Replica resummation of the Baker-Campbell-Hausdorff series

Abstract

We developed a novel perturbative expansion based on the replica trick for the Floquet Hamiltonian governing the dynamics of periodically kicked systems where the kick strength is the small parameter. The expansion is formally equivalent to an infinite resummation of the Baker-Campbell-Hausdorff series in the un-driven (non-perturbed) Hamiltonian, while considering terms up to a finite order in the kick strength. As an application of the replica expansion, we analyze an Ising spin 1/2 chain periodically kicked with magnetic field of strength $h$, which has both longitudinal and transverse components. We demonstrate that even away from the regime of high frequency driving, the heating rate is nonperturbative in the kick strength bounded from above by a stretched exponential: $e^{-{\rm const}\,h^{-1/2}}$. This guarantees existence of a very long pre-thermal regime, where the dynamics is governed by the Floquet Hamiltonian obtained from the replica expansion.