Current operators in Bethe Ansatz and Generalized Hydrodynamics: An exact quantum/classical correspondence

Generalized Hydrodynamics is a recent theory that describes large scale transport properties of one dimensional integrable models. It is built on the (typically infinitely many) local conservation laws present in these systems, and leads to a generalized Euler type hydrodynamic equation. Despite the successes of the theory, one of its cornerstones, namely a conjectured expression for the currents of the conserved charges in local equilibrium has not yet been proven for interacting lattice models. Here we fill this gap, and compute an exact result for the mean values of current operators in Bethe Ansatz solvable systems, valid in arbitrary finite volume. Our exact formula has a simple semi-classical interpretation: the currents can be computed by summing over the charge eigenvalues carried by the individual bare particles, multiplied with an effective velocity describing their propagation in the presence of the other particles. Remarkably, the semi-classical formula remains exact in the interacting quantum theory, for any finite number of particles and also in the thermodynamic limit. Our proof is built on a form factor expansion and it is applicable to a large class of quantum integrable models.