Holographic shear correlators at low temperatures, and quantum $η/s$

TL;DR

The paper investigates quantum corrections to IR dynamics in a holographic, near-extremal AdS$_4$ setup by coupling a 3D finite-density CFT to an AdS$_2$ throat whose nearly gapless Schwarzian modes control low-energy fluctuations. Using a Schwarzian path-integral treatment, it derives quantum corrections to the IR Green's function and, via UV–IR matching, determines the quantum-modified shear viscosity $\eta$ and the ratio $\eta/s$ across multiple regimes defined by $\omega$, $T$, and the gap $E_{gap}$. In the semiclassical limit, $\eta^{\text{sc}} = s^{\text{sc}}/(4\pi)$, but quantum Schwarzian effects produce substantial deviations: $\eta^{\text{qu}}/\eta^{\text{sc}}$ grows like $1/\sqrt{CT}$ for $CT\ll1$ and approaches the classical value for $CT\gg1$, with $CT=T/E_{gap}$. Notably, $\eta/s$ dips below $1/(4\pi)$ when $E_{gap}\ll T\ll \mu$ and diverges as $T\to0$, signaling a transition to glassy low-energy dynamics; these results connect black-hole quantum fluctuations to transport in strongly coupled quantum liquids and provide a framework for exploring quantum near-extremal holography.

Abstract

The strongly-coupled 3-dimensional theory, holographically dual to black branes at fixed chemical potential $\muext$ and temperature $T \ll μ$ is considered in AdS$_4$ Einstein-Maxwell theory. The retarded Green's functions at frequency $ω$ is calculated using holography in the regime $ω, T \ll \muext$ but otherwise arbitrary. When the transverse space has finite volume, there is a non-zero energy scale $E_\text{gap}$, scaling as $1/μ$ for large $μ$, below which quantum-gravitational corrections due to the fluctuations of the nearly-gapless Schwarzian modes become important. Such corrections to the retarded Green's function are calculated at different relative values of $ω$, $T$, and $E_\text{gap}$. The $ω\to 0$ limit is used to define the shear viscosity $η$. As the temperature is lowered below $μ$, quantum corrections are found to increase the value of $η$ with respect to its semiclassical value. The quantum-corrected result for $η$ diverges as $\sqrt{E_\text{gap}/T}$ at $T \ll E_\text{gap}$, in accord with corresponding results for the absorption cross section. The quantum result for the ratio $η/s$, where $s$ is the entropy density, dips below the semiclassical limit of $1/4π$ when $E_\text{gap} \ll T \ll μ$, then turns back to increase towards lower temperatures, and finally diverges at temperatures much below $E_\text{gap}$.

Figures (6)

• Figure 1: Semiclassical entropy $S^\text{s.c.}$ and quantum entropy $S^\text{qu}$ as a function of $CT$. The quantum entropy is calculated in the exact Schwarzian theory as in Iliesiu:2020qvm. In this plot we choose $\mu=10$. At large values of $CT$ the quantum entropy reaches the semiclassical limit, while it approaches zero at small $CT$. It can be trusted at the lower end when $CT$ is much larger than the cutoff scale when the above quantum entropy curve hits zero. This scale ($\approx e^{-\frac{2}{3} S_0} \approx 10^{-29}$) is where non-perturbative effects come into play.
• Figure 2: Plot of $\eta^\text{qu}/\eta^\text{s.c.}$ calculated in the exact Schwarzian theory with $\mu=10$. At large values of $CT$ this reaches the semiclassical limit of 1, while at small $CT$ there is a divergence of the form $1/\sqrt{CT}$. This result can be trusted at the lower end when $CT$ is much above the non-perturbative scale $\approx e^{-\frac{2}{3} S_0} \approx 10^{-29}$.
• Figure 3: Plot of $4 \pi \, \eta^\text{qu}(CT)/s^\text{qu}(CT)$ calculated in the exact Schwarzian theory with $\mu=10$. This ratio reaches the value 1 asymptotically as $CT\gg 1$. There is a minimum value at $CT \approx 15$. At very small values of $T/E_\text{gap}$ (but still larger than the non-perturbative scale $\approx e^{-\frac{2}{3} S_0} \approx 10^{-29}$), the curve has a divergence of the form $\sqrt{E_\text{gap}/T}$.
• Figure 4: Plots of $g_{_T}(k)$ calculated with $CT=0.1, 1, 4$. Note that the ranges of the vertical axes in the three plots are different. In each case there is a bell-shaped region, but the maximum value increases rapidly with $T$.
• Figure 5: Plot of $f(k,\omega)$ and $1+ \frac{k}{\pi C\omega}$ calculated with $C\omega=0.1$. $f$ is approximately constant for small values of $k$, and $f$ is approximately a linear function at large values of $k$.