Van der Waals-like Behaviour of Charged Black Holes and Hysteresis in the Dual QFTs

Using the rules of the AdS/CFT correspondence, we compute the spherical analogue of the shear viscosity, defined in terms of the retarded Green function for the stress-energy tensor for QFTs dual to five-dimensional charged black holes of general relativity with a negative cosmological constant. We show that the ratio between this quantity and the entropy density, $\tilde\eta/s$, exhibits a temperature-dependent hysteresis. We argue that this hysteretic behaviour can be explained by the Van der Waals-like character of charged black holes, considered as thermodynamical systems. Under the critical charge, hysteresis emerges owing to the presence of two stable states (small and large black holes) connected by a meta-stable region (intermediate black holes). A potential barrier prevents the equilibrium path between the two stable states; the system evolution must occur through the meta-stable region, and a path-dependence of $\tilde\eta/s$ is generated.