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

A bstract The strongly-coupled 3-dimensional theory, holographically dual to black branes at fixed chemical potential μ and temperature T ≪ 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 ≪ μ but otherwise arbitrary. When the transverse space has finite volume, there is a non-zero energy scale E 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 gap . The ω → 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 $$ \sqrt{{\mathrm{E}}_{\mathrm{gap}}/T} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>gap</mml:mi> </mml:msub> <mml:mo>/</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> </mml:msqrt> </mml:math> at T ≪ E 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 1/4 π when E gap ≪ T ≪ μ , then turns back to increase towards lower temperatures, and finally diverges at temperatures much below E gap .