MSc Physics
Track: Theoretical Physics
Master Thesis
Entanglement Entropy, Differential Entropy and the Dynamics
of Spacetime Geometry
In AdS/CFT by Alex Kieft 5836956 60 ECTS September 2013 - August 2014 Supervisor: Prof. dr. J. de Boer Examiner: dr. A. CastroAbstract
A convenient measure for the amount of entanglement in a quantum mechanical system is the entanglement entropy. In conformal field theories, the entanglement entropy can be calculated using the replica trick, which we apply to several two-dimensional systems. Whenever the CFT has a holographic dual, the entanglement entropy is proportional to a minimal surface area ending on the asymptotic boundary of the dual AdS spacetime − a statement which is encoded in the Ryu-Takayanagi (RT) formula S = A/4G. We derive a new result for the holographic entanglement entropy in a three-dimensional conical deficit spacetime using the RT formula and show that it matches the associated CFT2 result for a certain excited state. Whereas
the entanglement entropy is related to minimal surfaces ending on the boundary of AdS, the “differential entropy” is related to more general bulk surfaces, i.e. those which do not nesessarily extend all the way to the boundary. We review the defini-tion of differential entropy and the closely related “residual entropy” within quantum mechanical systems and offer illustrative examples, before reviewing its application within AdS3/CFT2. Elaborating upon the philosophy that geometry comes from
en-tanglement, it has recently been shown that the linearized vacuum Einstein equations are equivalent to a “first law” of entanglement entropy. We review this derivation and compare it with Jacobson’s derivation of the full non-linear Einstein equations from the first law of thermodynamics. Based on the similarities between these two approaches, we present some preliminary ideas towards a direct holographic deriva-tion of Jacobson’s original argument using the “first law of differential entropy”, although we raise questions to the viability of such an attempt.
Acknowledgements
I would like to express my deep gratitude to my advisor Jan de Boer for guiding me throughout this project. I would also like to thank my examiner dr. Alejandra Castro for her comments on the preliminary version of this thesis, and dr. Diego Hofman for valuable conversations and for bringing useful literature to my attention. I am greatly endebted to Benjamin Mosk, who has been a big help with his careful explanations on differential geometry. Furthermore, I am very greatful to my fellow students and friends Jochem Knuttel, Eva Llabr´es and Gerben Oling for lots of valu-able conversations. Special thanks goes to Manus Visser for innumervalu-able invaluvalu-able discussions and a very fruitful collaboration over the past two years. I also wish to thank the rest of my fellow master students for their pleasant company.
Contents
1 Introduction 5
2 Background 9
2.1 Anti-de Sitter space . . . 9
2.2 Entanglement entropy . . . 12
3 Entanglement Entropy in CFT2 16 3.1 Replica trick . . . 16
3.2 Entanglement entropy of the ground state . . . 20
3.3 Entanglement entropy for excited states . . . 22
4 Holographic Entanglement Entropy 26 4.1 Heuristic derivation of RT formula . . . 26
4.2 Some properties of holographic entanglement entropy . . . 28
4.3 Examples of HEE in AdS3 . . . 30
4.4 Holographic entanglement entropy in a conical metric . . . 34
5 Hole-ography 37 5.1 Spherical Rindler-AdS space . . . 37
5.2 Residual entropy . . . 38
5.3 Holographic holes in AdS3 and differential entropy . . . 42
6 Linearized Gravity from the First Law of Entanglement Entropy 48 6.1 The ‘first law’ of entanglement entropy . . . 48
6.2 Holographic interpretation of the first law . . . 51
6.3 First law of black hole thermodynamics and the Iyer-Wald theorem . . . 54
6.4 Linearized gravity from the first law . . . 55
7 First Law of Differential Entropy 60 7.1 Jacobson’s derivation . . . 60
7.2 Local Rindler horizons and differential entropy . . . 62
8 Conclusion 67
Appendix A Scalar curvature of conical spaces 69
1
Introduction
How to formulate a consistent theory of quantum gravity is one of the major outstanding prob-lems in contemporary theoretical physics. Our ordinary theory of gravity is Einstein’s general relativity, which is a classical field theory. On the other hand, the other fundamental forces of nature are all formulated in terms of local quantum field theories. Attempts to treat general relativity on an equal footing with the other forces − by quantizing it on the nose − quickly run into trouble because general relativity appears to be non-renormalizable. As such, infinitely many counter-terms are required to obtain finite answers for the one-loop (i.e. quantum) correc-tions, rendering the theory unpractical if not unphysical. Since the direct quantization of general relativity appears to be a fruitless effort, a radically different approach is needed to arrive at a consistent description of gravity at the quantum level.
As is well known, the leading candidate for a consistent theory of quantum gravity is string theory, most notably because it naturally encompasses a spin-two particle representing the graviton. In contrast to the point particles which occur in ordinary quantum field theories, the fundamental constituents of string theory are one-dimensional objects of finite extent. As such, string theory is UV-finite and hence does not suffer the same fate as general relativity upon quantization: it is perfectly renormalizable! Another success of string theory is that it has allowed for a precise comparison between the microscopic and macroscopic entropy of a black hole, as was famously demonstrated by [1]. The macroscopic entropy of a black hole is given by the celebrated Bekenstein-Hawking formula [2, 3],
S = kBc
3
~ A
4G, (1.1)
where A is the area of the horizon and G, kB, c and ~ are the usual fundamental constants of
nature.1 The Bekenstein-Hawking formula presents remarkable connection between quantum theory, gravity and statistical mechanics. It is therefore generally believed that black hole horizons provide a window into the nature of quantum gravity.
Certainly, the most important lesson that the Bekenstein-Hawking formula has had to offer is that the information about the black hole’s interior is fully encoded on the surface that forms the event horizon. This realization sparked the idea of holography [4, 5]. The holographic principle states that the degrees of freedom of quantum gravity are, as opposed to an ordinary local quantum field theory, not localized within a given spacetime region, but are instead localized on the boundary surface of that spacetime region. More precisely, a certain gravitational theory in a d + 1-dimensional spacetime M can be fully described by a non-gravitational quantum field theory on the d-dimensional boundary spacetime ∂M. Currently, the most concrete and well-understood implementation of the holographic principle is the AdS/CFT correspondence [6]. It states that there exists an exact equivalence between certain gravitational theories on a d + 1-dimensional anti de-Sitter space and certain conformal field theories living on the d-1-dimensional asymptotic boundary of AdS. Although it is generally believed that we do not live in an anti-de Sitter universe, the AdS/CFT corresponanti-dence forms an excellent theoretical playground to
1
study the properties of holographic theories. It is hoped that these lessons will one day help us understand more realistic settings like de Sitter space, which is considered as a toy model for our own universe.
Although AdS/CFT was originally formulated as a duality between a certain realization of string theory on AdSd+1 and a supersymmetric conformal gauge theory [6], the correspondence
is more general. In particular, AdS/CFT is an example of a strong-weak duality: when the gravitational theory is strongly coupled, the field theory is weakly coupled, and vice versa. Focusing on the weak coupling regime of string theory, the theory becomes approximately equal to semi-classical (super-)gravity. Thus, semi-classical general relativity on d + 1-dimensional AdS is dual to a strongly coupled d-dimensional CFT without supersymmetry. In the present thesis, we will be interested in this regime of the AdS/CFT correspondence. Despite its relative simplicity (compared to string theory realizations of the correspondence), its study allows us to gain deep insights in the fundamental structure of spacetime.
One of the deepest conceptual implications of the holographic duality is the emergence of spacetime. Emergence here means that smooth classical spacetime does not exist at a fun-damental level, but rather that it arises as a thermodynamic phenomenon in a coarse-grained description. The idea of emergence can be made precise within the context of AdS/CFT. Here the d-dimensional CFT describes the behaviour of the fundamental degrees of freedom of the quantum gravity theory. When we ‘zoom out’, meaning that we consider the theory at larger distance scales or, equivalently, at lower energy scales, another spacetime dimension arises in such a way that the full spacetime is now d + 1-dimensional AdS. The question that arises, then, is how the extra dimension emerges from field theory data. Recently, it has been suggested that entanglement plays a central role in the emergence of spacetime. Ryu and Takayanagi have made this connection precise by their proposal that the area of a certain minimal surface in AdS is proportional to the entanglement entropy SAin the boundary CFT [7],
SA=
A(γA)
4G , (1.2)
where γA is a surface anchored at the boundary, that stretches into AdS in such a way that
its shape minimizes the area functional. Aside from these technicalities, the Ryu-Takayanagi formula indeed conveys a deep relationship between gravitional spacetime and entanglement, as we will try to illuminate by the following thought experiment.
Consider splitting the entire space supporting the CFT into two parts, denoted by A and its complement B. The entanglement entropy SA then quantifies the amount of entanglement
between these two field theory regions. According to the Ryu-Takayanagi formula, there exists a certain minimal surface γAin the dual AdS spacetime that divides the manifold into two parts
¯
A and ¯B and whose area A(γA) is directly related to the value of SA. Suppose now that we
change the amount of entanglement between the regions A and B, yielding a different value for the entanglement entropy. The geometric quantity A(γA) in the dual spacetime therefore
changes accordingly by virtue of (1.2), which serves as a heuristic confirmation that the emergent spacetime geometry indeed comes from entanglement [8, 9, 10]. We can stretch this thought experiment a bit further by decreasing the value of SA all the way down to zero, such that the
Figure 1: Both figures represent a timeslice of AdSd+1. The conformal field theory lives on the boundary,
which is split into two parts A and B. The holographic entanglement entropy is given by the area of the minimal surface shown in orange, which splits gravitational spacetime into two pieces ¯A and ¯B. When the entanglement between A and B vanishes, the associated spacetime regions become disconnected. The drawback of this illustration is that the boundary geometry seems to change, which is not true. Figure based on [8, 9].
surface in the dual gravitational theory has vanishing size, as illustrated in figure 1. Hence, the associated regions of spacetime ¯A and ¯B become disconnected. We can therefore, again very heuristically, interpret entanglement as the “glue” of spacetime. Extended to more general settings, the idea is that smooth classical spacetime emerges from the entanglement between the microscopic degrees of freedom of quantum gravity.
Now, if geometry comes from entanglement, what are the corresponding dynamical laws gov-erning fluctuations in the geometry? Of course, we already know that fluctuations are governed by the Einstein equations at the classical level. Thus, the question becomes how the Einstein equations are encoded in the dual CFT. In an attempt to answer this question, it was shown by [11] that the entanglement entropy obeys a first law-like relation δSA = δE, where δE is a
small perturbation of the energy of the quantum state. Using the Ryu-Takayanagi formula (1.2) for the holographic entanglement entropy, the first law holographically translates into a relation between energy and geometry, which schematically is the content of the Einstein equations. The details, however, are quite different, because the holographic energy does not correspond to a local bulk matter field. Nevertheless, it has been shown that the entanglement first law is indeed equivalent to the gravitational field equations [12, 13], although only the linearized vacuum Einstein equations have been derived holographically. A holographic derivation of the full non-linear Einstein equations remains elusive. The main obstruction is that the holographic approach uses global rather than local Rindler horizons [13]. Nearly twenty years ago, before the era of holography, Jacobson showed that applying the first law of thermodynamics to a local Rindler horizon, the full Einstein equations emerge as a consequence of the entropy/area relation S = A/4G [14]. Inspired by his success, we would like to reconstruct his argument from a holographic viewpoint using local Rindler horizons.
The fact that the holographic derivation of the Einstein equations makes use of global rather than local Rindler horizons is closely related to the observation that the RT formula picks out a very special class of surfaces, namely minimal surfaces that extend to the asymptotic boundary. An interesting question is whether a similar holographic correspondence exists between entropy and more general surface areas. In particular, one might ask which field theoretical quantity is associated to the area of a spherical surface centered inside global AdS, i.e. a “hole”. This
question was addressed by [15] within the context of AdS3. The authors argue that the relevant
field theory observable is the “differential entropy”, which is an alternating sum of entanglement entropies around the full field theory (we will make this more precise later). Although this is an interesting topic its own right, we will primarily be interested in the dynamics around such bulk surfaces. Certainly, these bulk surfaces are much more local objects than the minimal RT surfaces which are used to evaluate the entanglement entropy. The differential entropy thus appears as a suitable candidate to replicate Jacobson’s argument in a holographic manner. How-ever, it appears that the Einstein equations are already needed to compute of the variation of the hole’s area, which means that we are regressing into circular reasoning when attempting to holographically derive the Einstein equations.
The thesis is outlined as follows. We start with a brief introduction to anti-de Sitter space and entanglement entropy in chapter 2. The calculation of the entanglement entropy in simple two-dimensional conformal field theories is reviewed in chapter 3. In chapter 4, we present an introduction to the Ryu-Takayanagi (RT) formula (1.2) and discuss various examples in AdS3,
displaying exact agreement with the CFT2results from chapter 3. In addition, we present a novel
result at the end of chapter 4, finding agreement with the dual CFT calculation for an excited state characterized by twist operators. Chapter 5 is dedicated to the residual entropy and the differential entropy, where we present new independent results for the residual entropy of a three-qubit system. In chapter 6, we review the derivation of the linearized Einstein equations from the first law of entanglement entropy. In chapter 7, we recall Jacobson’s derivation of the Einstein equations. We then give some initial steps towards a holographic derivation by proposing a first law of differential entropy. Finally, we shall discuss the aforementioned circularity when trying to derive the Einstein equations from the first law of differential entropy.
2
Background
In the current chapter we will present some relevant background material. Section 2.1 provides a review of anti-de Sitter space and its use in the AdS/CFT correspondence. The focus lies on three dimensions, but most observations and derivations straightforwardly generalize to higher dimensions. In section 2.2, we will review the definition of entanglement entropy and discuss some of its properties.
2.1 Anti-de Sitter space
Anti-de Sitter space (AdS) is a solution to the Einstein equations in the presence of negative cosmological constant. In particular, it is a maximally symmetric spacetime, which means that every point in spacetime is equivalent to every other point. As an immediate consequece, such spacetimes must have constant curvature, but the conditions on the curvature tensors are more stringent than that. The curvature tensors for maximally symmetric spaces are then given by
Rµνρσ = 1 L2(gµσgνρ− gµρgνσ) (2.1) Rµν = − d L2gµν (2.2) R = −d(d + 1) L2 , (2.3)
where L is a constant with dimension length and is known as the AdS-radius and d is the number of spatial dimensions of the AdSd+1 spacetime. One can easily verify that this spacetime is a
solution to the vacuum Einstein equations sourced by a cosmological constant Rµν−
1
2Rgµν+ Λgµν = 0. (2.4)
Taking the trace of this equation and comparing with (2.3) fixes the value of the cosmological constant as
Λ = −d(d − 1)
2L2 . (2.5)
2.1.1 Three-dimensional gravity
The most striking feature of three-dimensional gravity is that is has no local degrees of freedom, which follows from a simple counting argument. Notice that the metric has six components, three of which are fixed by diffeomorphism (i.e. gauge) invariance, whereas the other three are fixed by the Einstein field equations. Therefore, the metric has no free parameters and thus no local degrees of freedom.Therefore, the curvature of a three-dimensional spacetime has to be constant. More precisely, every three-dimensional spacetimes is maximally symmetric, which means any three-dimensional spacetime with negative cosmological constant is locally equivalent to AdS3. The theory is therefore easy to handle, but not completely trivial. Different global
identifications give rise to a variety of spacetimes, including one with black hole-like features. In this section, we will review some properties of AdS3.
2.1.2 Global coordinates
It is convenient to represent anti-de Sitter space as a hypersurface in a higher dimensional em-bedding space. This is how one typically deals with maximally symmetric spaces. For instance, the 2-dimensional Euclidean sphere can be defined as the solution to the constraint ~x2= R2 in flat 3-dimensional Euclidean space R3, whose coordinates are known as embedding coordinates. Introducing the usual parametrization in terms of angular coordinates θ, φ for the constraint equation ~x2 = R2 allows one to calculate the induced metric on the sphere.
We apply the same procedure to find the AdS3 metric. A negatively curved object with
constant curvature is known as a hyperboloid. Since we are looking for a negatively curved space in Lorentzian signature, the embedding space is R2,2, with metric
ds2= −dT12− dT22+ dX12+ dX22. (2.6)
The constraint equation for the Lorentzian hyperboloid is
− T12− T22+ X12+ X22= −L2, (2.7)
which can be parametrized with the following set of coordinates T1 = L cosh ρ sin τ
T2 = L cosh ρ cos τ
X1= L sinh ρ cos φ (2.8)
X2= L sinh ρ sin φ.
To find the induced metric, we take the differentials of (2.8) and plug these into metric (2.6), yielding
ds2 = L2 − cosh2ρ dτ2+ dρ2+ sinh2ρ dφ2 . (2.9)
What we have obtained here is AdS3 in global coordinates. It is quite special that one
can cover a whole spacetime with a single coordinate system (generically one needs multiple coordinate patches to cover the whole space).
Note that there is a subtlety concerning τ . As can be seen from (2.8), τ is periodic on the hyperboloid, which would produce unphysical features like closed timelike curves. This issue is easily remedied by reminding ourselves that the embedding structure is just a helpful tool to arrive at metric (2.9), which ultimately defines anti-de Sitter space with non-periodic τ . However, the embedding coordinates have other computational benefits, such as finding geodesics in AdS3, as we shall do in section 4.3.
Next, consider the coordinate transformation ρ = arcsinh tan ξ with 0 ≤ ξ ≤ π/2. In these coordinates the metric reads
ds2 = L
2
cos2ξ −dτ
Figure 2: Global AdS3 compactified as a solid cylinder, where the physical distance to the boundary is
infinite. Each timeslice corresponds to a hyperbolic disk with radial coordiate ρ.
Setting aside the overall conformal factor L2/ cos2ξ, (2.10) corresponds to the metric of a solid cylinder, leading to the pictorial reprentation of AdS in figure 2. The dual field theory resides on the conformal boundary of this cylinder (at ξ = π/2), but it should be remembered that this boundary is actually infinitely far away.
Another convenient way to represent AdS in global coordinates is to introduce the coordinates r = L sinh ρ, t = Lτ and φ = ˜φL : ds2 = − 1 + r 2 L2 dt2+ 1 + r 2 L2 −1 dr2+ r2dφ2, (2.11)
where we have dropped the tilde on φ. In this thesis we shall mostly make use of (2.11), although we will also make use of metric (2.9).
2.1.3 Poincar´e coordinates
There exists another convenient coordinate system for AdS. Going back to the hyperboloid (2.7), we parametrize it differently as T1 = t/z X1 = x/z X2+ T2 = −L/z (2.12) X2− T2 = L z −t 2+ x2+ z2 ,
for which the metric reads
ds2 = L
2
z2 −dt
2+ dz2+ dx2 . (2.13)
The asymptotic boundary is at z → 0, which is simply two-dimensional Minkowski space. Poincar´e coordinates are therefore highly convenient for AdS/CFT calculations. The price we
pay, however, is that these coordinates cover only half of the full AdS space.2 To see this, note that the third and fourth coordinate in (2.12) are essentially light-like coordinates. In particular, from the third line it then follows that z itself is light-like, which splits the space into two charts: z > 0 translates into −X2 < T2, corresponding to one half of the hyperboloid, whereas the other
half z < 0 gives the second Poincar´e chart, i.e. −X2 > T2. When we refer to the Poincar´e patch
of AdS, we refer to the region of AdS covered by one of these charts (z > 0 is typically chosen), where z is now chosen to be spacelike. The locus that is not covered is z = 0, which corresponds to the boundary of AdS.
Another way to think about the Poincar´e patch is as the asymptotic limit, r → ∞, of global AdS (2.11). This means that we can neglect the contant O(r0) term, giving
ds2 = −r 2 L2dt 2+L2 r2dr 2+ r2dφ2.
Making the coordinate redefinition z = L2/r and x = Lφ gives us again the Poincar´e metric (2.13). Hence, the Poincar´e metric can be thought of as the near-boundary metric of global AdS. 2.1.4 BTZ black holes
Einstein’s equations in three dimensions yield a family of solutions with black hole-like features [17], known as BTZ black holes. Since there is no local curvature in three dimensions, the BTZ black hole has no curvature singularity like its higher dimensional relatives. On the other hand, the BTZ black hole has an horizon, which justifies its status as a black hole. Just like higher dimensional black holes, its metric is completely determined by the black hole’s mass, angular momentum (but no electric charge).
The metric of the neutral, non-rotating BTZ black hole is given by [17],
ds2= − r 2− r2 + L2 dt2+ r 2− r2 + L2 −1 dr2+ r2dφ2. (2.14)
Here r+ is the radial coordinate of the horizon, which is related to the mass M as and inverse
temperature β as M = r 2 + 8GL2 and β = 2πL2 r+ . (2.15) 2.2 Entanglement entropy
In classical physics the concept of entropy quantifies the uncertainty we have about the exact state of a physical system at hand. Classical entropy is always related to ignorance about the state of the system, not to some fundamental randomness of nature. In quantum mechanics, the situation is radically different, where positive entropies may arise due to the probabilistic nature of quantum mechanics, i.e. even without an objective lack of information. The entropy
2
This is only true if we do not decompactify τ , i.e. keep τ periodic. If we decompactify τ , the Poincar´e patch fits infinitely many times in the full AdS space. This is understood by looking at the Penrose diagrams of the spaces, which is discussed in e.g. [38].
we are talking about is known as the entanglement entropy, whose definition we will review in the current section.
Consider a quantum mechanical system described by a single state vector |Ψi, which is called a pure state. In contrast, a mixed state cannot be described by a single state vector. The density matrix of a pure state is defined as
ρ = |Ψi hΨ| , (2.16)
which has an associated Von Neumann entropy, defined as
S ≡ − Tr ρ log ρ, (2.17)
which vanishes for pure states (we consider a simple example shortly hereafter). In contrast, the density matrix of a mixed state can be expressed as ρ = P
ipi|ψii hψi|, where pi is the weight
of each state, yielding a non-zero von Neumann entropy.
Focusing on a pure state, suppose that we factorize the total Hilbert space as a tensor product of two subspaces
H = HA⊗ HB.
Now, if there exist vectors |ΨAi ∈ HA and |ΨBi ∈ HB such that |Ψi = |ΨAi ⊗ |ΨBi, then
|Ψi is untentangled; otherwise, it is entangled. A convenient way of quantifying the amount of entanglement between A and B is given by first writing down the reduced density matrix ρAof
subsystem A,
ρA= TrBρ, (2.18)
where the trace is over the Hilbert space of subset B. The entanglement entropy is then defined as the von Neumann entropy of ρA,
SA= − TrAρAlog ρA, (2.19)
which vanishes (is positive) in the absence (presence) of entanglement. Another way of writing the entanglement entropy is in terms of the eigenvalues of ρA, namely SA= −Piλilog λi, which
follows from the cyclicity of the trace.
As an intuitive example, consider a system consisting of two qubits. The first entry in the two-qubit state vector denotes the state of qubit A, the second that of B. If we consider the following state, then the reduced density matrix of qubit A is easily constructed by tracing over B:
|Ψi = cos θ |00i + sin θ |11i , ⇒ ρA=
cos2θ 0 0 sin2θ
! .
For θ = 0, we obtain the unentangled state |00i, whose reduced density matrix is equivalent to that of a pure state, with one eigenvalue equal to 1 and the other 0, yielding SA = 0. On the
other hand, the state with coefficients cos θ = sin θ = 1/√2 is the maximally entangled state, whose density matrix is diag{12,12}, yielding the maximum value for the entanglement entropy, SA= log 2.
Figure 3: A timeslice of the d-dimensional spacetime which harbors the quantum field theory. The Hilbert space is factorized as a tensor product of the interior A and the exterior B of an imaginary sphere. The entanglement entropy is proportiotional to the area of the boundary ∂A, shown in red.
2.2.1 Properties of entanglement entropy
We now discuss some general properties of entanglement entropy, which will be relevant when considering the holographic entanglement entropy from chapter 4 onwards.
Strong subadditivity
The von Neumann entropy satisfies a relation known as strong subadditivity. For two overlapping subsets A and B, the following inequality holds [18],
S(A ∪ B) ≤ S(A) + S(B) − S(A ∩ B). (2.20)
The proof is not elementary, so we will not review it here. However, in chapter 4 we shall see that (2.20) is easily proven in a holographic context.
Area law for QFT ground state
Given the previous discussion on entanglement entropy, it is natural to ask about its relation to thermodynamic entropy. The latter is usually an extensive quantity, meaning that it scales with the volume of a system. In constrast, a simple argument convinces us the entanglement entropy cannot be extensive. Consider a quantum field theory, which we split into the interior A and exterior B of an imaginary sphere, as in figure 3. The Hilbert space then factorizes as a tensor product of A and B, i.e.
Htot= HA⊗ HB, (2.21)
so that the state in the interior A is described by the the reduced density matrix ρA= TrBρ, and
similarly ρB= TrAρ for the exterior B. Now, it can easily be shown that if the total system is in a
pure state, then these two density matrices have identical eigenvalues, plus some additional zeroes for the density matrix of the larger Hilbert space. Consequently, the entanglement entropy of is equal for both regions, SA= SB, which suggests that the entanglement entropy can depend only
on properties that are shared by the two regions, and since Vol(A) 6= Vol(B), the entanglement entropy cannot be extensive. On the other hand, the regions do have a shared boundary, i.e. ∂A = ∂B. Combining this observation with the locality of the quantum theory, the conjecture
is that the entanglement entropy of a pure state in a local QFT is proportional to the area3 of the boundary surface [19]
SA=
Area(∂A)
δd−1 + ..., (2.22)
where ∂A is the boundary of region A (e.g. the interior of the sphere), δ is a cut-off to regulate the answer and the dots represent subleading terms which come from correlations across larger scales. The area law is naturally explained by the fact that in a local quantum field theory, short-distance correlations are the most dominant. Thus, the degrees of freedom which are localized close to surface on one side are entangled predominantly with those close to the surface on the other side. Since local quantum field field theories are UV divergent, there are in fact infinitely many modes that contribute to the entanglement across the surface, rendering the entanglement entropy infinite. When the theory is regularized on a lattice with spacing δ, the area law (2.22) can be interpreted as ‘the number of entanglement lines being cut by the surface ∂A’.
Note that the area law does not hold if the system is in a mixed state. Mixed states are characterized by an extensive contribution to the entanglement entropy, i.e. they contain a term proportional to the volume of the region.
3
It should be noted that the area law is slightly violated in two-dimensional CFTs, where SA ∼ logδ`, with ` the length of the interval. This will become clear in the next chapter.
3
Entanglement Entropy in CFT
2The main goal of this chapter is to calculate the entanglement entropy of a single interval in two-dimensional conformal field theories. The most direct calculation of the entanglement entropy for a system with a finite number of degrees of freedom is to explicitly compute the reduced density matrix (exactly or numerically) and then calculate SA= − Tr ρAlog ρA= −P λilog λi,
where λi are the eigenvalues of ρA. However, a direct reconstruction of the full reduced density
matrix of a generic interacting QFT is a notoriously difficult task. We therefore review the approach of [22, 23], known as the “replica trick” (earlier work found in [24]), to obtain the entanglement entropy of the ground state of a two-dimensional conformal field theory in section 3.2, including generalization to finite size and finite temperature. In section 3.3, we review the calculation of the entanglement entropy for excited states.
3.1 Replica trick
The crucial observation that makes the replica trick possible is that the reduced density matrix ρA is positive semi-definite, meaning that its eigenvalues satisfy λi ∈ [0, 1] and P λi = 1.
Therefore, the convergence and analyticity of the quantity Tr ρnA =P
iλni holds for n ≥ 1, so
that derivative with respect to n exists and is analytic for any n > 1. Therefore, one may infer that the entanglement entropy SA= −P λilog λi is equivalent to
SA= − lim n→1 ∂ ∂nTr ρ n A. (3.1)
so that the computation of SA now amounts to calculating Tr ρnA. For a general n ≥ 1, this will
only make our lives harder instead of easier. However, for positive integral n, the quantity Tr ρnA can in some cases be computed by a suitable path integral. One then analytically continues the result to a general (complex) value of n, such that the limit n → 1+is well-defined, from which the entanglement entropy follows using (3.1).4
It turns out that the calculation of Tr ρnAis tantamount to computing the partition function on an n-sheeted Riemann surface Rn, as will be discussed below based on [22, 23, 25].
3.1.1 Path integral formulation
Consider the ground state of a two-dimensional CFT, where the subset A consists of the degrees of freedom within a single interval x ∈ [u, v] at τ = 0, with coordinates (τ, x) ∈ R2. For simplicity, we consider a CFT that contains just one field φ(τ, x). The ground state wave functional can then be found by path-integrating the Euclidean action S from τ = −∞ to τ = 0 [22], Ψ φ−(x) = Z φ(0,x)=φ−(x) τ =−∞ Dφ e−S(φ). (3.2) 4
There are cases where the analytic continuation to non-integer n is ill-defined, a phenomenon known as “replica symmetry breaking” [20, 21].
The total density matrix ρ is then defined as ρφ−,φ+ = Ψ (φ−(x)) Ψ†(φ+(x0)), where Ψ† can be obtained by path-integrating from τ = ∞ to τ = 0, giving
ρφ−,φ+ = Z−1 Z τ =∞ τ =−∞ Dφ e−S(φ)Y x δ φ(0−, x) − φ−(x) Y x δ φ(0+, x) − φ+(x). (3.3)
where Z = Tr ρ is the partition function that ensures normalization.
Pictorially, there is a cut at τ = 0, coming from the fact that the fields φ−(x) and φ+(x) are not yet identified at τ = 0. Taking the trace makes the identification, which means setting φ−(x) = φ+(x) and integrating over these variables in the path integral (3.3), which clearly reproduces Tr ρ = 1 due to the normalization factor Z−1. Now, to obtain the reduced density matrix of region A, one traces over the degrees of freedom in its complement. In the path integral, we sew together the fields at all points x /∈ A, leaving an open cut along the interval A at τ = 0. The reduced density matrix is then given by expression (3.3), where x is now restricted to the interval A [ρA]φ−,φ+ = Z−1 Z τ =∞ τ =−∞ Dφ e−S(φ) Y x∈A δ φ(0−, x) − φ−(x) Y x∈A δ φ(0+, x) − φ+(x), (3.4)
where Z is the full partition function, so that tracing over A gives TrAρA= 1.
Now we would like to construct ρn
A, obtained by making n copies of the above
ρnA= [ρA]φ− 1,φ + 1[ρA]φ − 2,φ + 2 . . . [ρA]φ − n,φ+n, (3.5)
and sewing them together cyclically along the cuts for 1 ≤ j ≤ n,
φj(0+, x) = φj+1(0−, x). (3.6)
where the final identification φn(0+, x) = φ1(0−, x) comes from the trace Tr ρnA. As a result,
we obtain a structure known as an n-sheeted Riemann surface Rn, illustrated in figure4. The
quantity Tr ρnA is then obtained through the path-integral
Tr ρnA= Z−n Z
(τ,x)∈Rn
Dφ e−S(φ)≡ Zn
Zn, (3.7)
where Zn is the partition function on Rn.
3.1.2 Twist operators
Although it is sometimes possible to directly calculate the partition function (3.7) on Rn, such
a direct approach becomes difficult for Riemann surfaces with complicated topology [22]. We therefore consider moving the problem to the complex plane C, where the structure of the Riemann surface can be implemented through appropriate boundary conditions at the branch points (u, v) (the boundary points between region A and its complement). However, there is a subtlety involved in this procedure, as discussed in [25]. It is argued that the fields φ become non-local on the plane, because whereas the space on which the fields live has grown by a factor
Figure 4: Each sheet is associated to the two-dimensional plane and the cut arises because the fields are not identified on each sheet separately. Instead, the sheets are sewed together cyclically, giving rise to the n-sheeted Riemann surface shown in blue. Here n = 3 for simplicity. This figure is based on [22].
of n, the number of fields has remained the same. To remedy the problem, we instead consider n fields φk, i.e. one on each sheet, where they are now local. The boundary conditions (3.6)
which encode the structure of the Riemann surface then read
φk(e2πi(w − u)) = φk+1(w − u), φk(e2πi(w − v)) = φk−1(w − v), (3.8)
where w = x + iτ is the complex coordinate on the plane (the anti-holomorphic coordinate ¯
ω = x − iτ is implicit everywhere). It is imporant to note that the coordinate w is multi-valued on the n-sheeted Riemann surface, in the sense that each value of w actually represents n different points, one for each sheet.
Equivalently, we can regard the twisted boundary conditions (3.8) as the insertion of two twist operators Φ+(k)n and Φ
−(k)
n at w = u and w = v, respectively, for each k-th sheet. Twist
operators can formally be defined through the path integral as [25], hΦ+ n(u)Φ − n(v)iC= Z C(u,v) Dφ exp − Z C d2xL(φ) , (3.9)
where RC(u,v) denotes the restricted path integral with constraints (3.8). Since the complete Riemann surface Rnconsists of n sheets, we conclude that the full partition function Zncan be
written as Zn= n Y k=1 hΦ+(k)n (u)Φ−(k)n (v)iC, (3.10)
from which it follows that (3.7) can be written as
Tr ρnA=
n
Y
k=1
hΦ+(k)n (u)Φ−(k)n (v)iC. (3.11)
We conclude that the calculation of the entanglement entropy is now tantamount to the compu-tation of a two-point function of the twist fields Φ+n and Φ−n. It should be stressed that the twist operators are primary operators, which means that their two-point function is fully contrained
by conformal symmetry to be of the form (see e.g. [26]) hΦ+n(u, ¯u)Φ−n(v, ¯v)i ∝ 1
|u − v|2∆, (3.12)
where ∆ = h + ¯h is the scaling dimension of the twist fields. Here we have written the anti-holomorphic part explicitly for clarity, but we will omit it hereafter. The proportionality constant is infinite in a continuum field theory, but may be regularized with a UV cutoff, as already mentioned in section 2.2.1.
We conclude that in order to evaluate (3.11), we have to determine the scaling dimension ∆ of the twist fields. In order to do so, it is note (3.9) implies that a general correlation function on the Riemann surface Rn can equivalently be written as a correlation function on the plane
C with appropriate boundary conditions:
hO(k)(w) . . . i Rn = R C(u,v)Dφ O(w) . . . e−S(φ) R C(u,v)Dφ e−S(φ) = hΦ +(k) n (u)Φ−(k)n (v)O(w) . . . iC hΦ+n(u)Φ−n(v)iC , (3.13)
where O(k)(w) . . . represents any number of operators on the k-th sheet and φ is (again) short-hand notation for n different fields.
3.1.3 Scaling dimension of twist fields
We will now determine the scaling dimension ∆ of the twist fields, by first making the mapping from Rn to C explicit5 [22], Rn→ C : w → z = w − u w − v n1 , (3.14)
which maps the branch points (u, v) to (0, ∞). Whereas w is multi-valued on the entire Riemann surface because it takes the same value for identical points on each sheet, the mapping (3.14) produces a coordinate z which is single-valued on the entire plane. Now, under any conformal mapping, the (holomorphic) stress tensor transforms accordingly as (see e.g. [26]),
T (w) = dz dw
2
T (z) + c
12{z, w}, (3.15)
where c is the central charge and
{z, w} = z
000z0− (3/2)(z00)2
(z0)2 (3.16)
is the Schwarzian derivative.
Now, the crucial observation is that the expectation value of the holomorphic stress tensor vanishes on the plane due to translational and rotational symmetry, hT (z)iC= 0. Therefore, we 5The power 1/n makes sure that the argument of z (i.e. the complex angle in z = |z|eiθ) has periodicity 2π instead of 2πn.
can calculate its expectation value on the Riemann surface using (3.14) and (3.15), yielding hT (w)iRn = c 12{z, w} = c(1 − n−2) 24 (u − v)2 (w − u)2(w − v)2. (3.17)
In the previous section, we noted that correlators on the Riemann surface may equivalently be written as a correlation function on the plane involving the twist fields Φ±n. Therefore, the right hand side of (3.17) is equal to the left hand side of (3.13), i.e.
hT (w)Φ+n(u)Φ−n(v)iC= c(1 − n −2) 24 (u − v)2 (w − u)2(w − v)2hΦ + n(u)Φ − n(v)iC. (3.18)
We can evaluate the left hand side of (3.18) using the conformal Ward identity (see e.g. [26]), hT (w)Φ+n(u)Φ−n(v)i = h (w − u)2 + ∂u w − u+ h (w − v)2 + ∂v w − v hΦ+n(u)Φ−n(v)i, (3.19)
where the derivatives may be calculated using (3.12), giving
hT (w)Φ+ n(u)Φ−n(v)iC= h(u − v)2 (w − u)2(w − v)2hΦ + n(u)Φ−n(v)iC. (3.20)
By combining (3.18) and (3.20) and using h = ¯h for the twist fields [25], we conclude that the scaling dimension is given by
∆ = c(1 − n
−2)
12 . (3.21)
3.2 Entanglement entropy of the ground state
Now that we know the value of ∆, it is easy to compute Tr ρnA, since this is simply given by n powers of the two-point function hΦ+n(u)Φ−n(v)i. From (3.11), (3.12) and (3.21) we obtain
Tr ρnA= ` µ
−6c(n−1/n)
, (3.22)
where ` = |u − v| is the size of interval A and µ is the UV cutoff. Plugging (3.22) into (3.1) gives the famous result of [24] for the ground state entanglement entropy on an infinite line6
SA=
c 3log
`
µ, (3.23)
where the presence of the UV cutoff µ reflects that the entanglement entropy is physically divergent in a local quantum field theory.
3.2.1 Generalizations to finite size or finite temperature
We have learned that Tr ρnA transforms as the two-point function of primary operators Φ±n. The upshot is that we can easily compute the two-point function of primary operators in other
geometries M, as long as M is related to the plane by a conformal mapping C → M : w → z(w), under which the two-point function transforms as
hΦ+ n(z1)Φ−n(z2)i = dw(z1) dz h dw(z2) dz h hΦ+ n(w1)Φ−n(w2)iC. (3.24)
This observation allows us to quickly calculate the entanglement entropy in different geometries. Consider a system of finite size Lcy, which is a cylinder when Euclidean time is included.
The cylinder is related to the plane via
w = e2πiz/Lcy, (3.25)
so the two-point function on the cylinder can be calculated from (3.24), giving hΦ+n(z1)Φ−n(z2)icyl = w0(z1)w0(z2) ∆ hΦ+n(u)Φ−n(v)iC = L 2πie 2πi(z1+z2)/Lcy ∆ e2πiz1/Lcy− e2πiz2/Lcy µ −2∆ = Lcy πµ sin π` Lcy −2∆ , (3.26)
where now ` = z2 − z1 is the size of the interval on the cylinder. To get the entanglement
entropy, we use the result for the scaling dimension (3.21) and subsequently take the derivative with respect to n and set n = 1 [22], yielding
SA= c 3log Lcy πµ sin π` Lcy . (3.27)
In the limit where the subsystem is much smaller than the total system, i.e. ` L, we recover the result on the line (3.23).
Similarly, we can compute the entanglement entropy of a thermal state on an infinitely long system, represented by a cylinder with compactified (Euclidean) time direction τ ∼ τ + β, where β is the inverse temperature T−1. We map the plane to this cylinder via
w = e2πz/β. (3.28)
Observing that the current cylinder (3.28) is related to the cylinder (3.25) through Lcy → iβ,
we can substitute this in (3.27) to obtain,
SA= c 3log β πµsinh π` β . (3.29)
In the low temperature limit ` β, (3.29) again reduces to the entanglement entropy of the ground state on a line (3.23), as expected. In the high temperature limit, ` β, we find S ∼ 3βπc`, which is extensive (linear in the volume `). In this limit, SA agrees with the ordinary
3.3 Entanglement entropy for excited states
The entanglement entropy for excited states on a system of length Lcy was first calculated by
[27]. The excited state is denoted by Υi and it is defined through the insertion of an operator in the infinite past τ → −∞ acting on the vacuum7 0i,
Υi = lim
τ →−∞Υ(τ, x)
0i. (3.30)
The wave functional is, just like the ground state, given by a path integral (cq. (3.2)), but with an extra factor due to the operator Υ(x, −∞)
Ψ(φ, Υ) =
Z φ(0,x)=φ−(x)
τ =−∞
Dφ e−S(φ)Υ(−∞, x). (3.31)
The associated density matrix reads ρΥ= CZ1−1 Z Dφ e−S(φ)Y x δ φ(0−, x) − φ−(x) Y x δ φ(0+, x) − φ+(x)Υ(−∞, x)Υ†(∞, x), (3.32) where the fields are not yet identified at τ = 0, Υ†(∞, x) is the conjugate operator inserted in the infinite future and Z is the partition function for the degrees of freedom φ in the absence of Υ. To fix the normalization constant C, we take the trace and demand that Tr ρΥ = 1. The
trace produces a factor Z from φ and a two-point function due to the two inserted operators Υ, Υ†, yielding
C = 1
hΥ(−∞, x)Υ†(∞, x)i. (3.33)
The reduced density matrix ρΥ(A) is then obtained by tracing over the degrees of freedom in
B, giving an expression identical to (3.32), but with the cut restricted to x ∈ A.
We now again employ the replica trick. This entails making n copies of the above and sewing them together cyclically along the cuts, yielding an n-sheeted Riemann surface Rn, which now
includes 2n operator insertions Υk and Υ †
k, one for each sheet 1 ≤ k ≤ n. To calculate the
entanglement entropy, we are interested in the trace Tr ρnΥ(A). Similar to the above, we get a contribution Zn from the field degrees of freedom, times by a 2n-point function from the
operators, Tr ρnΥ(A) = Zn Zn 1 hQn k=1Υk(−∞, x)Υ † k(∞, x)iRn hΥ1(−∞, x)Υ†1(∞, x)inR 1 , (3.34)
where the subscripts 1 and n refer to the single- and multi-sheeted model, respectively. Observing that the factor Zn/Z1n is precisely Tr ρngs(A) of the ground state (see eq. (3.7)), we can define
7
This is known as the state-operator correspondence. It crucially depends on the conformal invariance of the theory, which allows us to employ radial quantization. The infinite past corresponds to the limit z, ¯z → 0, where z = exp (2π(τ + ix)/Lcy). Since this point is the origin of the z-plane, we can associate a local operator to it. For an ordinary QFT (one without conformal symmetry), we cannot define a local operator in the infinite past since we cannot regard it as a single point in space.
the ratio between the excited state and the ground state by FΥ= Tr ρnΥ(A) Tr ρn gs(A) = h Qn k=1Υk(−∞, x)Υ†k(∞, x)iRn hΥ1(−∞, x)Υ†1(∞, x)inR1 . (3.35)
The entanglement entropy may then be calculated as follows. First, note that the R´enyi entropy is given by SA(n)= 1−n1 log Tr ρnA, from which we can define the quantity
FΥ= exp
h
(1 − n)SAexc,(n)− SAgs,(n)i. (3.36)
Taking the derivative with respect to n and analytically continuing to n → 1 then gives the entanglement entropy SAexc= SAgs−∂FΥ ∂n n=1 . (3.37) 3.3.1 2n-point function
Our aim is now to calculate the 2n-point function in (3.35). As remarked in the previous section, the correlation functions are difficult to calculate on Rn. The solution again lies in
shifting the system to the complex plane. The complicated topology of the Riemann surface is then encoded in the transformation factor of the correlation functions. Let the entanglement interval be A = [u, v] and introduce complex coordinates w = x + iτ . Instead of immediately mapping to the single complex plane, we first map Rn to n copies of the plane via w → ζ, with
ζ = e
iπ(w−u)/Lcy− e−iπ(w−u)/Lcy eiπ(w−v)/Lcy− e−iπ(w−v)/Lcy
!
, (3.38)
The branch points w = (u, v) get mapped to ζ = (0, −∞), whereas the points in the infinite past w = w−∞and future w = w+∞, corresponding to the coordinates of Υ and Υ†, respectively, get
mapped to
w±∞→ ζ±∞= e±iπ∆φ, (3.39)
as can easily be verified using (3.38). Here ∆φ = |v − u|/Lcy is angular size of the interval.
Next, we map these n copies of the plane to a single plane C via ζ → z = ζ1/n, i.e.
z = e
iπ(w−u)/Lcy− e−iπ(w−u)/Lcy eiπ(w−v)/Lcy− e−iπ(w−v)/Lcy
!1/n
. (3.40)
Similar to the discussion of the previous section, the complicated topology of the Riemann surface now gets encoded by imposing twisted boundary conditions on the fields. On the plane, these conditions may be written as Υk(z) = Υk+1(e2πik/nz), for 1 ≤ k ≤ n. It follows that the
two operator insertion points ζ±∞ get mapped to the points
zk,n− = exp iπ n(∆φ + 2k) , zk,n+ = exp iπ n(−∆φ + 2k) , (3.41)
where zk,n− (z+k,n) are the points coming from the infinite past (future). Under the mapping (3.40), the fields Υ(w) scale as
Υ(w) = dz dw
h
Υ(z), (3.42)
where the derivative is given by dz dw = z n 4π Lcy sin(π∆φ) e iπ Lcy(w−u)− e− iπ Lcy(w−u) −1 e iπ Lcy(w−v)− e− iπ Lcy(w−v) −1 , (3.43)
as can be verified either by hand or using mathematica. Evaluated at the coordinates (3.41), this corresponds to dz dw z=zk,n− = z − k,n n Λ and dz0 dw z=zk,n+ = z + k,n n Λ,¯ (3.44) where Λ = 4π Lcy sin(π∆φ)e−2π|w|/Lcyeiπ(u+v)/Lcy. (3.45)
For the quantity FΥ, the Λ’s cancel in the ratio (3.35), giving
FΥ = n−2n(h+¯h) Qn k=1(z − k,n)h(z + k,n)h (z1,1− )h(z+ 1,1)h n hQn k=1Υk(zk,n− )Υ † k(z + k,n)iC hΥ1(z1,1− )Υ†1(z1,1+ )in C , (3.46)
which is difficult to evaluate due to the explicit dependence on the coordinates z±k,n. We can get rid of the powers of zk,n by transforming back to the cylinder, which is realized by z = eit.
The coordinates of the operators are then given by t±k,n= πn(∓∆φ + 2k), as can be read off from (3.41). Importantly, the operators transform accodingly as
Υ(t) = (iz)hΥ(z). (3.47)
Inserting (3.47) transformation in (3.46), the powers of zk,n cancel as promised, yielding
FΥ(n)= n−2n(h+¯h)h Qn
k=1Υk(t−k,n)Υ†k(t+k,n)icy
hΥ1(t−1,1)Υ†(t+1,1)incy
. (3.48)
which is a 2n-point function on a cylinder of circumference Lcy.
3.3.2 Small interval limit
While it is in principle possible to evaluate (3.48) exactly for certain models [27, 28, 29, 30], such an endeavor lies outside the scope of this thesis. We are interested in its universal results by considering the limit l L, or equivalently ∆φ 1, so that we can approximate the terms
ΥΥ†appearing in (3.48) by an operator product expansion (OPE). The result is cited from [27], FΥ(n)' 1 +h + ¯h 3 1 n − n (π∆φ)2+ O ∆φ3 , (3.49)
where we have included the anti-holomorphic weight ¯h for completeness. Using equation (3.37), we obtain the following universal result for the small interval limit of the entanglement entropy of an excited state ∆SA' 2π2 3 (h + ¯h) ` Lcy 2 . (3.50)
In the next chapter, we will see that when the excited state is a twist field, expression (3.50) may be matched to the holographic entanglement entropy in a conical defect spacetime.
4
Holographic Entanglement Entropy
When we divide the total space of a field theory into two parts A and B, the entanglement entropy SA measures how much information about the entanglement is hidden in B from an
observer in A. Given the AdS/CFT conjecture, we would like to know what the corresponding division is in the dual bulk theory. In particular, one might ask whether it is possible to pinpoint a well-defined division surface γAseparating two bulk regions A0 and B0. Rather remarkably, it
turns out that such a surface exists, which asymptotically coincides with the surface that divides the field theory regions A and B, i.e. ∂γA= ∂A, because the field theory lives on the boundary
of AdS. This, however, still gives infinitely many choices for the surface. The proposal of Ryu and Takayanagi is that we should choose the minimal surface area. The entanglement entropy SA is then holographically computed by the Ryu-Takayanagi (RT) formula [7, 31, 32]
SA= min ∂γ=∂A Area(γA) 4G , (4.1)
where G is Newton’s constant. Formula (4.1) was historically motivated by the analogy with black holes, whose horizons can also be thought of as a surface beyond which information is hidden.
4.1 Heuristic derivation of RT formula
The main idea behind the holographic computation of entanglement entropy is the holographic analog of the replica trick outlined in the previous chapter. That is to say, we make n copies of the gravitational spacetime and sew them together cyclically, defining an n-sheeted geometry Sn.
We then evaluate the gravitational partition function on this space and apply formula (3.1). The reason we expect this procedure to yield the entanglement entropy is due to the fundamental principle of AdS/CFT, namely the bulk to boundary relation, which entails an equivalence of the partition functions,
ZCFT= ZAdS. (4.2)
The strong coupling regime of the CFTdis dual to classical GR plus matter terms8 on AdSd+1.
Hence, the right hand side of (4.2) is just the classical partition function of the Einstein-Hilbert action ZAdS = e−SEH, (4.3) where SEH= − 1 16πG Z M dd+1x√g(R + 2Λ), (4.4)
where we omitted the matter terms in the action.
The boundary Riemann surface Rn is characterized by the presence of a deficit angle 9
8
Here we focus on the case without supersymmetry; with supersymmetry the CFT is dual to SUGRA plus matter terms on AdSd+1.
9
To see this, note that one would have to go around a branch point n times to return to the first sheet. This means that the period of the angular coordinate on the complex plane (i.e. θ in the parametrization x = ρ cos θ and τ = r sin θ) is 2πn. Normally, the periodicity is 2π, so the deficit angle is 2π − 2πn = 2π(1 − n).
Figure 5: A timeslice of the near-boundary region of AdSd+1. When the field theory is bipartitioned
into region A and its complement B, the entanglement entropy is proportional by the area of the minimal surface ˜A that coincides with ∂A on the boundary. This figure is based on [31].
δ = 2π(1 − n) on the surface ∂A. If we were sufficiently dilligent, we could find a back-reacted (d + 1)-dimensional geometry Sn whose metric that approaches that of Rn at the boundary
r → ∞. In three dimensions, we can safely assume that the back-reacted Sn is given by an
n-sheeted AdS3 geometry, because locally every solution of the Einstein equations is still AdS3.
Similar to the n-sheeted Riemann surface on the boundary, the n-sheeted AdS3 geometry is
characterized by a deficit angle δ = 2π(1 − n) along some one-dimensional surface γA. It
should be noted that the conical deficit metric does not satisfy the Einstein equations, but approximately does so in the limit where n → 1 [33]. Inspired by the AdS3 result, [7] makes
the natural assumption that this result generalizes to general d, where Sn is now an n-sheeted
AdSd+1 geometry and γA is a codimension two surface.
As reviewed in appendix A, the Ricci scalar for conical spaces behaves like a delta function
R = 4π(1 − n)δ(γA) + R(0), (4.5)
where the delta function is localized on the entire codimension two surface, i.e. δ(γA) = ∞
for x ∈ γA and δ(γA) = 0 otherwise, and R(0) is the Ricci scalar of pure AdSd+1. Plugging
(4.5) in expression (4.4) for the gravitational action and evaluating the integral yields a term proportional to the area of γA,
SEH= −
(1 − n)Area(γA)
4G + . . . , (4.6)
where the dots refer to terms linear in n coming from the part of AdS away from the deficit angle. By virtue of the equivalence of the partition functions, we can compute the entanglement
entropy via10 SA= − ∂ ∂n[log Zn− n log Z1] n=1, (4.7)
where Z is the gravitational partition function defined in (4.3). Plugging (4.6) into (4.7) and observing that the contributions from pure AdS cancel because these are linear in n, we find
SA= − ∂ ∂n (1 − n)Area(γA) 4G n=1 = Area(γA) 4G . (4.8)
That γA is the minimal surface can heuristically be understood from expression (4.6). By
invoking the action principle in the gravity theory, expression (4.6) tells us that γA should be
the minimal area surface.
4.2 Some properties of holographic entanglement entropy
Here we review some properties of the holographic entanglement entropy. In particular, we show that strong subaddivity is satisfied and that the CFT area law is reproduced, both of which were already discussed in the previous chapter. Notice that these are not the only properties of the entanglement entropy, but merely the ones that will be useful in the remainder of the thesis. For a full account of its properties, see [31, 32].
4.2.1 Strong subadditivity
As discussed in section 2.2.1, the entanglement entropy satisfies an inequality known as strong subadditivity. It is given by
S(I1∪ I2) + S(I1∩ I2) ≤ S(I1) + S(I2), (4.9)
where I1 and I2 are two overlapping regions in a QFT. We will now show that the holographic
entanglement entropy given by (4.1) satisfies this inequality in AdS3. The arguments
straight-forwardly apply to higher dimensions, but AdS3 is particularly suitable because its minimal
surfaces are geodesics.
Proof: Consider for simplicity a timeslice of AdS3 in Poincar´e coordinates. Figure 6a shows
the boundary intervals I1 and I2, as well as the corresponding blue bulk geodesics used to
calculate the entanglement entropies, which intersect in the bulk at point p. Also shown are the two green geodesics, associated to the entanglement entropies S(I1∪ I2) (big) and S(I1 ∩ I2)
(small). In the current state of affairs, it is not obvious that the inequality (4.9) is satisfied. To proceed, we re-arrange the original blue geodesics at the intersection point p to form two different curves, shown in red and yellow in figure 6b. This re-arrangement leaves the sum of the areas unchanged. Thus, we can express the right hand side of (4.9) in terms of the areas of these new curves. The crucial observation is that the red (yellow) curve is homologous to
10
In the previous chapter, we expressed this equivalently as SA= − ∂ ∂nTr ρ n A n=1, Tr ρ n A≡ Zn Zn 1 . Using Tr ρA= 1, we find ∂nlog Tr ρnA|n=1= (∂nTr ρnA)/(Tr ρ
n
Figure 6: (a) The geodesics associated to the boundary intervals I1 and I2 are shown in blue, whereas
the geodesics associated to I1∪ I2 and I1∩ I2 are shown in green. In figure (b), the blue geodesics are
recombined at the intersection point p into the red and yellow geodesic, which in general are not minimal surfaces. Since the big (small) green arc is a geodesic, its length is smaller than that of the red (yellow) curve, which proves the inequality (4.9).
(correspond to the same boundary interval as) the big (small) green geodesic. However, the latter are geodesics due to the minimality condition in the RT prescription (4.1), which means have the smallest area within their respective homology classes. Thus, the sum of the two green surface areas is smaller than the combined area of the red and yellow curve. Going back to the original configuration with blue geodesics, it follows that the inequality (4.9) is satisfied, as desired.
4.2.2 Relation to CFT area law
It is not hard to convince oneself that the RT formula gives rise to the area law (2.22) in the boundary field theory. First, note that since the AdS metric is divergent at z = 0, we have to introduce a cutoff z = µ. The near-boundary region z = µ then gives the dominant divergent contributions to the area of the minimal surface, so from the RT formula (4.1) we find S ∝ Area(∂A)/µ, in accordance with the area law.
The correspondence between the RT formula (4.1) and the area law (2.22) is related to the observation that areas and volumes are proportional on large enough scales, which is the fundamental property of anti-de Sitter space which makes holography possible. In order to see that area ∝ volume on large scales, consider a very large sphere (i.e. with radius R → ∞) in AdSd+1 and compute its area and volume:
Area ∼ Rd−1 Volume ∼ Z L R0d−1dR0 p1 + R02/L2 R→∞ −−−−→ LR d−1 d − 1 , from which we conclude that indeed
Figure 7: The figures shown in (a), (b) and (c) represent a timeslice of Poincar´e, global and BTZ coordinates, respectively. The boundary intervals are shown in green and the associated geodesic is shown in orange. The cutoff used to regulate the results is represented by the small gap between the geodesic and the boundary.
4.3 Examples of HEE in AdS3
We will now explicitly consider some examples of the RT prescription (4.1) in three-dimensional gravity theories, namely in Poincar´e, global and BTZ coordinates. Here minimal surfaces are spatial geodesics, whose lengths are relatively easy to compute in an AdS3 metric. These
ex-amples are also instructive because the results can be compared to the CFT2 expressions from
the previous chapter. The agreement is perfect, which historically provided one of the first compelling evidences for the validity of (4.1).
4.3.1 Poincare coordinates
We shall start with the calculation in Poincar´e coordinates (2.13), whose metric is reproduced here for convenience:
ds2 = L
2
z2 −dt
2+ dz2+ dx2 , (4.11)
where L is the radius of AdS3. The boundary is at z = 0, where the metric diverges. Hence,
the length of a spatial geodesic ending on the boundary is actually infinite, which is consistent with the field theory result. To get a finite answer, we place the boundary at a finite distance z = µ. This setup is represented in figure 7a.
We will now show that spatial geodesics with both endpoints on the boundary at z = µ are semi-circles. To construct the geodesic, we minimize the distance functional
Length(γA) = Z ds q det(gabx˙ax˙b) = L Z dz z s 1 + dx dz 2 , (4.12)
leading to the Euler-Lagrange equation L z
x0
p1 + (x0)2 = constant, (4.13)
geodesic has a turning point at some maximal distance z = z∗, where the derivative diverges, i.e.
x0(z∗) → ∞. Plugging this into (4.13) informs us that the constant is equal to L/z∗. Therefore,
the Euler-Lagrange equation may be written as the following differential equation dx dz = z pz2 ∗ − z2 . (4.14)
Integrating this expression, we find that geodesics must satisfy x2+ z2= z∗2, justifying the claim
about semi-circles.
For a boundary interval of size `, with coordinates x ∈ [−`/2, `/2], we can parametrize the semi-circles as
(x, z) = `
2(cos s, sin s), ( ≤ s ≤ π − ) (4.15)
where = 2µ/l is the parametrized UV cutoff, which consistently reproduces lims→0z = µ.
Using the parametrization (4.15), the length of this geodesic can obtained as
Length(γA) = 2L Z π/2 ds sin s = 2L log ` µ, (4.16)
from which the entanglement entropy can be found by dividing by 4G. We wish to relate (4.16) to the CFT result (3.23). Note that the cutoffs can simply be identified, since the AdS/CFT correspondence says that the UV of the CFT is dual to the IR of the gravitational theory. Using the standard AdS/CFT dictionary c = 3L/2G [31], we finally obtain
Svac= c 3log ` µ. (4.17) 4.3.2 Global coordinates
We will now calculate the length of a spatial geodesic ending on the boundary of global AdS3,
as in figure 7b, with metric
ds2 = L2 − cosh2ρ dτ2+ dρ2+ sinh2ρ dφ2 . (4.18)
In order to evaluate the entanglement entropy, we use the embedding construction of AdS3
(following [34]), which we reviewed in section 2.1. That is, we embed the hyperboloid in the embedding space R2,2, with metric g
αβ = diag(−1, −1, +1, +1) and embedding coordinates
T1 = L cosh ρ sin τ, X1= L sinh ρ cos φ
T2= L cosh ρ cos τ, X2 = L sinh ρ sin φ, (4.19)
For convenience, we combine these coordinates into a single vector Xα= (T1, T2, X1, X2), whose
inner product is defined with respect to the embedding metric. Hence, the hyperboloid is given by X2 = −T12− T2
To find a spacelike geodesic, one looks for solutions which minimize the following functional δI = 1 2 Z ds gαβ dXα ds dXβ ds = Z ds L(X, ˙X), (4.20)
where gαβ is the embedding metric. To ensure we stay on the hyperboloid, it is helpful to
introduce a Lagrange multiplier, so that the Lagrangian reads L = 1
2 ˙
X2+ λ(X2+ L2). (4.21)
The Euler-Lagrange equations for this Lagrangian are ¨Xα = 2λXα. To solve for λ, we make use of the fact that X2 = −L2, so that
d2X2
ds2 = ¨X · X + ˙X · ˙X = 0. (4.22)
Combining (4.22) with the equations of motion, it follows that λ = ˙X2/2L. Therefore, the equations of motion take the rather simple form
L2X¨α= ˙X2Xα. (4.23)
For spacelike geodesics, we have ˙X2 = 1, so the solution to (4.23) is
Xα(s) = mαes/L+ nαe−s/L, m2= n2 = 0, 2m · n = −L2 (4.24)
where mα and nα are constant vectors chosen which enure X2 = −L2. We can then compute the proper distance between two geodesic points Xα(s1) and Xα(s2) via
X(s1) · X(s2) = m · n
e(s2−s1)/L+ e−(s2−s1)/L
= −L2cosh(∆s/L), (4.25)
where ∆s is the proper distance between the two points on the geodesic. In particular, if these are its endpoints, the proper distance equals the geodesic length, i.e. ∆s = Length(γA).
We now have all the means to evaluate the length of a spatial geodesics whose endpoints lie on the boundary of global AdS3. These points are given by (τ, ρ, φ) = (0, ρ0, −`/2L) and (τ, ρ, φ) =
(0, ρ0, `/2L), where ` is the size of the boundary interval and 2πL = Lcy is the circumference11
of the boundary cylinder and the radial coordinate ρ0 is a large distance cutoff. In the
embed-ding coordinates, we denote these points by Xα(s
1) and Xα(s2), respectively. Inserting these
coordinates in the left hand side of (4.25), we obtain the following expression for the geodesic distance,
cosh(∆s/L) = 1 + 2 sinh2ρ0sin2
`
2L. (4.26)
Assuming that the IR cutoff ρ0 is large, eρ0 1, we can then approximate the geodesic length
11
This means that the boundary field theory lives on a circle of circumference Lcy= 2πL. It may seem odd that the AdS radius L is directly related to the circumference of the boundary. However, taking a different compactification amounts to a simple rescaling of L, which is irrelevant.
as Length(γA) = ∆s ≈ L arccosh 1 2e 2ρ0sin2 ` 2L ≈ 2L log eρ0sin ` 2L , (4.27)
which when divided by 4G gives the entanglement entropy. In order to match the CFT result (3.27), we relate the cutoffs via eρ0 = 2L/µ and use the standard AdS/CFT dictionary c = 3L/2G, giving Svac= c 3log 2L µ sin ` 2L , (4.28)
which agrees with (3.27) (where Lcy= 2πL).
Alternatively, (4.28) can be obtained from the map between global and Poincar´e coordinates given in [35],
1
z = cosh ρ cos τ + sinh ρ cos φ
t = z cosh ρ sin τ (4.29)
x = z sinh ρ sin φ
which in particular allows us to map the boundary points from Poincar´e to global coordinates. To wit, we map (t, z, x) = (0, µ, `/2) to (τ, ρ, φ) = (0, ρ0, `/2L), yielding the following relation
`
2 = µ sinh ρ0sin(`/2L) = L sin(`/2L), (4.30)
where we again used eρ0 = 2L/µ for the cutoff. Substituting relation (4.30) into the Poincar´e result (4.17) reproduces the global result (4.28).
4.3.3 BTZ coordinates
We now consider the BTZ black hole, whose metric is written as
ds2= − r 2− r2 + L2 dt2+ r 2− r2 + L2 −1 dr2+ r2dφ2. (4.31)
Changing coordinates to r = r+cosh ρ, t = L
2
r+τ, φ =
L r+
˜
φ, we may write its metric equivalently as
ds2 = L2(− sinh ρ2dτ2+ dρ2+ cosh2ρd ˜φ2). (4.32) The strategy which will give us the entanglement entropy is to exploit the high degree of sim-ilarity with the global metric (4.18). First, move to Euclidean signature via τ = iτE. Now, in
order to obtain a smooth geometry, the Euclidean time is compactified as τE ∼ τE+ β, where
