Abstract
The Rényi entropies as a generalization of the entanglement entropy imply much more information. We analytically calculate the Rényi entropies (with a spherical entangling surface) by means of a class of neutral hyperbolic black holes with scalar hair as a one-parameter generalization of the MTZ black hole. The zeroth-order and third-order phase transitions of black holes lead to discontinuity of the Rényi entropies and their second derivatives, respectively. From the Rényi entropies that are analytic at n = ∞, we can express the entanglement spectrum as an infinite sum in terms of the Bell polynomials. We show that the analytic treatment is in agreement with numerical calculations for the low-lying entanglement spectrum in a wide range of parameters.
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