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Generalized Hydrodynamics is a recent theory that describes large scaletransport 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 thesuccesses of the theory, one of its cornerstones, namely a conjecturedexpression for the currents of the conserved charges in local equilibrium hasnot yet been proven for interacting lattice models. Here we fill this gap, andcompute an exact result for the mean values of current operators in BetheAnsatz solvable systems, valid in arbitrary finite volume. Our exact formulahas a simple semi-classical interpretation: the currents can be computed bysumming over the charge eigenvalues carried by the individual bare particles,multiplied with an effective velocity describing their propagation in thepresence of the other particles. Remarkably, the semi-classical formula remainsexact in the interacting quantum theory, for any finite number of particles andalso in the thermodynamic limit. Our proof is built on a form factor expansionand it is applicable to a large class of quantum integrable models.

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