Holographic entanglement entropy and generalized entanglement temperature

In this work we study the flow of holographic entanglement entropy in dimensions $d\ge 3$ 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 $E=(\frac{d-1}{d})T_g{S}_{\mathrm{REE}}$, which becomes an exact relation in the entire region of the subsystem size $l$, including both the infrared ($l\to \infty$) as well as the ultraviolet ($l\to 0$) 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 $d\ge 3$, 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 Bañados, 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 ${\mathrm{AdS}}_{3+1}$ and ${\mathrm{AdS}}_{4+1}$ black holes. We observe that this critical value $l_c$ where the crossover takes place decreases with increase in the dimension of the spacetime.

Figure 1

Variation of the subsystem size $\frac{l}{z_h}$ with respect to $\frac{z_t}{z_h}$.

Figure 2

Variation of $\frac{{\beta }_g}{z_h}$ with respect to $\frac{l}{z_h}$ in ${\mathrm{AdS}}_{3+1}$ black hole.

Figure 3

Variation of $\frac{d(\frac{{\beta }_g}{z_h})}{d(\log (\frac{l}{z_h}))}$ with respect to $\frac{l}{z_h}$ in ${\mathrm{AdS}}_{3+1}$ black hole.

Figure 4

Variation of $\frac{{\beta }_g}{z_h}$ with respect to $\frac{l}{z_h}$ in ${\mathrm{AdS}}_{4+1}$ black hole.

Figure 5

Variation of $\frac{d(\frac{{\beta }_g}{z_h})}{d(\log (\frac{l}{z_h}))}$ with respect to $\frac{l}{z_h}$ in ${\mathrm{AdS}}_{4+1}$ black hole.