Abstract
We calculate the Rényi entropy S q (μ, λ), for spherical entangling surfaces in CFT’s with Einstein-Gauss-Bonnet-Maxwell holographic duals. Rényi entropies must obey some interesting inequalities by definition. However, for Gauss-Bonnet couplings λ, larger than specific value, but still allowed by causality, we observe a violation of the inequality $$ \frac{\partial }{\partial q}\left(\frac{q-1}{q}{S}_q\left(\mu, \lambda \right)\right)\ge\ 0 $$ , which is related to the existence of negative entropy black holes, providing interesting restrictions in the bulk theory. Moreover, we find an interesting distinction of the behaviour of the analytic continuation of S q (μ, λ) for imaginary chemical potential, between negative and non-negative λ.
Highlights
An alternative root was provided in [28] where they used conformal transformations to relate the entanglement entropy across a spherical entangling surface to the thermal entropy in a hyperbolic cylinder R × Hd−1
As in previous calculations [30, 44], the entanglement Renyi entropy for a spherical entangling surface in Minkowski space can be calculated as the thermal entropy in the hyperbolic cylinder
The latter can be connected with the thermal entropy of asymptotically AdS topological black holes with hyperbolic horizons via the AdS/CFT dictionary
Summary
A problem that usually appears is that one does not have a good way to represent the operator ln ρA and to calculate entanglement entropy. If one represents the density matrix of the full system by a path integral (as in the vacuum or a thermal ensemble, for example), the reduced density matrix ρA and its positive integer powers ρqA can be represented by path integrals If those path integrals can be computed explicitly for all q, one can obtain a generalization of Entanglement entropy called Entanglement. The entanglement entropy can be indirectly calculated as the limit q → 1 of the Renyi entropy. One can recover the entanglement entropy from the Renyi entropies as an appropriate limit. Apart from providing an indirect method to calculate entanglement entropy, are interesting for other reasons. It can be shown that the knowledge of all Renyi entropies for all integers q > 0 is sufficient to recover the whole spectrum of the density matrix ρA [37]. When we proceed to calculate Renyi entropies of CFTs from holographic dual theories, these inequalities can provide interesting constraints in the former, as in this case, the calculation is not performed directly on a probability distribution, but rather based on black hole thermodynamics, making the validity of the above inequalities non-trivial
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