Abstract
In this work we study the flow of holographic entanglement entropy in dimensions $d \geq 3$ in the gauge/gravity duality set up. 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 $E= \left(\frac{d-1}{d}\right)T_g~S_{REE}$, which becomes an exact relation in the entire region of the subsystem size $l$, including both the infrared ($l\rightarrow\infty$) as well as the ultraviolet ($l\rightarrow 0$) 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 $d\geq 3$, 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 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.
Highlights
The von Neumann entropy or the entanglement entropy (EE) is one of the fundamental and well-studied entities of quantum physics [1]
We observe that a generalized entanglement temperature Tg can be defined which gives the Hawking temperature TH in the infrared region and leads to a generalized thermodynamics like law E 1⁄4 ðd−d1ÞTgSREE, which becomes an exact relation in the entire region of the subsystem size l, including both the infrared (l → ∞) as well as the ultraviolet (l → 0) regions
We define a generalized entanglement temperature Tg using the first law of black hole thermodynamics (E 1⁄4 ðd−d1ÞTHSBH) and we show that Tg produces the exact Hawking temperature in the IR domain of the dual field theory
Summary
The von Neumann entropy or the entanglement entropy (EE) is one of the fundamental and well-studied entities of quantum physics [1]. In this work we observe the evolution of HEE with the subsystem size in dimensions d ≥ 3 in both UV and IR domain of the theory by incorporating all the terms in the expansion Such a study was carried out earlier in [24] in the case of the 2 þ 1-dimensional BTZ black hole which corresponds to a 1 þ 1-dimensional CFT. We define a generalized entanglement temperature Tg using the first law of black hole thermodynamics (E 1⁄4 ðd−d1ÞTHSBH) and we show that Tg produces the exact Hawking temperature in the IR domain of the dual field theory. We represent this for AdS3þ1 and AdS4þ1 black holes.
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