  We study the fate of interacting quantum systems which are periodically
driven by switching back and forth between two integrable Hamiltonians. This
provides an unconventional and tunable way of breaking integrability, in the
sense that the stroboscopic time evolution will generally be described by a
Floquet Hamiltonian which progressively becomes less integrable as the driving
frequency is reduced. Here, we exemplify this idea in spin chains subjected to
periodic switching between two integrable anisotropic Heisenberg Hamiltonians.
We distinguish the integrability-breaking effects of resonant interactions and
perturbative (local) interactions, and illustrate these by contrasting
different measures of energy in Floquet states and through a study of level
spacing statistics. This scenario is argued to be representative for general
driven interacting integrable systems.
