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We study the non-equilibrium phase diagram of a fully-connected Ising$p$-spin model, for generic $p>2$, and investigate its robustness with respectto the inclusion of spin-wave fluctuations, resulting from a ferromagnetic,short-range spin interaction. In particular, we investigate the dynamics of themean-field model after a quantum quench: we observe a new dynamical phasetransition which is either first or second order depending on the even or oddparity of $p$, in stark contrast with its thermal counterpart which is firstorder for all $p$. The dynamical phase diagram is qualitatively modified by thefluctuations introduced by a short-range interaction which drive the systemalways towards various paramagnetic phases determined by the strength of timedependent fluctuations of the magnetization.

Changing the order of a dynamical phase transition through fluctuations in a quantum p-spin model