Holographic entanglement entropy and generalized entanglement temperature

In this work we study the flow of holographic entanglement entropy in dimensions d >= 3 in the gauge/gravity duality setup. We observe that a generalized entanglement temperature T-g can be defined which gives the Hawking temperature T-H in the infrared region and leads to a generalized thermodynamics like law E = (d-1/d)TgSREE, which becomes an exact relation in the entire region of the subsystem size l, including both the infrared (l -> infinity) as well as the ultraviolet (l -> 0) regions. Furthermore, in the IR limit, T (g) produces the Hawking temperature T (H) along with some correction terms which bears the signature of short distance correlations along the entangling surface. Moreover, for d >= 3, the IR limit of the renormalized holographic entanglement entropy gives the thermal entropy of the black hole as the leading term, however, does not have a logarithmic correction to the leading term unlike the Banados, Teitelboim, Zanelli (BTZ) black hole (d = 2) case. The generalized entanglement temperature T (g) also firmly captures the quantum mechanical to thermal crossover in the dual field theory at a critical value l (c) of the subsystem size in the boundary which we graphically represent for AdS(3+1) and AdS(4+1) black holes. We observe that this critical value l (c) where the crossover takes place decreases with increase in the dimension of the spacetime.