Mesoscopic Impurities in Generalized Hydrodynamics

Abstract

We study 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 state on the left and right side. We find that in interacting models it is not possible to disentangle incoming and outgoing states, which means that it is not possible to think of scattering as a mapping which maps the incoming state to the outgoing state. We then introduce a novel class of impurities, dubbed mesoscopic impurities, whose spatial size is mesoscopic (i.e.\ their size $L_{\mathrm{micro}} \ll L_{\mathrm{imp}} \ll L$ is much larger than the microscopic length scale $L_{\mathrm{micro}}$, but much smaller than the macroscopic scale $L$). Due to 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, for instance non-uniqueness of solutions and a non-analytic dependence on the impurity strength. In models with one quasi-particle species and a scattering phase shift that depends on the difference of momenta only, we find that one can describe the scattering using an effective Hamiltonian. This Hamiltonian is dressed due to the interaction between particles and satisfies a self consistency fixed point equation. On the example of the hard rods model we demonstrate 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 which can be solved numerically.