The connection between cMERA and holography

(1)

University of Amsterdam

Institute of Theoretical Physics

Faculty of Science

Master Thesis

The connection between cMERA and holography

Emma Loos

10558616

Supervisors:

Prof. Jan de Boer

Dr. Michael Walter

(2)

(3)

(4)

(5)

Abstract

MERA is an entanglement renormalization scheme originally developed to approximate ground states of quantum many-body interacting systems. cMERA is a continuous analogy of this scheme that has been proposed to approximate ground states of QFTs. It was conjectured by Swingle in 2009 and later by Nozaki et al. in 2012 that there exists a relation between (c)MERA and holography. They find that the geometry of the (c)MERA network of critical theories is that of AdS space and therefore suggest that (c)MERA can perhaps be considered as an alternative realization of the AdS/CFT correspondence. From the point of view of holographic renormalization it is however not clear how the (c)MERA net-work of a CFT can see the AdS geometry. A CFT sits in a fixed point in QFT space so that the dual bulk theory will not evolve into the bulk. Hence if (c)MERA is indeed an alternative realization of the AdS/CFT correspondence one would suspect that the (c)MERA network of a CFT is not able to register the AdS geometry. Recent results by Milsted et al. affirm this view. They showed that the geometry of the MERA of a CFT is that of a light sheet and also elaborated on how the MERA network can be adjusted so that it does have an AdS geometry.

In the light of the ongoing discussion about the connection between (c)MERA and the AdS/CFT cor-respondence we propose an alternative approach to the problem. In 2018 it was shown by McGough et al. that T ¯T deformed CFTs have a gravity dual that lives in pure AdS space with a radial cutoff. In contrast to a true CFT, a T ¯T deformed CFT follows a RG flow and it corresponds to moving the geometrical cutoff of the gravitational theory inwards. Therefore it would be interesting to consider instead of the cMERAs of a true CFT the cMERA of a T ¯T deformed CFT and find out if and how it is connected to the AdS/CFT correspondence. Instead of assigning a geometry to a (c)MERA network we propose to consider the gravitational dual of a T ¯T deformed CFT and to find out if in its spacetime a cMERA-like structure can be distinguished. In particular we propose to compare the field content at two different geometrical cutoff surfaces in the bulk theory and to see if this map is generated by a cMERA-like operator that also prepares the ground state of a T ¯T deformed CFT. If it does, it seems as if (c)MERAs might indeed be interpreted as an alternative realization of the AdS/CFT correspondence and in that case we hope to make clearer how the scale transformator and the entangler in the cMERA setup are related to objects in AdS space. If it does not, we might have to consider a continuous version of the MERA modification that Milsted et al. proposed and then try to relate objects in this alternative cMERA setup and holography.

(6)

Introduction

Whenever I tell people that I study physics they often say - after they have somewhat surprisedly remarked that I am a girl - to be quite fascinated by physics. Many people either refer to quantum mechanics or to Einstein’s theory of general relativity. It is not strange that people do so, because both are very succesful theories. But it is also coincedent that people refer to precisely these two theories because it has appeared to be very difficult to unite the two. Physicists have been able to find a quan-tum mechanical description of the other fundamental forces such as electromagnetism, but finding a description of quantum gravity is still a great challenge today. Another big open problem in modern physics is that of strongly coupled theories. Due to their exponentially large Hilbert spaces it is very difficult to model and make predictions about these theories.

A very promising candidate to study both quantum gravity and strongly coupled interacting systems is holography, and in particular its best-known example, the AdS/CFT correspondence. The AdS/CFT correspondence states that a string theory living in a (d + 1)-dimensional Anti de Sitter (AdS) space can be dual to a conformal field theory (CFT) in d dimensions [1]. Two theories being dual means that they, although they are completely different, describe the same physics. In particular it means that objects in one theory can be related to objects in its dual theory. This can be very useful because quan-tities that are difficult to calculate in one theory might be easier to compute in the other theory. The AdS/CFT correspondence is in this aspect especially useful because of two important features. Firstly, every string theory contains the graviton, the quantum mechanical particle of gravity. In string theory itself it is hard to describe the strongly coupled regime where quantum mechanical effects truly come into play, but since we understand CFTs reasonably well, the AdS/CFT correspondence might help us out here. Secondly, the AdS/CFT correspondence is an example of the weak/strong duality, meaning that whenever the string theory in AdS space is in its perturbative regime, its dual CFT is strongly coupled, and vice versa. Indeed, in the perturbative regime we are able to desrcibe string theory, so this might be a chance to learn something about strongly coupled CFTs. The AdS/CFT correspondence is thus a powerful framework to study both quantum gravity and strongly coupled QFTs.

The AdS/CFT correspondence in principle gives us a framework to describe a UV complete theory of quantum gravity in terms of its dual CFT, so why don’t we already have a theory of quantum grav-ity? The reason for this is that the AdS/CFT correspondence is still not fully understood. More specific, it is not always straightforward how objects on one side of the correspondence should be translated to

(9)

(a) _(b)

Figure 1.1: Graphical representation of tensors. (a), every different tensor, Tij _{and A}

klmn, is represented

by a different geometrical object and the number of indices denotes the number of legs. (b), graphical representation of the tensor product Tij_A

jlmn. Legs are glued together when indices are contracted.

objects in the dual theory. One of the biggest questions is how to reconstruct (d + 1)-dimensional local gravitational physics directly from the dual d-dimensional field theory point of view. To gain more in-sight in this and other unsolved topics it would be very useful to have a simplified version of holography. And this is precisely the subject of this work: we focus on the question whether tensor networks and in particular (c)MERAs can be interpreted as such a simplified version of the AdS/CFT correspondence. Tensor networks

Tensor networks were originally developed to approximate ground states of quantum many-body inter-acting systems. As we pointed out above, the Hilbert spaces of such systems are in general exponentially large preventing a full simulation of the Hamiltonian already for a relatively small number of particles. To nonetheless make predictions about many-body systems physicists often rely on using simplifying assumptions. Tensor networks are an example of such a simplifying assumption: they target that regime of the Hilbert space that corresponds to low energy states and by accordingly using a variational proce-dure an approximation of the ground state of the theory can be found.

One might wonder how tensor networks precisely target that regime of the Hilbert space that corre-sponds to the low energy states. It can be intuitively understood from the observation that many natural quantum lattice systems have ground states that are little entangled because the entanglement entropy satifies an area law behaviour [2]. Consider for example a gapped lattice system with a unique ground state so that this ground state is a pure state. Cutting the system into two regions A and B, the Von Neumann entropy, SA= TrρAlog ρA, with ρA= TrBρ the reduced density matrix, measures the

en-tanglement between the region A and B. If the two regions are not entangled at all SA= 0 and its value

will increase when the regions get more entangled, being bounded from above by the maximum value S = |A| log₂(d). In thermodynamics entropy is known to be an extensive quantity. The Von Neumann entropy however does not scale with the volume of region A but instead with the area of that volume and we often refer to this scaling behavior as the area law behavior. Due to the area law behavior ground states of gapped systems are far less entangled than they could be when their entanglement entropy would be extensive, so we might characterize the ground states of these systems by their (relatively) little entanglement. The pure states exhibiting low entanglement in the sense of satisfying an area law

(10)

Figure 1.2: The MPS network (left) describes a one-dimensional spin chain via virtual entanglement degrees of freedom. The PEPS network (right) describes a two-dimensional spin system in the same manner. The free legs in these networks represent physical degrees of freedom, e.g. spins on a chain. The row of contracted tensors above are virtual and account for the entanglement structure in this chain.

constitute a very small subset of all pure states and the (small) region of the Hilbert space where these states live is sometimes referred to as the ‘natural corner of Hilbert space’. As we will see in chapter 4 tensor network states also satisfy an area law behaviour. Tensor networks therefore prepare precisely those states that are relatively little entangled and hence target the ‘natural corner of the Hilbert space’. Upon optimization accordingly the ground state can be found.

Tensor networks are collections of tensors and using the graphical rules to represent tensors [3] they can easily be graphically depicted. Every different tensor is represented by a different geometrical object and every tensor-index is illustrated by a “leg” coming out of the geometrical object, see figure 1.1a. Legs of different tensors are connected if the indices of the tensors are contracted, see figure 1.1b. The number of values that each of the indices, i.e. legs between the tensors, can take, represents quantitative measure of the amount of quantum correlation in the wave function. The indices are usually called bond or ancillary indices, and the amount of possible values are referred to as bond dimensions. Using the rules described above complicated correlation structure can be formed and moreover can be depicted relatively easily.

Different types of tensor networks can be used to describe different theories. The Matrix-Product State (MPS) for example describes a one-dimensional spin chain via virtual entanglement degrees of freedom. We show an MPS network on the left side of figure 1.2. The open legs represent the spin chain and the row of connected tensors accounts for the entanglement structure in the chain. A two-dimensional spin system can be represented by a Projected Entangled Pair States (PEPS) network [4] which we depict on the right side of figure 1.2. In both tensor networks the ground state can be found by a variational optimization.

In this thesis we will focus on another type of tensor network, the Multiscale Entanglement Renormal-ization Ansatz (MERA) network [5] and especially on its continuous analogy, the cMERA network1

[6]. (c)MERAs distinguish themselves from MPS and PEPS networks in that they consist of multiple layers, see figure 1.3. In the MERA procedure the ground state is gradually built from a relatively

(11)

Figure 1.3: A MERA network consists of multiple layers. The dangling legs represent the physical degrees of freedom and these are prepared by a tensor network of entanglers and scalers.

simple state without an entanglement structure. Every layer, which consists of two different tensors, endows the state with some entanglement - after all, a ground state in general has some structure. In fact, the two tensors that make up a layer are an entangling unitary and an isometry that acts as a scale transformator, thereby stretching the correlation that the entangling unitary applied over longer distance scales. The successive action of the entangling unitary and the stretching isometry of different layers creates a complex state in which both short- and long-range correlations are present. The ground state of the system is prepared by that combination of entanglers and scalers that minimizes the energy. The layers of the (c)MERA network can be parametrized by some variable w and if we do so we explicitly see an extra spatial direction emerging. Keeping the rescaling characteristic of the isometry in mind we might interpret this extra spatial dimension as parametrizing different length, or, equivalently, different energy scales in the system. Therefore, (c)MERAs are believed to be closely related to renor-malization group flow [7], [8] in which theories are also considered at different energy scales.

Connection of (c)MERA networks and the AdS/CFT correspondence

The emergence of an extra spatial direction in (c)MERA networks is reminiscent of the extra spatial dimension in the AdS/CFT correspondence. Moreover, one of the entries of the AdS/CFT dictionary is that of holographic renormalization group flow [9] which states that the RG flow in the boundary theory is related to the radial evolution of the on-shell action in the bulk. Motivated by the close connection between (c)MERA and RG flows on the one hand, and the connection between holography and RG flows on the other hand, Swingle investigated [10] if there also exists a connection between MERA and holography. He observed that entanglement entropy in a MERA setup is computed in a way that is reminiscent of the way entanglement entropy is computed in a holohraphic setup: in both setups it is computed by the some minimal surface that hangs in the network/bulk. Moreover, he pointed out that in the MERA network the separation distance of the lattice sites decrease by a factor of e2w, where w parametrizes the layers in the network, and therefore assigned a metric to the network,

ds2= dw2+ e2wd~x2

which can be written as the Poincar´e metric if we change coordinates via w = log z. This suggests that the MERA state in the UV should be interpreted as living in the boundary theory of the AdS/CFT correspondence and and that the MERA network itself should be interpreted as the bulk. If the theory whose ground state is being approximated by a UV MERA state is a holographic CFT, this picture at first sight seems to agree with that of holographic renormalization: we have a CFT - a MERA state in

(12)

the UV - and if we follow its RG flow we are “walking” into the gravitational bulk theory - into the MERA network.

Inspired by Swingle’s work Nozaki et al. [11] performed a more detailed calculation. Their work relies on extracting a network geometry via the Ryu-Takayanagi formula, SA = Area(γ_4G A)

N where γA is a min-imal surface which ends on the boundary of a region A that biparts the system at the AdS boundary. The Ryu-Takayanagi formula shows that entanglement entropy in the boundary theory is related to the geometry of the bulk. By calculating the entanglement entropy in a (c)MERA setup and accord-ingly reading off the metric in the “bulk” they find a network geometry that matches to that of AdS space. Again it is suggested that the cMERAs might be interpreted as an alternative realization of the AdS/CFT correspondence.

Recently Milsted and Vidal [12] abolished the idea that the MERA network in its current formula-tion can be regarded as resembling the bulk AdS spacetime. They show that the geometry of a true CFT is that of a light sheet meaning that the network of a MERA of a CFT has a flat metric. We can understand this if we take in mind the principle of holographic renormalization: the RG flow in the boundary theory is related to a radial evolution into the bulk. In the case that a (c)MERA network approximates the ground state of a CFT there however is no flow because the theory is by definition sitting in a fixed point in QFT space so that when we consider the theory at a different energy scale we have not moved at all in QFT space. Therefore it is to be expected that also in the gravitational bulk theory there has not taken place any radial evolution. Accordingly the question arises how the (c)MERA in that case can “feel” the hyperbolic structure of AdS space. A more logical outcome would be that the metric of the MERA of a CFT is flat.

If correct, the argument of [12] seems to rule that (c)MERAs and holography are connected. The authors however proposed a way to alter the MERA formulation in such a way that the geometry of the MERA network indeed becomes that of AdS space. They suggest to add a non-unitary part to the originally unitary quantum circuit that prepares the MERA state. The authors call the non-unitary part a layer of Euclideons, which refers to the fact that the it should be of the form e−H. From the point of view of holographic renormalization this seems to be a reliable idea: adding a layer of Euclideons to the network reminds of as a first step deforming the CFT slightly such that the theory is brought out of its fixed point in QFT space so that a RG flow can take place.

We see that different physicists have proposed different relations between (c)MERA and holography. In this project we have thought about alternative ways to test if and how the two subjects are related. Firstly we noted that it would be interesting to test the relation that Swingle and Nozaki et al. con-jecture for a different boundary theory. In particular one that has a non-trivial RG flow such that its (c)MERA network can see the AdS geometry. We propose to consider T ¯T deformed CFTs. It was shown [13] that T ¯T deformed CFT2s have a gravity dual that lives in pure AdS3 with a radial cutoff.

The value of the geometrical cutoff, rc, shrinks as the deformation parameter µ on the boundary side

grows. This means that the more deformed the boundary theory becomes - the further the theory flows away from the CFT point in QFT space - the deeper we flow into the bulk while the geometry inside the compact bulk spacetime remains that of pure AdS. Therefore, if we would compute the cMERA of a T ¯T deformed CFT we expect to find the metric of pure AdS if the proposal of Swingle and Nozaki et

(13)

al. is correct. The improvement with respect to their works would be that the T ¯T deformed CFT has a non-trivial RG flow that is dual to a radial evolution in the bulk such that its (c)MERA will be able to detect the structure of AdS space.

During this project we have tried to establish the cMERA of a T ¯T deformed CFT. The cMERA formu-lation that was developed by [6] however only works well for free theories whereas T ¯T deformed CFTs have an interaction term. The perturbative approach to weakly interacting theories of [8] so far did not yield a satisfying result. Along the way we however noticed that perhaps developing a strategy that does not rely on determining a geometry would be beneficial. Both Swingle and Nozaki et al., and Milsted and Vidal try to indicate how (c)MERAs are related to holography by assigning a geometry to a (c)MERA network. They however use a different technique to do this and it seems as if this step is ambiguous. Towards the end of this work we therefore propose an alternative approach to test how (c)MERAs are related to holography. In particular we suggest to start from AdS space and to see if a cMERA structure can be discovered. More specific, if we can find a map that maps the (bulk) field content at a radius r to the field content at a radius r + δr we might be able to extract a cMERA-like operator and this could perhaps reveal more precisely how (c)MERAs are related to holography.

This thesis is organized as follows. First, in chapter 2 we will give a short review of the AdS/CFT correspondence. This knowledge is necessary to follows the discussion if and how cMERAs and holog-raphy are related. After that, we introduce MERAs and in particular cMERAs in chapter 3. Chapter 3 is mainly devoted to the cMERAs of free theories, but we will give an overview of currently available techniques to compute the cMERAs of interacting theories in Appendix ??. We will dive into the dis-cussion about cMERA networks having a AdS or a flat metric in chapter ??. After that in chapter 5 we will switch gears and make the reader familiar with the operator T ¯T , T ¯T deformed CFTs and their gravity dual, AdS space with a radial cutoff. In the outlook, chapter 6, we will summarize where and why the attempts that we did to establish the cMERA of the T ¯T deformed CFT broke down.

(14)

Chapter 2

Preliminaries

2.1 AdS/CFT correspondence

The Anti-deSitter/Conformal Field Theory (AdS/CFT) correspondence was first conjectured by Mal-dacena in 1997 [1]. It is a duality between a (d + 1)-dimensional Anti de Sitter (AdS) space which contains quantum gravity and a d-dimensional Conformal Field Theory (CFT) without gravity. It is usually stated as,

Type IIB string theory on AdS5× S5' d = 4, N = 4 SU(N) super-Yang-Mills theory (2.1)

{ls, gs} {N, gY M}

where one should note that super-Yang-Mills theory is in fact a (super)conformal field theory which is the reason that we speak of the AdS/CFT correspondence. The free parameters on the gravitational side are the string length lsand the string coupling gs. On the field theory side N denotes the number

of supercharges, SU(N ) the gauge group and the free parameters are N and the coupling constant gY M.

The free parameters on either sides of the correspondence are related via g2Y M = 2πgs, 2g2Y MN = L

4

/l4s (2.2)

where L is the radius of curvature of the AdS space. We usually refer to the AdS space as the bulk and to the CFT as the boundary because in some sense the CFT lives on the boundary of the AdS space, of the bulk1_{. The ' sign means that the two theories are dual.}

Dualities, quantum gravity and the holographic principle

Two theories being dual means that they give a different description of the same physics. This is possible because the theories are mathematically identical, in particular, the Hilbert spaces and the dynamics of the theories agree. Physically two dual theories however differ, they may be described by another Lagrangian. Indeed, as we will see the mathematical properties of the theory that lives in AdSd+1 space

and CFTd are similar, but already at this point we observe that string theories and CFTs physically

seem to be completely different. Every string theory contains the graviton and hence is a gravitational

1_{The correspondence comes down to a mapping of operators that live on the boundary of the AdS space to operators}

(15)

theory whereas naively QFTs on flat spacetime do not appear to be a theory of quantum gravity. This is however precisely where the power of this duality lies. Due to the large amount of symmetries, CFTs are understood reasonably well. The duality implies that being able to describe one of the two theories well means that in principle one is also able to describe its dual theory. Therefore it is believed that the AdS/CFT correspondence can help us to reason about a description of quantum gravity.

Besides the at first glance preposterous statement that a non-gravitational theory can teach us some-thing about quantum gravity, the AdSd+1/CFTd correspondence is nontrivial in another sense. The

string theory that lives in the bulk is a (d + 1)-dimensional theory whereas the conformal boundary theory is a d-dimensional theory. i.e. the two theories live in different dimensions. The AdSd+1/CFTd

correspondence is therefore a realisation of the holographic principle. The holographic principle was motivated by the Bekenstein bound [14], S ≤ Ad/(4G), which implies that the entropy that is stored in

a (d + 1)-dimensional volume is encoded in its boundary area Ad. More generally we can say that the

holographic principle states that in a gravitational theory the number of degrees of freedom in a given volume V scales as the surface area ∂V of that volume. Up until today it however has not fully been understood how a d-dimensional theory can capture all the information of a (d + 1)-dimensional theory. Different limits, the strong/weak duality and strongly coupled field theories

When stating the correspondence (2.1) we specified the free parameters of the bulk and the boundary theory, {ls, gs} and {N, gY M} respectively. Different limits of these parameters specify different forms

of the correspondence. As will become clear in section 2.1.3 it is convenient to define a new coupling in the gauge theory, the ’t Hooft coupling, λ ≡ g2

Y MN . The strongest form of the correspondence is the

one stated in (2.1) with free parameters N and λ on the boundary side. In that case the gravitational bulk theory is a quantum string theory with gs 6= 0 and l2s/L2 6= 0. A slightly weaker but still strong

form of the correspondence is when we take N → ∞ and λ a fixed but arbitrary value. In that case we find via the relations (2.2) that gs→ 0 while L4/l4s 6= 0. The coupling constant gs→ 0 indicates that

the gravitational bulk theory in this limit, known as the ’t Hooft limit, is a classical string theory and that string lengths ls have an arbitrary value. Since string theory is currently best understood in the

perturbative regime, the AdS/CFT correspondence is usually evaluated in the large N limit. A weak form of the correspondence is when on the boundary side we take N → ∞ and λ large. In that case we find via (2.2) that gs→ 0 and ls2/L

2 _{→ 0. Hence we see that not only the string coupling is small}

but also the string size compared to the radius of curvature is infinitesimal. This is therefore the point particle limit of type IIB string theory which is given by type IIB supergravity.

The different forms of the AdS/CFT correspondence are summarized in Table 2.1. From its weak form we learn that the AdS/CFT correspondence is an example of the strong/weak coupling duality, which means that if we consider the regime that one of the theories is weakly coupled, the dual theory is strongly coupled. Indeed, we see that a weakly coupled gravitational theory is dual to a strongly coupled field theory. Strongly coupled field theories are generally very hard to describe due to their exponentially large Hilbert space. Due to the strong/weak duality that the AdS/CFT correspondence exhibits and the fact that we are doing well describing string theory in the perturbative regime, it is believed that the AdS/CFT correspondence can serve as a tool to study strongly coupled field theories.

(16)

N = 4 Super Yang-Mills theory type IIB string theory on AdS5× S5

Strongest form any N and λ Quantum string theory, gs6= 0, l2s/L26= 0

Strong form N → ∞, λ fixed but arbitrary Classical string theory, gs→ 0, l2s/L26= 0

Weak form N → ∞, λ large Classical supergravity, gs→ 0, l2s/L2→ 0

Table 2.1: Different forms of the AdS/CFT correspondence

coupled field theories - two big open problems in modern physics - are encouraging and have sparked enthusiasm in the physics community for over two decades2_{. It has been thoroughly studied and}

al-though a formal proof of the correspondence is still lacking, over the years an extensive dictionary has been established that relates objects of the gravitational bulk theory in AdS space to objects in dual boundary CFT, indicating that the correspondence between the two theories exists.

To get all readers on the same page in this chapter we will give a short review of the AdS/CFT correspondence. To avoid a story that requires a string theoretical background, we will not present Maldacena’s original derivation here (but one could consult [15], or [16] for a rough idea). Instead, we will repeat the argument that was given by ’t Hooft [17] in section 2.1.3 to create some intuition why a string theory and a gauge theory such as a CFT can be dual. This automatically gives us the chance to elaborate on the large N limit, the regime where the AdS/CFT correspondence is best tractable and therefore at the moment is mostly studied. This story is mainly based on [18], [19] and [20]. After that we will list some entries of the dictionary in section 2.1.4. Before we get there we will however first introduce the fundaments of the AdS CFT correspondence: CFTs in section 2.1.1 and AdS space in section 2.1.2.

Many overviews of AdS/CFT have been written. A fine summary about the subject is given by [15] and a more extensive and educational source is [21]. Useful and comprehensive references about CFTs are [22], [23] and [24]. To learn more about AdS space we refer to [25].

2.1.1 Conformal Field Theory

Let us consider a Quantum Field Theory (QFT) with some Lagrangian. The Lagrangian of a QFT can in general be separated in a kinetic and an interaction part, and the interaction term comes with a coupling constant that determines the strength of the interaction. Despite its name the coupling constant is not a constant number. Its value depends on the energy scale, µ, at which we consider the theory. Hence instead of writing it as g we should actually write it as g(µ). The change of its value at different energy scales is usually referred to as the “running of the coupling”. How the coupling runs, is governed by the beta function,

β(g) = µ∂g ∂µ =

∂g ∂ ln µ.

The beta function is a tool to track how a field theory changes when it is considered at different (energy) scales. This change of the theory describes a flow in QFT space and we call this the Renormalization Group (RG) flow.

2_{There are even reports of participants of the Strings ’98 conference dancing the ’Maldacena’ at the conference dinner}

(17)

Some theories are connected to special points in QFT space, fixed points. They are called fixed points because whenever a theory has flowed to this point it can not move away from it. The reason for this is that when we consider the theory at another energy scale we recover precisely the same theory, i.e. a theory with the same symmetries and the same value of the coupling constant, such that it covers the same point in QFT space. Because at these points the coupling constant does not change anymore we can characterize them as those points where the beta function vanishes. Moreover, because the theory does not change at all we say that at this point the theory is scale invariant. In two dimensions it has been proven [26], [27] that theories at these points are even conformally invariant. Therefore, we often think of conformal field theories to live in the fixed points of QFT space.

2.1.1.1 Symmetries

A Conformal Field Theory (CFT) is a relativistic QFT with an extended symmetry group. A CFT is not only invariant under the usual Poincar´e transformations, translations and rotations, but also under scaling transformations and special conformal transformations. The conformal group is generated by conformal transformations, xµ_{→ x}0µ_{, where,}

1. x0µ= xµ_{+ a}µ_{, for translations,}

2. x0µ= Λµ

νxν, for Lorentz transformations,

3. x0µ= λxµ_{, for scaling transformations,}

4. x0µ= xµ+aµx2

1+xνaν+a2x2, for special conformal transformations Note that in d dimensions µ = 0, ..., d − 1.

The conformal transformations preserve angles but not necessarily lengths. In particular, one can check that under a conformal transformation the metric transforms up to a factor,

gµν(x) → Ω2(x)gµν(x).

The generators of the symmetries are, 1. Pµ = −i∂µ

2. Mµν = i(xµ∂ν− xν∂µ)

3. D = −ixµ∂µ

4. Kµ= i(xνxν∂µ− 2xµxν∂ν)

The algebra that these generators obey can conveniently be summarized if we first relabel the generators as Jab for a, b = 0, ..., d + 1, Jµν = Mµν; Jµd= 1 2(Kµ− Pµ); Jµ(d+1)= 1 2(Kµ+ Pµ); J(d+1)d= D. The algebra that the generators Jab obey is then,

(18)

where ηab = diag(−1, 1, ..., 1. − 1). We can conclude that the conformal group in d dimensions is

iso-morphic to SO(2, d). As we will encounter this is the same symmetry group as that of AdSd+1 space.

2.1.1.2 Two-dimensional CFTs

The complex plane and the De Witt algebra

Let us consider the Euclidean plane with coordinates (z0_{, z}1_{). Under a change of coordinates z}µ_{→ w}µ_(x)

we know that the metric transforms as,

gµν → ∂w µ ∂zα ∂wν ∂zβ gαβ.

If the coordinate transformation is a conformal mapping then we also know that the metric simply obtains a conformal fator, gµν_{(w) = Λ(z)g}µν_{(z). If we substitute the metric of the Euclidean plane}

on the right-hand side of these expressions and accordingly combine them, it can be shown that the following relations hold,

∂w1 ∂z0 = ∂w0 ∂z1, and, ∂w0 ∂z0 = − ∂w1 ∂z1, or, ∂w1 ∂z0 = − ∂w0 ∂z1, and, ∂w0 ∂z0 = − ∂w1 ∂z1.

These are the Cauchy-Riemann equations for respectively holomorphic and anti-holomorphic equations. This means that for each (z0, z1) we might as well use the complex coordinates z = z0+ iz1, ¯z = z0− iz1.

This is special for two-dimensional CFTs and allows us to do complex analysis.

On the complex plane, ds2 _{= dzd¯}_{z, we can use the De Witt algebra. Because the Cauchy-Riemann}

equations are satisfied the conformal transformations coincide with the analytic coordinate transforma-tions,

z → f (z), z → ¯¯ f (¯z) which we infinitesimally can write as,

z → z0= z + n(z) z → ¯¯ z0 = ¯z + ¯(¯z) (2.3)

where n(z) = −zn+1 and ¯(¯z) = −¯zn+1. These transformations are induced by the infinitesimal

generators,

ln= −zn+1∂z ¯ln = −¯zn+1∂z¯

that satisfy an algebra,

[lm, ln] = (m − n)lm+n [¯lm, ¯ln] = (m − n)¯lm+n [lm, ¯ln] = 0

This is known as the De Witt algebra. We see that it consists of two independent copies of the same algebra and therefore it is often useful to regard z and ¯z as independent coordinates.

The local conformal group is generated by the full set of generators {ln, ¯ln} for n ∈ Z. The global

(19)

Figure 2.1: Mapping the cylinder R × S1to the complex z-plane. Past and future infinity τ = ±∞ are mapped to z = 0 and z = ∞ and equal time slices Στ to slices of constant radius Σr.

we see that l−1 and ¯l−1 can be identified as the generators of translations, l0+ ¯l0 and i(l0− ¯l0) as

the generators of dilatations and rotations respectively. And it can be shown that l1 and ¯l1 are the

generators of the special conformal transformations (see for example [24] for details). The generators of the global conformal algebra are also useful to characterize properties of physical states. Suppose that we work in a basis of eigenstates of the operators l0 and ¯l0 with eigenvalues h and ¯h. We call these

eigenvalues conformal weights of the state and because l0+ ¯l0generated dilations and i(l0− ¯l0) rotations,

the conformal or scaling dimension ∆ and the spin s are given by ∆ = h + ¯h and s = h − ¯h. In what follows we will generalize these ideas to the full quantum realization of the De Witt algebra.

Quantization procedure and the Virasoro algebra

Let us consider flat two-dimensional Euclidean spacetime, ds2_{= dτ}2_{+ dx}2_{. We change to the Euclidean}

equivalent of light-cone coordinates in Minkowski space, w = τ + ix, ¯w = τ − ix. To eliminate the IR divergences we accordingly compactify the spatial coordinate, x = x + 2π, so that space becomes a cylinder, R × S1. Now we consider a conformal map w → z = ew= eτ +ixthat maps the cylinder to the complex plane coordinatized by z. The infinite past and future, τ = ±∞ are mapped to the points z = 0 and z = ∞, and equal time surfaces correspond to circles of constant radius, see figure 2.1. Usually to move between two different surfaces of equal time, Στ and Στ0, we use the Hamiltonian. In that case the unitary operator eiHt governs the time evolution between two states living on Στ and Στ0. To move between circles of constant radii in the z-plane, Σr and Σr0, we need to dilate the circle of radus r, i.e. we need the dilatation operator. Therefore we can regard the dilatation generator on the conformal plane as the Hamiltonian of the system. Accordingly the Hilbert spaces are built up on the surfaces Σr and thus we use the unitary operator eiDτ to capture the evolution of states. This procedure for

defining a quantum theory on the plane is known as radial quantization.

Because we interpret the dilatation operator as the Hamiltonian of the system it is convenient to consider representations of the conformal group that involve operators (or fields), O∆, that are eigenfunctions of

(20)

the scaling operator D,

[−iD, O∆(0)] = ∆O∆(0) (2.4)

where ∆ is the conformal dimension of the operator O∆. The conformal dimension is sometimes

also called the scaling dimension because under a rescaling, x → x0 _{= λx, a field transforms as,}

O(x) → O0(x0) = λ∆_{O(x). With the interpretation of D as Hamiltonian in the back of our minds,}

equation (2.4) might be interpreted as an equivalent of the time-independent Schr¨odinger equation. Using that analogy the conformal dimensions can be regarded as the energy eigenvalues of the theory. Indeed, just as we in quantum mechanics aim at finding the full energy spectrum, in conformal field theory we search for a full spectrum of conformal dimensions.

The analogy with quantum mechanics can be expanded further if we consider the effect of the operators P and K on the operator O∆,

[−iD, PµO∆(0)] = (∆ + 1)O∆(0)

[−iD, KµO∆(0)] = (∆ − 1)O∆(0)

We see that the operator P and K respectively increase and descrease the conformal dimension of the operator, i.e. that they act as some kind of lowering and raising operator. In fact, just as in quantum mechanics a ground state that is annihilated by the annihilation operator, a |0i = 0, can be found, in a CFT we can find operators of lowest conformal dimension,

[Kµ, O∆(0)] = 0.

We call these operators primary operators. Primary operators are the fundaments of the so-called descendant operators,

O0_∆0 = Pµ1...PµnO∆

where ∆0 = ∆ + n. We see that descendants can be built form primaries via the “raising” operator P . The collection of a primary and its descendants is called a conformal tower.

Perhaps you noted that although in quantum mechanics we speak about energy states, we have only been talking about operators on the complex plane. The reason for this is that states of energy ∆ at τ = 0 on R × S1 _{are intimately related to a local operator with dimension ∆ at ρ = 0 on the complex}

plane. This is called the state-operator correspondence and we can capture the general idea using the following definition,

hφ(x1)φ(x2)...O∆(0)i = h0| φ(x1)φ(x2)... |∆i

i.e. we can interpret every “energy” eigenstate |∆i as being created by the operator O∆.

Having defined the quantum theory on the complex plane, we should also introduce the quantum me-chanical version of the Witt algebra: the Virasoro algebra. It is the central extension of the Witt algebra and is formed out of the generators Ln and ¯Ln, for n ∈ Z, which are in fact the mode operators of the

(21)

on the Hilbert space and satisfy commutation relations, [Ln, Lm] = (n − m)Ln+m+ c 12(n 3_{− n)δ} n+m,0, [ ¯Ln, ¯Lm] = (n − m) ¯Ln+m+ ¯ c 12(n 3_{− n)δ} n+m,0, [Ln, ¯Lm] = 0

where c is the central charge of the theory that can be interpreted as keeping track of the degrees of freedom in the theory. We see that we again get two independent copies of the same algebra and moreover that if c = ¯c = 0 the Virasoro algebra reduces to the classical De Witt algebra. By Noether’s theorem we see that because there are infinitely many generators and thus infinitely many symmetries two-dimensional CFTs also have an infinite number of charges.

Noether’s theorem states that for every continuous symmetry in a field theory there is a current Jµ

that is conserved, ∂µJµ = 0. We consider theories with a conformal symmetry xµ → xµ+ µ(x) and

therefore the conserved current corresponding to these symmetries can be written as, Jµ= Tµνν

where Tµνis the symmetric energy-momentum tensor. It can be shown that conservation of the dilatation

currrent implies that the energy-momentum tensor is traceless, Tµ

µ = 0. For two-dimensional CFTs we

can express the stress-energy tensor components on the complex plane in terms of the stress-energy tensor components on the Euclidean plane via Tzz = 1₄(T00− 2iT10− T11), Tz ¯¯z= 1₄(T00+ 2iT10− T11)

and Tz ¯z = T¯zz = 1₄(T00+ T11) = 1₄Tµµ = 0. Combining these equations with the equation ∂µTµν = 0

that follows from translational invariance, it can be shown that ∂z¯Tzz = 0 and that ∂zTz ¯¯z = 0, i.e.

that the two non-vanishing components of the stress-energy tensor are a chiral and an anti-chiral field, Tzz(z, ¯z) =: T (z) and T¯z ¯z(z, ¯z) =: ¯T (¯z). These fields can be expanded as,

T (z) =X n∈Z z−n−2Ln where Ln= 1 2πi I dz zn+1T (z)

and similarly for the anti-chiral field. We see that the Virasoro generators indeed give rise to an infinite number of charges.

2.1.2 Anti de Sitter space

Anti de Sitter space is a maximally symmetric solution to Einstein’s equations with a constant, negative cosmological constant. That is, given the (vacuum) Einstein equation,

Rµν−1₂R gµν+ Λgµν = 0,

the Riemann tensor and cosmological constant for AdS space are, Rµναβ= −1 l2 AdS (gµαgνβ− gµβgνα) Λ = −d(d − 1) 2l2 AdS

where lAdSis the radius of curvature of AdS space. The form of the Riemann tensor indicates that AdS

(22)

The nature of lAdS as being the curvature radius of AdS space becomes more clear if we actually

try to build an AdS space. A (d + 1)-dimensional AdS, AdSd+1, can be obtained from a submanifold of

(d + 2)-dimensional Minkowski space which has a metric,

ds2= −dx20+ d

X

i=1

(dxi)2− dx2d+1.

The submanifold that defines AdS space is given by the (d + 1)-dimensional hyperboloid,

−x2 0+ d X i=1 x2i − x2d+1= −l2AdS (2.5)

From this description it is clear that lAdS can be interpreted as the curvature radius of the manifold.

We also see that the symmetry group of AdSd+1is SO(2, d) which is indeed similar to that of CFTs. A

metric that describes AdS space can be obtained from the manifold by simply parametrizing x0, xi, xd+1

such that (2.5) is obeyed. Different choices of coordinate systems reveal different aspects of AdS space and in what follows we will treat the most important ones.

Global coordinates

A convenient choice to examine AdS space is to consider a set of coordinates, the so called global coordinates, that cover the whole of AdS space. The global coordinates are defined by,

x0= p l2_{+ r}2_cos(t/l) xd+1 = p l2_{+ r}2_sin(t/l) d X i=1 x2_i = r2

Note that for convenience we wrote lAdS = l and that the coordinates xi parametrize a Sd−1 sphere.

Using the global coordinates the metric that describes AdS space becomes, ds2= −(1 +r 2 l2)dt 2₊ dr 2 1 + r_l22 + r2dΩ2d−1 (2.6)

where t ∈ (−∞, +∞), r ∈ [0, ∞) and where dΩ2

d−1 is the metric on a S

d−1 _{sphere. This global cover}

of AdS shows that deep in the bulk, r l, AdS space reduces to Minkowski space in spherical coor-dinates. At the boundary, r → ∞, however, AdS space looks very different from Minkowski space. To get a better understanding of the boundary structure of AdS we will therefore choose a slightly different parametrization.

Cylinder coordinates

Taking r = l tan ρ the boundary becomes more tractable because the radial coordinate now lies in a finite range, ρ ∈ [0, π/2) . Under this change of coordinate the metric becomes,

ds2= l

2

cos2_ρ(−dt

2_{+ dρ}2_{+ sin}2

(23)

The terms within parentheses define the metric of an infinite solid cylinder with ρ being the radial parameter: ρ = 0 describes the center and as ρ → π/2 the boundary is approached. We can therefore conclude that the boundary of global AdS is conformal to a cylinder R × Sd−1. Recall from the previous section that a CFT on a Euclidean plane is conformal to a cylinder and that hence the boundary of AdS space is a suitable place for a CFT to live, or, stated more carefully, that it seems to be possible to map quantities on the boundary of AdS space to quantities in a CFT.

Poincar´e coordinates

Global coordinates and the cylinder coordinates cover all of AdS and are useful to learn about the properties of AdS space. For later use we however also define Poincar´e coordinates,

x0= z 2 1 + 1 z2(l 2_{+ ~}_y2_{− t}2₎ x1= l zt xj= l zyj xd+1= z 2 1 − 1 z2(l 2_{+ ~}_y2_{− t}2₎

where t, yj ∈ (−∞, ∞) and z ∈ [0, ∞). Note that j = 2, ..., d now and that only half of space is covered

by these coordinates. This choice of parametrization yields the metric, ds2= l

2

z2( dz 2_{+ dy}

µdyµ)

We see that this metric is conformally equivalent to a flat half-space Minkowski spacetime.

2.1.3 ’t Hooft large N limit

The goal of introducing the ’t Hooft large N limit is two-fold. Firstly it specifies the regime in which the AdS/CFT correspondence is best tractable. Secondly, the argument that ’t Hooft gave was the first one that indicated that string theories and gauge theories might be dual to each other. The main observation was that SU(N) gauge theory in the large N limit with effective coupling λ = g2N can be described by a summation over topologies of two-dimensional surfaces, just like in a string theory with coupling gsand string lentgh ls. The argument took place in a completely different context though: the

context of strong interactions.

Quantum Chromo Dynamics (QCD) is the non-Abelian SU (3) gauge theory that describes the strong interaction. The theory is asymptotically free meaning that at high energies the coupling, g, is very small so that the theory has a nice perturbative description. At low energies however, the theory is strongly coupled making it very diffcult to make accurate predictions. Using gauge theory we can thus only say something about the strong interaction for high energies. ’t Hooft came up with an approach that allows us to say something about strongly coupled SU (N ) gauge theories as well.

Consider a SU (N ) gauge theory with a coupling constant g. In principle this theory is described by two parameters, N and g. ’t Hooft however decided to consider the theory as being described by the

(24)

Figure 2.2: The propagators in QCD, the Dirac field in fundamental representation (a), and the spin bosons in adjoint (top) and fundamental × anti-fundamental (bottom) representation.

parameters N and the effective coupling λ ≡ g2N , which is called the ’t Hooft coupling. For a theory we typically want to compute the partition function and correlation functions and in the perturbative limit calculations simplify a lot because an expansion in Feynman diagrams up to some low order can be done. The perturbative limit of the theory at hand is g =pλ/N → 0. If we however simultaneously take the large N limit, N → ∞, the effective coupling λ remains finite and in principle can even be large. The regime N → ∞ and λ large would approximately describe the strong coupling limit. The upshot of introducing the ’t Hooft coupling is thus that although we sit in the effectively strongly coupled regime for correlators a diagrammatic expansion in the small parameter λ/N can be made. To consider the diagrams we supply the Feynman graphs with an additional structure: the double line notation. Double line notation

QCD consists of quarks and gluons. The former are massive spin 1/2 fermions that carry a color and an electric charge, and are represented by Dirac fields in the fundamental representation 3 of the gauge group SU (3), ψa(x), where a is the color index. The latter are spin 1 bosons, AA_µ(x), which in fact also carry color charges, since they lie in the adjoint representation 8 of SU(3). The index A is an index in the adjoint representation. Their propagators are respectively denoted as,

hψa_{(x) ¯}_ψb_{(y)i = δ}ab_{S(x − y);} _hAA µ(x)A B ν(y)i = δ AB_D µν(x − y)

The first of the two equations, the one that carries a color index, is denoted by a straight line as shown in figure 2.2 (a) (top), and from the delta-function it can be seen that the color at the beginning of the line is the same as at the end of it. The pictorial propagator corresponding to the second of these two equations, the one carrying a index in the adjoint representation, is shown in figure 2.2 (b) (top). Recall however that the gluon also carries color. Therefore it is possible to write the gluon propagator in terms of color indices as well,

hAa

µb(x)Acνd(y)i = Dµν(x − y)(

1 2δ a dδcb− 1 2Nδ a bδdc)

Instead of one index A the gluon now carries two indices a, b. An intuitive way to explain the fact that we need two color indices is that we have to maintain electric charge neutrality. Before we graphically depicted a propagator with one color index as one straight line. The fact that we can denote the propagator of the spin one bosons as carrying two color indices implies that instead of drawing one spiralling line, two straight lines in opposite direction can be drawn. This is shown in figures 2.2 and 2.3 and is called the double line notation. Note that the last term of the gluon propagator, the one that contains a factor _2N1 , corresponds to letting lines cross in double line notation, but that we can ignore it in the large N limit so that we do not have to deal with diagrams that have crossing lines.

(25)

Figure 2.3: The propagator of the gauge field (wavy line) and the interaction vertex by double-line notations (top, middle). The bottom figure is an example of a vacuum amplitude. Source [18].

A more formal version of this story is one that makes use of representations. From the propagators we have seen that fundamental (and anti-fundamental) representations are represented by single straight lines and adjoint representations by wiggly lines. It is however possible to write an adjoint representation in terms of fundamental and anti-fundamental representations. In fact, for a U (N ) gauge group we have that,

adjoint = fundamental × anti-fundamental (2.7)

which can easily be checked by the trivial identity N2 _{= N × N , where N}2 _{is the dimension of the}

adjoint representation and N of the (anti-)fundamental representation. For a SU(N) gauge group the relation (2.7) is not entirely true: for the SU(N) group the dimension of the adjoint representation is N2_{− 1. In the limit N → ∞ this effect is however negligible such that we can safely use (2.7) for a}

SU(N) gauge theory in the large N limit as well.

Now that we know how to represent Feynman diagrams in a double line representation, let us see with what strength each diagram comes. To do this, let us derive the Feynman rules. The Lagrangian of a gauge theory can schematically be written as,

L = 1

g2 Y M

[∂A∂A + A2∂A + A4] =N λ[...] We see that the rules can be summarized as,

• Associate a factor λ/N to each propagator • Associate a factor N/λ to each interaction vertex

(26)

Figure 2.4: Examples of vacuum amplitudes. Diagram (a) is characterized by (V, E, F ) = (2, 3, 3), diagram (b) by (V, E, F ) = (4, 6, 4), and diagram (c) by (V, E, F ) = (4, 6, 2) so we find that diagram (a) and (b) both are characterized by an Euler characteristic χ = 2 and diagram (c) by an Euler characteristic χ = 0. Indeed, diagrams (a) and (b) are planar diagrams, diagram (c) is a nonplanar diagram.

• Associate a factor N to each loop

A factor N is associated to each loop because of the summation over the colors N .

If we now consider a diagram in which we denote the number of vertices by V , the number of propagators by E and the number of loops by F , we find that the amplitude of a diagram is given by,

amplitude ∼ N λ V_λ N E NF = λE−VNV −E+F = fh(λ)Nχ

where χ = V − E + F is the Euler characteristic, a topological invariant quantity, and where h is a new measure of the number of loops in the diagram. Usually the Euler characteristic is a combination of the number of vertices V , the number of edges E and the number of faces F so one might be surprised that we identified the number of propagators with E, the number of edges, and the number of loops with F , the number of faces. However, if we fill in the loops by surfaces we find that each vertex can still be interpreted as a vertex, V , that each propagator can be interpreted as an edge, E, and that each loop becomes a face, F . Rather surprisingly we have thus found that the amplitude of a Feynman diagram is determined by its topology. Indeed if we consider the examples in figure 2.4 we find that the diagrams (a) and (b) have an Euler characteristic χ = 2, i.e. g = 0, and that diagram (c) has an Euler characteristic χ = 0, i.e. g = 2. The genus number is reflected in the fact that the diagrams (a) and (b) can be drawn in the plane, whereas diagram (c) can not (due to the crossing leg). Therefore we call diagrams with g = 0 planar diagrams and diagrams with g > 0 non-planar. In the large N limit the partition function, that has a schematic form,

log Zgauge= X g N2−2g ∞ X i=1 λifi(...)

is therefore dominated by planar diagrams.

One might wonder what this all has to do with string theory and especially why this analysis makes the relation between gauge and string theories plausible. To see this, consider the scattering of four closed strings. String theory allows us to calculate the S-matrix of this process perturbatively and this is done

(27)

Figure 2.5: Sum over topologies in the scattering of four closed strings.

by summing over all possible world sheet topologies. Schematically we can write it as, S-matrix = X

genus g

g_s2g−2F (ls)[...]

and this expansion is represented in Figure 2.5. We see that in the limit gs→ 0 the world sheets with

low genus number are dominant, which is reminiscent of the planar diagrams being dominant in the large N limit of gauge theory. Both in gauge theory and in string theory we can do an expansion over topologies. Furthermore, we see that lowest genus diagrams dominate in the large N limit in gauge theories and in the limit gs→ 0 in string theories. The similarities between these two expansions are

remarkable and leads to suspect that SU(N ) gauge theory and string theory can be related to each other under the identification N ∝ 1/gs.

2.1.4 Dictionary

The AdS/CFT correspondence proposes a map between the gravitational bulk theory and the boundary field theory. Accordingly objects of one of the two theories can be mapped to objects of the other, dual theory. The collection of these relations between the boundary and the bulk theory together form what we call a dictionary. A very important entry of the AdS/CFT dictionary is the field-operator map,

lim

r→∞r

∆_{φ(t, r, Ω) ' O(t, Ω)}

where φ is a bulk scalar field and O a primary operator both of conformal dimension ∆. We see that bulk fields of a particular conformal dimension evaluated at the boundary correspond to primary operators in the CFT with that same conformal dimension. We often refer to the CFT as the boundary theory and here we see that the reason for that is because the field-operator map holds only at the boundary of AdS space. Another entry of the dictionary relates the mass of a free massive scalar field in AdS space to the conformal dimension ∆ of a scalar operator in the boundary CFT,

∆ = d 2+ 1 2 p d2_{+ 4m}2_.

An entry that will be important in the rest of this thesis is the Ryu-Takayanagi proposal [29] [30]. The Ryu-Takayanagi proposal considers the holographic interpretation of entanglement entropy. If we have a CFT that lives on R × Sd−1_{we can consider a quantum state |ψi living on a Cauchy slice with topology}

Sd−1_{. Upon decomposing the slice into two regions A and its complement B we can compute the Von}

Neumann entropy,

(28)

where ρA= TrBρ is the reduced density matrix to region A. The Von Neumann entropy is often used as

a measure for the entanglement between two subsystems A and B. Ryu and Takayanagi proposed that in a holographic setup the Von Neumann entropy of the region A in some state |ψi is given by the area of the minimal codimension two extremal area surface, γAwhose asymptotic boundary is the boundary

of A in the geometry dual to |ψi, i.e. for which we have ∂γA= ∂A. To leading order in 1/N ,

SA=

A(γA)

4G

(29)

Chapter 3

cMERA

The Multi-scale Entanglement Eenormalization Ansatz (MERA) was developed as a tool in approximat-ing the ground state of lattice Hamiltonians in the study of interactapproximat-ing quantum many-body systems [5]. Describing an interacting many-body system is still a great challenge today and to make progress one often relies on making use of a simplifying variational ansatz of which MERA is an example. In general a ground state has a (complex) entanglement structure. In the MERA procedure a state with a complex structure is prepared by systematically applying entanglement over different length scales to a state |Ωi without any entanglement. Every layer of the MERA network consists of an entangler and a scale transformator. Every layer thus endows the state with some entanglement and accordingly stretches this correlation over a longer distance. Successive application of these layers therefore creates a state with correlation over different scales. The ground state is accordingly found by minimizing the energy which fixes the form of the tensors in the network.

Over the years MERA was successfully applied to study the physics of a wide variety of strongly in-teracting systems in low dimensions [6] and when this success became apparent the desire to develop a similar method that approximates ground states of QFTs arised. Haegeman et al. [6] succeeded in formulating such a continuous analogy for free theories and they called it cMERA. In this continuous entanglement scheme the sequence of tensors that prepare a state is replaced by a unitary,

U (s1, s2) = Pe−i R_s2

s1 ds (K(s)+L)_,

that is a path-ordered exponential consisting of an entangler K(s) and a scale transformator L. Their action in the cMERA formulation is similar to that in the MERA formulation. The (main) difference is that the entangling does not occur in discrete steps, but continuously.

Although the continuous formulation functions well for free field theories [11], [6], [32], it has appeared to be difficult to apply it to interacting theories. A natural way to understand this, is to notice that (c)MERA can be considered as an alternative way to follow the RG flow of a system [5], [8]. Moreover, the ground state of an interacting theory in general is non-Gaussian. The RG structure of non-Gaussian states are however often non-trivial. Therefore it is not surprising that it is also difficult to establish the cMERA of interacting systems.

(30)

In this chapter we will introduce the (c)MERA philosophy. We will start with an intuitive explana-tion of MERA in secexplana-tion 3.1 which is mainly based on [8]. In secexplana-tion 3.2 we turn to the continuous analogy, cMERA, for which we have used [8], [6], [32], [33], [34]. After that we will show how to con-struct the cMERA of a free scalar field theory in section 3.3 which is based on [6] and [34]. As we noted above constructing cMERAs of interacting theories is very challenging. Two works that recently made progress on this subject are [8] and [35]. We review the work of [8] in appendix A to give a taste of future directions and because we tried to apply the method that is developed in this work. The rest of this thesis can however be understood without having read appendix A.

3.1 MERA

Say we want to find the ground state of a translationally invariant system that consists of 2n_{lattice sites}

in one spatial dimension where each site has a local dimension d. Another way of saying this is that we consider a system of 2n _{qudits. The Hilbert space H}

2n of the system can be written as, H2n = Cd⊗ ... ⊗ Cd' Cd

2n

To approximate the ground state of this system we start from a sublattice comprised of 2m_{sites, where}

m is an integer such that m < n. The Hilbert space H2m of the sublattice is, H2m = Cd⊗ ... ⊗ Cd' Cd

2m

and we consider the state |Ω2mi ∈ H₂m given by,

|Ω2mi = |0i ⊗ |0i ⊗ ... ⊗ |0i (3.1)

Since |Ω2mi is the tensor product of 2mtimes |0i, we imagine that each of the 2msites is in a state |0i. Therefore, if the sublattice is in a state |Ω2mi none of its sites are entangled. The state |Ω₂mi is thus a state without an entanglement structure living in a Hilbert space H2m.

A ground state generally has an entanglement structure and it moreover lives in the Hilbert state H2n. Therefore, we apply an isometry V2m to the state |Ω2mi. This isometry is a collection of local isometries acting on the 2m _{lattice sites separately and it embeds the state |Ω}

2mi ∈ H₂m into a larger Hilbert space H2m+1, V2m : H₂m ,→ H₂m+1. It satisfies V₂†mV2m = 1_H

2m and acts as, V2m|Ω₂mi =

d

X

i1,...,i2m

ci1,i2,...,i2m|0i ⊗ |i1i ⊗ |0i ⊗ |i2i ⊗ ... ⊗ |0i ⊗ |i2mi (3.2) We see that between each two lattice sites a new site is added such that the system now consists of 2m+1

lattice sites. Even though the amount of sites has doubled we keep the spacial volume in which the system lives constant. Therefore we can think of a new lattice site being added in the “space between” each pair of existing sites. As such the distance between two neighbouring sites has decreased by a factor 1/2. Additionally, the state is not a product state and thus has an entanglement structure. In particular, because the isometry V2m is a collection of local tensors we believe that in the state (3.2) for example the first and the second site, that are represented by |0i and |i1i, are entangled, and the third

(31)

other we endow the state with some extra entanglement by acting with a unitary U2m+1 that acts on V2m|Ω₂mi as, U2m+1V₂m|Ω₂mi = d X i1,...,i2m ci1,i2,...,i2m|i1i ⊗ |i2i ⊗ ... ⊗ |i2m+1i (3.3) The unitary U2m again consists of a collection of local tensors of which the first only acts on lattice sites two and three, the second only on sites four and five, etc.. Accordingly in the state (3.3) all nearest neighbors are entangled with each other and because of this feauture we often call the unitary U2u the entangler.

Although we have obtained a state with an entanglement structure now, it still does not live in the right Hibert space. Therefore we further transform (3.3) by acting with another isometry V2m+1 that embeds the state into a Hilbert space H2m+2 by again adding a new site in between each two existing sites. To also entangle all nearest neighbors in this new state we act with the unitary U2m+2. We repeat this procedure until we end up with a state that lives in the full Hilbert space H2n and we find that it has the form,

|ΨMERAi = U2nV₂n−1U2n−1...V2m+1U2m+1V2m|Ω₂mi ∈ H₂n (3.4) We see that the state contains both short- and long-distance correlations. In fact the isometry V2m and the unitary U2m+1which acted locally on the 2mand 2m+1sites respectively are responsible for the long-range correlations since the initital 2m _{and 2}m+1 _{sites are now 2}n−m_{− 1 and 2}n−m−1_{− 1 sites apart.}

Similarly the last isometry V2n−1 and unitary U₂n are responsible for the short-distance correlations. These short-range correlations are as small as the distance between neighbouring sites. In general, we can say that some unitary U2k, for m ≤ k ≤ n, is responsible for correlations across a scale of about 2n−k sites in the final MERA state. One can understand this hierarchical structure by taking into account the effect of the isometries: in each step the previous correlations are stretched over a longer distance. The collection of unitaries/entanglers and isometries together form a quantum circuit. The layers of the network are parametrized by u = 2m, 2m+1, ..., 2n and hence we think of one layer as consisting of the combination of an entangler and a rescaler. Note that the depth of the network in fact is some emergent extra direction. A graphical representation of a quantum circuit with similar entanglers and isometries at each layer is shown in Fig. 3.1. The isometries are represented by the green rectangles and the entanglers by the pink squares. The depth in this figure is parametrized by u.

Instead of writing the MERA state as (3.4) we can also write it fully in terms of unitaries which will turn out to be useful when turning towards cMERA. We can do this by initially starting with a product state that already lives in the full Hilbert space, i.e. |Ω2ni ∈ H₂n. In that case the product state is given by,

|Ω2ni = |0i ⊗ |0i ⊗ ... ⊗ |0i

We define the unitary ˜U2m to act only on sites positioned at multiples of 2n−m, thereby only entangling sites that are 2n−m_{− 1 sites apart. Accordingly the unitary ˜}_U

2m+1 acts only on those sites that are multiples of 2n−m−1_{, creating correlation across distance scales of 2}n−m−1_{− 1 sites.} _{Stated more}

(32)

Figure 3.1: The MERA quantum circuit. Starting from a product state |Ω2mi at the top a state with an entanglement structure, |ψMERAi, is built by acting on |Ω2mi with isometries (green rectangles) and entanglers (blue squares).

generally, a unitary ˜U₂m+k will act only on sites positioned at multiples of 2n−m−k and “skips over” the other sites. In this reformulation the isometries V in fact “hide” in the unitaries: the unitaries do not simply generate entanglement blindly across all available sites, but take into account the scale at which it acts. The sites that are temporarily unused are called ancillae. In this language we can write the MERA state as,

|ΨMERAi = ˜U2nU˜₂n−1... ˜U₂m+1U˜2m|Ω₂ni (3.5) Note that in this setup the unitaries ˜U2m are responsible for the long range correlations and the unitaries

˜

U2n for the short range correlations.

We see that the MERA state depends on the choice of unitaries ˜U2nU˜₂n−1... ˜U₂m+1U˜2m and on the choice of m. To determine the best estimate of the ground state of the translationally invariant lattice system we have to determine the set of unitaries that minimizes the ground state energy. If the set {au},

for u = 2m_{, 2}m+1_{, ..., 2}n_{, parametrizes the unitaries then the ground state energy is,}

Emin= min{ai}hΨMERA({ai})| H |ΨMERA({ai})i

The set {au} that minimizes the ground state energy substituted in (3.4) yields the MERA state that

approximates the ground state. Renormalization Group flow

An interesting feature of MERA is its close relation to the Renormalization Group (RG) flow. In QFT the RG flow shows the changes of a physical system as viewed at different energy scales. In renormaliz-able theories the system at some scale will generally consist of smaller, self-similar copies. That is, after performing a scale transformation the system will have the same symmetries as the initial system. This

The connection between cMERA and holography

University of Amsterdam

Institute of Theoretical Physics

Master Thesis

The connection between cMERA and holography

Emma Loos

Supervisors:

Prof. Jan de Boer

Dr. Michael Walter

Contents

Chapter 1

Introduction

Chapter 2

Preliminaries

2.1

AdS/CFT correspondence

2.1.1

Conformal Field Theory

2.1.2

Anti de Sitter space

2.1.3

’t Hooft large N limit

2.1.4

Dictionary

Chapter 3

cMERA

3.1

MERA