Dynamics of quantum integrable models via coupled Heisenberg equations

Dynamics of a quantum system can be described by coupled Heisenberg equations. In a generic many-body system these equations involve an exponentially large hierarchy of operators that is intractable without approximations. In contrast, in an integrable system one may hope to find a relatively small subset of relatively simple operators closed with respect to commutation with the Hamiltonian and analytically solve the corresponding system of linear differential equations. We have successfully applied this idea to models where the Hamiltonian is an element of the Onsager algebra (such as the transverse-field Ising spin-(1/2) chain and the superintegrable chiral n-state Potts models), and to the Kitaev model on the honeycomb-like Bethe lattice. For these models, we present analytical results for the quench dynamics inaccessible by other state-of-the art methods. Further prospects of this approach and a number of related open problems will be outlined and discussed.