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Description
In this work, we calculate the eigenvalues of the probe (overlap) Dirac operator on thermal gauge ensembles of $2+1$ flavor QCD generated using domain wall fermions as well as pure $SU(3)$ gauge theory on the lattice. Focusing on the infrared part of the eigenspectrum that lies within the non-perturbative magnetic scale, we propose suitable observables that allow us to categorize different regions of the eigenspectrum unambiguously. While most of these eigenmodes are completely delocalized and chaotic, i.e. their nearest-neighbor level spacing fluctuations are similar to random matrices of a Gaussian unitary ensemble (GUE), we show that a classical non-thermal state of SU(3) gauge theory consisting of magnetic gluons is also chaotic, thus a non-trivial realization of the Bohigas-Giannoni-Schmit conjecture. This allows us to estimate an upper bound on the thermalization time $\sim 1.44$ fm/c of magnetic gluons by matching the magnetic scales in these two regimes. Furthermore we also observe the appearance of a few eigenmodes in the deep-infrared part of the spectrum having level spacing ratios that are intermediate between Poissonian ensemble and the GUE. Whereas close to the chiral crossover transition, their fractal dimensions carry information about the universality class, at higher temperatures when $U_A(1)$ is effectively restored their origin is due to the disorder present in gauge fields.