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arXiv:cond-mat/0210592 · Workshop on Integrable Theories, Solitons and Duality

Conductance from Non-perturbative Methods II

Olalla A. Castro-Alvaredo, Andreas Fring

Published Dec 1, 2002• 0 citations
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Abstract

This talk provides a natural continuation of the talk presented by Andreas Fring in this conference.Part I was focused on explaining how the DC conductance for a free Fermion theory in the presence of different kinds of defects can be computed by evaluating the Kubo formula.In this talk I will focus on an alternative method for the computation of the same quantity, that is the evaluation of Landauer formula.Once again, the integrability of the theories under consideration will be exploited, since a thermodynamic Bethe ansatz analysis provides all the input needed in that case, apart from the corresponding reflection and transmition amplitudes of the defect.The basic conclusion of our analysis will be the perfect agreement between the two different theoretical descriptions mentioned.The results I will talk about are contained on a series of papers [1,2,3,4,5,6], with emphasis on the first two, resulting mainly from a collaboration with Andreas Fring, who presented the first part of the work.

DOI
10.22323/1.008.0010

Related papers

Conductance from Non-perturbative Methods I

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We investigate different methods to compute the DC conductance in a quantum wire doped with some impuritied by exploiting the integrability of the theories under consideration.As an essential ingredient in all methods we evaluate the reflection and transmission amplitudes of the impurities for a variety of defects.When the impurities in the wire are coupled to an external three dimensional laser field, we predict the generation of harmonic emission spectra.We propose a modified version of the well-known Kubo formula, which incorporates the impurities of the system and evaluate the current-current two-point correlation function it involves with the help of a form factor expansion.A comparison with the corresponding quantities computed in a Landauer transport theory picture is carried out in part II.The work I want to report about is based on a series of papers [1,2,3,4,5,6] with an emphasis on the first two.Olalla Castro-Alvaredo will present the second part of this talk.

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This talk provides a natural continuation of the talk presented by Andreas Fring in this conference.Part I was focused on explaining how the DC conductance for a free Fermion theory in the presence of different kinds of defects can be computed by evaluating the Kubo formula.In this talk I will focus on an alternative method for the computation of the same quantity, that is the evaluation of Landauer formula.Once again, the integrability of the theories under consideration will be exploited, since a thermodynamic Bethe ansatz analysis provides all the input needed in that case, apart from the corresponding reflection and transmition amplitudes of the defect.The basic conclusion of our analysis will be the perfect agreement between the two different theoretical descriptions mentioned.The results I will talk about are contained on a series of papers [1,2,3,4,5,6], with emphasis on the first two, resulting mainly from a collaboration with Andreas Fring, who presented the first part of the work.
We investigate different methods to compute the DC conductance in a quantum wire doped with some impuritied by exploiting the integrability of the theories under consideration.As an essential ingredient in all methods we evaluate the reflection and transmission amplitudes of the impurities for a variety of defects.When the impurities in the wire are coupled to an external three dimensional laser field, we predict the generation of harmonic emission spectra.We propose a modified version of the well-known Kubo formula, which incorporates the impurities of the system and evaluate the current-current two-point correlation function it involves with the help of a form factor expansion.A comparison with the corresponding quantities computed in a Landauer transport theory picture is carried out in part II.The work I want to report about is based on a series of papers [1,2,3,4,5,6] with an emphasis on the first two.Olalla Castro-Alvaredo will present the second part of this talk.
We compute the DC conductance with two different methods, which both exploit the integrability of the theories under consideration. On one hand we determine the conductance through a defect by means of the thermodynamic Bethe ansatz and standard relativistic potential scattering theory based on a Landauer transport theory picture. On the other hand, we propose a Kubo formula for a defect system and evaluate the current-current two-point correlation function it involves with the help of a form factor expansion. For a variety of defects in a fermionic system we find excellent agreement between the two different theoretical descriptions.
We investigate different methods to compute the DC conductance in a quantum wire doped with some impuritied by exploiting the integrability of the theories under consideration. As an essential ingredient in all methods we evaluate the reflection and transmission amplitudes ofthe impurities for a variety of defects. When the impurities inthe wire are coupled to an external three dimensional laser field, we predict the generation of harmonic emission spectra. We propose a modified version of the well-known Kubo formula, which incorporates the impurities of the system and evaluate the current-current two-point correlation function it involves with the help of a form factor expansion. A comparison with the corresponding quantities computed in a Landauer transport theory picture is carried out in part II.
Interesting discoveries have recently been made in the area of solid state physics and nano technology regarding electronic systems that are confined to reduced spatial dimensions. Under these circumstances the quantum nature of the electrons begins to surface. In the one-dimensional case this class of systems is therefore called quantum- or nano wires. The conductance in such quantum wires becomes quantized and is dominated by electronic correlations. The goal of this thesis was to develop a fast and efficient procedure to describe a realistic transport measurement for such systems. The method presented here is in- spired by the work of Dan Bohr et al. who used the Density Matrix Renormalization Group algorithm(DMRG) to evaluate the Kubo formula for the linear response to an applied potential. In a similar way the procedure of this thesis also computes the Kubo formula using the dynamical DMRG in order to calculate the low frequency response to an AC source-drain voltage of finite systems. Making use of a special finite size scaling allows to extrapolate the results into the thermodynamic limit and to finally obtain the linear DC-conductance. First, the theoretical framework for the new method was developed. This was fol- lowed by testing the new approach in different model systems. The first model com- prised of a tight-binding chain that was filled with spinless, noninteracting fermions. In subsequent steps the model was expanded by introducing a homogeneous in- teraction in the Luttinger liquid regime followed by inhomogeneous interaction for different chain sections. The potential impact of impurities was assessed in the above models. Lastly, the two cases of the Hubbard chain of spin- 2 1 electrons and of the Holstein model in one dimension, where Einstein phonons interact with spinless fermions, were studied. The results of the proposed method are consistent with Luttinger liquid and Lan- dauer scattering theory. In the Luttinger liquid regime the influence of phonons to the charge structure factor was confirmed. A dependency on the length of the quantum wire in the form of dissipation effects for the conductance was not observed.
We improve the density-matrix renormalization group (DMRG) evaluation of the Kubo formula for the zero-temperature linear conductance of one-dimensional correlated systems. The dynamical DMRG is used to compute the linear response of a finite system to an applied ac source-drain voltage; then the low-frequency finite-system response is extrapolated to the thermodynamic limit to obtain the dc conductance of an infinite system. The method is demonstrated on the one-dimensional spinless fermion model at half filling. Our method is able to replicate several predictions of the Luttinger liquid theory such as the renormalization of the conductance in a homogeneous conductor, the universal effects of a single barrier, and the resonant tunneling through a double barrier.
We derive a Kubo-type formula that describes proper non-equilibrium stationary states for finite-size systems. We first argue that the usual Kubo formula considers only equilibrium states of the perturbed system, which are inappropriate to describe transport properties. Moreover, we show that the standard Kubo formula for the dc conductance is only appropriate in the thermodynamic limit. We demonstrate that taking into consideration explicitly the coupling to the leads/baths solves both problems. This approach results in well-behaved response functions, without the singular contributions from degenerate states encountered when Kubo formulae for infinite-size systems are inappropriately used for finite-size systems. We also derive a second, more efficient formulation which is valid only for a set of special physical quantities, which includes the charge current density operator.