arXiv:1105.5883 · Physical Review B

Transport properties of a Y junction connecting Luttinger liquid wires

D. N. Aristov, P. Wölfle

Published Oct 17, 2011• 31 citations

Abstract

We study the transport properties of three Luttinger liquid wires (with possibly different interaction strength), connected through a Y junction, within the scattering state formalism. We first formulate the problem in current algebra language and focus on the case of a symmetric setup, for which the scattering matrix and the matrix of conductances is parametrized by two variables. For these, we derive coupled renormalization group equations, first in a ladder summation up to infinite order in the interaction. The fixed-point structure and the implicit solution of these equations is presented. It is shown that higher-order terms beyond the ladder approximation do not change the scaling behavior near the fixed points. For sufficiently strong attractive interaction, a new fixed point with unusual properties is found.

DOI: 10.1103/physrevb.84.155426

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arXiv:1105.5883 · Physical Review B

Transport properties of a Y junction connecting Luttinger liquid wires

D. N. Aristov, P. Wölfle

Published Oct 17, 2011• 31 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevb.84.155426

Related papers

Transport through asymmetric two-lead junctions of Luttinger liquid wires

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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.

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Chiral Y junction of Luttinger liquid wires at strong coupling: Fermionic representation

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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.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.

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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.

g,{g}_{3}>0

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

g<0

and/or

{g}_{3}<0

) four fixed points are found, whose stability depends on the interaction strength. For special values of the interaction parameters (a)

g=0

{g}_{3}<0

or (b)

{g}_{3}+{g}^{2}/2=0

g<0

we find whole lines of stable fixed points. We conjecture that lines of fixed points appear generically at interfaces in the interaction coupling constant space separating two regions with different stable fixed points. We also find that in certain regions of the

g

{g}_{3}

plane the RG flow is towards a fixed point without chirality, implying that the effect of the magnetic flux is completely screened.

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 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 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

{g}_{2}

. 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.