Entanglement entropy and the first law at third order for boosted black branes

Sabyasachi Maulik,Harvendra Singh

doi:10.1007/jhep04(2021)065

Abstract

Gauge/gravity duality relates an AdS black hole with uniform boost with a boosted strongly-coupled CFT at finite temperature. We study the perturbative change in holographic entanglement entropy for strip sub-region in such gravity solutions up to third order and try to formulate a first law of entanglement thermodynamics including higher order corrections. The first law receives important contribution from an entanglement chemical potential in presence of boost. We find that suitable modifications to the entanglement temperature and entanglement chemical potential are required to account for higher order corrections. The results can be extended to non-conformal cases and AdS plane wave background.

Highlights

In most of the works on the first law of entanglement so far, the bulk excitation is considered only at leading order and there is no general consensus on its validity beyond the leading order
We study the perturbative change in holographic entanglement entropy for strip sub-region in such gravity solutions up to third order and try to formulate a first law of entanglement thermodynamics including higher order corrections
We find that suitable modifications to the entanglement temperature and entanglement chemical potential are required to account for higher order corrections

Summary

Minimal area functional for boosted AdS black hole

For static space-time the HEE can be calculated using the Ryu-Takayanagi formula [4, 5], which asserts that the entanglement entropy associated with a region A with boundary ∂A on the asymptotic boundary of the (d + 1)-dimensional space-time is dual to the area of a minimal co-dimension 2 surface γA in the bulk such that ∂γA = ∂A. where GN is the Newton’s constant in (d + 1)-dimensions. The space-time of our interest is a boosted AdS black hole described by the line element ds. The RT surface is conveniently described by the function x1 (z) and its area is given by the integral. Where Vd−2 = 2πryLd−3 is the overall volume of the extended directions and the y-circle on the boundary and is a UV cut-off put to protect the integral from divergence as z → 0. For ease of calculation we will often assume R = 1 Upon extremization this area functional gives a first integral of motion.

Perturbative calculation of entanglement entropy

First law of entanglement thermodynamics

Large boost limit and AdS plane wave

Extension to non-conformal Dp branes

Conclusion

A List of integrals