Critical properties of the prethermal Floquet time crystal

The critical properties characterizing the formation of Floquet time crystal in prethermal phase are investigated analytically periodically driven $O(N)$ model. In particular, we focus on line separating trivial with period synchronized dynamics and absence long-range spatial order from non-trivial where is accompanied by period-doubling dynamics. vicinity line, a combination dimensional expansion exact solution for $N\to\infty$, determine exponent $\nu$ that characterizes divergence correlation length equal-time functions, $\beta$ growth amplitude order-parameter, as well initial-slip $\theta$ aging when quench performed deep to line. exponents $\nu, \beta, \theta$ found be identical those drive. addition, functional form depend whether system probed at times small or large compared drive period. structure two-point obtained linear response perturbing potential show algebraic decays longer ranged than drive, besides being period-doubled, also oscillate space wave-vector $\omega/(2 v)$, $v$ velocity quasiparticles, $\omega$ frequency.