Perturbative entanglement thermodynamics for AdS spacetime: renormalization

Abstract

Abstract:We study the effect of charged excitations in the AdS spacetime on the first law of entanglement thermodynamics. It is found that 'boosted' AdS black holes give rise to a more general form of first law which includes chemical potential and charge density. To obtain this result we have to resort to a second order perturbative calculation of entanglement entropy for small size subsystems. At first order the form of entanglement law remains unchanged even in the presence of charged excitations. But the thermodynamic quantities have to be appropriately 'renormalized' at the second order due to the corrections. We work in the perturbative regime where T thermal ≪ T E .

Holographic entanglement first law for d + 1 dimensional rotating cylindrical black holes

“…However, V ∆P is an essential term in the first law when there is another variable in addition to r d + and a i in the metric. For example this is the case for the boosted black brane [27] and some other backgrounds. When we fix the size of the subsystem (l), the entanglement temperature and angular velocity can be obtained for d + 1 dimensional rotating cylindrical black holes to satisfy the entanglement first law (50).…”

Entanglement entropy at higher orders for the states of a = 3 θ = 1 Lifshitz theory

“…These results are on expected lines. Quite importantly, the (KK) charge contribution in the first law is present at the first order itself, unlike in the relativistic cases studied in [7] where the contibution of the charges appears only at the second order. At the second order, once again we find that the first law relation requires the entanglement temperature (and strip width) to be suitably corrected or renormalised.…”

“…The boundary of such extremal surfaces coincides with the boundary of the subsystem in the CFT. Recently it has been observed that the excitations in the CFT follow entanglement laws similar to the black hole thermodynamic laws [5,6,7]; see also [9], [12], [10], [13]. It is understood now that the entanglement entropy (S E ) and the energy of small excitations (E) in AdS spacetime obey a definite relation…”

“…Both the solutions (finite temperature and the zero temperature one) allow us to embed codimension-2 strip like surfaces (at constant lightcone time) inside the bulk. We evaluate the area of small strips using perturbative method up to second order, by using the procedure introduced in [7]. We find that for small strips (but bigger than some critical size) the leading term in the entanglement entropy has indeed logarithmic dependence on strip width l. Whereas the subleading term which accounts for the entropy of excitations goes as l 4 .…”

“…This equation is described as the first law of entanglement thermodynamics. The charge contributions can simply arise for a boosted black-brane vacua [7], where the charged excitations could be either Kaluza-Klein (KK) momentum modes along a compactified brane direction or the dual winding modes of a string.…”