Dynamical Crossovers in Prethermal Critical States

We study the prethermal dynamics of an interacting quantum field theory with an $N$-component order parameter and $O(N)$ symmetry, suddenly quenched in the vicinity of a dynamical critical point. Depending on the initial conditions, the evolution of the order parameter, and of the response and correlation functions, can exhibit a temporal crossover between universal dynamical scaling regimes governed, respectively, by a quantum and a classical prethermal fixed point, as well as a crossover from a Gaussian to a non-Gaussian prethermal dynamical scaling. Together with a recent experiment, this suggests that quenches may be used in order to explore the rich variety of dynamical critical points occurring in the nonequilibrium dynamics of a quantum many-body system. We illustrate this fact by using a combination of renormalization group techniques and a nonperturbative large-$N$ limit.

Figure 1

A dynamical transition separating a noncritical relaxation from coarsening occurs at $r=r_c(c_0,{\mathrm{\Omega }}_0,u)$. For $d<2$ (left panel), a dynamical transition occurs only for ${\mathrm{\Omega }}_0=0$, while any finite value of ${\mathrm{\Omega }}_0$ prevents it; for $d>2$, instead, the dynamical transition takes place also for finite values of ${\mathrm{\Omega }}_0$.

Figure 2

Correlation length $\xi$ for the large-$N$ limit of Eq. (1), as a function of the distance from the critical point $\delta r\equiv |r-r_c|$. Results for $d=2.5$, $u=10$, $c_0=100$, and various values of ${\mathrm{\Omega }}_0$ (${\mathrm{\Omega }}_0^2=0.1$, 0.5, 1, 2, 5, 10, from the lowermost to the uppermost curves, respectively). The dashed line is proportional to $\delta r^{-2/3}$, while the dotted one is proportional to $\delta r^{-2}$, corresponding to the quantum and classical scaling, respectively. Inset: ${\mathrm{\Omega }}_0\xi$ vs $\delta r/{\mathrm{\Omega }}_0^{\alpha }$, based on the same data as the main plot, with $\alpha \approx 1.5$ determined numerically.

Figure 3

Magnetization $M(t)$ for the large-$N$ limit of Eq. (1), as a function of time $t$ for $d=2.5$, $u=1$, $c_0=100$, $r=r_c$, and for several values of ${\mathrm{\Omega }}_0$, reported in the legend of the main plot. Inset: ${\mathrm{\Omega }}_0^{\alpha }M$ vs ${\mathrm{\Omega }}_{0}t$, based on the same data as the main plot, with $\alpha \approx 0.12$ determined numerically.