Holography of negative energy states

Quantum states with negative energy densities have been long known to exist in quantum field theories. We explore the structure of such states for holographic theories using quantum information theory tools and show how certain negative energy states are naturally captured by the thermodynamics of black holes with hyperbolic horizon at zero temperature, suggesting that they provide a dual description of those states. Our results give a satisfying field theory understanding of the distinct thermodynamics of such black holes.


I. INTRODUCTION
Classical energy conditions are local inequalities involving the energy momentum tensor T µν which constraint the allowed matter in a classical theory, e.g. the Null Energy Condition is given by T µν u µ u ν ≥ 0 with u µ any null vector. Inequalities such as this one where first proposed in General Relativity in order to neglect unphysical solutions to Einstein gravity equations. They allow to exclude exotic geometries such as wormholes [1], time machines [2] and warp drives [3,4], while are a key ingredient for proving some strong results such as singularity theorems [5][6][7] and topological censorship [8], among others.
When introducing quantum fields it has been long known that such classical constraints fail to be true [9,10], since there are states in the Hilbert space with negative energy densities. In fact, the energy density at any given point in space-time can be made arbitrarily negative by choosing a suitable quantum state [11,12]. One is then lead to consider weaker energy constraints such as the Averaged and Quantum Null Energy Conditions [13][14][15]. In order to better understand the origin and relevance of such quantum bounds, it seems important to understand the structure of these negative energy states. In this work we focus on their holographic description. To do so, we use quantum information theory techniques which have been previously shown to be very useful in the study of negative energy [14][15][16][17][18][19].
In the following section we start by defining the modular vacuua of any global state reduced to a space-time region, as the states with minimum expectation value on the modular hamiltonian of the reduced system. Using relative entropy, we show their similarities to the global vacuum of the theory. In section III we consider the ground state of a CFT reduced to a ball and show that the modular vacuua maximize the amount of negative energy inside the ball and provide with a sharp energy bound.
In section IV we present our main result, where we show that for holographic CFTs the negative energy ex- * felipero@usc.edu citations and degeneracy of the modular vacuua are naturally captured by the thermodynamics of black holes with hyperbolic horizon at zero temperature. The thermodynamics of such black holes has been long know to have some odd features whose interpretation has been for the most part unclear [20][21][22]. Our results give a natural understanding of such behavior and suggest that these black holes provide with a holographic description of the modular vacuua of this setup.

II. MODULAR VACUUA
We start with a general discussion regarding reduced states in which the modular vacuua naturally appear. Consider an arbitrary quantum field theory in d dimensional space-time and a fixed global state described by the density operator ρ. For any smooth and space-like region A we can define the reduced state as where K A is the modular hamiltonian andĀ the complementary region with Hilbert space HĀ. The operator ρ A usually has a more complex structure than the global state ρ, but describes the same physics when calculating the expectation value of an observable in A. More precisely, it verifies where O A is any operator in the causal region of A. By considering ρ A instead of ρ we become independent of the degrees of freedom inĀ at the expense of considering a more complicated density operator. In this context, a natural question that arises is what is the pure state |ψ we can construct in the Hilbert space H A that is the most "similar" to ρ A . Relative entropy, defined as S(ρ 1 ||ρ 2 ) . = Tr (ρ 1 ln(ρ 1 )) − Tr (ρ 1 ln(ρ 2 )) for any density operators ρ 1 and ρ 2 , seems to be particularly well suited to answer such question since it is a measure of the statistical distance between ρ 1 and ρ 2 in the following sense: given the state ρ 1 the probability of confounding it with state ρ 2 after N trials of some measurement decays as e −N S(ρ1||ρ2) for large N [23]. It therefore allows for a precise quantification on how "similar" a state |ψ is to ρ A . We then consider (2) with ρ 2 = ρ A and ρ 1 = |ψ ψ|. Writing ρ A in terms of its modular hamiltonian K A and using that the entanglement entropy of ρ 1 vanishes since it describes a pure state, we find To calculate the first term we use that K A is a hermitian operator, meaning that it will be diagonalized by a complete and orthonormal set {|ψ w (u) } with real eigenvalues k(u), where u and w are parameters which label the eigenspace and its degeneracy respectively. Expanding |ψ in this set the relative entropy in (3) becomes (4) We can further simplify this expression by writing k(u) in terms of the Renyi entropies of ρ A , defined as with q ∈ N 0 . The following values of q are particularly useful where S q=1 (ρ A ) = S(ρ A ) is the entanglement entropy and k 0 . = k(u min ) is the minimum eigenvalue of K A , which can be written as Since (4) will be minimum when k(u) = k 0 , we can use expression (6) and find where |ψ min w are the eingestates of K A with minimum eigenvalue k 0 . We conclude that any linear combination of |ψ min w minimizes the statistical distance to ρ A over the set of pure states in H A . Just from the definition of the modular hamiltonian (1), this is a very natural result and is in accordance with the behavior of a thermal state e −βH /Z β , where the ground state |0 (which has the minimum eigenvalue of energy H |0 = 0) is also the closest pure state.
This analogy is in fact quite precise as can be seen from defining the following unitary operator U (s) = e isKA /Z.

Considering the action O
for any operators O A and O A and inverse temperature β = 1. It is then reasonable not only to refer to the states |ψ min w in (7) as the modular vacuua of the reduced system, but also to call the expectation value K A the modular energy.
The modular vacuum energy is given by k 0 (6) and provides with a sharp bound for the expectation value of K A on any state Calculating k 0 explicitly for a particular system gives an inequality that can supply interesting information about the field theory under consideration. In the following we will consider this inequality for a particular system and show that it gives a constraint on the negative energy excitations on the causal domain of A.

III. NEGATIVE ENERGY BOUND
The previous discussion was done in full generality for any state ρ and quantum field theory. To further investigate the structure of the modular vacuua we consider the global ground state ρ = |0 0| of a CFT in d dimensional Minkowski space-time and take the region A as a ball of radius R, so that the modular hamiltonian is given by [24,25] where dΣ ν = dΣ n ν , with n ν a unit vector normal to any (d − 1) dimensional space-like surface C A in the causal domain of the ball whose boundary is at t = 0 and | x| = R. The conformal Killing vector ξ ν generates a flow that keeps the sphere fixed and is given by It can be interpreted as an inverse local temperature vector which can be defined and calculated for much more general systems [26,27]. 1 The KMS periodicity condition provides with a formal definition of a thermal state for operators in infinite dimensional space. To show it holds with β = 1, notice that Considering different surfaces C A will change the explicit expression of K A but will leave its spectrum unchanged 2 . For definiteness, we may take C A at t = 0 so that the modular hamiltonian can be written as This operator gives the energy density in the ball as weighted by the inverse local temperature, which is a positive function. Due to local negative energy excitations, we expect this operator to have some negative eigenvalues in its spectrum. The modular vacuua correspond to a very special set of states, given by the ones which maximize the amount of negative energy in the ball. From (6) we already see that their modular energy k 0 will be negative, since the Renyi entropy is a decreasing function of q and K A ρ = 0| K A |0 = 0. Moreover, from (8) we have the following inequality which holds for the expectation value of any state and surface C A , and the bound is sharp for the modular vacuua. The modular vacuum energy k 0 gives a bound on the negative energy excitations in the causal domain of A. The fact that (12) holds for an infinite set of surfaces C A is specially interesting. The right hand side of this inequality will not only be negative but also divergent, due to the infinite entanglement contributions captured by the Renyi entropies on the boundary of the ball. Just from the integral expression on the left hand side such a behavior is not a surprise and can be expected.
The key observation is the fact that when considering averages of energy densities, the weight function should be defined in a complete Cauchy surface 3 . Therefore, in order to recover the integral expression in (12), such function must be equal to zero outside the ball and given by the inverse local temperature inside. Since the conformal Killing vector (10) vanishes at the boundary | x| = R, the resulting weight function is continuous, but non differentiable. This apparently minor and technical detail is the reason the integral (12) is able to capture infinite negative energy excitations on the boundary of the ball and become divergent for certain quantum states. This was explicitly shown by Fewster and Hollands (section 4.2.4 of [12]) and Verch (Prop. 3.1 of [29]) for two dimensional CFTs and we will provide additional evidence in appendix A 4 .
Apart from having an understanding of the divergence on both sides of (12) we learn that both have their origin in the sharp localization of boundary of the region. On this boundary, the Renyi entropy captures infinite entanglement contributions while the integral, infinite negative energy excitations.
Despite this divergent behavior, the derived energy bound is still an interesting quantity to study, specially because it is sharp for the modular vacuua. We will illustrate this in the following section by showing how nontrivial information can be extracted from it. There are other energy inequalities, such as the Quantum Null Energy Condition [13,15] which are useful and conceptually interesting despite of the fact that for certain states they involve divergent quantities [32].
In appendix A we use an independent approach to re derive, generalize and calculate explicitly the inequality (12) for two dimensional CFTs.
The modular vacuua seem to be given by a complex set of states which are very difficult to study using standard field theory tools. In the following section we will show that when considering holographic CFTs these states are captured in a very simple way by hyperbolic black holes at zero temperature.

IV. HOLOGRAPHY OF THE MODULAR VACUUA
We now explicitly compute the modular vacuum energy k 0 for this system. To do so we use the construction developed in [24], where it was shown that the reduced ground state on the ball ρ A can be conformally mapped to a thermal state with temperatureT = 1/(2πR) on a background geometry R × H d−1 , where H d−1 is a hyperbolic plane with curvature scale R. Given that ρ A and the thermal state are related by a unitary conformal transformation, the Renyi entropy (5) is invariant and can be calculated from the free energy of the thermal state as [33] where F (T ) . = E(T ) − T S(T ), with E(T ) and S(T ) the energy and entropy of the thermal state. In particular, the entanglement entropy and infinite Renyi entropy are given by Using these expressions in (12), k 0 can be written as For an arbitrary CFT this result is not particularly useful, since the calculation of the energy of a thermal state in a hyperbolic geometry is still a very difficult computation. However, if we restrict to holographic CFTs, the AdS/CFT dictionary [34][35][36] suggests that the thermal state will be dual to a black hole in asymptotic AdS with a hyperbolic horizon. This means that the energy of the thermal state is mapped to the mass of the black hole E(T ) → M BH (T ), a quantity that can be obtained from a standard computation. For a generic temperature, the mass of the hyperbolic black hole will depend on the gravity theory the specific CFT is dual to. However, in [24] it was shown that for T =T the thermal state is described by a hyperbolic slicing of AdS which has a finite temperatureT associated to an acceleration horizon analogous to Rindler's in Minkowski space-time. Since pure AdS is a solution to any covariant theory of gravity with negative cosmological constant, the above result is completely general. Moreover the "mass" of pure AdS vanishes M BH (T ) = 0, meaning that the modular vacuum energy can be computed holographically as where M BH (0) is the zero temperature mass of the black hole solution with hyperbolic horizon in the dual gravity theory. This expression might seem peculiar, given that in the previous section we argued that k 0 should not be only negative but divergent, which seems a curious thing to expect from the zero temperature mass of a black hole.
However, it has been long known that black holes in asymptotic AdS with hyperbolic horizon have an exceptional thermodynamics in which their zero temperature mass has exactly these characteristics: it is negative and divergent [20][21][22]. The most negative value of mass allowed by the black hole thermodynamics is given by M BH (0), in exact correspondence with the maximum amount of negative energy allowed by the theory inside the ball according to (12). We have therefore found a very satisfying holographic explanation for the unusual thermodynamics of hyperbolic black holes in asymptotic AdS.
We can also investigate how the degeneracy of the modular vacuua Ω 0 is encoded in the black hole thermodynamics. This was already considered in section 5 of [33] by comparing the large q expansion of the Renyi entropy expressions (5) and (13), where a simple calculation shows This means that if we consider a flat superposition of the modular vacuua we have S(ρ 0 ) = −Tr (ρ 0 ln(ρ 0 )) = S BH (0) .
We emphasize that these expressions hold for any holographic CFT and therefore suggest the following: the hyperbolic black holes at zero temperature provide with a holographic description of a flat superposition of the modular vacuua of the ground state of a CFT reduced to a ball (16). This is in line with the field theory discussion of section II where we pointed out the similarities between the modular vacuua and the ground state; both their holographic duals, pure AdS and the hyperbolic black hole, are at zero temperature. For a specific gravity theory the mass and entropy of the black hole can be computed and written in terms of field theory quantities through standard methods. In appendix B we briefly review the calculation for Einstein gravity. The procedure is similar to the ones presented in [33] and in fact some results can already be extracted from their equations through (12).
By considering the hyperbolic black hole solution in Einstein gravity [20] and using (17) we find where S(ρ A ) is the entanglement entropy of ρ A . As expected, the modular vacuum energy is negative and divergent since it is proportional to S(ρ A ). The degeneracy is also divergent apart from the d = 2 case where the modular vacuum is unique, in agreement with [37].
Since not all holographic field theories will be dual to Einstein gravity, we can also consider the Gauss-Bonnet hyperbolic black hole [21,22] for d ≥ 4, which allows for field theories with a more complicated structure. Although the mass and entropy can be computed analytically for generic d the expressions are quite complicated, so we only present the d = 4 results which are given by where n c . = c/a with a and c the central charges in d = 4, defined in the usual way from the trace of T µν . The allowed range of n c is given by n c ∈ [2/3, 1 + 2/3] (see appendix B for details).
We can consider the behavior of these quantities for a fixed value of a and variable c. Since the entanglement entropy is independent of c [24], from (19) we can directly analyze how the modular vacuum energy and degeneracy behave as a function of c. As c increases so does the modular vacuum energy while its degeneracy decreases until it becomes equal to one in the limiting case. This behavior together with the energy inequality (12) means that while CFTs with larger c allow for more negative energy inside the ball, the number of states with this critical behavior decreases. This is a non trivial statement that we where able to extract from the bound (12) despite its divergent nature.

V. DISCUSSION
In this work we have explored the holographic description of the modular vacuua of the ground state of a CFT reduced to a ball, which contain maximum amount of negative energy inside this region. Despite of the fact that such states seem to have a very complicated structure which makes them difficult to study using field theory techniques, we have shown through (17) that their holographic counterpart seems quite simple and given by hyperbolic black holes at zero temperature. The negative mass of such black holes played a crucial role in our discussion.
Though our analysis was made entirely for zero temperature black holes, we can speculate on the holographic meaning of finite temperatures. Pure AdS (which has zero temperature) is dual to the ground state of the CFT, while thermal excitations are described by a black hole at finite temperature. Given the similarities between the ground state and the modular vacuua discussed in section II we might consider an analogous situation; the modular vacuua are dual to the zero temperature hyperbolic black hole while excitations of that modular vacuua are described by the finite temperature black hole. Since its mass will be negative for temperatures between zero and T (where the mass vanishes M BH (T ) ≡ 0) such range could correspond to other states in the CFT with negative energy inside the ball. For small perturbations of thẽ T case towards smaller temperatures, a simple argument suggests that this is indeed so (see section 4.2 of [38]).
A crucial step for making the connection at zero temperature was the large q expansion of the Renyi entropy. A further analysis of the subleading contributions of the expansions obtained from its usual definition (5) and the thermodynamic expression (13), might shed some light into the meaning of hyperbolic black holes at small but finite temperature.
From the field theory perspective it is also interesting to continue the study of the modular vacuua for systems in which the modular hamiltonian has non local contributions. Though it is unclear whether such states will still have negative energy density inside the region, inequality (8) might contain interesting physical information. A good starting point for this analysis is to consider a two dimensional chiral fermion or scalar field reduced to two disjoint intervals, where the exact modular hamiltonian contains non-local terms and can be computed from the results in [28,39]. a segment of length ℓ = 2R [40,41]. The constant term is a non universal contribution which can be absorbed into a redefinition of the regulator according to ǫ → e 2 ǫ. We then have the following result This inequality agrees with the one obtained by calculating the right hand side of (12) using that S ∞ (ρ A ) = S(ρ A )/2 from [41,42]. It also matches with our the holographic calculation in (18a).
This procedure for calculating the modular vacuum energy will be useful whenever the modular hamiltonian is proportional to the energy momentum tensor. For a global thermal state reduced to an interval of length ℓ = 2R this is also the case, but with inverse local temperature equal to [42][43][44] f β (x) = 2β sinh π(R − x)/β sinh π(R + x)/β sinh (2πR/β) .
(A6) Considering (A1) with h(x) = f β (x) inside the interval and zero outside, we get the modular hamiltonian on the left hand side and an integral on the right, which after the change of variables z = coth(πR/β)/ coth(πx/β) is reduced to Once again we obtain a divergent integral due to the non differentiability of the function at the boundary. To regulate such divergence we introduce a regulator which takes into account the change of coordinates, z max = coth(πR/β)/ coth(π(R − ǫ)/β) so that integral can be easily solved and gives The first term between square brackets we recognize as the entanglement entropy of the thermal state reduced to a segment of length ℓ = 2R [41], where we identify the same non universal constant factor we had for the ground state.
The second term can be correctly identified as K β A β after solving a simple integral and using that the energy density of a thermal state is given by T 00 (x) β = cπ/6β 2 [45] 6 . We then find the following inequality Comparing with the general expression of k 0 given in (6) and using that S ∞ (ρ β A ) = S(ρ β A )/2 from [41,42] we find perfect agreement with our previous discussion.
The divergent contribution to the modular vacuum energy k 0 (β) in both the zero (A4) and finite temperature (A7) case is independent of β and therefore exactly the same. Having argued that such divergence has its origin in the non differentiability of the weight function at the boundary, we expect f β (x) to be independent of β near x = ±R. Taylor expanding (A6) we find that this is indeed so

Appendix B: Black hole thermodynamics
In this appendix we briefly review the calculation of the zero temperature mass and entropy of the hyperbolic black hole for Einstein gravity in (d + 1) space-time dimensions. The black hole solution is given by [20] ds 2 = −V (r) (dtL/R) 2 + dr 2 /V (r) + r 2 dH 2 d−1 , where dH 2 d−1 is the unit metric on the (d − 1) hyperbolic plane and L the AdS radius. The time coordinate has been rescaled so that in the limit r → +∞ the boundary metric R × H d−1 is recovered with curvature scale R.
The function V (r) = (r/L) 2 − 1 − µ/r d−2 determines the horizon radius r + from V (r + ) = 0, while the black hole mass is related to the factor µ according to where w d−1 is the infinite volume of the unit hyperbolic plane, ℓ p Planck's length and in the second equality we have written µ = µ(r + ) from V (r + ) = 0. The temperature of the black hole can be computed from the surface gravity κ as where it is equal toT = 1/(2πR) for r + = L. From the first law of black hole thermodynamics dS = dM/T , we can compute its entropy as S = 2πw d−1 (r + /ℓ p ) d−1 . From (B1) we can solve for the zero temperature horizon radius and find (r 0 + /L) 2 = (d − 2)/d, so that the zero temperature mass and entropy are given by where we have written everything in terms S(T ). From (14) we see that the black hole entropy atT is mapped to the entanglement entropy S(ρ A ) (after proper regularization of w d−1 [24]), so that we recover (18).
For the hyperbolic black hole in Gauss-Bonnet gravity [21,22,46], the calculation is completely analogous but more involved. Following a similar procedure as in the Einstein case (and using the convenient conventions of [33]) both the zero temperature mass and entropy can be computed analytically for arbitrary d.
For d = 4 the allowed range of n c is usually taken as n c ∈ [2/3, 2] [33]. However, this does not take into account the fact that any physical black hole solution must have non-negative entropy. With this under consideration, we find n c ∈ [2/3, 1 + 2/3] where for n max c (19b) vanishes.