Hydrodynamics of cold holographic matter

TL;DR

This work demonstrates that hydrodynamics describes the low-energy transverse excitations of the energy-momentum tensor and current in a strongly coupled field theory dual to planar RN-AdS$_4$ for all relevant energy scales, including the zero-temperature limit. Using a matched asymptotic analysis, the authors show a diffusion mode with diffusion constant $D=\eta/(\varepsilon+P)$ persists beyond the usual $\omega\ll T$ regime, with the decay rate controlled by the near-horizon IR geometry. At $T=0$, the diffusion is governed by an IR CFT in the AdS$_2$ near-horizon region, while for small $T$ the Schwarzschild-AdS$_2$ region provides the leading corrections; these IR data fix the $q$-dependence of the dispersion. Numerical studies of poles and spectral functions corroborate the analytic results, confirming the diffusion mode and its hot/cold temperature behavior, and highlighting the role of the IR geometry in holographic hydrodynamics.

Abstract

We show that at any temperature, the low-energy (with respect to the chemical potential) collective excitations of the transverse components of the energy-momentum tensor and the global U(1) current in the field theory dual to the planar RN-AdS4 black hole are simply those of hydrodynamics. That is, hydrodynamics is applicable even at energy scales much greater than the temperature. It is applicable even at zero temperature. Specifically, we find that there is always a diffusion mode with diffusion constant proportional to the ratio of entropy density to energy density. At low temperatures, the leading order momentum and temperature dependences of the dispersion relation of this mode are controlled by the dimension of an operator in the thermal CFT1 dual to the near-horizon Schwarzschild-AdS2 geometry.

Figures (4)

• Figure 1: The locations of the purely imaginary 'poles' nearest the origin as determined by Leaver's method when $q/\mu=0.1$ (shown as dots), as a function of the order of the power series expansion used $N_\text{max.}$. The red line shows the analytic result (\ref{['eq:ZeroTResultforDispersionRelation']}) for the $T=0$ diffusion mode.
• Figure 2: Dispersion relations of the purely imaginary 'poles' nearest the origin as determined by Leaver's method, at fixed $N_\text{max.}=300$ (shown as dots). The red line shows the analytic result (\ref{['eq:ZeroTResultforDispersionRelation']}) for the $T=0$ diffusion mode. The data on both plots is the same -- the left hand plot is a magnified version of the near-origin region of the right hand plot.
• Figure 3: The numerical values of the dimensionless diffusion constant $\mu\mathcal{D}$ (shown as dots) as a function of $T/\mu$ as computed by $N_\text{max.}=300$ Leaver's method (left hand plot) and direct numerical integration from the horizon (right hand plot). These were extracted by fitting the function $\omega\left(q\right)=-i\mathcal{D}q^2-i\Delta q^4$ to the numerical location of the pole $\omega\left(q\right)$ at $q/\mu=\left\{0.01,0.05,0.1,0.2,0.3,0.4,0.5\right\}$. The red lines show the hydrodynamic result for $\mu\mathcal{D}$, given by equations (\ref{['eq:ExpectedHydroDiffusionResult']}) and (\ref{['eq:IntroEtaOverS']}), which according to our previous analysis should be valid for all $T/\mu$ shown.
• Figure 4: Numerical results for the $q/\mu=0.1$ spectral functions $\chi_{T^{ty}T^{ty}}$ (shown as dots) and $\chi_{J^yJ^y}$ (shown as crosses), computed via direct numerical integration from the horizon, for $T/\mu\approx0.005$ (left hand plot) and $T/\mu\approx0.16$ (right hand plot). $\chi_{T^{ty}T^{ty}}$ is plotted in units of $r_0^3/2\kappa_4^2L^4$ and $\chi_{J^yJ^y}$ is plotted in units of $r_0/2\kappa_4^2$. The analytic results (\ref{['eq:HydroGreensFunctionsResult']}) for $\chi_{T^{ty}T^{ty}}$ (dashed line) and $\chi_{J^yJ^y}$ (solid line) are plotted in red.