Shear Modes, Criticality and Extremal Black Holes

TL;DR

The paper investigates a (2+1)-dimensional strongly coupled boundary theory at zero temperature and finite density, using its holographic dual extremal Reissner–Nordström AdS$_4$ black hole. By analyzing odd-parity bulk fluctuations through gauge-invariant master fields, it reveals an emergent IR scaling governed by an AdS$_2$ near-horizon region, and derives the structure of retarded Green functions in the shear channel, including a branch cut along the negative imaginary frequency axis and a discrete set of damped quasinormal modes. The authors develop a two-region matching framework to extract IR data, demonstrate the IR scaling exponents depend on a single IR CFT operator, and confirm, via Leaver-type numerics, that all QNMs lie in the lower-half plane, signaling stability. Upon introducing finite temperature, the branch cut dissolves into a sequence of poles with negative imaginary parts, yielding a hydrodynamic diffusion pole in the small-$q$ limit and recovering the universal shear viscosity result $\eta/s=1/4\pi$; as $T\to 0$ this finite-temperature spectrum smoothly approaches the zero-temperature branch-cut structure. Overall, the work clarifies how emergent IR quantum criticality manifests in vector and tensor retarded correlators and maps the analytic structure of these correlators to the near-horizon geometry of extremal black holes.

Abstract

We consider a (2+1)-dimensional field theory, assumed to be holographically dual to the extremal Reissner-Nordstrom AdS(4) black hole background, and calculate the retarded correlators of charge (vector) current and energy-momentum (tensor) operators at finite momentum and frequency. We show that, similar to what was observed previously for the correlators of scalar and spinor operators, these correlators exhibit emergent scaling behavior at low frequency. We numerically compute the electromagnetic and gravitational quasinormal frequencies (in the shear channel) of the extremal Reissner-Nordstrom AdS(4) black hole corresponding to the spectrum of poles in the retarded correlators. The picture that emerges is quite simple: there is a branch cut along the negative imaginary frequency axis, and a series of isolated poles corresponding to damped excitations. All of these poles are always in the lower half complex frequency plane, indicating stability. We show that this analytic structure can be understood as the proper limit of finite temperature results as T is taken to zero holding the chemical potential fixed.

Figures (7)

• Figure 1: Electromagnetic and gravitational quasinormal frequencies (in the shear channel) of extremal Reissner-Nordström AdS$_4$ black hole. Plot (a) shows the quasinormal frequencies of $\Phi_{+}$ and plot (b) shows the quasinormal frequencies of $\Phi_{-}$. For both plots, $\textswab{q}=1$ and $M=100$. As $M$ is increased the poles on the negative imaginary axis get closer to one another and form a branch cut in the $M\to\infty$ limit.
• Figure 2: Quasinormal frequencies of $\Phi_\pm$ (in the extremal case) for $\textswab{q}=1/\sqrt{3}$. (a) Quasinormal frequencies of $\Phi_{+}$. (b) Quasinormal frequencies of $\Phi_{-}$. $M=100$ for both plots.
• Figure 3: Quasinormal frequencies for $\textswab{q}=0$ of $a_y$ and ${h^x}_y$ in the extremal Reissner-Nordström AdS$_4$ black hole background. Plot (a) shows the quasinormal frequencies of $a_y$ and plot (b) are the quasinormal frequencies of ${h^x}_y$. We set $M=100$ for both plots.
• Figure 4: Quasinormal frequencies of $\Phi_{\pm}$ in non-extremal Reissner-Nordström AdS$_4$ black hole background. Plot (a) shows the quasinormal frequencies of $\Phi_{+}$ while plot (b) shows the quasinormal frequencies of $\Phi_{-}$. For both plots, $\textswab{q}=1$, $M=100$, and $T=0.09\mu$.
• Figure 5: The lowest quasinormal frequency of $\Phi_{-}$ as function of $q$. The temperature is $T=0.09\mu$, and $M=100$. The solid black curve is the quadratic fit: $i \textswab{w}\simeq 0.17\textswab{q}^2$.