Emergence of Floquet time crystals in infinite-range spin systems with multi-body interactions*

Time crystals are out-of-equilibrium phases of matter emerging as a consequence of time-translation symmetry breaking. In periodically driven systems, discrete time-translation symmetry breaking gives rise to Floquet time crystal (FTC) phases. A FTC phase is characterized by the emergence of ordering of the Floquet eigenstates, which manifests as a subharmonic system response. In this work we study infinite range spin systems with multi-body interactions, admitting an exact mean-field description, in presence of a periodic drive. Going beyond the usual case of two-body interactions, we show that the higher multi-body interactions give rise to FTC phases showing eigenstate ordering leading to system responses with periods as large as the degree of the interaction. Using the mean-field description of these models, and relying on the phenomenology of dynamical flows and area preserving maps, we construct a classical picture of FTC phases in driven Hamiltonian systems. Finally, in systems hosting more than one FTC phase, we explore control protocols to extract rigid system responses carrying more than one subharmonic frequency.