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Journal of the Physical Society of Japan

On the Renormalization of Conductance in Tomonaga-Luttinger Liquid

Arisato Kawabata

Published Jan 15, 1996• 83 citations
Publisher

Abstract

A simple explanation is given for the renormalization factor of the conductance in one-dimensional interacting electron systems. Within the Tomonaga-Luttinger model, it is shown that the factor arises from the renormalization of electric potential, and that the conductance is not renormalized if we define it correctly. The theory is valid even when the interaction is present over the entire system.

DOI
10.1143/jpsj.65.30

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A simple explanation is given for the renormalization factor of the conductance in one-dimensional interacting electron systems. Within the Tomonaga-Luttinger model, it is shown that the factor arises from the renormalization of electric potential, and that the conductance is not renormalized if we define it correctly. The theory is valid even when the interaction is present over the entire system.
The conductance of one-dimensional interacting electron systems is calculated in a manner similar to Landauer's argument for non-interacting systems. Unlike in previous studies in which the Kubo formula was used, the conductance is directly evaluated as the ratio of current JJJ to the chemical potential difference ΔμΔμΔμ between right-going and left-going particles. It is shown that both JJJ and ΔμΔμΔμ are renormalized by electron-electron (e-e) interactions, but their ratio, the conductance, is not renormalized at all if the e-e interactions are the only scattering mechanism. It is also shown that nonequilibrium current fluctuation at low frequency is absent in such a case. These conclusions are drawn for both Fermi liquids (in which quasi-particles are accompanied with the backflow) and Tomonaga-Luttinger liquids.
The conductance of one-dimensional interacting electron systems is calculated in a manner similar to Landauer's argument for non-interacting systems.Unlike in previous studies in which the Kubo formula was used, the conductance is directly evaluated as the ratio of current J to the chemical potential difference ∆µ between right-going and left-going particles.It is shown that both J and ∆µ are renormalized by electron-electron (e-e) interactions, but their ratio, the conductance, is not renormalized at all if the e-e interactions are the only scattering mechanism.It is also shown that nonequilibrium current fluctuation at low frequency is absent in such a case.These conclusions are drawn for both Fermi liquids (in which quasi-particles are accompanied with the backflow) and Tomonaga-Luttinger liquids.
We studied the conductance renormalization and conductivity of multisubband Tomonaga-Luttinger models with intersubband interactions. We found that, as in single-band systems, the conductance of a multisubband system with an arbitrary number of subbands is not renormalized due to interaction between electrons. We derived a formula for the conductivity in multisubband models. We applied it to a simplified case and found that intersubband interaction enhances the conductivity, which is contrary to the intrasubband repulsive interaction, and that the conductivity is further enhanced for a larger number of subbands.
We investigate the system-size dependence as well as the temperature dependence of the conductance in 1D electron systems, paying particular attention to the effect of Umklapp scattering. By taking into account the renormalization of the external potential due to electron-electron interaction, the correction to the conductance, 2e^2/h, due to Umklapp scattering is estimated perturbatively. The Tomonaga-Luttinger liquid parameter appears in the conductance via the system-size and temperature dependence.
The finite-size Tomonaga-Luttinger Hamiltonian with an arbitrary potential is mapped onto a noninteracting Fermi gas with renormalized potential. This is done by means of flow equations for Hamiltonians and is valid for small electron-electron interaction. This method also yields an alternative bosonization formula for the transformed field operator which makes no use of Klein factors. The two-terminal conductance can then be evaluated using the Landauer formula. We obtain similar results for infinite systems at finite temperature by identifying the flow parameter with the inverse squared temperature in the asymptotic regime.