Dynamics of quantum integrable systems 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 that is closed with respect to commutation with the Hamiltonian. In this case, there is a chance to analytically solve the corresponding system of Heisenberg equations.
We successfully apply this idea
(1) to models where the Hamiltonian is an element of the Onsager algebra (such as the transverse-field Ising chain and the superintegrable chiral $n$-state Potts (also known as clock) models), and
(2) to the Kitaev model on the honeycomb-like Bethe lattice. In the latter case, a new algebra containing the Hamiltonian is discovered. 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. In particular, we will explain how, in the framework of the method, a progress in understanding underlying algebras will almost automatically lead to new results for quench dynamics.

References

1. O. Lychkovskiy, “Closed hierarchy of Heisenberg equations in integrable models with Onsager algebra”, SciPost Phys., 10 (2021), 124
2. O. Gamayun, O. Lychkovskiy, “Out-of-equilibrium dynamics of the Kitaev model on the Bethe lattice via a set of Heisenberg equations”, SciPost Phys., 12 (2022), 175