Hydrodynamics and beyond in the strongly coupled N=4 plasma

TL;DR

This work examines how hydrodynamic modes and higher quasinormal modes shape the real-time response of the strongly coupled N=4 plasma via AdS/CFT. By numerically computing residues of retarded Green functions for energy-momentum and R-charge sectors, it reveals that diffusion residues decay and decouple at short wavelengths, while sound residues persist, effectively behaving like higher resonances. The authors demonstrate causality by showing front velocities approach the light cone (v_F = 1) and define hydrodynamic time and length scales that delineate the regime of validity for hydrodynamics. They also show that including higher resonances, properly regularized by analytic terms, is essential to accurately reconstruct spectral functions and understand the breakdown of hydrodynamics at finite momentum. Overall, the results inform how to extend hydrodynamic descriptions and interpret spectral data in strongly coupled plasmas.

Abstract

We continue our investigations on the relation between hydrodynamic and higher quasinormal modes in the AdS black hole background started in arXiv:0710.4458 [hep-th]. As is well known, the quasinormal modes can be interpreted as the poles of the retarded Green functions of the dual N=4 gauge theory at finite temperature. The response to a generic perturbation is determined by the residues of the poles. We compute these residues numerically for energy-momentum and R-charge correlators. We find that the diffusion modes behave in a similar way: at small wavelengths the residues go over into a form of a damped oscillation and therefore these modes decouple at short distances. The sound mode behaves differently: its residue does not decay and at short wavelengths this mode behaves as the higher quasinormal modes. Applications of our findings include the definition of hydrodynamic length and time scales. We also show that the quasinormal modes, including the hydrodynamic diffusion modes, obey causality.

Figures (11)

• Figure 1: Imaginary (left) and real (right) parts of the retarded $G_{tt}$ correlator as a function of the frequency at ${\mathfrak{q}}=0.2$ (black) and ${\mathfrak{q}}=0.4$ (grey). The dotted line is the hydrodynamic mode contribution, the solid line is the exact solution and the dashed line is the four-mode approximation.
• Figure 2: (Left) Real and imaginary parts of the residues for the first four complex momentum modes in the transverse component $\Pi_T$. (Right) Idem for longitudinal component $\Pi_L$. The numerical values are normalized by $N^2 T^2/8$ and the square of the mode number.
• Figure 3: Real and imaginary parts of the residues for the diffusion complex momentum mode. The numerical values are again expressed in terms of $N^2 T^2 /8$.
• Figure 4: (Left) The numerical value of the shear mode [solid black] compared with the correct expression [dotted gray] and the value from second order hydrodynamics [dashed gray]. (Middle) The numerical value of the real part of the sound mode [solid black] compared with the second order hydrodynamic approximation [dotted gray]. (Right) The numerical value of the imaginary part of the sound mode [solid black] compared with the second order hydrodynamic approximation [dotted gray].
• Figure 5: (Left) Real and imaginary parts of the first four complex momentum modes in the sound channel. (Right) Idem for sound mode.