Holography and the sound of criticality

TL;DR

This paper uses gauge/gravity duality with an extremal RN-AdS$_4$ background to study the longitudinal (sound) channel of a class of $(2+1)$-dimensional strongly coupled field theories at zero temperature and finite density. It introduces new gauge-invariant master fields $\Psi_{\pm}$ to obtain a clean holographic dictionary for retarded Green's functions, and analyzes the full QNM spectrum through a combination of analytical matched asymptotics (inner AdS$_2$ region) and Leaver's numerical method. The authors find two zero-temperature sound modes with dispersion $\omega_s(k)= c_s k - i\Gamma_s k^2$ and $c_s \approx 1/\sqrt{2}$ (numerically $c_s\approx 0.704$) and $\mu\Gamma_s \approx 0.083$, consistent with the zero-temperature hydrodynamic limit, while IR correlators exhibit emergent scaling dictated by the AdS$_2$ throat and a branch cut along the negative imaginary axis. Higher resonances are analyzed via residues and Green's functions, showing that the sound modes dominate the IR spectral function for $k \lesssim \mu$, and that the full correlators can be effectively captured by including the sound sector at low frequencies. The results illuminate quantum critical IR dynamics in holographic 2+1D systems and provide a framework for exploring emergent criticality and dissipative sound propagation in strongly coupled media.

Abstract

Using gauge/gravity duality techniques, we discuss the sound-channel retarded correlators of vector and tensor conserved currents in a class of $(2+1)$-dimensional strongly-coupled field theories at zero temperature and finite charge density, assumed to be holographically dual to the extremal Reissner-Nordström AdS$_4$ black hole. Using a combination of analytical and numerical methods, we determine the quasinormal mode spectrum at finite momentum for the coupled gravitational and electromagnetic perturbations, and discuss the appropriate choice of gauge-invariant variables (master fields) in order for the black hole quasinormal frequencies to reproduce the field theory spectrum. We discuss the role of the near horizon AdS$_{2}$ geometry in determining the low-frequency behavior of retarded correlators in the boundary theory, and comment on the emergence of criticality in the IR. In addition, we establish the existence of a sound mode at zero temperature and compute the speed of sound and sound attenuation constant numerically, obtaining a result consistent with the expectations from the zero temperature limit of hydrodynamics. The dispersion relation of higher resonances is also investigated.

Figures (6)

• Figure 1: Sound-channel electromagnetic and gravitational quasinormal frequencies for the extremal Reissner-Nordström AdS$_4$ black hole. Plot (a) shows the quasinormal frequencies of $\Psi_{+}$, with the sound modes depicted in blue. Plot (b) shows the quasinormal frequencies of $\Psi_{-}$. For both plots, $\textswab{q}=0.5$ and $M=300$. Plot (c) shows the spectra of $\Psi_{+}$ and $\Psi_{-}$ superimposed.
• Figure 2: Dispersion relation of the sound modes. Plot (a) depicts the imaginary part of $\textswab{w}_{s}$ as a function of momenta, while plot (b) shows the real part of $\textswab{w}_{s}$. The dots represent the actual values obtained numerically, while the solid line results from fitting the data by a smooth curve. The inset plots compare the expected dispersion relation \ref{['expected sound dispersion']} (red line) with the values obtained numerically (black line), for small values of the momenta. For large momentum, the slope of the real part becomes unity within numerical precision.
• Figure 3: Dispersion relation of the first five overtones of $\Psi_{+}$. Plot (a) shows the variation of the imaginary part of the corresponding quasinormal frequencies as we vary the momentum $\textswab{q}$. Plot (b) shows the corresponding change in the real part of the frequency.
• Figure 4: Dispersion relation of the first five overtones of $\Psi_{-}$. Plot (a) shows the variation of the imaginary part of the corresponding quasinormal frequencies as we vary the momentum $\textswab{q}$. Plot (b) shows the corresponding change in the real part of the frequency.
• Figure 5: Absolute value of the sound pole residue as a function of momentum. Plot (a) shows the residue of $G_{tt,tt}$ ; plot (b) shows the residue of $G_{x,x}$.