Bosonic excitations of the AdS4 Reissner-Nordstrom black hole

TL;DR

This work analyzes the long-lived excitations in the strongly coupled, finite-density theory dual to RN-$AdS_4$ by studying the poles of longitudinal Green's functions and their spectral functions. For $q\ll\mu$, it identifies a sound mode with $v_s\approx1/\sqrt{2}$ and a purely imaginary diffusion-like mode, mapping how their residues and lifetimes evolve with temperature and momentum. The energy-density spectrum remains dominated by the sound mode across temperatures, while the charge-density spectrum crosses from sound to diffusion dominance at a crossover $T_{\text{cross}}\sim\mu^2/q$. An effective hydrodynamic scale is proposed to explain the propagation window, and the results are contrasted with the D3/D7 system, showing that zero-sound-like behavior is not universal among holographic finite-density theories.

Abstract

We study the long-lived modes of the charge density and energy density correlators in the strongly-coupled, finite density field theory dual to the AdS4 Reissner-Nordstrom black hole. For small momenta q<<μ, these correlators contain a pole due to sound propagation, as well as a pole due to a long-lived, purely imaginary mode analogous to the μ=0 hydrodynamic charge diffusion mode. As the temperature is raised in the range T\lesssimμ, the sound attenuation shows no significant temperature dependence. When T\gtrsimμ, it quickly approaches the μ=0 hydrodynamic result where it decreases like 1/T. It does not share any of the temperature-dependent properties of the 'zero sound' of Landau Fermi liquids observed in the strongly-coupled D3/D7 field theory. For such small momenta, the energy density spectral function is dominated by the sound mode at all temperatures, whereas the charge density spectral function undergoes a crossover from being dominated by the sound mode at low temperatures to being dominated by the diffusion mode when T μ^2/q. This crossover occurs due to the changing residue at each pole. We also compute the momentum dependence of these spectral functions and their corresponding long-lived poles at fixed, low temperatures T<<μ.

Figures (21)

• Figure 1: The sound attenuation $\Gamma$ in an LFL as a function of temperature, at fixed $\omega$ and $\mu$. A is the collisionless quantum regime, B is the collisionless thermal regime and C is the hydrodynamic regime.
• Figure 2: Variation of the real part of the sound mode as the temperature is increased. The crosses mark the $T=0$ numerical results, the dots are the numerical results for $T>0$, and the solid lines are the $\mu=0$ analytic result (\ref{['eq:herzogsound']}).
• Figure 3: Variation of the imaginary part of the sound mode as the temperature is increased. The crosses marks the $T=0$ numerical results, the dots are the numerical results for $T>0$, and the solid lines are the $\mu=0$ analytic result (\ref{['eq:herzogsound']}).
• Figure 4: Variation of the imaginary part of the sound mode as the temperature is increased, in the regime $T<\mu$. The dots are the numerical results for $T>0$, and the two dashed lines on each plot denote $T/\mu=q/\mu$ and $T/\mu=\sqrt{q/\mu}$ as one moves to the right along the plot.
• Figure 5: A superposition of the plots of the temperature dependence of the normalised imaginary part of the sound mode when $q/\mu=0.2$ for both the D3/D7 theory and the RN-$AdS_4$ theory. Crosses denote the D3/D7 numerical results Davison:2011ek and circles denote the RN-$AdS_4$ results. Moving from left to right, the dotted lines mark the transition points between the quantum and thermal collisionless regimes, and the thermal collisionless regime and the hydrodynamic regime, in the D3/D7 theory. These occur when $\omega\sim T$ and $\omega\sim T^2/\mu$ respectively. There are no results for the D3/D7 sound mode in the hydrodynamic regime since the hydrodynamic sound mode is suppressed in the probe brane limit. We refer the reader to Davison:2011ek for a more detailed discussion of these features.