Tensor Networks As Probes of Near Horizon Geometry.

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Universiteit Van Amsterdam

Master project

Tensor Networks As Probes of

Near Horizon Geometry.

Shustrov Yaroslav

supervised by Dr. Jan de Boer

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Chapter 1 Introduction

One of the greatest puzzles of modern high energy physics is problem of unification of General Relativity and Quantum Field Theory. It is also known as problem of “quantization of gravity”.

Many different approaches exist, however so called AdS/CFT correspondence introduced almost 20 years ago is considered to be one of the most notable among them. This proposal suggests equivalence between a (d+1)-dimensional theory of quantum gravity in anti-de Sitter space and a conformal field theory in d dimensions. Although this statement still doesn’t have strict mathematical proof, huge number of explicit calculations suggest validity of this proposal.

According to AdS/CFT correspondence bulk physics may be recovered solely from information on the boundary. Bulk non-local quantum gravity is implied to have local limit at low energies. It is important to understand how do bulk degrees of freedom organise themselves to produce local field theory in the aforementioned limit.

Explicit realisation of this duality encounters number difficulties, many of which have been successfully resolved within past years. For instance certain paradoxes associated with reconstruction of local bulk operators from part of the boundary were resolved by means of so called quantum error correction code. In particular It was shown that bulk operators may be reconstructed from multiple boundary regions.

All observables in the bulk are implied to have interpretation in terms of bound-ary information. One of the most notable realisations of such statement is so called Ryu-Takayanagi proposal. It relates two concepts from seemingly disconnected the-ories. In particular this proposal states that entanglement entropy of boundary region is equal(up to constant) to the area of bulk minimal surface, bounding this region. First quantity is characteristic value from quantum mechanics while the notion of minimal surfaces is used in general relativity.

In some particular cases[1] it may be shown that just knowing entanglement entropy of the boundary is enough to obtain explicitly bulk solution. This became one of reasons to believe that bulk space-time is an emergent phenomena [2].

One of the most widely used probes of bulk geometry is minimal surface anchored on the AdS boundary. For instance it appears in RT formula and AdS-Rindler reconstruction. In case of AdS3 this minimal surface is reduced to geodesic. This

geodesic bounds bulk region which is known as causal wedge. One may show that linearized version of Einstein equations holds in this region[3].

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However depending on amount of matter(entanglement) in the bulk, minimal surfaces may fail to reach all internal regions. This happens not only in presence of horizons in the bulk but also for regular geometries such as dense enough stars. The regions that can’t be reached by these minimal surfaces are called holographic shadows. It is very important to find interpretation of these bulk regions in terms of boundary information. For instance this information will shed light on near horizon geometry. Many crucial questions about in-falling observer may find explicit answers. Although different kinds of probes were considered in the literature, all of them have holographic shadows of certain size(which may be as large as lAdS.).

Purpose of our work is to propose new tool, namely tensor networks, as a toy model of AdS/CFT which may be used to find interpretation of holographic shadow regions in terms of boundary information.

Initially introduced in condensed matter physics tensor networks turned out to be also very useful in area of AdS/CFT. It was later realised by Swinger[26] that tensor network MERA(originally designed to represent ground state of certain hamiltonians) satisfies formula, analogous to RT formula for empty AdS. This was first motivation to try tensor networks in general and MERA in particular as a toy model of AdS/CFT. A lot of progress has been done in this area within past years. Many nontrivial properties of AdS/CFT correspondence such as quantum error correction properties found their reflection in different tensor networks.

There are two main directions in incorporation of tensor networks and hologra-phy. First one is to provide different modifications of MERA itself so they could capture more aspects of AdS/CFT. The second approach is to build other types of tensor networks from different types of tensors which basically define properties of obtained object. Each of these approaches has its own advantages and disadvan-tages. However for purposes of our work only second one may be used, because only this approach so far may introduce sub-AdS scale resolution(describe physics at distances smaller than lAdS).

In this thesis we overview different kinds of tensor networks and argue whether each of them may be used for the problem of interest. Although much work re-mains to be done our conclusion is that random tensor network seems to be the most promising one among considered. Firstly, it possesses sub-AdS scale resolu-tion. Secondly, RT formula can be proven for highly entangled bulk states. These properties are needed in order to obtain TN representation of constant time slice of BTZ black hole which captures nontrivially geometry of holographic shadow region. With this TN one may potentially find interpretation of aforementioned region in terms of boundary information.

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Chapter 2 General Concepts

2.1 AdS Space-time.

In this section Anti-de Sitter space-time will be introduced. In particular we will consider different coordinate systems used to represent this space-time and connec-tions between them. Most of explicit calculaconnec-tions will be made for AdS3, because in

our work we focus mainly on this particular case. Following section is based on [4].

Global coordinate systems.

AdS space-time is maximally symmetric solution of Einstein equations with neg-ative cosmological constant Λ = −d(d−1)_2l2

AdS

, where lAdS is radius of curvature and d is

dimension of space-time.

One can view AdS(p, q) as a hyperboloid, embedded into space-time Rp,q+1_{. In}

particular let us consider flat space-time with two spacial dimensions (X1, X2) and

two time-like directions (X0, X3):

ds2 = −dX₀2− dX₃2+ dX₁2+ dX₂2 (2.1)

Now to obtain AdS3 space-time we take locus of points at fixed time-like distance

from the origin:

−X₀2− X₃2+ X₁2+ X₂2 = −l_AdS2 (2.2) The induced metric will indeed have constant negative curvature. One can also check that it is maximally symmetric by listing all d(d+1)₂ Killing vectors.

We can introduce change of coordinates:

        

X0 = lAdScosh ρ cos τ

X3 = lAdScosh ρ sin τ

X1 = lAdS sinhρ sin θ

X2 = lAdS sinhρ cos θ

(2.3)

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ds2 = l2_AdS(−cosh2ρdτ2+ dρ2+ sinh2ρdθ2) (2.4) where ρ ∈ R+ _{and ρ → ∞ corresponds to the boundary of AdS space-time.}

τ ∈ [0, 2π]. To avoid closed time-like curves one should redefine τ in a way that τ ∈ R.

Figure 2.1: AdS3 space-time. Constant time slice is given by hyperbolic disc. Image

form Wikipedia.

One can produce change of coordinates to obtain yet another global covering of AdS space-time:

r ≡ lAdS · sinhρ

t ≡ lAdS· τ

Which will lead to expression:

ds2 = −f (r)dt2+ 1 f (r)dr 2_{+ r}2_dθ2_; _{f (r) = 1 +} r 2 l2 AdS (2.5)

Let us define one more global covering(coordinate system, which covers the whole manifold). We will also consider AdS3 case, but now its Euclidean version.

−X2 0 + X 2 3 + X 2 1 + X 2 2 = −l 2 AdS (2.6) ds2 = −dX₀2+ dX₃2+ dX₁2+ dX₂2 Change of variables may be introduced:

x = X1 X0+ lAdS y = X2 X0+ lAdS z = X3 X0+ lAdS (2.7) ρ2 ≡ x2 _{+ y}2_{+ z}2 ₌ X0− lAdS X0+ lAdS

Once put together one gets:

ds2 = 4l 2 AdS (1 − ρ2₎2(dx 2 + dy2+ dz2) (2.8) Or in spherical coordinates: ds2 = 4l 2 AdS(dr2+ r2(dθ2+ sin 2_θdφ2₎₎ (1 − r2₎2 (2.9)

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Which for θ = Π₂; 0 < r < 1 in known as Poincare disc metric.

Last of global coverings that we will introduce may be obtained by following change of coordinates in 2.8:

r = tanh y 2

(2.10)

New metric will look like:

ds2 = l_AdS2 (dy2+ sinh2ydΩ2) (2.11)

Incomplete coordinates patches.

There exist coordinate patches, which cover only parts of AdS space-time, yet which are widely used. We will also consider only AdS3 case:

−X2 0 − X32+ X12+ X22 = −lAdS2 (2.12) ds2 = −dX₀2− dX2 3 + dX 2 1 + dX 2 2

After change of coordinates:

             X1 = l2 AdS 2r 1 + _l4r2 AdS (α2_{+ x}2_{− t}2₎ X2 = _l r AdSt X3 = _l r AdSx X4 = l2 AdS 2r 1 − _l4r2 AdS (α2− x2_{+ t}2₎ (2.13) we will obtain: ds2 = − r 2 l2 AdS dt2+l 2 AdS r2 dr 2₊ r2 l2 AdS dx2 (2.14)

where r ≥ 0, r → ∞ corresponds to boundary of AdS. If we introduce yet another coordinate transformation:

z ≡ 1 u · lAdS

we will obtain another wildly used coordinate patch:

ds2 = l

2 AdS(dz

2_{− dt}2_{+ dx}2₎

z2 (2.15)

where z =0 corresponds to conformal boundary. There are several notable properties of this metric. It is conformally equivalent to half space of Minkowski space-time. Or put it in different way we can see that in the limit z → 0 there is Minkowski space-time on the boundary. Any constant space-time slice of this metric gives hyperbolic half-plane also known as Poincare half-half-plane metric. (We also discuss certain properties of geodesics in these hyperbolic half-planes in appendix A.)

If one makes transformation to Euclidean time this metric becomes full covering of Euclidean AdS space-time.

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Figure 2.2: Green region represents patch 2.15. Full AdS(given by 2.4) is represented by cylinder. Image from [5].

2.2 Holographic shadow of BTZ black hole.

Holographic shadow[6] is the region in the bulk which can’t be reached by any minimal surface anchored on the boundary. This type of regions exists both for bulk geometries with horizons and for regular ones. The example of latest type of geometry is dense star in AdS space-time. Even though from point of view of general relativity nothing distinguishes holographic shadow region from rest of the bulk a lot of interesting things happen there. For instance it is unclear how to reconstruct bulk operators which lie inside this region. In case of BTZ black hole holographic shadow region surrounds horizon. Presence of this region is the reason why it is difficult to get answers to number of questions about observer falling into black hole. This section gives an explicit derivation of this region. In particular we show the connection between size of holographic shadow and radius of black hole. Following part of section summarises certain results from articles [6], [7] .

First let us give brief sketch of Ryu-Takayanagi(RT) proposal. Deeper analysis of certain aspects of this proposal will be given in next sections. For now we are interested in case of BTZ black hole, which metric is given by[7]:

ds2 = −f (r)dt2+ dr

2

f (r) + r

2

dθ2, f (r) = r2− r2₊ (2.16)

where r+ is radius of black hole.

RT proposal connects entanglement entropy of boundary region with area of minimal surface anchored on this region.

Suppose we have constant time slice of BTZ bh. Let us consider simply connected boundary region A. Now we take some bulk region R, which matches A on the boundary. This statement may be rewritten in following way:

∂R = EA∪ A; ∂EA = ∂A

We need to change size of R in such a way so EA becomes minimal surface,

ho-mologous to the boundary region A(can be continuously transformed to region A). Length(area in higher dim) of this minimal surface will be equal(up to constant) to entanglement entropy of boundary region A.

Depending on size of A the solution may be given by two different families of bulk curves:

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Figure 2.3: Left - connected family: EA is given by geodesic matching A; Right

- disconnected family. EA is given by sum of geodesic and circumference of BH.

Homology constraint on minimal surface is crucial.

Equation for geodesics is given by[7]:

n (r, θ) : r = γ(θ, θ∞, r+) ≡ r+ 1 − cosh 2 (r+θ) cosh2(r+θ∞) −1/2o (2.17)

In the above expression r+is radius of BTZ black hole, θ∞is parameter associated

to each geodesic(see Fig 2.3), θ is angle parameter along the geodesic. lAdS is set to

be equal to 1 and may be restored from dimensional analysis.

According to RT proposal in case of multiple families of minimal surfaces we pick one with the smallest length(area).

Hence if we start from small region A and then gradually increase its size then entropy of region A will be given at first by family of connected minimal surfaces and starting from some point by disconnected ones.This indicates that phase transition between between two families takes place.

Figure 2.4: r+= 0.2 At some point two geodesics from different families

correspond-ing to orange region will have the same length. Image from [7]

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means of minimal geodesics. Following notation from article [6] we will call this region holographic shadow.

Now based on approach introduced in this article we will evaluate size of holo-graphic shadow for given radius of black hole.

Boundary region is defined as θ∞; r → ∞. Length of geodesic in this metric is

given by: γA = Z ds = Z p grr(r0)2+ gθθdθ = Z s (r0₎2 (r2_{− r} +)2 + r2_dθ, _r0 _≡ dr dθ (2.18) Integrand is invariant under translations in θ direction. So we can consider it to be some kind of Lagrangian. In this case r corresponds to spacial coordinate and θ corresponds to time. Then we can define hamiltonian corresponding to it:

H = −r 2 q (r0₎2 (r2_−r +)2 + r 2

In general H is some constant. We may fix it by considering some boundary con-dition. For instance we may note that geodesic reaches its minimum value r∗ (in

radial direction) when r0 = 0. This leads to equation:

dr dθ = r r∗ p r2_{− r}2 ∗ q r2_{− r}2 + (2.19)

We may integrate it to obtain:

θ∞= Z ∞ r∗ drdθ dr = 1 2r+ cosh−1 r2 ∗+ r2+ r2 ∗− r2+ (2.20) Hence r∗ = r+ tanh(θ∞r+) (2.21) Now let us find θswitch at which phase transition takes place. It is defined by

condi-tion:

γ(θswitch) = γ(π − θswitch) + 2πr+ (2.22)

where γ(θ) is length of geodesic defined by θ.

γ(θ∞) = 2 Z ∞ r∗ rdr pr2 _{− r}2 ∗pr2− r+2 (2.23)

Once we put these equations together we will obtain:

θswitch = π 2 + 1 2r+ logcosh(πr+) (2.24) Hence: rmin = r+ tanh πr+ + r+e −πr+ sinh(πr+) (2.25) And finally size of holographic shadow is given by:

4r0 ≡ rmin− r+ =

2r+e−πr+

sinh(πr+)

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One can see that size of holographical shadow indeed depends on radius of black hole. In particular : r+ lAdS 4 r0 ∼ lAdS r+ lAdS 4 r0 ∼ e −#r+ lAdS

Figure 2.5: Size of holographic shadow rmin as a function of horizon radius r+ of

BTZ black hole. Image from [6]

One may also show [6] that holographical shadow also exists if we consider dense enough star in Ads.

There are other probes of geometry near BTZ bh. However for all of them shadow still takes place.

2.3 Important concepts of Quantum Mechanics.

This section aims not to give strict definitions but rather to provide summary of useful notions and different connections between them, which will be used in future investigations. It is mainly based on article[8].

Hilbert space is a complex-valued vector space which defines quantum system. One may say that Hilbert space is quantum analogue of phase space of classical system. However we must stress out that the principal difference is that unlike phase space Hilbert space is vector space. That leads to the superposition principle, which states that superposition of two states of quantum system is also state of this system. It is obviously not true for classical system and exactly because of aforementioned reason.

As was mentioned before quantum system is defined by its state. If one knows quantum state together with its hamiltonian then one can predict evolution of system by means of Schrodinger equation:

ˆ

H |ψ(t)i = i~∂

∂t|ψ(t)i (2.27) Quantum state may be either pure or mixed. Let us first introduce notion of pure quantum state [8]. Pure state is a state that can be written as a ket vector in Hilbert space. Let us consider Hilbert space which factorises into two subspaces:

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H = H1⊗ H2. Suppose {|aii} is basis in H1, {|bji} is basis in H2. The most general

pure state |ψi ∈ H may be written as:

|ψi =X i,j Ci,j(|aii ⊗ |bji) = X i,j Ci,j|aibji (2.28)

If one can rewrite state |ψi as:

|ψi =X

i,j

C_iA|aii ⊗ CjB|bji (2.29)

then we call this state separable(not entangled). This means that two parts of this state are uncorrelated. If the state can’t be written in aforementioned form then it is entangled.

In general the problem to say whether some pure state is separable or not is quite difficult. So called Schmidt decomposition is used to tackle this problem [9].

Let H1 and H2 be Hilbert spaces of dimensions n and m respectively.

Sup-pose n ≥ m. Then for ∀w ∈ H1 ⊗ H2 there exist orthonormal sets {u1, ...un} ⊂

H1, {v1, ...vn} ⊂ H2 such that: w = m X i=1 αiui⊗ vi (2.30)

where αi are real, non-negative and uniquely determined by w.

Not all states in Hilbert space may be represented as ket vectors. Suppose we have system which is in state |ψii with probability pi,P pi = 1. We will call such

states mixed states. They may be represented by means of density matrix:

ρ =X

i

pi|ψii hψi| (2.31)

where we imply that {|ψii} is orthonormal basis.

Suppose we have Hilbert space H = HA⊗ HB. {|ai} is basis in HA and {|bi} is

basis in HB correspondently. One may describe state of subsystem A(analogously

of subsystem B) in the following way:

ρA≡ Tr(ρ) =

X

i

hai| ρ |aii (2.32)

ρA is called reduced density matrix of subsystem A.

Two mixed states may have the same density matrix. In this case it is impossible to distinguish them. This may be seen from following consideration.

Let us start from mixed state given by ρ(2.31). Suppose we have orthonormal basis |βki. Then one may construct set of projectors { ˆPj}:

ˆ

Pj = |βji hβj|

The probability to measure |βji is given by:

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One of particular examples of mixed states is so called thermal state. In this case probability distribution pi is Boltzmann distribution:

pi = e−βEi, β =

1 T

where T − temperature, Ei− energy of particular state |ψii.

Unlike mixed states when system is in pure state(no matter entangled or sepa-rable) it is in this state with probability 1. Thus pure state may also be represented by means of density matrix:

ρpure = |ψi hψ|

Measurement of a pure state will always yield results related to only one quan-tum state, whereas with mixed state one cannot know beforehand what state will be measured. We will always consider normalised states, so T r(ρ) = 1. Density matrices gives us one more way to distinguish pure and mixed states[8]:

pure: ρ2 = ρ, T r(ρ2) = 1 mixed: ρ2 6= ρ, T r(ρ2) < 1 T r(ρ2_{) is called purity of state.}

Note that closed system may be both in pure and mixed state. However if the closes system was initially in pure state it can’t evolve to mixed state by means of unitary process. To produce mixed state from pure one, one needs to introduce some temporary external interaction.

Mixed state can also be either separable or entangled. Mixed state is called separable [9] if it can be written as a convex combination of pure product states:

ρ =X i pi|aii hai| ⊗ |bii hbi| = X i piρai ⊗ ρ b i (2.33)

where {|aii} and {|bii} are(not necessary orthogonal) bases of Ha and Hb

corre-spondingly. H = Ha⊗ Hb. Term ”convex” means 0 6 pi 6 1,P i

pi = 1.

Example of mixed separable state is:

ρ = 1

2(|↑↑i h↑↑| + |↓↓i h↓↓|) If ρ is not separable then it is entangled.

Now let us consider Hilbert space of dimension d with basis {|ψii}. One may

define mixed state which consists from mixture of states |ψii all distributed with the

same probability pi = 1_d. We will call such state maximally mixed. Density matrix

of such state is given by:

ˆ ρmm =

ˆ I

d (2.34)

There is also different concept of maximal entanglement. We will introduce this concept for pure states only. However the same thing may be done for mixed states as well [10].

Suppose we have a bipartite pure state |ψi ∈ HA⊗ HB, dim(HA) 6 dim(HB).

Then |ψABi is maximally entangled iff[11]:

ρA= 1

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It is said to be maximally entangled if reduced density matrices are maximally mixed.

Example of maximally entangled states is set of Bell states[12].Bell states are 4 specific maximally entangled states of 2 qubits which form basis in two-qubit Hilbert space:

φ+ = √1

2(|0iA⊗ |0iB+ |1iA⊗ |1iB)

φ− = √1

2(|0iA⊗ |0iB− |1iA⊗ |1iB) (2.35)

ψ+ = √1

2(|0iA⊗ |1iB+ |1iA⊗ |0iB)

ψ− = √1

2(|0iA⊗ |1iB− |1iA⊗ |0iB)

Another important quantity is entanglement entropy also known as von Neumann entropy. This quantity is a measure of entanglement between subsystem and rest of the system:

S(ρ) = −T r( ˆρ log ˆρ) (2.36) Since we can diagonalise density matrix(because it is unitary) we may rewrite the expression in different way:

S(ρ) = −

d

X

k=1

λklog λk (2.37)

where λk are nonnegative eigenvalues of ˆρ. Entropy gives yet another criterion to

distinguish between pure and mixed states:

pure state: S( ˆρ) = 0

mixed state: S( ˆρ) > 0

If the quantum system is known to be in pure state, then entanglement entropy of any subregion of this system is equal to entanglement entoropy of complementary part of this subregion. However in the case when the whole system is in mixed state these two entropies are not equal anymore.

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Chapter 3 AdS/CFT

3.1 Introduction

Let us present short summary of basic aspects of AdS/CFT correspondence. Fol-lowing section is based on [13, 14, 25].

The origin of AdS/CFT correspondence lies in the fact that there exists duality between description of some system in terms of closed and open strings. In particular Maldacena considered in his work[13] stack of D-3 branes[15]. At certain low energy limit the description of system in terms of closed strings is reduced to string theory on AdS5×S5. At the same time description of initial system in terms of open strings

corresponds to some particular CFT on the boundary of aforementioned manifold, namely N=4 super Yang-Mills theory.

AdS/CFT correspondence states that two Hilbert spaces are equivalent to each other[25]:

HCFT = HAdS-QGrav (3.1)

and all symmetries can be matched between two theories.

Suppose we have field φ in bulk(string theory living of manifold M), which has boundary value φ(0). Then one can define partition function in the bulk as:

Zstring[φ(0)] =

Z

φ(0)

Dφexp(−S[φ]) (3.2)

AdS/CFT correspondence states that:

Zstring[φ(0)] = ZCFT[φ(0)] = hexp[

Z

∂M

ddx√gφ(0)(x)O(x)]iCFT (3.3)

where O(x) is some CFT operator on the boundary ∂M . Hence φ(0)(x) plays role

of source for correlation function of operator O(x). Now let us consider some limit of this correspondence, namely bulk side being in supergravity limit. In this case at low energies Zstring[φ(0)] is mainly defined by saddle-point approximation. Hence

one can write:

Zsugra[φ(0)] = exp(−S[φcl(φ(0))]) (3.4)

where φcl are fields satisfying low energy equations of motion with b.c.:

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Several examples of explicit calculations for AdS/CFT will be introduced in this thesis. Notion of bulk reconstruction will be introduced in following sections and two-point functions on the boundary will be evaluated in appendix B.

3.2 Holographic Derivation of Entanglement

En-tropy from AdS/CFT

AdS/CFT correspondence is so far the best attempt to build precise theory of quan-tum gravity. It implies that any bulk observables should have interpretation in terms of boundary information. Possibly the most notable example of this principle is so called Ryu-Takayanagi formula. It connects two concepts from seemingly different theories. It shows deep connection between quantum theory on the boundary and gravitational theory in the bulk. Although this formula was derived for arbitrary dimension case, we will focus of case of 3 bulk dimensions.

In this section RT proposal will be shown to hold for case of empty AdS. In the beginning we provide explicit calculation of length of geodesic matching some boundary region. After that we obtain expression for entropy of boundary region A by means of so called ”Replica Trick”. Eventually equivalence of two expressions will be shown. We base our derivations on [17].

Let us consider AdS3/CF T2. Anti de Sitter space is presented by metric

d2s = R2(− cosh ρ2dt2+ dρ2+ sinh ρ2dθ2) (3.5) We will introduce cutoff at ρ = ρ0 because this metric is divergent at ρ → ∞.

Figure 3.1: (a) AdS3, (b) slice of bulk at fixed t. The picture is taken from [17]

Let us consider subregion A of boundary at fixed t. It is defined by θ ∈ [0,2πl_L ] at fixed t.

Ryu-Takayanagi proposal states that entropy of boundary region A is given up to constant by length of minimal surface(geodesic in case of AdS3) bounding this

region:

SA=

Length(γA)

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where G stands for gravitational constant in the bulk(AdS).

Let us first evaluate length of γA. It is possible to do these calculations using

metric mentioned above, however it’s much easier to consider Poincare covering. This metric does not cover Anti de Sitter space-time completely. At the same time minimal geodesic fully belongs to this region.That’s why we can use it for this prob-lem. The calculation will be done in similar manner as the one in section 2.2. Poincare covering metric of AdS3 looks like:

ds2 = R2dz

2_{− dt}2_{+ dx}2

z2 (3.7)

where R is radius of AdS and which at t fixed will give an expression:

d¯s2 = R2dz

2_{+ dx}2

z2 (3.8)

Expression for length of static geodesic will look like:

γA= Z d¯s = Z s gzz( dz dλ) 2 + gxx( dx dλ) 2 dλ = Z Rp( ˙z)2_{+ 1} z dx (3.9) Boundary conditions in new coordinates will look like:

Oi : (z = a(cutoff), x = − l 2) Of : (z = a(cutoff), x = l 2) (3.10)

It is useful to take into account some specific properties of integrand. Remem-bering that z = z(x) one can see that the whole function is invariant under trans-formations x → x + 4, 4 - const. We can define integrant to be some kind of ”Lagrangian” in analogy with classical mechanics:

S = Z

L(q, ˙q, t)dt; t → x, q → z, ˙q → ˙z. (3.11)

The integral is invariant under transformation x → x + 4, hence one can define corresponding ”Hamiltonian”:

H = ∂L

∂ ˙qq − L =˙

R

zp( ˙z)2_{+ 1} (3.12)

which doesn’t depend on x.

Corresponding ”equations of motion” will look like: d

dx(H ˙z) + R2

Hz3 = 0 (3.13)

which may be solved by following expression:

z = z(x) = r 2R Hl l2 4 − x 2 _(3.14)

Now let us plug this expression into integral for geodesic length and introduce new coordinates:

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x = l

2cosα z = l

2sinα (3.15) where α ∈ (, π − ), = 2a_l [19].

Finally expression for entanglement entropy is given by:

SA= Length(γA) 4G(3)_N = c 3log l a ; c = 3R 2G(3)_N (3.16) Replica Trick

Now let us obtain explicit expression for entanglement entropy of boundary re-gion A. We will use only information from boundary itself. This will be done by means of so called Replica trick [18]. If the expression we obtain will be equal to expression form eqn. (3.16), that will mean that Ryu-Takayanagi proposal holds(at least in this particular setup).

Suppose we are given 1+1 lattice QFT at zero temperature living on a line. Let lattice spacing be equal to a(UV cut-off). For simplicity we will consider subregion of the system to be given by single interval A = [u1, v1]. This setup captures major

aspects of the approach. More complicated cases can be seen in [17]. General expression for entanglement entropy for subregion A looks like:

SA = −T r(ρAlog ρA)

where ρA is reduced matrix.

This expression is difficult to evaluate explicitly because log will lead to infinite series. One can rewrite expression of entropy in different way.

Let us consider an expression: d dax

a ₌ d

dae

alnx _{= x ln x}

hence we can state:

− lim

a→1(

d dax

a_{) = −x ln x}

This result may be used do redefine entanglement entropy in new way[18]:

SA= − lim n→1 ∂ ∂nT r(ρ n A) ! (3.17)

Now instead of infinite series we need to calculate T r(ρn

A) for some particular n.

This formula works in principle for arbitrary real n. Now let us briefly sketch or strategy[18]:

· evaluate T r(ρn

A) only for integer n > 1 (This case has clear physical

interpre-tation, which will be shown within calculations)

· make analytical continuation of this expression for all complex n · recover SA as a limit n → 1

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To calculate T r(ρn_A) we will first define ρ and then will arrive to expression for ρA. In our calculations we define region A to be some spacial region. That is why

it is preferable to write ρ in basis of states that have explicit spacial dependence. Let us start by reminding ourselves how to define density matrix in terms of wave functions(which is used generally in QM). After that we will follow similar procedure for our case. It will help us to obtain expression for density matrix of our QFT state in basis which has explicit spacial dependence and hence is easier to operate with.

One can define amplitude for point particle to evolve from initial to final state in following way: hf |ii = xf(t=tf) Z x−(t=ti) dxe−SE _(3.18)

If we set tf = +∞ in Euclidean signature |f i ≡ |0i. It takes place because evolution

operator in euclidean signature decays exponentially when euclidean time goes to infinity.

So ground state wave function may be defined as:

ψ0(x) ≡ h0|ii (3.19)

Conjugated wave function of ground state can be defined in the same way:

ψ₀∗(x) =

x+

Z

−∞

dxe−SE _(3.20)

Now we can use above equations to obtain explicit expression for density matrix of ground state:

ρ(x−, x+) = ψ∗0(x)ψ0(x) (3.21)

As was mentioned above one can use aforementioned procedure to define density matrix for ground state of QFT. However in this case we will have field dependent wave functionals: ψ0(ϕ) = tE=+∞ Z ϕ=ϕ+,(t=+0) Dϕe−SE _(3.22)

Hence one can write density matrix of ground state in this basis [18]:

ρ(ϕ+, ϕ−) ≡ ψ0∗(x)ψ0(x) = (3.23) = +∞ Z −∞ Dϕe−SEY x δ(ϕ(x, +0) − ϕ+)δ(ϕ(x, −0) − ϕ−)

where we took into account that:

t=+∞ Z t=+0 Dϕ t=−0 Z t=−∞ Dϕ ≡ t=+∞ Z t=−∞ Dϕ (3.24)

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Once traced out complementary of region A we will obtain: ρA(ϕ+, ϕ−) = (Z1)−1 +∞ Z −∞ Dϕe−SE Y x∈A δ(ϕ(x, +0) − ϕ+)δ(ϕ(x, −0) − ϕ−) (3.25)

where Z1 is vacuum partition function on R2 needed for normalisation of ρA.

This can be represented in this way:

Figure 3.2: ρA. The picture is taken from [19]

For our problem we need to obtain expression for Trρn

A, which can be represented

graphically by means of gluing n aforementioned surfaces in particular way:

ϕ+(i − 1) → ϕ−(i); ϕ+(i) → ϕ−(i + 1)... For the case n = 3 this Riemannian

surface will look like:

Figure 3.3: Riemannian surface R3,1 where 3 stands for number of copies, 1 - number

of intervals.The picture is taken from [20]

So T r(ρn

A) may be represented in terms of path integral on n-sheeted Riemann

surface Rn: T r(ρn_A) = (Z1)−n Z (tE,x)∈Rn Dφe−S(φ)≡ Zn (Z1)n (3.26)

At this point we have QFT living on some complicated Riemannian surface Rn,1.

Problem of evaluation of T r(ρn

A) in general is not solvable. However if we consider

CFT instead(and this is exactly what we are interested in) then it is possible to get explicit expression for trace of reduced density matrix. First step is to map Rn,1 to

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ω = x + it → ζ = ω − u ω − υ ζ → z = ζ1/n = (ω − u

ω − υ)

1/n

We can write down expression for transformation of T (ω)[18]:

T (ω) = (dz dω) 2_{T (z) +} c 12{z, ω}; {z, ω} = (z000z0− 3 2z 002₎ z02 (3.27)

where c is central charge of conformal filed theory on a complex plane.

Stress-energy tensor of CFT on a complex plane C possesses rotational and translational invariance hence:

hT (z)ic= 0

Taking this into account we can write:

hT (ω)iRn,1 =

c

12{z, ω} =

c(1 − (1/n)2_{)(υ − u)}2

24(ω − u)2_{(ω − υ)}2 (3.28)

Now follows very important moment of our discussion. We claim that CFT on sur-face Rn,1is equivalent to CFT on n disjoint complex planes C, yet in the presence of

some primary operators φn and φ−n living at points ui and υi correspondingly(i ∈

1, ..n). One can also generalise above procedure for greater number of disjoint in-tervals. These operators are known as twist fields. Generally speaking these are local fields which contain in themselves information about nontrivial topology(in our case) of points ui and υi of surface Rn,1. There are n pairs of them exactly

because Rn,1 contains n complex surfaces glued together. More detailed definition

of them is given in [18]. At this stage conformal dimensions of these operators are yet to be defined. To do so let us first write down conformal Ward identity [18]:

hT (z)φn(u)φ−n(υ)ic= |υ − u|−24n−2 ¯4n (3.29)

Equivalence of two descriptions mentioned above implies:

hT (ω)iRn,1 = n · hT (z)φn(u)φ−n(υ)ic (3.30)

Both φn and φ−nhave the same conformal dimension due to the properties of

trans-formation of correlation function of 2 dimensional CFT under conformal transfor-mations. Taking into account 3.28 we may deduce that they are given by:

4n = 4n = c 12 n − 1 n (3.31)

where c is central charge of CFT on a complex plane C and n is number of these planes in Rn,1.

It may be shown[18] based on properties of twist fields that T r(ρn

A) behaves(up to

constant) under scale and conformal transformations exactly as two-point function of twist fields. Hence:

ZA (n)

Zn ∝ T rρ n

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where UV cut-off parameter a is needed to make expression 3.32 dimensionless as it should be[17].

We can now use this result to obtain expression for entropy. Yet in our initial problem we have CFT with spacial dimension x of a final size L.

One may use property that T rρn

A is transformed as two-point correlation

func-tion. In this case we need to use transformation ω → z = _2πL log ω which will transform each sheet of Rn,1 into cylinder. We need to orient branch cut

perpendic-ular to the axis of the cylinder. This will be the right way to get expression which will correspond to subsystem of length l = (υ1− u1) of system with length of spatial

dimension L. [18] T rρn_A= cn L πasin πl L −c(n−1/n)/6 (3.33) Once we plug this result into formula for SA we will get:

SA= c 3log L πasin πl L + c0₁ (3.34)

We will take c0_n ≡ log cn

1−n , and c1 = 1 because of consistency condition. [18]

This formula in case l L will indeed give the result obtained previously:

SA= c 3log l a (3.35)

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3.3 Bulk reconstruction

3.3.1 Global bulk reconstruction

Let us consider procedure of reconstruction of local bulk fields in AdS from CFT [21]. We will perturbatively construct operators in CFT which obey bulk equations of motion and satisfy boundary conditions dictated by ”extended dictionary” [22]:

lim

r→∞r 4

φ(r, x) = O(x) (3.36)

The case of free bulk fields will be considered. In particular the case where inter-actions of fields are suppressed by _N1 will be of interest. At leading order of _N1 for bulk scalar field we will have:

( − m2)Φ(r, x) = 0 (3.37) Solution of this equation may be found by means of mode expansion:

Φ(r, x) = Z

dkakΦk(r, x) + h.c. (3.38)

where ak, a †

k - annihilation/creation operators and integration is taken over

momen-tum.

This expression on the boundary will look like:

Φ(x) = Z dkakϕk(x) + h.c. (3.39) where lim r→∞r 4 φk(r, x) = ϕk(x)

If ϕk(x) are orthogonal, then we can obtain ak from above equation:

ak =

Z

d_exϕ∗_k(_ex)Φ(_ex) (3.40)

Putting everything together we will obtain:

Φ(r, x) = Z dk hZ dxϕ_e ∗_k(x)Φ(_e _ex) i Φk(r, x) + h.c. (3.41)

In the case when integrations over k and _ex are exchangeable we may obtain:

Φ(r, x) = Z d_exK(r, x,x)Φ(_e x)_e (3.42) where K(r, x,x) =_e Z dkϕ∗_k(_ex)Φk(r, x) + h.c. (3.43)

We may rewrite this expression in way, more often used in literature [21]:

Φ(r, x) = Z

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Where Y defines conformal boundary Sd−1_{× R, with R being time direction.}

K(r, x; Y ) is known as ”smearing function”, which obeys bulk equations of motion. It may be chosen [21] to have support (depend only on) in the region, where x and Y are space-like separated. In case of AdS3 this region will look like:

Figure 3.4: Green region defines area of dependence of K for bulk field in the point x. Image from [22]

3.3.2 AdS-Rindler reconstruction

This type of reconstruction realises the assumption that if one wishes to reconstruct certain bulk field placed just near the boundary one in principle doesn’t have to use information from the whole boundary, but rather only from some small region. To define this kind of reconstruction, we need to introduce notion of bulk causal wedge[22].

Let us consider Cauchy surface which will denote as P and boundary subregion A: A ∈P

There exist number of points on the boundary with the property that all possible time-like and null-like curves crossing those points will intersect region A as well. We will call union of these points as boundary domain of dependence of A and denote it as D[A].

Let us consider some boundary region R and all bulk time-like and null-like curves which start on R. We will call union of them bulk causal future J+_{[R]. In}

the same way one may define bulk causal past.

Now one can define causal wedge of CFT subregion A: Wc[A] = J+[D[A]] ∩ J−[D[A]]

We will also introduce for future use so called causal surface of A, denoted by χA.

It is defined as part of the intersection of boundaries of J+_{[D[A]], which does not}

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Figure 3.5: AdS-Rindler wedge for AdS3 spacetime. Image from [22]

Once taken into account AdS-Rindler reconstruction claims [22] that ∀ bulk fields Φ(x, r) ∈ Wc[A] may be reconstructed by means of boundary operators O(Y ) ∈ D[A]

Φ(r, x) = Z

dY K(r, x; Y )O(Y ) + O(1/N ) (3.45)

It is important to note that in this case K doesn’t exist as a function and should be considered as a distribution for integration against CFT expectation values [22].

3.3.3 Paradoxes of AdS-Rindler reconstruction

According to aforementioned procedure when one considers bulk field which is placed nearer to the boundary, then one needs to use smaller boundary region for recon-struction. If we follow this logic naively we will run into several paradoxes.

For instance, suppose we have some bulk field φ(x) and a point Y on the bound-ary. One may pick boundary for reconstruction of φ(x) to be P /Y . In this case bulk operator is mapped to boundary O(φ(x)) which has to commute with any local operator eO(Y ). Choice of Y is arbitrary. If we admit that φ(x) is always mapped to the same boundary operator then we obtain that O(φ(x)) commutes with all local operators on the boundary and hence by Schur’s lemma must be proportional to unity operator.

Le’t consider another example. Suppose we have bulk operator φ(x) which lies both in causal wedge of boundary region A and B. In this case this operator may be reconstructed from both Wc[A] and Wc[B]. But if we claim that corresponding

boundary operator O(φ(x)) is unique then we must demand O(φ(x)) to have support only in Wc[A]T Wc[B]. But we can have case as described in following image, when

x is located outside Wc[A]T Wc[B].

Finally we may consider setup of right part of image. In this case point x lies outside Wc[A], Wc[B], Wc[C]. This means that φ(x) can be reconstructed from either

AS B or B S C or A S C. Hence O(φ(x)) should have support only on 3 points of the boundary. Suppose we now have 3 other regions of the same size but slightly

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rotates with respect to initial ones. Then we will have to arrive to the conclusion that φ(x) can’t be reconstructed at all.

All these paradoxes appear because we demand O(φ(x)) to be unique operator on the boundary for particular φ(x). We may resolve them by claiming that there may exist different ways to reconstruct the same bulk operator. This idea may be explicitly realised by means of so called quantum error correction code.

Figure 3.6: Setups of three aforementioned paradoxes taking place for Ads-Rindler reconstruction in AdS3 . Image from [22]

3.3.4 AdS/CFT as quantum error correction code

In this section we will consider the simplest example of quantum error correction code (QECC) and then we will point out its connection with AdS/CFT. More de-tailed analysis may be found in [22].

Suppose we have 1 qutrit(system that has 3 dimensional Hilbert space) living in the bulk(which will correspond to logical Hilbert space) and 3 qutrits living on the boundary(and forming 27 dimensional physical Hilbert space).

Figure 3.7: Setup: 3 physical qutrits on the boundary, one logical qutrit in the bulk.

Now let us consider subspace(so called code subspace) of Hphysical of dimension

3 formed by vectors |ψi. We may define encoding operator E:

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|0i → f|0i = |000i + |111i + |222i

|1i → f|1i = |012i + |120i + |201i (3.47) |2i → f|2i = |021i + |102i + |210i

Encoding map can be explicitly defined:

E =X

j

f

|ji hj| (3.48)

This map is an isometry:

E†E = Ilogical

In the case of interest states from code subspace satisfy important restriction. Any collection of qutrits from code subspace either contain all information about logical state |ψi or no information about it at all. In our setup it means(analogous condition may be written for any other subsystem):

T rAB h ψe E D e ψ i = 1 3I (3.49)

In particular this condition means that for any state ψe

E

reduced density matrix of any of qutrits will be maximally mixed. Hence one can’t obtain information about the state from any single qutrit. Now let us summarise scheme of QECC:

Enc: Hlogical → Hphysical

Noise: Hphysical → Hphysical

Decoder: Hphysical → Hlogical

Code can reconstruct logical state is following condition is satisfied:

Dec ◦ Noise ◦ Enc(|ψi hψ|) = |ψi hψ| (3.50) Operator of noise may be in principle different. We are interested in case of erasure of one of qutrits(for instance C):

Noisec(ρ) = trc(ρ), where ρ = Enc(|ψi hψ|)

Decoding operator which acts only on non-erased physical qudits: Decc(ρ) = trA(UABρU

† AB)

where for case of interest:

UAB|a, bi = |a, b − ai

Let us consider simple example:

|ψi = |1i

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Noise(ρ) = (|01i + |12i + |20i)(h01| + h12| + h20|) Dec(Noise(ρ)) = K(|1i + |1i + |1i)(h1| + h1| + h1|)

K is just normalisation constant. We also took into account that (0 − 2)mod3 = 1. We may immediately see similarity between aforementioned set up and 3d para-dox of AdS-Rindler reconstruction. For instance in this case operators with support on AS B correspond to Decoder operator. Operator in the bulk corresponds to logical qutrit. General idea is that this operator may be protected against erasure of one of boundary qutrits.

This simple example illustrates why it is natural to connect AdS/CFT and QECC. Depending on choice of logical qudits(systems with dim(H) = d) in the bulk we obtain different corresponding code subspaces on the boundary. So AdS/CFT may be viewed as many QECCs at once[22].

The fact of AdS-Rindler reconstruction that the same operator may be recon-structed by means of different causal wedges corresponds to freedom to choose dif-ferent code subspaces. Depending on position of this bulk operator it may be either well protected against different kinds of erasures of boundary information or not. Now let us make a few final comments comments about this topic.

Figure 3.8: Operator in the centre(far from the boundary) is protected against erasure of any of green regions. Operator near the boundary will be lost after erasure of small red region. Image from [22]

Bulk reconstruction(both global and Rindler) relies on _N1 order approximation. The fundamental limitation on them appears when we have to take into account backreaction[22].

Procedure of definition of code subspace Hcas stated in [22] looks like this. One

should consider local bulk operators φi(x) which acting on bulk vacuum state(in

region of order of lAdS) produce certain set(with some subtleties[22] these states

may define certain subspace of bulk Hilbert space):

|Ωi , φi(x) |Ωi , φi(x1)φj(x2) |Ωi , ...

These states will correspond to boundary code subspace Hc. One would like to know

how large this code subspace can be in principal. To answer this question we need to realise that each φi(x) acting on the bulk state arises its energy. At some point

one will have to take into account backreaction. In this case we will not be able to use AdS-Rindler reconstruction which relies on perturbation theory approxima-tion around a fixed background. As a result one can state that interpretaapproxima-tion of AdS/CFT as QECC also fails at high enough energies.

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Chapter 4 Tensor Networks and AdS/CFT

4.1 Introduction

Tensor networks theory is relatively young and dynamically developing topic. It has many implications in areas like condensed matter and others. Within past few years however it was also realised that one can use TNs as toy models of AdS/CFT correspondence. The most advanced of them already capture a lot of non trivial aspects of holography. For instance certain types of TN allow representation of logical operators on multiple boundary regions. This mimics so called AdS-Rindler reconstruction. Quantum error-correcting properties may also be explicitly realised in this set-up.

Although it is believed by many that TN may become powerful tool for better understanding AdS/CFT correspondence there are still plenty open questions which are needed to be solved. One of the most notable among them is the fact that so far no TN can describe dynamical processes. It is only developed in a way to map states between boundary and bulk Hilbert spaces. Suppose one takes boundary Hamiltonian and maps it to bulk. Usually one will obtain highly nonlocal(in terms of bulk degrees of freedom) Hb. In low energy limit it is reduced to local Hamiltonian

only for very specific boundary Hamiltonians. It is not yet known how to find them. We introduced this example to show that a lot of work remains to be done before we can truly say that TN approach realizes most properties of holography. However we will show in following chapters that question of holographic shadows can already be addressed by formalism of tensor networks.

4.2 MERA and Geometry.

Tensor networks like multi-scale entanglement renormalization ansatz (MERA) were initially introduced as an advanced numerical approach to study strongly entangled

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quantum many-body systems. However it was later realised by Swingle [26] that MERA has some properties similar to AdS/CFT.

Since then a lot of work[27, 28] was done to modify MERA in such a way so it could capture essential points of AdS/CFT correspondence. Although plenty of progress has been done in this field, MERA is not of a much interest for purposes of current work because it doesn’t possess sub-Ads scale resolution. This property is essential for us because the largest size of holographic shadow is of order of lAdS.

Yet we believe that this section plays its role from pedagogical point of view. Let us first say few words about MERA in general.

As we already said MERA was initially used for variational estimation of the ground state of the critical(gapless) systems in D dimensions. In particular we will consider 1+1 dim case.

It is possible to view MERA in 2 different ways. One of them is so called

Figure 4.1: a) Example of MERA 1+1(k=2) b) Disentanglers which are used to decrease amount of entanglement from layer to layer c) Isometries which are used for coarse graining procedure. Picture from [29].

Down to up:

One starts from ”boundary state” and renormalises it from layer to layer by means of coarse-graining procedure. It is explicitly defined by so called rescaling factor k (k=2 in our case) which tells how many sites will be coarse grained per block. Once going from certain layer to upper one we obtain renormalised state in the Hilbert space of smaller dimension. This state will still have some resemblance to the previous one. However because of removed entanglement the new one is much simpler to deal with.

At the same time we can view this procedure in opposite direction. It may be shown[29] that one can use MERA to build an approximation of the ground state of CFT on the boundary from simple initial input(upper layer state).

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Now we would like to introduce concept of ”geometry” for MERA TN. At this point one should understand it as set of all possible cuts through TN. These cuts will play role of different curves in some real geometry. Certain ”length” will be associated to each cut, depending on how many crossed lines it contains. As was shown by Swinger minimal cuts in MERA behave like geodesics in empty AdS. Our goal will be to show that there is one in one correspondence between ”geometry” in MERA and geometry of AdS. To do so we will introduce two length scales in given tensor network. Then by matching ”lengths” of certain cuts with lengths of corresponding curves in AdS we will find explicit expressions for these two scales in terms of radius of AdS. It will be shown that distance between nearest sites in MERA can’t be smaller than lAdS and no change of coordinates can fix it. Our

derivations are mainly based on [29].

Figure 4.2: Left image shows MERA TN(where disentanglers and isometries are suppressed for convenience), right one shows time slice of Poincare AdS coordinates. Pictures are taken from [29].

Constant time slice of Poincare metric is given by:

ds2 = (dx

2_{+ dz}2₎

z2 L 2

AdS (4.1)

Lengths of curves are given by:

|γ1| = LAdS Z x0 0 q (dz dx) 2_{+ 1} z dx = x0 z0 LAdS (4.2) |γ2| = 2LAdSlog x₀ a

Now let us find lengths of appropriate curves in MERA:

We define z0 = kma (Can be seen from picture). Number of bonds between two

sites is given by x0

km_a

|γ1M ERA| = L1(number of bonds) = L1

x0

z0

|z0=kma (4.3)

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LAdS = L1

Now let us find length of second curve:

|γ2M ERA| = 2L2(number of vertical bonds) (4.4)

In this case number of vertical bonds between zero and m-th level is equal to m:

x0 = z0 = kma m = log_kz0 a = log_kx0 a (4.5) |γ2M ERA| = |γ2AdS| L2 = LAdSlog k

4.2.1 Limitations on sub-AdS scale resolution of MERA.

To use tensor network as a probe of bulk physics we would like it to have sub-AdS scale resolution. It means that distance between nearest nodes should be smaller than LAdS. Using arguments from [29] we will show that no allowed change of

coordinates can lead to sub-AdS scale resolution.

In previous section we considered xM ERA, zM ERAto be trivially related to xAdS, zAdS.

Suppose now that:

xM ERA= f (xAdS)

zM ERA= g(zAdS)

In new coordinates:

f (a) - UV-most lattice spacing(in x direction) g(a) - UV cutoff in the holographic(z) direction.

Now let us compare γ1 in two coordinate systems. Because of transformation

we’ll get:

xM ERA₀ = f (xAdS₀ )

z₀AdS = kma → g(z₀AdS) = kmg(a) (4.6)

km = g(z AdS 0 ) g(a) Number of nodes: x0 km_a → f (xAdS 0 ) km_{f (a)}

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|γ1AdS| =

LAdSxAdS0

zAdS 0

|γ1M ERA| = L1(number of nodes) = L1

xAdS 0 f (a) g(a) g(zAdS 0 ) = x AdS 0 LAdS zAdS 0 g(z₀AdS) ∂γ1 ∂xAdS 0 = L1f0(xAdS0 ) g(a) f (a) = LAdS g(zAdS₀ ) zAdS 0 (4.7)

The middle expression depends only on x0 and the right one on z0. The fact

that we can vary these 2 parameters independently and that these expressions are equal to each other gives restrictions on f(x) and g(x). We can only allow:

f (x) = xx and g(z) = zz where x and z are some constants.

Now we can get:

|γ1M ERA| = L1 xxAdS0 xa za zz0AdS = LAdS xAdS₀ zAdS 0 (4.8) L1 = LAdS

Now let us consider case γ2:

Number of bonds at zero level: xM ERA0

f (a)

We can calculate number of bonds in geodesic γ2:

2 log_k(xM ERA₀ /xa) γ2AdS = 2 log xAdS 0 a = 2LAdS xM ERA 0 xa (4.9) Hence L2 = L log k

So it was shown that no change of coordinates is able to give possibility to describe sub-AdS scale. We shall consider different tensor network for that.

4.3 Isometric Tensors

We have shown in previous section that MERA TN(at least at level it is currently de-veloped) doesn’t have sub-AdS scale resolution. Before we introduce different kinds of tensor networks which may resolve this problem we need to introduce building blocks which will be used in future investigations. This review is mainly based on articles [30] and [31].

Let us introduce concepts of isometry and perfect tensors:

Definition: Map T is called isometry between Hilbert spaces HA and HB,

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There are some useful properties of isometry: 1. dim(A) ≤ dim(B).

Simple example: Isometry preserves angles between vectors. If one wishes to find isometry from 3d vector space to 2d one, he will face the fact that there are no 3 vectors(apart from ~0) in 2d vector space orthogonal to each other. 2. Unitary transformation is particular case of isometry when dim(A) = dim(B). 3. If T isometry from HA to HB then:

(a) T†T = 1HA

(b) T T† - is projector from Hilbert space HB to subspace of dim(HA)

Suppose we have isometric tensor T. If some operator O is acting on its incoming legs one can ”push it through T” by means of following procedure:

T O = T OT†T = (T OT†)T ≡ O0T

Figure 4.3: Process of pushing operator O through isometric tensor T. Picture from [31]

4.4 Perfect tensors and RT formula.

Tensor networks formed by perfect tensors share many important properties with AdS/CFT correspondence. We will see in next sections that they do not possess sub-AdS scale resolution. However they may be generalised to other types of tensor networks which will have this property. That is why it is important to understand explicit construction of this type of TN.

Let us consider some tensor[31] Ta1...a2n. Its indices(legs) form Hilbert space of

dim D2n_{, where D is dim of each leg. We may split this Hilbert space in to two}

fractions, one of which containing |A| and other |Ac_|.

Definition: We will call 2n legged tensor Ta1...a2n perfect if for any bipartition

|A| 6 |Ac_{| the obtained tensor is proportional to isometric map from |A| to |A}c_|.

Important property of this kind of tensors is that density matrix, formed from any n legs of such tensor is maximally mixed.

Ryu-Takayanagi formula.

Based on [31] we will derive RT formula for tensor network, built from identical perfect tensors. This tensor network(TN) represents holographic state:

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Note: In general sets {|Pii} and {|Qii} don’t have to be normalised or

orthogo-nal.

Figure 4.4: Tensor network, split into two subregions. Picture from [31]

We want to get restriction on amount of entropy contained in the boundary region A. To do so we should obtain reduced density matrix of this region first:

ρA= trAc(|ψi hψ|) =

X

ij

hQj|Qii |Pii hPj| (4.11)

Its rank is defined by number of terms in the sum. Each leg i has dim ν (bond dim). Number of legs is defined by number of cuts |c| . Hence, rank of ρA is less or

equal to ν|c|.

Maximal amount of entropy corresponds to maximally mixed density matrix. It can be proven that for such matrix S = log N , where N - rank of density matrix.

Taking into account this statement and the fact that rank of ρAcan’t be greater

than νc, one can see that the best bound on entropy of region A is obtained if we consider cut c be minimal one(which will be denoted as γA):

SA ≤ |γA| log ν (4.12)

Note: It is worth to mention that we do not specify particular tensors at each site. We only claim that all of them are perfect and that each of them has the same amount of legs. So far we haven’t specified properties of P and Q. Let us consider them to be isometries from i to a and b correspondently.

It may be shown(taking into account one of properties of isometry) that in this case {|Pii} and {|Qii} are orthonormal sets.

From this one can see that in this case ρA is indeed maximally mixed density

matrix. Hence RT formula holds exactly:

SA= |γA| log ν. (4.13)

So we have arrived to conclusion that in aforementioned setup entropy of bound-ary region is defined, up to constant, by ”length” of minimal cut bounding this region. This indeed resembles RT formula for AdS/CFT correspondence. It is im-portant to notice that 4.13 is proven only for single boundary regions. Discussion about entropy of several disconnected intervals may be found in [31].

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At last the theorem, which states when Paiis an isometry from {|ii} to {|ai} and

Qbi is isometry from {|ii} to {|bi} correspondently will be provided without prove:

Theorem: [31] Suppose that we have holographic state associated to a simply-connected planar tensor network of perfect tensors, whose graph has ”non-positive curvature”. Then for any connected region A on the boundary, we have SA =

|γA| log ν; in other words, the lattice RT formula holds.

Term ”simply-connected” means that tensor network doesn’t contain any graph loops or multiple edges. ”Non-positive curvature” is quite difficult to define in general but what we really need is to ensure that distance functional from one site of TN to another one doesn’t have interior local maximum.

4.5 Absence of sub-AdS scale resolution for TN

formed by perfect tensors.

Let us show why this particular type of TN can’t be used for purposes of our work. In particular we will show that distance between nearest sites can’t be smaller than lAdS or equivalently that one should associate area of order lAdS2 to each tensor in

TN.

We will consider one of setups introduced in [31] and then argue that similar arguments may be extended to different setups.

First let us make several assumptions. As was stated in previous chapters con-stant time slice of AdS3 is given by Poincare disc. It is defined by constant scalar

curvature:

R = 2 −l2

AdS

(4.14)

Note: In Riemannian geometry the scalar curvature(Ricci scalar) is a certain cur-vature invariant of Riemannian manifold. This quantity assigns a real number to each point of manifold. This number represents the amount by which the volume of geodesic ball in curved Riemannian manifold deviates from that of the standard ball in Euclidean space. In case of interest (2dim manifold) this value completely characterises the curvature of surface.

We will also need to introduce so called Gaussian curvature which is for our purposes may be viewed as some intrinsic measure of curvature which is case of 2d manifold reduces to R/2. More deep introduction may be found here[32]. Very good resource about hyperbolic geometry [33].

Now let us consider uniform tessellation of Poincare disc. Tessellation means that we take certain types of polygons(which in general may differ from each other) and fill our spacetime by them. In case of uniform tessellation all polygons are taken to be identical.

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First tessellation of interest is {5, 4}. Here 5 stands for number of sides of polygon(pentagon in this case) and 4 stands for number of polygons meeting at each vertex.

Now our goal is to evaluate area associated to each polygon. This may be done by means of Gauss-Bonnet theorem, which applied to some region R states that:

Z R KdA + Z ∂R kgds = 2πχ(R) (4.15)

where K - is Gaussian curvature, kg is the geodesic curvature and χ(R) is the Euler

characteristic of the residual region.

This theorem connects local notion of geometry of certain manifold with it’s topology.

In case of interest for certain polygon with m sides(remember that each side is part of some spacial geodesic) this formula may be used to obtain area of polygon:

Am = {a(m − 2)π − (a(1) + a(2)... + a(m))}

1

−K (4.16) where a(i) stands for inner angle of polygon(see image).

Figure 4.5: Uniform tessellation of hyperbolic plane 5,4. Image from internet

Let us consider pentagon tessellation {5, 4}. In this case 4 polygons meet together in each vertex. Sum of corresponding angles is equal to 2π. Hence a(i) = π₂. Each polygon is pentagon, so m=5. Once we plug everything to 4.16 we obtain:

A5 =

π 2l

2

AdS (4.17)

We can vary tessellation in 2 different ways. First is to change number of sides of polygons, or change number of polygons meeting at each vertex. However it may be shown by explicit calculation that in any setup area associated to polygon will be of order of l2

AdS. One possible way to obtain sub-AdS scale resolution will be to

introduce nonuniform tessellation. Even though this approach will not lead to any immediate problems(RT formula will still hold) if we use this new tessellation as was done in the article to build holographic code, then we will run into contradictions.

Let us sketch brief argument which explains why nonuniform tessellation leads to problems and can’t be used to obtain TN with sub-AdS scale resolution. Fol-lowing logic of [31] we build tensor networks by means of putting one tensor per polygon(pentagon) and connecting them together. Note that we use perfect tensors

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defined above, all legs have the same dimensionality. This may be done at least in two different ways. One way will be to pick tensors with number of legs equal to number of sides of corresponding polygon. In this case uncontracted legs will be only those on the boundary of tensor network. In this way we obtain TN describing single boundary state. Second way will be to pick tensors which have one more leg then number of sides of corresponding polygon. As a result we will get tensor net-work which realises map from set of boundary legs to rest ones, which will be called bulk legs. These two sets define 2 Hilbert spaces. We want obtained TN to mimic AdS/CFT correspondence, hence we can immediately conclude that dimensionality of bulk Hilbert space can’t be greater than that of boundary Hilbert space. But if we want to have sub-AdS scale resolution in this TN we will have to increase number of polygons, hence increase total number of bulk legs. It may be shown that almost instantly one will obtain bulk Hilbert space larger than that of boundary. As a consequence any similarity between this TN and AdS/CFT correspondence will be lost[31].

Figure 4.6: Left picture: Holographic code build on top of tessellation {5, 4} [31], red dots correspond to bulk legs. Right picture: Example of nonuniform tessellation with sub-AdS scale resolution[30].

Possible way to resolve this contradiction would be to modify properties of build-ing blocks - tensors. This leads us to another type of tensor networks.

4.6 Tensor network formed by pluperfect tensors.

4.6.1 Definition of pluperfect tensor.

As was mentioned in previous section TN formed by perfect tensors can’t be used to describe sub-AdS scale resolution because nonuniform tessellation will cause rapid

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growth of dimension of bulk Hilbert space. Once it becomes larger than boundary one we loose similarity between given TN and AdS/CFT. However we are going to show that if we change properties of tensors we use, then it is possible to avoid this problem. In case of tensor network formed by pluperfect tensors we start from bulk Hilbert space, which dimension is much larger than those of boundary. Yet it so happens that because of tuned properties of pluperfect tensors one can define physical subspace(defined by so called gauge invariance property) in the bulk Hilbert space. We will show that one can define isometry from this subspace to the boundary Hilbert space. This crucial property will give us sub-AdS scale resolution for the first time. However it will be shown that RT formula can be proven only for limited number of bulk states, namely for direct product states. That will be a problem, because bulk state corresponding to presence of BH is implied to be highly entangled. This will force us to move to yet another generalisation of this TN.

In this section notion of pluperfect tensor will be introduced. Then tensor net-work will built from them and some properties of this netnet-work will be presented. This section summarises results introduced in article [30].

In general pluperfect tensor should have odd number of indices. Without loss of generality one may introduce all definitions for particular case 2n + 1 = 5. In this case pluperfect tensor will look like TI

αβγδ. In-plane indices α, β, γ, δ ∈ 1,2,...D.

Index I (logical degree of freedom) ∈ 1, 2...D4_.

Pluperfect conditions.

1. T realises unitary map from indices α, β, γ, δ to index I.

T_αβγδI T_αβγδJ ∗ = δIJ (4.18) 2. There exists subset of indices of dim D2 _{among indices I, for which, for fixed}

I tensor TI

αβγδ is a perfect tensor. So, for fixed I ∈ 1, 2...D2 TαβγδI defines unitary

transformation from any two to other two inplane indices.

T_αβγδI T_µνγδI∗ = 1

D2δαµδβν, I ∈ 1, 2...D

2 _(4.19)

3. Let us consider I fixed ,I ∈ 1, 2...D2_{. For any inplane index α, T}I

αβγδ is a

unitary mapping from Iα to βγδ, up to normalisation.

T_αβγδI T_µβγδJ ∗ = 1

DδIJδαµ I, J ∈ 1, 2...D

2 _(4.20)

Tensor Networks As Probes of Near Horizon Geometry.

Universiteit Van Amsterdam

Master project