Understanding the approach to thermalization from the eigenspectrum of non-Abelian gauge theories

Abstract

We study some interesting aspects of the spectral properties of SU(3) gauge theory, both with and without dynamical quarks (QCD) at thermal equilibrium using lattice gauge theory techniques. By calculating the eigenstates of a massless overlap Dirac operator on the gauge configurations, we implement a gauge-invariant method to study spectral properties of non-Abelian gauge theories. We have unambiguously categorized Dirac eigenvalues into different regimes based on a quantity defined in terms of the ratios of nearest neighbor spacings. While majority of these eigenstates below the magnetic scale are similar to those of random matrices belonging to the Gaussian Unitary ensemble at temperatures much higher than the chiral crossover transition in QCD, a few among them start to become prominent only near the crossover. These form fractal-like clusters with the median value for their fractal dimensions hinting at the universality class of the chiral transition in QCD. We further demonstrate that momentum modes below the magnetic scale in a particular non-equilibrium state of QCD are classically chaotic and estimate an upper bound on the thermalization time $\sim 1.44$ fm/c by matching this magnetic scale with that of a thermal state at $\sim 600$ MeV.

Figures (4)

• Figure 1: Left panel: Average value $\langle \tilde{r} \rangle$ in different spectral windows defined in terms of the overlap Dirac eigenvalues at various temperatures $T$. Red and blue points correspond to bulk and intermediate eigenmodes, respectively. The results are compared with the theoretical expectations for the GUE and completely uncorrelated (Poisson) spectra, shown as red and black horizontal lines, respectively. Right panel: Probability distribution of ratios of consecutive level spacings for bulk eigenmodes at different temperatures with and without dynamical quarks. Note that for pure SU(3) gauge ensemble at $T=624$ MeV, the $T_c$ represents the deconfinement temperature $T_d$.
• Figure 2: Variation of first Rényi entropy (top panel) and $D_2$ (bottom panel), for eigenvectors binned in $\lambda/T$ for different temperatures, performed on the gauge configurations mentioned in table \ref{['tab:table1']}.
• Figure 3: Probability distribution of the fractal dimension $D_f$ at different temperatures and in eigenvalue intervals characterized as the intermediate (left panel) and the bulk eigenmodes (right panel).
• Figure 4: Scaling of the Lyapunov exponent $\gamma$ for SU(3) with the energy density $\varepsilon$. The exponential growth of the distance function $D(t)$ between two classical trajectories in the gauge space for different values of initial gluon density is shown in the inset.