{4\mathit{g}\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}\mathrm{\ensuremath{-}}3}I4g\ifmmode~\else˜\fi\ensuremath−3. Both the I(V) characteristic and the voltage fluctuations exhibit universal power-law behavior at the superfluid insulator transition where g\ifmmode \tilde{}\else ~{}\fi{}=3/2. The possible application of these results to the nonlinear transport properties of gated Hall bars is discussed.
The temperature-dependent nonlinear conductance for transport of a Luttinger liquid through a barrier is calculated in the nonperturbative regime for g=1/2-\ensuremath{\epsilon}, where g is the dimensionless interaction constant. To describe the low-energy behavior, we perform a leading-logarithmic summation of all diagrams contributing to the conductance which is valid for \ensuremath{\Vert}\ensuremath{\epsilon}\ensuremath{\Vert}\ensuremath{\ll}1. With increasing external voltage, the asymptotic low-temperature behavior displays a turnover from the ${\mathit{T}}^{2/\mathit{g}\mathrm{\ensuremath{-}}2}$ to a universal T2{\mathit{T}}^{2}T2 law.
We describe the relationship between quantum Hall edge states and the one-dimensional Luttinger liquid model. The Luttinger liquid model originated from studies of one-dimensional Fermi systems, however, it results that many ideas inspired by such a model can find applications to phenomena occurring even in higher dimensions. Quantum Hall systems which essentially are correlated two-dimensional electronic systems in a strong perpendicular magnetic field have an edge. It turns out that the quantum Hall edge states can be described by a one-dimensional Luttinger model. In this work, we give a general background of the quantum Hall and Luttinger liquid physics and then point out the relationship between the quantum Hall edge states and its one-dimensional Luttinger liquid representation. Such a description is very useful given that the Luttinger liquid model has the property that it can be bosonized and solved. The fact that we can introduce a simpler model of noninteracting bosons, even if the quantum Hall edge states of electrons are interacting, allows one to calculate exactly various quantities of interest. One such quantity is the correlation function which, in the asymptotic limit, is predicted to have a power law form. The Luttinger liquid model also suggests that such a power law exponent should have a universal value. A large number of experiments have found the quantum Hall edge states to show behavior consistent with a Luttinger liquid description. However, while a power law dependence of the correlation function has been observed, the experimental values of the exponent appear not to be universal. This discrepancy might be due to various correlation effects between electrons that sometimes are not easy to incorporate within a standard Luttinger liquid model.
We describe the relationship between quantum Hall edge states and the one-dimensional Luttinger liquid model. The Luttinger liquid model originated from studies of one-dimensional Fermi systems, however, it results that many ideas inspired by such a model can find applications to phenomena occurring even in higher dimensions. Quantum Hall systems which essentially are correlated two-dimensional electronic systems in a strong perpendicular magnetic field have an edge. It turns out that the quantum Hall edge states can be described by a one-dimensional Luttinger model. In this work, we give a general background of the quantum Hall and Luttinger liquid physics and then point out the relationship between the quantum Hall edge states and its one-dimensional Luttinger liquid representation. Such a description is very useful given that the Luttinger liquid model has the property that it can be bosonized and solved. The fact that we can introduce a simpler model of noninteracting bosons, even if the quantum Hall edge states of electrons are interacting, allows one to calculate exactly various quantities of interest. One such quantity is the correlation function which, in the asymptotic limit, is predicted to have a power law form. The Luttinger liquid model also suggests that such a power law exponent should have a universal value. A large number of experiments have found the quantum Hall edge states to show behavior consistent with a Luttinger liquid description. However, while a power law dependence of the correlation function has been observed, the experimental values of the exponent appear not to be universal. This discrepancy might be due to various correlation effects between electrons that sometimes are not easy to incorporate within a standard Luttinger liquid model.
The nonlinear transport in a Luttinger liquid with finite-range interaction is considered for a general time-dependent driving electric field, in the presence of an impurity. In the static limit, the nonlinear current-voltage characteristic depends only on the integral of the driving field. For a time-dependent field, a relation between the currents at different sites is derived which depends only on the nonlocal ac conductivity of the Luttinger system without impurity, but is independent of the strength and the shape of the electric field. Experimental consequences are discussed.
The nonlinear Luttinger liquid phenomenology of one-dimensional correlated Fermi systems is an attempt to describe the effect of the band curvature beyond the Tomonaga-Luttinger liquid paradigm. It relies on the observation that the dynamical structure factor of the interacting electron gas shows a logarithmic threshold singularity when evaluated to first order perturbation theory in the two-particle interaction. This term was interpreted as the linear one in an expansion which was conjectured to resum to a power law. A field theory, the mobile impurity model, which is constructed such that it provides the power law in the structure factor, was suggested to be the proper effective model and used to compute the single-particle spectral function. This forms the basis of the nonlinear Luttinger liquid phenomenology. Surprisingly, the second order perturbative contribution to the structure factor was so far not studied. We first close this gap and show that it is consistent with the conjectured power law. Secondly, we critically assess the steps leading to the mobile impurity Hamiltonian. We show that the model does not allow to include the effect of the momentum dependence of the (bulk) two-particle potential. This dependence was recently shown to spoil power laws in the single-particle spectral function which previously were believed to be part of the Tomonaga-Luttinger liquid universality. Although our second order results for the structure factor are consistent with power-law scaling, this raises doubts that the conjectured nonlinear Luttinger liquid phenomenology can be considered as universal. We conclude that more work is required to clarify this.
The nonlinear Luttinger liquid phenomenology of one-dimensional correlated Fermi systems is an attempt to describe the effect of the band curvature beyond the Tomonaga-Luttinger liquid paradigm. It relies on the observation that the dynamical structure factor of the interacting electron gas shows a logarithmic threshold singularity when evaluated to first order perturbation theory in the two-particle interaction. This term was interpreted as the linear one in an expansion which was conjectured to resum to a power law. A field theory, the mobile impurity model, which is constructed such that it provides the power law in the structure factor, was suggested to be the proper effective model and used to compute the single-particle spectral function. This forms the basis of the nonlinear Luttinger liquid phenomenology. Surprisingly, the second order perturbative contribution to the structure factor was so far not studied. We first close this gap and show that it is consistent with the conjectured power law. Secondly, we critically assess the steps leading to the mobile impurity Hamiltonian. We show that the model does not allow to include the effect of the momentum dependence of the (bulk) two-particle potential. This dependence was recently shown to spoil power laws in the single-particle spectral function which previously were believed to be part of the Tomonaga-Luttinger liquid universality. Although our second order results for the structure factor are consistent with power-law scaling, this raises doubts that the conjectured nonlinear Luttinger liquid phenomenology can be considered as universal. We conclude that more work is required to clarify this.