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Журнал Экспериментальной и Теоретической Физики

On the Exact Solution for a Luttinger Liquid with Repulsion and a Single Point Impurity

V. V. Afonin, Victor Yu. Petrov

Published Dec 15, 2023• 0 citations
Publisher

Abstract

The expression for the conductance of a 1D channel, which has been obtained using the well-known exact solution, is analyzed. It is shown that in the case of strong electron—electron interaction, the slowest (linear in frequency) asymptotics of the conductance is determined by the behavior of electron—electron interaction in the region of transition from 1D to 3D motion realized near the impurity.

DOI
10.31857/s0044451023090134

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The expression for the conductance of a 1D channel, which has been obtained using the well-known exact solution, is analyzed. It is shown that in the case of strong electron—electron interaction, the slowest (linear in frequency) asymptotics of the conductance is determined by the behavior of electron—electron interaction in the region of transition from 1D to 3D motion realized near the impurity.
We show that the conductance of a 1D channel with one point-like impurity critically depends on the asymptotic behavior of the e - e interaction at low momenta k (about the inverse length of the channel). The conductance reemerges (contrary to the case of a point-like repulsive potential) if the potential satisfies the condition V ( k = 0) = 0. For example, this happens if the bare e - e interaction is screened by the charges in the bulk. The relation of this phenomenon to the long-range order present in the Luttinger model is discussed. We consider spinless electrons but generalization is straightforward.
Properties of one-dimensional (1D) transport strongly depend on the proper choice of boundary conditions. It has been frequently stated that the Luttinger-liquid (LL) conductance is renormalized by the interaction as ge2{ge}^{2}ge2/h. To contest this result I develop a model of 1D LL wire with the interaction switching off at the infinities. Its solution shows that there is no renormalization of the universal conductance while the electrons have a free behavior in the source and drain reservoirs.
A new approach describing Luttinger Liquid with point-like impurity as field theory for the phase of scattering is developed. It based on a matching of the electron wave functions at impurity position point. As a result of the approach, an expression for non-local action has been taken. The non-locality of the theory leads to convergence of the observed values in an ultraviolet region. It allows studying conductance of the channel up to electron-electron interaction strength of the order of unit. Expansion of the non-local action in small frequency powers makes possible to develop a new approach to the renormalization group analysis of the problem. This method differs from the "poor man's"\ approach widely used in solid-state physics. We have shown, in the Luttinger Liquid "poor man's"\ approach breaks already in two-loop approximation. We analyse the reason of this discrepancy. The qualitative description of the phenomenon is discussed in detail.