Equilibration and prethermalization in the Bose-Hubbard and Fermi-Hubbard models

We study the Bose-Hubbard and Fermi-Hubbard models in the (formal) limit of large coordination numbers Z >> 1. Via an expansion into powers of 1/Z, we establish a hierarchy of correlations which facilitates an approximate analytical derivation of the time evolution of the reduced density matrices for one and two sites, etc. With this method, we study the quantum dynamics (starting in the ground state) after a quantum quench, i.e., after suddenly switching the tunneling rate J from zero to a finite value, which is still in the Mott regime. We find that the reduced density matrices approach a (quasi) equilibrium state after some time. For one lattice site, this state can be described by a thermal state (within the accuracy of our approximation). However, the (quasi) equilibrium state of the reduced density matrices for two sites including the correlations can not be described by a thermal state. Thus, real thermalization (if it occurs) should take a much longer time. This behavior has already been observed in other scenarios and is sometimes called "prethermalization". Finally, we compare our results to numerical simulations for finite lattices in one and two dimensions and find qualitative agreement.