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arXiv:0709.1562 · Europhysics Letters (EPL)

Transport of interacting electrons through a potential barrier: Nonperturbative RG approach

D. N. Aristov, P. Wölfle

Published Mar 18, 2008• 20 citations
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Abstract

We calculate the linear response conductance of electrons in a Luttinger liquid with arbitrary interaction g 2 , and subject to a potential barrier of arbitrary strength, as a function of temperature. We first map the Hamiltonian in the basis of scattering states into an effective low-energy Hamiltonian in current algebra form. Analyzing the perturbation theory in the fermionic representation the diagrams contributing to the renormalization group (RG) β-function are identified. A universal part of the β-function is given by a ladder series and summed to all orders in g 2 . First non-universal corrections beyond the ladder series are discussed. The RG-equation for the temperature-dependent conductance is solved analytically. Our result agrees with known limiting cases.

DOI
10.1209/0295-5075/82/27001

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We calculate the linear response conductance of electrons in a Luttinger liquid with arbitrary interaction g 2 , and subject to a potential barrier of arbitrary strength, as a function of temperature. We first map the Hamiltonian in the basis of scattering states into an effective low-energy Hamiltonian in current algebra form. Analyzing the perturbation theory in the fermionic representation the diagrams contributing to the renormalization group (RG) β-function are identified. A universal part of the β-function is given by a ladder series and summed to all orders in g 2 . First non-universal corrections beyond the ladder series are discussed. The RG-equation for the temperature-dependent conductance is solved analytically. Our result agrees with known limiting cases.
We calculate the linear response conductance of electrons in a Luttinger liquid with arbitrary interaction g2{g}_{2}g2​, and subject to a potential barrier of arbitrary strength, as a function of temperature. We map the Hamiltonian in the basis of scattering states into an effective low energy Hamiltonian in current algebra form. First the renormalization group (RG) equation for weak interaction is derived in the current operator language both using the operator product expansion and the equation of motion method. To access the strong coupling regime, two methods of deducing the RG equation from perturbation theory, based on the scaling hypothesis and on the Callan-Symanzik formulation, are discussed. The important role of scale-independent terms is emphasized. The latter depend on the regularization scheme used (length versus temperature cutoff). Analyzing the perturbation theory in the fermionic representation, the diagrams contributing to the renormalization group $\ensuremath{\beta}$-function are identified. A universal part of the $\ensuremath{\beta}$-function is given by a ladder series and summed to all orders in g2{g}_{2}g2​. First nonuniversal corrections beyond the ladder series are discussed and are shown to differ from the exact solutions obtained within conformal field theory which use a different regularization scheme. The RG equation for the temperature-dependent conductance is solved analytically. Our result agrees with known limiting cases.
We calculate the conductance of a system of two spinless Luttinger liquid wires with different interaction strengths g1 , g2 , connected through a short junction, within the scattering state formalism. Following earlier work we formulate the problem in current algebra language, and calculate the scale dependent contribution to the conductance in perturbation theory keeping the leading universal contributions to all orders in the interaction strength. From that we derive a renormalization group (RG) equation for the conductance. The analytical solution of the RG-equation is discussed in dependence on g1 , g2 . The regions of stability of the two fixed points corresponding to conductance G = 0 and G = 1,respectively, are determined.
We calculate the conductance of a system of two spinless Luttinger liquid wires with different interaction strengths g1 , g2 , connected through a short junction, within the scattering state formalism. Following earlier work we formulate the problem in current algebra language, and calculate the scale dependent contribution to the conductance in perturbation theory keeping the leading universal contributions to all orders in the interaction strength. From that we derive a renormalization group (RG) equation for the conductance. The analytical solution of the RG-equation is discussed in dependence on g1 , g2 . The regions of stability of the two fixed points corresponding to conductance G = 0 and G = 1,respectively, are determined.
The electrical current through an arbitrary junction connecting quantum wires of spinless interacting fermions is calculated in fermionic representation. The wires are adiabatically attached to two reservoirs at chemical potentials differing by the applied voltage bias. The relevant scale-dependent contributions in perturbation theory in the interaction up to infinite order are evaluated and summed up. The result allows one to construct renormalization group equations for the conductance as a function of voltage (or temperature, wire length). There are two fixed points at which the conductance follows a power law in terms of a scaling variable $\ensuremath{\Lambda}$, which equals the bias voltage VVV, if VVV is the largest energy scale compared to temperature TTT and inverse wire length ${L}^{\ensuremath{-}1}$, and interpolates between these quantities in the crossover regimes.
Scattering from a localized potential of arbitrary strength is found for weakly interacting electrons. Poor man's renormalization group procedure allows us to calculate the renormalized transmission amplitudes at any energies. Simple formulas that describe the conductance at any temperature are derived. In the presence of the electron-electron backscattering, the low-temperature conductance deviates from the results of Luttinger liquid theory. In particular, the temperature dependence of the conductance may become nonmonotonic. In the presence of a magnetic field, backscattering gives rise to a peak in the differential conductance at bias equal to the Zeeman splitting.
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.