Quantum impurity in a Luttinger liquid: Exact solution of the Kane-Fisher model

In this work, we show in pedagogical detail that the most singular contributions to the slow part of the asymptotic density-density correlation function of Luttinger liquids with fermions interacting mutually with only short-range forward scattering and also with localised scalar static impurities (where backward scattering takes place) has a compact analytical expression in terms of simple functions that have second order poles and involve only the scale-independent bare tra...

Recently the authors developed a scattering approach that allows for a complete description of the steady-state physics of quantum-impurities in and out of equilibrium. Quantum impurities are described using scattering eigenstates defined ab initio on the open, infinite line with asymptotic boundary conditions imposed by the leads. The scattering states on the open line are constructed for integrable quantum-impurity models by means of a significant generalization of the Beth...

We study the influence of electron-phonon coupling on electron transport through a Luttinger liquid with an embedded weak scatterer or weak link. We derive the renormalization group (RG) equations which indicate that the directions of RG flows can change upon varying either the relative strength of the electron-electron and electron-phonon coupling or the ratio of Fermi to sound velocities. This results in the rich phase diagram with up to three fixed points: an unstable one ...

We calculate the density of states for an interacting quantum wire in the presence two impurities of arbitrary potential strength. To perform this calculation, we describe the Coulomb interactions in the wire within the Tomonaga-Luttinger liquid theory. After establishing and solving the Dyson equation for the fermionic retarded Green's functions, we study how the profile of the local density of states is affected by the interactions in the entire range of impurity potentials...

The response of a one-dimensional fermion system is investigated using Density Functional Theory (DFT) within the Local Density Approximation (LDA), and compared with exact results. It is shown that DFT-LDA reproduces surprisingly well some of the characteristic features of the Luttinger liquid, namely the vanishing spectral weight of low energy particle-hole excitations, as well as the dispersion of the collective charge excitations. On the other hand, the approximation fail...

We consider a linearly-dispersing quantum impurity interacting through a contact density-density term with a one-dimensional (1D) superfluid described by the Tomonaga-Luttinger liquid theory. Using a linked cluster expansion we characterize the impurity dynamics by calculating approximate expressions for the single-particle Green's function and for the time evolution of the density profile. We show the existence of two different dynamical regimes: (i) a $\textit{quasiparticle...

We establish the equivalence between the problem of a solitary potential scatterer in a Luttinger liquid of spinless repulsive fermions and an anisotropic Kondo model where the value of the impurity spin id determined by the scaling dimension of the scattering potential. The Bethe ansatz is used to derive non-perturbative expressions for the capacitance and conductance.

In one-dimensional quantum wires the interplay of electron correlations and impurities strongly influences the low-energy physics. The diversity of energy scales and the competition of correlations in interacting Fermi systems can be treated very efficiently with the functional renormalization group (fRG), describing the gradual evolution from a microscopic model Hamiltonian to the effective low-energy action as a function of a continuously decreasing energy cutoff. The fRG p...

Various sophisticated approximation methods exist for the description of quantum many-body systems. It was realized early on that the theoretical description can simplify considerably in one-dimensional systems and various exact solutions exist. The focus in this introductory paper is on fermionic systems and the emergence of the Luttinger liquid concept.

Quantum impurity models are prevalent throughout many body physics, providing some prime examples of strongly correlated systems. Aside from being of great interest in themselves they can provide deep insight into the effects of strong correlations in general. The classic example is the Kondo model wherein a magnetic impurity is screened at low energies by a non interacting metallic bath. Here we consider a magnetic impurity coupled to a quantum wire with pairing interaction ...