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Workshop on Integrable Theories, Solitons and Duality

Conductance from Non-perturbative Methods I

Andreas Fring, Olalla A. Castro-Alvaredo

Published Dec 1, 2002• 0 citations
Publisher

Abstract

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.

DOI
10.22323/1.008.0015

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Conductance from Non-perturbative Methods I

Olalla A. Castro-Alvaredo, Andreas Fring

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.

arXiv:cond-mat/0210599•Oct 28, 2002•0 citations

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Castro-Alvaredo, O A, Fring, A

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.

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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.

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The transport in a pure one-dimensional quantum wire is investigated for any range of interactions. First, the wire is connected to measuring leads. The transmission of an incident electron is found to be perfect, i.e., the conductance is not renormalized by the interactions. Either Landauer's approach or the Kubo formula can be used as long as the reservoirs impose the boundary conditions. Second, the Kubo formula as a response to the local field is reconsidered in a generic Luttinger liquid: the ``intrinsic'' conductance thus obtained is determined by the same combination of interaction parameters as that which renormalizes the current.

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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.
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.
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.
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.
The transport in a pure one-dimensional quantum wire is investigated for any range of interactions. First, the wire is connected to measuring leads. The transmission of an incident electron is found to be perfect, i.e., the conductance is not renormalized by the interactions. Either Landauer's approach or the Kubo formula can be used as long as the reservoirs impose the boundary conditions. Second, the Kubo formula as a response to the local field is reconsidered in a generic Luttinger liquid: the ``intrinsic'' conductance thus obtained is determined by the same combination of interaction parameters as that which renormalizes the current.
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 review some recent results from the mathematical theory of transport of charge and spin in gapped crystalline quantum systems. The emphasis will be on transport coefficients, such as conductivities and conductances. As for the former, those are computed as appropriate expectations of current operators in a non-equilibrium almost-stationary state (NEASS), which arises from the perturbation of an equilibrium state by an external electric field. While for charge transport the usual double-commutator Kubo formula is recovered (also beyond linear response), we obtain formulas for appropriately defined spin conductivities, which are still explicit but more involved. Certain “Kubo-like” terms in these formulas are also shown to agree with the corresponding contributions to the spin conductance. In addition to that, we employ similar techniques to show a new result, namely that even in systems with non-conserved spin, there is no generation of spin torque, that is, the spin torque operator has an expectation in the NEASS which vanishes faster than any power of the intensity of the perturbing field.