Non-analyticity of holographic Rényi entropy in Lovelock gravity
Received: May
Non-analyticity of holographic Renyi entropy in Lovelock gravity
V. Giangreco M. Puletti 0 1
Razieh Pourhasan 0 1
0 Dunhaga 3 , 107 Reykjav k , Iceland
1 Science Institute, University of Iceland
We compute holographic Renyi entropies for spherical entangling surfaces on the boundary while considering third order Lovelock gravity with negative cosmological constant in the bulk. Our study shows that third order Lovelock black holes with hyperbolic event horizon are unstable, and at low temperatures those with smaller mass are favoured, giving rise to rst order phase transitions in the bulk. We determine regions in the Lovelock parameter space in arbitrary dimensions, where bulk phase transitions happen and where boundary causality constraints are met. We show that each of these points corresponds to a dual boundary conformal eld theory whose Renyi entropy exhibits a kink at a certain critical index n.
Black Holes; AdS-CFT Correspondence
1 Introduction 2 3
Holographic Renyi entropy
Thermodynamics of Lovelock black holes
3.1
3.2
3.3
Topological Lovelock black holes
Causality constraints on the Lovelock parameters
Phase transitions for
2=3
4
Results and discussion
1
Introduction
We consider a quantum eld theory in a d-dimensional Minkowski spacetime when at t = 0
the system gets separated in two parts, A and its complement B, by a (d
ln tr ( nA) :
The whole set of eigenvalues of the reduced density matrix
A can be reconstructed by
knowing the RE for all the indices n. For CFT's in
at space, RE exhibits a universal
relation to the central charges of the theory, in particular the derivative of RE with respect
to n evaluated at n = 1 is proportional to the coe cient of the stress tensor two-point
function [5, 6]. Moreover, in the limit where n ! 1 RE reduces to EE.
In general, RE and EE are rather di cult to compute and measure, although
remarkable progress in this direction has been made recently [7{9]. In quantum
eld theory RE is
mainly computed by means of the so-called replica method [1, 10{12]. Here, one replaces
the computation of the n-th power of the density matrix (and thus the corresponding
parin [21] can be arduous to handle in general cases.
Another remarkable approach to compute holographic EE for spherical entangling
surfaces is the one proposed by Casini, Huerta, and Myers (CHM) [22], extended to the
holographic RE in [23]. It consists of a conformal mapping on the CFT which takes us from
an Euclidean conically singular geometry to an Euclidean smooth thermal hyperboloid.
The gravity dual of such a thermal CFT (if it exists) is a black hole with hyperbolic event
horizon in asymptotically AdS (AAdS) spacetimes. Hence, the CHM map relates the RE of
the original CFT to the free energy of AAdS hyperbolic black holes. The index of the RE
is translated into the inverse of the black hole temperature (compared to some reference
temperature). Therefore, the knowledge of RE at any n (quantum entanglement spectrum)
requires the knowledge of free energy (and thus thermal entropy) of a hyperbolic black hole
in AdS at any temperature. We will review the crucial steps of the CHM map in section 2.
The advantages of CHM approach are twofold. First of all, it avoids conical
singularities and related problems [13, 24], by working on a thermal ensemble which makes the
boundary geometry perfectly smooth and straightforwardly treatable via standard
holographic techniques. Second, it applies to any gravity theory (assuming they have a CFT
dual) and in particular to higher derivative gravities [23, 25{28], unlike the RT formula
which needs to be corrected [29{35].
In this manuscript we apply the CHM approach to study RE of holographic CFT,
dual to higher derivative gravity theories, in particular the so-called third order Lovelock
gravity [36, 37], in an asymptotically AdS spacetime. Lovelock gravities are interesting
generalizations of Einstein gravity, which are ghost-free and living in dimensions (strictly)
greater than four with small coupling constants, i.e. small corrections to general relativity.
In third order Lovelock gravity the Einstein-Hilbert action is corrected with terms
proportional to R
2 (with R the curvature scalar), also known as Gauss-Bonnet gravity,2 and R
3
with dimensionless coupling constants
and , respectively. We will review basic aspects
of Lovelock gravity in section 3.1. These theories have proven useful in exploring various
properties of holographic theories, as for example the viscosity bounds [40{42], although at
intermediate energy scales they might become problematic [43].3 However, in this work we
always assume that Lovelock couplings are small positive numbers, satisfying constraints
1The formula has recently been proved in [15, 16], a rst attempt to prove it was presented in [13],
cf. [17] for a recent review on holographic EE.
tant but the theory is still weakly coupled [43].
2For the relation between Gauss-Bonnet gravity and string theory see for (...truncated)