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

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.

💡 Research Summary

In this work the authors investigate the spectral properties of SU(3) gauge theory, both in the presence and absence of dynamical quarks, by computing the eigenvalues of a massless overlap Dirac operator on lattice gauge configurations. The overlap operator preserves an exact chiral symmetry on the lattice, making its eigenvalues gauge‑invariant probes of the underlying gauge fields. Using a large set of thermal ensembles generated with Möbius domain‑wall fermions (for 2+1‑flavor QCD) and pure Wilson gauge action (for the quenched case), they cover temperatures from just below the chiral crossover (≈149 MeV) up to a very hot gluonic plasma (≈624 MeV).

The central diagnostic is the ratio of consecutive level spacings, r_n = s_{n+1}/s_n with s_n = λ_{n+1}−λ_n, and its symmetrized average ⟨˜r⟩ = ⟨min(r_n,1/r_n)⟩. This quantity does not require unfolding of the spectrum and can be directly compared with the universal predictions of random matrix theory (RMT). The authors scan the spectrum in small λ/T bins and identify two distinct regimes: (i) “bulk modes” where ⟨˜r⟩ matches the Gaussian Unitary Ensemble (GUE) value 0.60266, and (ii) “intermediate modes” where ⟨˜r⟩ lies between the GUE prediction and the Poisson (uncorrelated) limit 0.386. The bulk modes are present already at temperatures well above the deconfinement temperature and persist up to the highest studied temperature; the intermediate modes become prominent only as the system approaches the chiral crossover temperature T_c ≈ 156 MeV.

To verify the identification of bulk modes, the full nearest‑neighbour spacing ratio distribution P(r) is compared with the analytical GUE form, showing excellent agreement for all temperatures. The authors then examine the spatial structure of the eigenvectors ψ(x) using the Renyi entropy R_α (α=1) defined as R_1 = –∑_x p_x ln p_x with p_x = |ψ(x)|². For bulk modes, R_1/ln V lies in the range 0.92–1.0, approaching unity at the highest temperature, indicating that these eigenstates are fully delocalized (ergodic) across the four‑dimensional volume. This delocalization is a direct manifestation of the Eigenstate Thermalization Hypothesis (ETH) in a non‑Abelian gauge theory.

In contrast, intermediate modes exhibit lower R_1/ln V values, signalling partial localization. A multifractal analysis yields a fractal dimension D_f ≈ 2.5–2.8, i.e. between the extremes of full delocalization (D_f=3) and point‑like localization (D_f=0). The emergence of such fractal clusters near T_c suggests a connection with the universality class of the chiral phase transition.

Beyond equilibrium properties, the paper studies a highly occupied non‑thermal gluonic state. Classical SU(3) gauge fields are initialized on a three‑dimensional lattice (N=64) with momentum modes below the magnetic scale k_mag ≈ g²T over‑occupied. By evolving these configurations with Hamiltonian dynamics, the authors compute the Lyapunov exponent λ from the exponential growth of infinitesimal perturbations. They find λ ≈ 0.14 fm⁻¹, confirming that low‑momentum gluon modes are classically chaotic. Matching the magnetic scale of this chaotic state to that of a thermal ensemble at T ≈ 600 MeV yields an upper bound on the thermalization time τ ≈ 1/λ ≈ 1.44 fm/c.

The presence of dynamical quarks does not alter the GUE statistics of bulk modes; instead, quarks affect the running of the coupling and thus the magnetic scale, leading to a faster thermalization of the low‑momentum gluons. Near T_c, however, quark effects become important for the structure of intermediate modes and for driving the chiral crossover.

In summary, the study combines three complementary tools—(1) level‑spacing ratio ⟨˜r⟩ to separate chaotic and non‑chaotic spectral windows, (2) Renyi entropy and multifractal analysis to quantify eigenstate localization, and (3) classical chaos diagnostics (Lyapunov exponent) to estimate thermalization times. It provides the first quantitative demonstration of the Bohigas‑Giannoni‑Schmit conjecture in a four‑dimensional non‑Abelian gauge theory, links fractal spectral features to the QCD chiral transition, and offers a concrete estimate of how quickly gluonic degrees of freedom below the magnetic scale can thermalize in heavy‑ion collisions.