Viscosity of gauge theory plasma with a chemical potential from AdS/CFT correspondence

TL;DR

Using the AdS/CFT correspondence, the paper studies the shear viscosity of a strongly coupled gauge theory plasma at finite chemical potential via quasinormal modes of the five-dimensional RN-AdS$_5$ black hole. By decoupling vector perturbations into $\Phi_\pm$ and locating the hydrodynamic diffusion pole, it extracts the diffusion constant $D_η$ and shows that the viscosity-to-entropy ratio satisfies $η/s = 1/(4π)$, independent of the chemical potential (within numerical error). The results support the universality of holographic transport at finite density and demonstrate how quasinormal-mode analysis links gravity perturbations to hydrodynamic coefficients in charged AdS black holes. The work reinforces the robustness of the KS bound in charged, strongly coupled plasmas and informs finite-density behavior in holographic models.

Abstract

We compute the strong coupling limit of the shear viscosity for the N=4 super-Yang-Mill theory with a chemical potential. We use the five-dimensional Reissner-Nordstrom-anti-deSitter black hole, so the chemical potential is the one for the R-charges U(1)_R^3. We compute the quasinormal frequencies of the gravitational and electromagnetic vector perturbations in the background numerically. This enables one to explicitly locate the diffusion pole for the shear viscosity. The ratio of the shear viscosity eta to the entropy density s is eta/s=1/(4pi) within numerical errors, which is the same result as the one without chemical potential.

Figures (3)

• Figure 1: (color online). QN spectrum of the perturbations $\Phi_-\:$ (left panel) and $\Phi_+\:$ (right). Dependence of $\omega/2 \pi T_{\text{H}}$ on $r_-/r_+ = 0$($\times$), $0.5$($+$), and $0.95$ ($\divideontimes$) is shown for spatial momentum $k/2 \pi T_{\text{H}} = 1.0$. On the left panel, the poles which are enclosed by the elongated circle are the hydrodynamic pole discussed in Sec. \ref{['sec:hydro']}. In contrast, there is no such poles for $\Phi_+$. All poles stay at a finite distance from the real axis.
• Figure 2: (color online). Dependence of $\omega_I/2 \pi T_{\text{H}}$ on $(k/2 \pi T_{\text{H}})^2$ for $r_-/r_+ = 0$($\times$), $0.25$($+$), $0.5$($\divideontimes$), $0.75$($\boxdot$), and $0.95$($\blacksquare$). The slope determines the diffusion constant $D_\eta$.
• Figure 3: (color online). Dependence of four low-lying QNM's on $k/2 \pi T_{\text{H}}$ for $r_-/r_+ = 0$($\times$), $0.5$($+$). The left panel shows the real part of $\omega/2 \pi T_{\text{H}}$ and the right shows the imaginary part. The hydrodynamic pole is the one which approaches $\omega\rightarrow 0$ as $k \rightarrow 0$ (the bottom one on the left panel and the top one on the right panel). As one can see, the other poles stay at a finite distance from the origin.