arXiv:1408.4914 · Physical Review B

Transport properties of a two-lead Luttinger-liquid junction out of equilibrium: Fermionic representation

D. N. Aristov, P. Wölfle

Published Dec 8, 2014• 11 citations

Abstract

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 $V$, if $V$ is the largest energy scale compared to temperature $T$ and inverse wire length ${L}^{\ensuremath{-}1}$, and interpolates between these quantities in the crossover regimes.

DOI: 10.1103/physrevb.90.245414

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arXiv:1408.4914 · Physical Review B

Transport properties of a two-lead Luttinger-liquid junction out of equilibrium: Fermionic representation

D. N. Aristov, P. Wölfle

Published Dec 8, 2014• 11 citations

arXiv HTML PDF

Abstract

\ensuremath{\Lambda}

, which equals the bias voltage

V

, if

V

is the largest energy scale compared to temperature

T

and inverse wire length

{L}^{\ensuremath{-}1}

, and interpolates between these quantities in the crossover regimes.

DOI: 10.1103/physrevb.90.245414

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arXiv:1304.5167Physical Review BAug 19, 201325 citations

\ensuremath{\Lambda}

, which equals the bias voltage

V

, if

V

is the largest energy scale compared to temperature

T

and inverse wire length

{L}^{\ensuremath{-}1}

, and interpolates between these quantities in the crossover regimes.

We develop the renormalization group theory of the conductances of

N

-lead junctions of spinless Luttinger-liquid wires as functions of bias voltages applied to

N

independent Fermi-liquid reservoirs. Based on the perturbative results up to second order in the interaction we demonstrate that the conductances obey scaling. The corresponding renormalization group $\ensuremath{\beta}$ functions are derived up to second order.

T

. Specifically, unlike in the two-terminal setup, the nonlinear conductances cannot be generically obtained from the linear ones by simply replacing

T

with the ``corresponding'' bias voltage or the largest one. Remarkably, a finite tunneling bias voltage prevents the interaction-induced breakup of the main wire into two disconnected pieces in the limit of zero

T

and a zero source-drain voltage.

The renormalization of conductances in a

Y

junction of spinless Luttinger-liquid wires additionally coupled to acoustic longitudinal phonons is investigated in fermionic representation. This system corresponds to geometry of a tunneling experiment and exhibits the interplay between the Coulomb repulsion and the attractive retarded interaction mediated by phonons. The retardation effects related to the propagation of phonons through the junction with arbitrary transmission and reflection amplitudes are taken into account. The appearing logarithmic corrections to conductances of the junction are treated in a renormalization group approach, and scaling exponents are calculated up to infinite order in the interaction after RPA-type summation. The fixed points and corresponding scaling exponents are considered in various nonequilibrium regimes. We show that the boundary exponent and the bulk anomalous dimension of fermion operator are characterized by two different Luttinger parameters, referring to the main wire, thanks to nonlocal character of phonon-mediated interaction. In the limiting case of the junction of only two wires, the scaling exponents found by our method are in exact correspondence with previous bosonization analysis.

g,{g}_{3}>0

) there is only one stable fixed point, corresponding to the complete separation of the wires. For attractive interaction (

g<0

and/or

{g}_{3}<0

) four fixed points are found, the stability of which depends on the interaction strength. We confirm our previous weak-coupling result of lines of fixed points for special values of the interaction parameters reaching into the strong-coupling domain. We find new fixed points not discussed before, even at the symmetric line

g={g}_{3}

, at variance with the results of M. Oshikawa et al. [J. Stat. Mech. (2006) P02008]. The pair-tunneling phenomenon conjectured by the latter authors is not found by us.