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AIP conference proceedings

Scaling of a Single Impurity Potential of Arbitrary Strength in a Tomonaga-Luttinger Liquid

Kenji Kamide, Yuji Tsukada, Susumu Kurihara

Published Jan 1, 2006• 2 citations
Publisher

Abstract

We study scaling of a single impurity potential in a Tomonaga‐Luttinger (TL) liquid for arbitrary impurity strength. Boson representation for the electron field operator including a phase shift by the impurity scattering is introduced, and used to calculate the temperature dependence of the local spectral density. The fixed line of an impurity potential is determined for TL liquid with spin dependent interactions (KS≠1), extending the results for weak and strong impurity limits to intermediate strength.

DOI
10.1063/1.2355223

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We study scaling of a single impurity potential in a Tomonaga‐Luttinger (TL) liquid for arbitrary impurity strength. Boson representation for the electron field operator including a phase shift by the impurity scattering is introduced, and used to calculate the temperature dependence of the local spectral density. The fixed line of an impurity potential is determined for TL liquid with spin dependent interactions (KS≠1), extending the results for weak and strong impurity limits to intermediate strength.
Transport through a spinless Tomonaga-Luttinger (TL) liquid is studied in the presence of a single impurity. A boundary bosonization technique is developed to treat an impurity of arbitrary strength. Our new bosonization formula (i) smoothly connects the ones for the strong and the weak impurity limits, and (ii) reproduces exact scaling equation for the transmission probability in the weak two-body interaction limit. With the use of our formula, TL parameter is determined from the long-distance correlation functions. The result indicates a possible renormalization of TL parameter due to the effect of the single impurity scattering.
Using bosonization and path-integral methods, we study general low-temperature behavior of nonmagnetic and magnetic impurity scattering in Tomonaga-Luttinger liquid, and calculate the electron Green function for a general backward scattering potential. We demonstrate that the electron density of state near the impurity site is suppressed by backward scattering, but it mainly remains invariant far away from the impurity, and that at zero temperature the electrons are completely reflected on the impurity site; the system breaks into two subsystems, but right- and left-moving electron fields have a twisted boundary condition. We also show that a testing charge can only be partially screened by conduction electrons, and in a strong interaction region the impurity susceptibility has a 1/T1/T1/T-type low-temperature behavior.
The single impurity problem in a spinful Tomonaga--Luttinger liquid is studied numerically using path-integral Monte Carlo methods. The advantage of our approach is that the system allows for extensive analyses of charge and spin conductance in the nonperturbative regime. By closely examining the behavior of conductances at low temperatures, in the presence of a finite backward scattering barrier due to the impurity, we identified four distinct phases characterized by either perfect transmission or reflection of charge and spin channels. Our phase diagram for an intermediate scattering strength is consistent with the standard perturbative renormalization group (RG) analysis in the limit of weak and strong backward scatterings, in the sense that all our phase boundaries interpolate the two limiting cases. Further investigations show, however, that precise location and form of our phase boundaries are not trivially explained by the standard RG analysis, e.g., some part of the phase diagram looks much similar to the weak backscattering limit, whereas some other part is clearly derived from the opposite limit. In order to give a more intuitive interpretation of such behaviors, we also reconsidered our impurity problem from the viewpoint of a quantum Brownian motion picture.