  We investigate both pure and mixed states Floquet dynamical quantum phase
transition (DQPT) in the periodically time-dependent extended XY model. We
exactly show that the proposed Floquet Hamiltonian of interacting spins can be
expressed as a sum of noninteracting quasi-spins imposed by an effective time
dependent magnetic field (Schwinger-Rabi model). The calculated Chern number
indicates that there is a topological transition from nonadiabatic to adiabatic
regime. In the adiabatic regime, the quasi-spins trace the time dependent
effective magnetic field and then oscillate between spin up and down states.
While in the nonadiabatic regime, the quasi-spins cannot follow the time
dependent effective magnetic field and feel an average magnetic field. We find
the range of driving frequency over which the quasi-spins experience adiabatic
cyclic processes. Moreover, we obtain the exact expression of the Loschmidt
amplitude and generalized Loschmidt amplitude of the proposed Floquet system.
The results represent that both pure and mixed states dynamical phase
transition occurs when the system evolves adiabatically. In other words, the
minimum required driving frequency for the appearance of Floquet DQPT is equal
to the threshold frequency needed for transition from nonadiabatic to adiabatic
regime.
