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Journal of Experimental and Theoretical Physics

Luttinger Liquid with Attraction and One Impurity: Exact Solution

V. V. Afonin

Published Feb 1, 2023• 1 citation
Publisher

Abstract

Abstract The exact solution to the problem of scattering of one-dimensional attracted fermions on a point impurity, which exists in the Kane–Fisher model is analyzed for a Luttinger parameter equal to 2. The expression obtained for the frequency dependence of the coefficient of fermion reflection from an impurity demonstrates impossibility of extrapolation of results for a relatively weak fermion–fermion interaction to the range of coupling constants on the order of unity.

DOI
10.1134/s1063776123020012

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Abstract The exact solution to the problem of scattering of one-dimensional attracted fermions on a point impurity, which exists in the Kane–Fisher model is analyzed for a Luttinger parameter equal to 2. The expression obtained for the frequency dependence of the coefficient of fermion reflection from an impurity demonstrates impossibility of extrapolation of results for a relatively weak fermion–fermion interaction to the range of coupling constants on the order of unity.
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 behavior of a single impurity in a one-dimensional Luttinger liquid is numerically investigated by means of the density-matrix renormalization group. By analyzing the finite-size scaling behavior of the low-energy spectrum, we confirm the theoretical prediction of Kane and Fisher [Phys. Rev. Lett. 68, 1220 (1992)] both for attractive and repulsive interactions. Moreover, we calculate the exponent of the orthogonality catastrophe, which gives a further support to the above theoretical prediction.