Nonthermal fixed points in quantum field theory beyond the weak-coupling limit

Quantum systems in extreme conditions can exhibit universal behavior far from equilibrium associated to nonthermal fixed points with a wide range of topical applications from early-Universe inflaton dynamics and heavy-ion collisions to strong quenches in ultracold quantum gases. So far, most studies have relied on a mapping of the quantum dynamics onto a classical-statistical theory that can be simulated on a computer. However, the mapping is based on a weak-coupling limit, while phenomenological applications often require moderate interaction strengths. We report on the observation of nonthermal fixed points directly in quantum field theory beyond the weak-coupling limit. For the example of a relativistic scalar $\mathrm{O}(N)$-symmetric quantum field theory, we numerically solve the nonequilibrium dynamics employing a $1/N$ expansion to next-to-leading order, which does not rely on a small coupling parameter. Starting from two different sets of overoccupied and of strong-field initial conditions, we find that nonthermal fixed points are not restricted to parameter ranges suitable for classical-statistical simulations but extend also to couplings of order 1. While the infrared behavior is found to be insensitive to the differences in the initial conditions, we demonstrate that transport phenomena to higher momenta depend on the presence or absence of a symmetry-breaking field expectation value.

Figure 1

Diagrammatic representation of the infinite series of contributions to the free energy density at NLO in the $1/N$ expansion of the 2PI effective action. The lines denote self-consistent propagators, solid circles denote vertices, and crosses denote field expectation values.

Figure 2

Illustration of the employed initial condition scenarios.

Figure 3

Particle number distribution for fluctuation initial conditions with $\lambda =0.01$, 0.1, 1 (top to bottom) at different times including the initial time (gray). The approximate power laws with exponents ${\kappa }_{\mathrm{N}}=5$ and ${\kappa }_{\mathrm{E}}=5/3$ characterize the dual cascade in the respective momentum ranges (dashed gray lines).

Figure 4

Dispersion relation for fluctuation initial conditions with $\lambda =0.01$ at $Qt=1200$ exhibiting an effective mass gap $m$. A fit to the quasiparticle dispersion relation $\sqrt{{\mathbf{p}}^2+m^2}$ is performed in good agreement with the numerical data for $m/Q\approx 0.6$. Additionally, a linear dispersion relation, $\omega =|\mathbf{p}|$, is shown for comparison.

Figure 5

Particle number distribution (left) and inverse slope parameter $\ln (1+1/f)$ (right) for fluctuation initial conditions with $\lambda =10$ at different times including the initial time (solid, gray). We observe a power-law behavior for intermediate times and the approach to thermal equilibrium for $Qt\gtrsim 600$. For comparison, the dashed gray line in 5 corresponds to a Bose-Einstein distribution with temperature $T\approx 1.6Q$ and chemical potential $\mu =0$.

Figure 6

Rescaled particle number distribution for fluctuation initial conditions at fixed time for different values of the coupling $\lambda$. In the range $\lambda \lesssim 2$, one observes that all curves collapse onto a single curve to good accuracy. In contrast, for stronger couplings, sizable deviations occur.

Figure 7

The ratio (24) for fluctuation initial conditions with $\lambda =0.01$, 1, 10 (top to bottom). In a perturbative regime, $I_F$ is expected to be dominated by the lowest one-loop contribution ${\mathrm{\Pi }}_F$ such that the ratio (24) is about 1. While at high momenta the ratio approaches unity, in the infrared regime, one observes very strong deviations from 1 for times where the transient inverse particle cascade occurs.

Figure 8

The sum $f(t,|\mathbf{p}|)+1/2$ for $\lambda =10$ at time $Qt=1000$ for the quantum (dashed line) and the classical (solid line) evolution for fluctuation initial conditions. In addition, we show the classical thermal function $T/(\omega (\mathbf{p})-\mu )$ (gray, dotted) for comparison, with parameters as described in the main text. One observes that, in contrast to the quantum evolution, the classical results for $f(t,|\mathbf{p}|)+1/2$ drop below $1/2$ at high momenta, showing the decay of the initial quantum-half.

Figure 9

Longitudinal ($\parallel$, left) and transverse ($\perp$, right) occupation number distributions for macroscopic field initial conditions with $\lambda =0.01$ at different times showing the emergence of a dual cascade picture.

Figure 10

As in Fig. 9, but for $\lambda =1$, for which the system is still showing a dual cascade picture.

Figure 11

As in Fig. 9, but for $\lambda =10$, where strong deviations from the dual cascade picture occur. The longitudinal spectrum approaches thermal equilibrium rather quickly. At the same time, the transverse occupation number distribution exhibits a cascadelike behavior for small momenta, while the ultraviolet already shows signs of thermalization.

Figure 12

Longitudinal ($\parallel$, left) and transverse ($\perp$, right) inverse slope parameters $\ln (1+1/f_{\parallel ,\perp })$ for macroscopic field initial conditions with $\lambda =10$ at different times. We observe the approach to thermal equilibrium. For comparison, the dashed line in each plot corresponds to a Bose-Einstein distribution with temperature $T\approx {\sigma }_0$ and chemical potential $\mu \approx 0.7{\sigma }_0$.

Figure 13

Rescaled longitudinal ($\parallel$, left) and transverse ($\perp$, right) occupation number distributions for macroscopic field initial conditions at fixed time ${\sigma }_{0}t=940$ for different values of the coupling $\lambda$.

Figure 14

Double-logarithmic plot of the time evolution of the envelope ${\varphi }_{\max }(t)$ normalized to its initial value ${\varphi }_0$ for $\lambda =0.01$. In the inset, the oscillating field is displayed. The field amplitude is $\sim t^{-c}$ with $c\approx 0.5$ at intermediate times and with a much smaller exponent of the order of a few percent during the turbulent stages of the evolution at later times.

Figure 15

For $\lambda =1$, the power law of the field amplitude ${\varphi }_{\max }(t)$ is $\sim t^{-c}$ with $c\approx 0.4$ for intermediate times (note the log-lin plot). At later times, an exponential damping with a small rate $\sim {10}^{-4}{\sigma }_0$ is found.

Figure 16

For $\lambda =10$, we observe an exponential decay of the field amplitude ${\varphi }_{\max }(t)$ with a rate of about $\sim {10}^{-3}{\sigma }_0$ (dashed line).

Figure 17

The ratio (24) for macroscopic field initial conditions with $\lambda =10$ for different times.