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arXiv:cond-mat/0311563 · Physical Review B

Renormalization group study of the conductances of interacting quantum wire systems with different geometries

Sourin Das, Sumathi Rao, Diptiman Sen

Published Aug 30, 2004• 39 citations
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Abstract

We examine the effect of interactions between the electrons on the Landauer-B\"uttiker conductances of some systems of quantum wires with different geometries. The systems include a long wire with a stub in the middle, a long wire containing a ring which can enclose a magnetic flux, and a system of four long wires which are connected in the middle through a fifth wire. Each of the wires is taken to be a weakly interacting Tomonaga-Luttinger liquid, and scattering matrices are introduced at all the junctions present in the systems. Using a renormalization group method developed recently for studying the flow of scattering matrices for interacting systems in one dimension, we compute the conductances of these systems as functions of the temperature and the wire lengths. We present results for all three regimes of interest, namely, high, intermediate, and low temperature. These correspond, respectively, to the thermal coherence length being smaller than, comparable to, and larger than the smallest wire length in the different systems, i.e., the lengths of the stub or each arm of the ring or the fifth wire. The renormalization group procedure and the formulas used to compute the conductances are different in the three regimes. In particular, the dimensionality of the scattering matrix effectively changes when the thermal length becomes larger than the smallest wire length. We also present a phenomenologically motivated formalism for studying the conductances in the intermediate regime where there is only partial coherence. At low temperatures, we study the line shapes of the conductances versus the energy of the electrons near some of the resonances; the widths of the resonances are found to go to zero with decreasing temperature. Our results show that the Landauer-B\"uttiker conductances of various systems of experimental interest depend on the temperature and lengths in a non-trivial way when interactions are taken into account.

DOI
10.1103/physrevb.70.085318

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We examine the effect of interactions between the electrons on the Landauer-B"uttiker conductances of some systems of quantum wires with different geometries. The systems include a long wire with a stub in the middle, a long wire containing a ring which can enclose a magnetic flux, and a system of four long wires which are connected in the middle through a fifth wire. Each of the wires is taken to be a weakly interacting Tomonaga-Luttinger liquid, and scattering matrices are introduced at all the junctions present in the systems. Using a renormalization group method developed recently for studying the flow of scattering matrices for interacting systems in one dimension, we compute the conductances of these systems as functions of the temperature and the wire lengths. We present results for all three regimes of interest, namely, high, intermediate, and low temperature. These correspond, respectively, to the thermal coherence length being smaller than, comparable to, and larger than the smallest wire length in the different systems, i.e., the lengths of the stub or each arm of the ring or the fifth wire. The renormalization group procedure and the formulas used to compute the conductances are different in the three regimes. In particular, the dimensionality of the scattering matrix effectively changes when the thermal length becomes larger than the smallest wire length. We also present a phenomenologically motivated formalism for studying the conductances in the intermediate regime where there is only partial coherence. At low temperatures, we study the line shapes of the conductances versus the energy of the electrons near some of the resonances; the widths of the resonances are found to go to zero with decreasing temperature. Our results show that the Landauer-B"uttiker conductances of various systems of experimental interest depend on the temperature and lengths in a non-trivial way when interactions are taken into account.
In this thesis, we calculate the linear dc conductance of two types of multi-terminal interacting systems: junctions of interacting quantum wires attached to Tomonaga-Luttinger liquid (TLL) leads, and closed and open long Aharonov-Bohm-Kondo (ABK) rings. In both cases, we obtain corrections to the non-interacting Landauer formula, arising from interactions in the TLL leads and the quantum dot in the Kondo regime respectively. In junctions of interacting quantum wires, if the wires are attached to Fermi liquid (FL) leads, the conductance is formally given by the Landauer formula with renormalized single-particle S-matrix elements. If, however, the wires are attached to TLL leads, i.e. the interaction does not vanish even in the leads, the conductance has an additional contribution dependent on the interaction strength in the leads. We calculate this additional contribution both at the first order in interaction and in the random phase approximation, and heuristically relate the FL conductance to the TLL conductance through a "contact resistance" between an FL lead and a TLL wire. In long ABK rings, where the interaction is due to spin-flip scattering at a quantum dot in the Kondo regime, the linear dc conductance consists of two parts: a disconnected part of the Landauer form, and a connected part that can be approximately eliminated at low temperatures. For a closed long ABK ring, where the electric current is conserved in the ring, the high-temperature conductance has qualitatively different behaviors for temperatures greater than and lower than the characteristic energy scale v_F /L, where v_F is the Fermi velocity and L is the ring circumference. Meanwhile, for an open long ABK ring where electrons may leak into the side leads coupled to ring arms, as long as the ring arms have both small transmission and small reflection, the ring behaves as a two-path Aharonov-Bohm interferometer, and we predict the observation of a π/2 phase shift due to scattering off the Kondo singlet formed at low energies around the impurity spin.
We use the technique of bosonization to understand a variety of recent experimental results on the conductivity of a quantum wire. The quantum wire is taken to be a finite-length Luttinger liquid connected on two sides to semi-infinite Fermi liquids through contacts. The contacts are modeled as (short) Luttinger liquids bounded by localized one-body potentials. We use effective actions and the renormalization group to study the effects of electronic interactions within the wire, the length of the wire, finite temperature, and a magnetic field, on the conductivity. We explain the deviations of the conductivity from 2Ne2/h{2Ne}^{2}/h2Ne2/h in wires that are not too short as arising from renormalization effects caused by the repulsive interactions. We also explain the universal conductance corrections observed in different channels at higher temperatures. We study the effects of an external magnetic field on electronic transport through this system and explain why odd and even spin-split bands show different renormalizations from the universal conductance values. We discuss the case of resonant transmission and of the possibility of producing a spin valve that only allows electrons of one value of the spin to go through. We compare our results for the conductance corrections with experimental observations. We also propose an experimental test of our model of the contact regions.
We study a system consisting of a junction of NNN wires, where the junction is characterized by a scalar SSS-matrix, and an impurity spin is coupled to the electrons near the junction. The wires are modeled as weakly interacting Tomonaga-Luttinger liquids. We derive the renormalization group (RG) equations for the Kondo couplings of the spin to the electrons on different wires. We analyze the RG flows and fixed points for different values of the initial Kondo couplings and of the junction SSS-matrix, such as the decoupled SSS-matrix and the Griffiths (connected) SSS-matrix. We find that the Kondo couplings flow either towards a ferromagnetic (FM) fixed point or towards large and antiferromagnetic (AFM) values in one of two ways. For the Griffiths SSS-matrix, one of the strong coupling flows is towards a FM fixed point with decoupled wires; this is seen by a perturbative analysis. Thus if we start with a system of connected wires with an AFM coupling to the spin impurity, the flow at large distances is towards a system of disconnected wires at the FM fixed point. For the decoupled SSS-matrix, the flow is either to a FM fixed point or to one of two strong coupling fixed points in which all the channels are strongly coupled to each other through the impurity spin. Strong interactions between the electrons with ${K}_{\ensuremath{\rho}}<N∕(N+2)$ can stabilize a multichannel fixed point in which the coupling between different channels goes to zero. We also study the temperature dependence of the conductance at the decoupled and Griffiths SSS-matrices.
We study the conductance of three or more semi-infinite wires which meet at a junction. The electrons in the wires are taken to interact weakly with each other through a short-range density-density interaction, and they encounter a general scattering matrix at the junction. We derive the renormalization-group equations satisfied by the S-matrix, and we identify its fixed points and their stabilities. The conductance between any pair of wires is then studied as a function of physical parameters such as temperature. We discuss the possibility of observing the effects of junctions in present day experiments, such as the four-terminal conductance of a quantum wire and crossed quantum wires.
We study the effect that resistive regions have on the conductance of a quantum wire with interacting electrons which is connected to Fermi liquid leads. Using the bosonization formalism and a Rayleigh dissipation function to model the power dissipation, we use both scattering theory and Green's function techniques to derive the DC conductance. The resistive regions are generally found to lead to incoherent transport. For a single wire, we find that the resistance adds in series to the contact resistance of h/e(2) for spinless electrons, and the total resistance is independent of the Luttinger parameter K-W of the wire. We numerically solve the bosonic equations to illustrate what happens when a charge density pulse is incident on the wire; the results depend on the parameters of the resistive and interacting regions in interesting ways. For a junction of Tomonaga-Luttinger liquid wires, we use a dissipationless current splitting matrix to model the junction. For a junction of three wires connected to Fermi liquid leads, there are two families of such matrices; we find that the conductance matrix generally depends on K-W for one family but is independent of K-W for the other family, regardless of the resistances present in the system. Copyright (c) EPLA, 2011
In a recent paper, we combined the technique of bosonization with the concept of a Rayleigh dissipation function to develop a model for resistances in one-dimensional systems of interacting spinless electrons [Europhys. Lett. 93, 57007 (2011)]. We also studied the conductance of a system of three wires by using a current splitting matrix MMM at the junction. In this paper, we extend our earlier work in several ways. The power dissipated in a three-wire system is calculated as a function of MMM and the voltages applied in the leads. By combining two junctions of three wires, we examine a system consisting of two parallel resistances. We study the conductance of this system as a function of the MMM matrices and the two resistances; we find that the total resistance is generally quite different from what one expects for a classical system of parallel resistances. We do a sum over paths to compute the conductance of this system when one of the two resistances is taken to be infinitely large. We study the conductance of a three-wire system of interacting spin-1$/$2 electrons, and show that the charge and spin conductances can generally be different from each other. Finally, we consider a system of two wires that are coupled by a dissipation function, and we show that this leads to a current in one wire when a voltage bias is applied across the other wire.