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arXiv:1002.0206 · Physical Review B

Off-critical Luttinger junctions

Brando Bellazzini, Mihail Mintchev, P. Sorba

Published Nov 10, 2010• 12 citations
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Abstract

We investigate Luttinger junctions of quantum wires away from criticality. The one-body scattering matrix, corresponding to the off-critical boundary conditions at the junction, admits, in general, antibound and/or bound states. Their contribution to the theory is fixed by causality. The presence/absence of bound states determines the existence of two different regimes with inequivalent physical properties. A scattering matrix without bound states defines an isolated equilibrium system. Bound states instead drive the system away from equilibrium, giving raise to nontrivial incoming or outgoing energy flows in the junction. We derive in both regimes and in explicit form the electromagnetic conductance tensor, pointing out the different impacts of bound and antibound states.

DOI
10.1103/physrevb.82.195113

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We investigate Luttinger junctions of quantum wires away from criticality. The one-body scattering matrix, corresponding to the off-critical boundary conditions at the junction, admits, in general, antibound and/or bound states. Their contribution to the theory is fixed by causality. The presence/absence of bound states determines the existence of two different regimes with inequivalent physical properties. A scattering matrix without bound states defines an isolated equilibrium system. Bound states instead drive the system away from equilibrium, giving raise to nontrivial incoming or outgoing energy flows in the junction. We derive in both regimes and in explicit form the electromagnetic conductance tensor, pointing out the different impacts of bound and antibound states.
Within the framework of boundary conformal field theory, we evaluate the conductance of stable fixed points of junctions of two and three quantum wires with different Luttinger parameters. For two wires, the physical properties are governed by a single effective Luttinger parameter for each of the charge and spin sectors. We present numerical density-matrix-renormalization-group calculations of the conductance of a junction of two chains of interacting spinless fermions with different interaction strengths, obtained using a recently developed method [Phys. Rev. Lett. 105, 226803 (2010)]. The numerical results show very good agreement with the analytical predictions. For three spinless wires (i.e., a Y junction) we analytically determine the full phase diagram and compute all fixed-point conductances as a function of the three Luttinger parameters.
We propose and investigate an exactly solvable model of non-equilibrium Luttinger liquid on a star graph, modeling a multi-terminal quantum wire junction. The boundary condition at the junction is fixed by an orthogonal matrix , which describes the splitting of the electric current among the leads. The system is driven away from equilibrium by connecting the leads to heat baths at different temperatures and chemical potentials. The associated non-equilibrium steady state depends on and is explicitly constructed. In this context, we develop a non-equilibrium bosonization procedure and compute some basic correlation functions. Luttinger liquids with general anyon statistics are considered. The relative momentum distribution away from equilibrium turns out to be the convolution of equilibrium anyon distributions at different temperatures. Both the charge and heat transport are studied. The exact current–current correlation function is derived and the zero-frequency noise power is determined.
We propose and investigate an exactly solvable model of non-equilibriumLuttinger liquid on a star graph, modeling a multi-terminal quantum wirejunction. The boundary condition at the junction is fixed by an orthogonalmatrix S, which describes the splitting of the electric current among theleads. The system is driven away from equilibrium by connecting the leads toheat baths at different temperatures and chemical potentials. The associatednon-equilibrium steady state depends on S and is explicitly constructed. Inthis context we develop a non-equilibrium bosonization procedure and computesome basic correlation functions. Luttinger liquids with general anyonstatistics are considered. The relative momentum distribution away fromequilibrium turns out to be the convolution of equilibrium anyon distributionsat different temperatures. Both the charge and heat transport are studied. Theexact current-current correlation function is derived and the zero-frequencynoise power is determined.
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.
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.