Mesoscopic impurities in generalized hydrodynamics

Abstract

This study focuses on the impurities in integrable models from the viewpoint of generalized hydrodynamics (GHD). An impurity can be thought of as a boundary condition for the GHD equation, relating the states on the left and right sides. It was found that, in interacting models, it is not possible to disentangle the incoming and outgoing states, which means that it is not possible to think of scattering as a mapping that maps the incoming state to the outgoing state. A novel class of impurities, dubbed mesoscopic impurities, was then introduced, whose spatial size is mesoscopic (i.e. their size L m i c r o ≪ L i m p ≪ L is much larger than the microscopic length scale L m i c r o but much smaller than the macroscopic scale L). Because of their large size, it is possible to describe mesoscopic impurities via GHD. This simplification allows one to study these impurities both analytically and numerically. These impurities show interesting non-perturbative scattering behavior, such as the nonuniqueness of the solutions and a nonanalytic dependence on the impurity strength. In models with one quasi-particle species and a scattering phase shift that depends only on the difference in momenta, the scattering can be described using an effective Hamiltonian. This Hamiltonian is dressed due to the interaction between the particles and satisfies a self-consistency fixed-point equation. In the example of the hard-rod model, it was demonstrated how this fixed-point equation can be used to find almost explicit solutions to the scattering problem by reducing it to a two-dimensional system of equations that can be solved numerically.