Discrete Floquet time crystals (DFTC) are characterized by the spontaneous breaking of the discrete time-translational invariance characteristic of Floquet driven systems. In analogy with equilibrium critical points, also time-crystalline phases display critical behaviour of different order, i.e. oscillations whose period is a multiple p > 2 of the Floquet driving period. Here, we introduce a new order parameter which is able to unambiguously detect crystalline phases regardless of the value of p and, at the same time, is a useful tool for chaos diagnostic. This new paradigm allows us to investigate the phase diagram of the long-range (LR) kicked Ising model to an unprecedented depth, unveiling a rich landscape characterized by self-similar fractal boundaries. Our solid theoretical picture describes the emergence of DFTCs phase both as a function of the strength and period of the Floquet drive, capturing the emergent Zp symmetry in the Floquet-Bloch waves.

We present a general and powerful numerical method useful to study the density matrix of spin models. We apply the method to finite dimensional spin glasses, and we analyze in detail the four dimensional Edwards-Anderson model with Gaussian quenched random couplings. Our results clearly support the existence of replica symmetry breaking in the thermodynamical limit.

We study the non-equilibrium phase diagram of a fully-connected Ising p-spin model, for generic p > 2, and investigate its robustness with respect to the inclusion of spin-wave fluctuations, described by a ferromagnetic, short-range spin interaction. We investigate the dynamics of the mean-field model after a quantum quench observing a new dynamical quantum phase transition which is either first or second order depending on the even or odd parity of p, in stark contrast with the static first order counterpart for all p. In addition, we find that the corresponding phase diagram is qualitatively modified by the fluctuations introduced by a short-range interaction which drive the system always towards various paramagnetic phases distinguished by the strength of time dependent fluctuations of the magnetization.