From strong to weak coupling in holographic models of thermalization

TL;DR

The paper studies how thermalization dynamics in strongly coupled gauge theories change as the coupling is reduced from infinity by computing finite-coupling corrections to holographic quasinormal spectra. Using higher-derivative gravity terms (R^4 and curvature-squared actions, including Gauss-Bonnet gravity), the authors analyze corrections to the three channels of metric perturbations and track the evolution of quasinormal poles, unveiling a coupling-dependent breakdown of hydrodynamics at a critical wave vector q_c. A key finding is that the hydrodynamic regime broadens with increasing coupling, while the ratio η/s times the relaxation time shows only modest variation away from its infinite-coupling value, suggesting a clumsy but useful link to kinetic theory intuition even at strong coupling. The work also reveals qualitatively distinct pole-density and spectral-function behaviors depending on whether η/s is above or below the universal infinite-coupling bound, and discusses implications for relaxation-time bounds and the viscosity–entropy connection in quantum systems.

Abstract

We investigate the analytic structure of thermal energy-momentum tensor correlators at large but finite coupling in quantum field theories with gravity duals. We compute corrections to the quasinormal spectra of black branes due to the presence of higher derivative $R^2$ and $R^4$ terms in the action, focusing on the dual to $\mathcal{N}=4$ SYM theory and Gauss-Bonnet gravity. We observe the appearance of new poles in the complex frequency plane at finite coupling. The new poles interfere with hydrodynamic poles of the correlators leading to the breakdown of hydrodynamic description at a coupling-dependent critical value of the wave-vector. The dependence of the critical wave vector on the coupling implies that the range of validity of the hydrodynamic description increases monotonically with the coupling. The behavior of the quasinormal spectrum at large but finite coupling may be contrasted with the known properties of the hierarchy of relaxation times determined by the spectrum of a linearized kinetic operator at weak coupling. We find that the ratio of a transport coefficient such as viscosity to the relaxation time determined by the fundamental non-hydrodynamic quasinormal frequency changes rapidly in the vicinity of infinite coupling but flattens out for weaker coupling, suggesting an extrapolation from strong coupling to the kinetic theory result. We note that the behavior of the quasinormal spectrum is qualitatively different depending on whether the ratio of shear viscosity to entropy density is greater or less than the universal, infinite coupling value of $\hbar/4πk_B$. In the former case, the density of poles increases, indicating a formation of branch cuts in the weak coupling limit, and the spectral function shows the appearance of narrow peaks. We also discuss the relation of the viscosity-entropy ratio to conjectured bounds on relaxation time in quantum systems.

Figures (31)

• Figure 1: The spectrum of a linear collision operator: a) discrete spectrum, b) continuous spectrum with a gap, realized for the interaction potential $U=\alpha/r^n$, $n>4$, c) gapless continuous spectrum, realized for the interaction potential $U=\alpha/r^n$, $n<4$, d) Hod spectrum (see text): $0 \leq \nu_{min} \leq \nu_c$. In all cases, $\nu =0$ is a degenerate eigenvalue corresponding to hydrodynamic modes (at zero spatial momentum).
• Figure 2: Phase diagram of the $d=1+1$ quantum Ising model sachdev-book-2. The relaxation time in the quantum critical region is determined by the lowest quasinormal frequency of the BTZ black hole.
• Figure 3: Singularities of a thermal two-point correlation function in the complex frequency plane at (vanishingly) small Hartnoll:2005ju (left panel) and infinitely large Starinets:2002br (right panel) values of the coupling.
• Figure 4: Singularities of thermal two-point correlation function of the energy-momentum tensor in the shear channel in the complex frequency plane at large coupling at $\eta/s>\hbar/4\pi k_B$ (left panel) and at $\eta/s<\hbar/4\pi k_B$ (right panel). Poles at infinitely large coupling are indicated by squares. At large but finite coupling, their new positions are shown by crosses.
• Figure 5: Poles (shown by squares) of the energy-momentum retarded two-point function of $\mathcal{N}=4$ SYM in the scalar channel, for various values of the coupling constant and $\mathfrak{q}=0.1$. From top left: $\gamma = \{10^{-5},\, 10^{-4}, \,10^{-3},\, 10^{-2}\}$ corresponding to values of the 't Hooft coupling $\lambda \approx \{609,\, 131, \, 28,\, 6\}$. Poles at $\gamma = 0$ ($\lambda\rightarrow \infty$) are shown by circles.