Universal self-similar dynamics of relativistic and nonrelativistic field theories near nonthermal fixed points

We investigate universal behavior of isolated many-body systems far from equilibrium, which is relevant for a wide range of applications from ultracold quantum gases to high-energy particle physics. The universality is based on the existence of nonthermal fixed points, which represent nonequilibrium attractor solutions with self-similar scaling behavior. The corresponding dynamic universality classes turn out to be remarkably large, encompassing both relativistic as well as nonrelativistic quantum and classical systems. For the examples of nonrelativistic (Gross-Pitaevskii) and relativistic scalar field theory with quartic self-interactions, we demonstrate that infrared scaling exponents as well as scaling functions agree. We perform two independent nonperturbative calculations, first by using classical-statistical lattice simulation techniques and second by applying a vertex-resummed kinetic theory. The latter extends kinetic descriptions to the nonperturbative regime of overoccupied modes. Our results open new perspectives to learn from experiments with cold atoms aspects about the dynamics during the early stages of our universe.

Figure 1

Schematic illustration of the occupation number distribution near a nonthermal fixed point, shown as a function of momentum $p$ for two subsequent times $t_1$ and $t_2>t_1$. The scaling exponents $\alpha$ and $\beta$ characterize the self-similar evolution according to Eq. (1).

Figure 2

The scaling function $f_S(\xi \equiv t^{\beta }|\mathbf{p}|)=t^{-\alpha }f(t,\mathbf{p})$, normalized to the amplitudes $A$ and $B$, exhibits accurate agreement between the nonrelativistic (circles) and relativistic (squares) simulation results.

Figure 3

Rescaled distribution function of the nonrelativistic theory as a function of the rescaled momentum for different times. The inset shows the original distribution without rescaling.

Figure 4

Evolution of the zero-momentum correlation divided by volume for the nonrelativistic Bose gas. A power law $\sim t^{\alpha }$ has been fitted in the time interval with self-similar evolution for the three largest volumes (dashed lines). Subsequently, the results for different volumes converge, signaling the formation of a coherent zero mode spreading over the entire volume.

Figure 5

Evolution of the condensate fraction for the nonrelativistic Bose gas for different volumes $V$. The different curves become approximately volume independent after rescaling of time by $V^{-1/\alpha }$, in agreement with (15).

Figure 6

Rescaled distribution function of the relativistic two-component theory as a function of the rescaled momentum for different times. The inset shows the original distribution without rescaling.

Figure 7

Rescaled distribution function of the relativistic four-component theory as a function of the rescaled momentum for different times. The inset shows the original distribution without rescaling.

Figure 8

The exponents $\alpha$ and $\beta$ with statistical error bars for the relativistic $N=4$ component field theory extracted at different reference times $t_{\text{ref}}$.

Figure 9

Dispersion relation for the relativistic $N=2$ component theory at different times for the same parameters as for Fig. 6. In the inset, the time evolution of the mass gap at zero momentum is shown, which is obtained from a $\sqrt{m^2+{\mathbf{p}}^2}$ fit to ${\omega }_{\mathbf{p}}$ at low momenta ($+$ symbol) and by measuring the oscillation frequency ${\omega }_c$ of the unequal-time correlation function (x symbol).

Figure 10

Evolution of the zero-momentum correlation divided by volume for the relativistic $N=4$ component theory, similar to Fig. 4 for the nonrelativistic case.

Figure 11

Illustration of different scattering channels. The vertex correction at NLO may be viewed as an effective interaction, which involves the exchange of an intermediate particle. Left: incoming particles with momenta $p$ and $l$ join into an intermediate particle that eventually splits. Middle and right: incoming particle with momentum $p$ emits the intermediate particle and becomes the final particle with momentum $q$ or $r$, respectively.

Figure 12

Integration area $\mathrm{\Delta }(\overline{p})$ as represented by the shaded region.