Abstract
Phys. Rev. B 94, 115142 (2016) 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 localised 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-Tunnelling 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 generalised 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 behaviour of the systems for both repulsive and
attractive interactions.