Abstract
We observe that in presence of excitation, a thermodynamic Smarr-like relation corresponding to a generalized entanglement temperature (${T}_{g}$) can be holographically obtained for the entanglement entropy of a subsystem. Such a relation emerges naturally by demanding that the generalized entanglement temperature produces the exact Hawking temperature as the leading term in the IR limit ($l\ensuremath{\rightarrow}\ensuremath{\infty}$). Remarkably, this relation has the same form as the Smarr relation in black hole thermodynamics. We demonstrate this for three spacetime geometries, namely, a background with a nonconformal factor, a hyperscaling violating geometry background, and a charged black hole background which corresponds to a field theory with a finite chemical potential.
Highlights
The entropy of a thermodynamical system counts the number of microstates of the system and is related to internal energy of the concerned system via the first law of thermodynamics dE 1⁄4 TdS
Such a relation emerges naturally by demanding that the generalized entanglement temperature produces the exact Hawking temperature as the leading term in the IR limit (l → ∞). This relation has the same form as the Smarr relation in black hole thermodynamics. We demonstrate this for three spacetime geometries, namely, a background with a nonconformal factor, a hyperscaling violating geometry background, and a charged black hole background which corresponds to a field theory with a finite chemical potential
Such a relation was first obtained in the context of gauge/gravity correspondence [3], where it was observed that the holographic entanglement entropy (HEE) of a system in the ultraviolet (UV) limit satisfies a thermodynamic like relation [4]
Summary
We observe that in presence of excitation, a thermodynamic Smarr-like relation corresponding to a generalized entanglement temperature (Tg) can be holographically obtained for the entanglement entropy of a subsystem. Such a relation was first obtained in the context of gauge/gravity correspondence [3], where it was observed that the holographic entanglement entropy (HEE) of a system in the ultraviolet (UV) limit satisfies a thermodynamic like relation [4] This observation led to the concept of entanglement temperature Tent, which has the universal behavior of being inversely proportional to the size of the concerned subsystem in the field theory. The static minimal surface corresponding to a thermally excited quantum field theory, wraps a portion of the event horizon, capturing flux of the Hawking radiation in the IR limit, and Tg on the static minimal surface should reduce to the Hawking temperature of the black hole in the IR limit This physical requirement immediately leads to a thermodynamics like Smarr relation obeyed by the HEE valid for the whole subsystem length. The importance of this background lies in the fact that the dual field theory of this gravity model is relativistic and
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