Holographic shear correlators at low temperatures, and quantum $\hat{I}\cdot /s$

Abstract

The strongly-coupled 3-dimensional theory, holographically dual to black branes at fixed chemical potential $\muext$ and temperature $T\ll \hat{I}¼$ is considered in AdS${}_4$ Einstein-Maxwell theory. The retarded Green's functions at frequency $\ddot{I}‰$ 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 $1/\hat{I}¼$ for large $\hat{I}¼$, 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 $\ddot{I}‰$, $T$, and $E_{\text{gap}}$. The $\ddot{I}‰\to 0$ limit is used to define the shear viscosity $\hat{I}\cdot$. As the temperature is lowered below $\hat{I}¼$, quantum corrections are found to increase the value of $\hat{I}\cdot$ with respect to its semiclassical value.The quantum-corrected result for $\hat{I}\cdot$ 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 $\hat{I}\cdot /s$, where $s$ is the entropy density, dips below the semiclassical limit of $1/4\ddot{I}€$ when $E_{\text{gap}}\ll T\ll \hat{I}¼$,then turns back to increase towards lower temperatures, and finally diverges at temperatures much below $E_{\text{gap}}$.