Current operators describe the flow of the conserved charges in integrable models. Whereas lots of information was known about the charges, surprisingly the current operators remained unexplored for a very long time. I review recent results in this topic, which include an exact finite volume formula for the mean values of the current operators, their embedding into the Quantum Inverse Scattering Approach (Algebraic Bethe Ansatz), and connections with long range deformations and TTbar deformations. --- Part of the London Integrability Journal Club. If you are a new participant, please register at integrability-london.weebly.com. The link will be emailed on Tuesday.

We review the recent progress regarding current operators in integrable models, focusing especially on integrable spin chains. These operators describe the flow of the conserved charges, and they are important for the construction of Generalized Hydrodynamics. They are also connected to long range deformations and TTbar-like deformations of the spin chains, and also to the theory of factorized correlation functions. We argue that these operators are very special, because their mean values can be computed relatively easily even in nested spin chains. This is rather unique because mean values in nested models are rather difficult to compute for generic operators. We review these various connections and also show how to construct current operators using the Quantum Inverse Scattering Approach, the canonical framework developed by the Leningrad school. [please email alejandro.cabo_bizet@kcl.ac.uk for the zoom link]