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Journal of Experimental and Theoretical Physics Letters

On Exact Solutions for a Luttinger Liquid with One Impurity

V. V. Afonin, Victor Yu. Petrov

Published Jun 1, 2019• 4 citations
Publisher

Abstract

Reasons for the existence of exact solutions of the Luttinger model have been discussed. It has been shown that scattering by impurities at certain electron-electron coupling constants can be written in terms of Fermi quasiparticles exactly taking into account the electron-electron interaction. This circumstance significantly simplifies the analysis and in some cases leads to the existence of exact solutions.

DOI
10.1134/s0021364019110031

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Reasons for the existence of exact solutions of the Luttinger model have been discussed. It has been shown that scattering by impurities at certain electron-electron coupling constants can be written in terms of Fermi quasiparticles exactly taking into account the electron-electron interaction. This circumstance significantly simplifies the analysis and in some cases leads to the existence of exact solutions.
A Luttinger liquid coupled to a quantum impurity describes a large number of physical systems. The Hamiltonian consists of left- and right-moving fermions interacting among themselves via a density-density coupling and scattering off a localized transmitting and reflecting impurity. We solve exactly the Hamiltonian by means of an incoming-outgoing scattering Bethe basis which properly incorporates all scattering processes. A related model, the weak-tunneling model, wherein the impurity is replaced by a tunnel junction, is solved by the same method. The consistency of the construction is established through a generalized Yang-Baxter relation. Periodic boundary conditions are imposed and the resulting Bethe ansatz equations are derived by means of the off-diagonal Bethe ansatz approach. We derive the spectrum of the model for all coupling constant regimes and calculate the impurity free energy. We discuss the low energy behavior of the systems for both repulsive and attractive interactions.
The Luttinger Theorem, which relates the electron density to the volume of the Fermi surface in an itinerant electron system, is taken to be one of the essential features of a Fermi liquid. The microscopic derivation of this result depends on the vanishing of a certain integral, the Luttinger integral I L , which is also the basis of the Friedel sum rule for impurity models, relating the impurity occupation number to the scattering phase shift of the conduction electrons. It is known that non-zero values of I L with I L = ± π /2, occur in impurity models classified as singular Fermi liquids. Here we show the same values, I L = ± π /2, occur in an impurity model in phases with regular low energy Fermi liquid behavior. Consequently the Luttinger integral can be taken to characterize these phases, and the quantum critical points separating them interpreted as topological.
This thesis addresses a theoretical study of the problem of a single impurity embedded in a one-dimensional system of interacting electrons in presence of electron-phonon coupling. First we consider a system with a featureless point-like potential impurity, followed by the case of a resonant level hybridised with a Luttinger Liquid. The stress is made on a more fundamental problem of a featureless scatterer, for which two opposite limits in the impurity strength are considered: a weak scatterer and a weak link. We have found that, regardless of the transmission properties of phonons through the impurity, the scaling dimensions of the conductance in these limits obey the duality condition, \\( \\triangle_{WS}\\) \\( \\triangle_{WL}\\) = 1, known for the Luttinger Liquid in the absence of phonons. However, in the case when the strength of phonon scattering is correlated with electron scattering by the impurity, we find a nontrivial phase diagram with up to three fixed points and a possibility of a metal-insulator transition. We also consider the case of a weakly interacting electron-phonon system in the presence of a single impurity of an arbitrary scattering potential. In the problem of a resonant level attached to the Luttinger Liquid we show that the electron-phonon coupling significantly modifies the effective energy-dependent width of the resonant level in two different geometries, corresponding to the resonant and anti-resonant transmission in the Fermi gas.