  The nonlinear effect of a driving force in periodically driven quantum
many-body systems can be systematically investigated by analyzing the effective
Floquet Hamiltonian. In particular, under an appropriate definition of the
effective Hamiltonian, simple driving forces may result in non-local
interactions. Here we consider a driven $ XXZ $ model to show that four-site
interactions emerge owing to the driving force, which can produce
state-selective localization, a phenomenon where some limited Ising-like
product states become fixed points of dynamics. We first derive the effective
Hamiltonian of a driven $ XXZ $ model on an arbitrary lattice for general spin
$ S $. We then analyze in detail the case of a one-dimensional chain with $
S=1/2 $ as a special case and find a condition imposed on the cluster of four
consecutive sites as a necessary and sufficient condition for the state of the
whole system to be localized. We construct such a localized state based on this
condition and demonstrate it by numerical simulation of the original
time-periodic model.
