Transport coefficients at zero temperature from extremal black holes

Using the AdS/CFT correspondence we study transport coefficients of a strongly-coupled (2 + 1)-dimensional field theory at zero temperature and finite charge density. The field theory under consideration is dual to the extremal Reissner-Nordstrom AdS(4) black hole in the bulk. We show that, like the cases of scalar and spinor operators studied in [1], the correlators of charge (vector) current and energy-momentum (tensor) operators exhibit scaling behavior at low frequency. The existence of such low frequency behavior is related to the fact that the near-horizon geometry of the extremal black hole background has an AdS(2) factor. We carefully calculate the shear viscosity (at zero temperature) and show that the ratio of the shear viscosity to the entropy density takes the value of 1/4 pi. Because of the AdS(2) factor, we argue that this result stays the same for all d-dimensional boundary field theories dual to the extremal Reissner-Nordstrom AdS(d+1) black holes. Also, we compute the charge conductivity at zero temperature. The limiting behavior of the conductivity for small frequencies is also attributed to the near horizon AdS(2) factor and is argued to hold regardless of the dimension of the zero-temperature boundary field theory. Finally, using the extremal dyonic AdS(4) black hole as the background, we extract the conductivity in the presence of a constant magnetic field.