Conductance of interacting systems

Start date: 30.09.1999
End date: 31.12.2009
Funded by: Universität Augsburg
Local head of project: Gert-Ludwig Ingold
External scientists / cooperations: Axel Freyn (Grenoble)
Rodolfo A. Jalabert (Strasbourg)
Rafael A. Molina (Madrid)
Jean-Louis Pichard (Saclay)
Peter Schmitteckert (Karlsruhe)
Dietmar Weinmann (Strasbourg)
Publications: Publication list


While the electric conductance of mesoscopic systems in the absence of electron-electron interactions is well understood, it is still a challenge to determine the conductance in presence of this interaction. In this project, the so-called embedding method was developped to determine the zero-temperature linear conductance through correlated electron systems and it was employed to improve our understanding of such systems.


Embedding method

The embedding method allows to numerically determine the linear conductance at zero temperature of a correlated electron system on a lattice. To this end, a region in which the electron-electron is taken into account and which is represented in red in the figure is closed to a ring by means of a segment without electron-electron interaction, represented in the figure in white. A magnetic flux threading the ring induces a persistent current which is related to the linear conductance. Within this project the presistent current is evaluated numerically (MWJIP03) by means of the density matrix renormalization group method (DMRG). In addition, the limit of the persistent current for infinite circumference of the ring has to be determined appropriately. In particular in the presence of resonances in the conductance, such an extrapolation has to be carried out carefully (MSWJIP04).

It is plausible that the embedding method through the persistent current accounts for two aspects of relevance for the conductance in a correlated electron system: the extension of the wave functions and how easy or difficult it is to bring an electron into the interacting region. Nevertheless it needs to be checked that the method provides quantitatively correct results. Seen from the interaction-free region, the interacting region represents a scatterer which can be described by means of its transmission. Then the Landauer formula provides a relation to the conductance. As the current for such a scattering system can be evaluated analytically as a function of the magnetic flux, numerically expensive DMRG calculations have been carried out to check the validity of this functional dependence. As was demonstrated in MSWJIP04, a one-parameter fit using the transmission amplitude works perfectly. This result then allows to determine the conductance by means of the charge stiffness which can be obtained more easily by numerially determining the persistent current for periodic and antiperiodic boundary conditions.

In MWJIP03 the embedding method was employed to determine the dependence of the conductance of spinless fermions at half filling as a function of the interacting region. While for an even number of lattice sites, the conductance is reduced with increasing interaction, this is not the case for an odd number of lattice sites. The reason is the energetic degeneracy of two electron states. An ideal conductance can also be obtained for an even number of lattice sites if the interaction strength is not abruptly brought to this maximum value within one lattice constant. In the presence of strong disorder it was shown that a moderate repulsive interaction can lead to an increase of the conductance. An extension of these studies away from half filling was carried out in MSWJIP04.

In order to study nonlocal effects induced by the interaction, in WJFIP08 an additional localized one-particle scatterer was placed in a variable distance from the correlated electron system. If the correlated electron system is perfectly conducting for arbitrary interaction strength, one observes a clear nonlocal effect manifesting itself in a change of the conductance as a function of the interaction strength due to the presence of the scatterer. Such an effect can also be achieved by means of a weak link where the hopping strength between two neighboring lattice sites is modified locally.

The results described so far referred all to one-channel problems. The extension to two channels in the form of two spin orientations can still be carried out easily for the Hubbard model where no spin flips do occur. Corresponding results as well as ideas for a generalization to the general two-channel case are described in FVSWIJP10. There the two-channel problem is expressed in terms of several one-channel problems which allow at least in principle to obtain the relevant parts of the scattering matrix of the correlated electron system. An explicit application of this method beyond the Hubbard model however does not yet exist. The results of this work also indicate that information on the conductance of a multi-channel problem can be obtained by studying the distribution of persistent currents under random channel mixing.