Generalized hydrodynamics is a recent theory that describes large-scale transport properties of onedimensional 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 semiclassical 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 semiclassical 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.

We consider the current operators of one dimensional integrable models. These operators describe the flow of the conserved charges of the models, and they play a central role in Generalized Hydrodynamics. We present the key statements about the mean currents in finite volume and in the thermodynamic limit, and we review the various proofs of the exact formulas. We also present a few new results in this review. New contributions include a computation of the currents of the Heisenberg spin chains using the string hypothesis, and simplified formulas in the thermodynamic limit. We also discuss implications of our results for the asymptotic behaviour of dynamical correlation functions.

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We construct families of exotic spin-1/2 chains using a procedure called "hard rod deformation". We treat both integrable and non-integrable examples. The models possess a large non-commutative symmetry algebra, which is generated by matrix product operators with fixed small bond dimension. The symmetries lead to Hilbert space fragmentation and to the breakdown of thermalization. As an effect, the models support persistent oscillations in non-equilibrium situations. Similar symmetries have been reported earlier in integrable models, but here we show that they also occur in nonintegrable cases.

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