#### Exploring the tensor networks/AdS correspondence

Received: June
Exploring the tensor networks/AdS correspondence
Arpan Bhattacharyya 0 1 2 5 6
Zhe-Shen Gao 0 1 5 6
Ling-Yan Hung 0 1 3 4 5 6
Si-Nong Liu 0 1 5 6
0 Nanjing , 210093 , China
1 220 Handan Road , 200433 Shanghai , China
2 Centre For High Energy Physics, Indian Institute of Science
3 State Key Laboratory of Surface Physics and Department of Physics, Fudan University
4 Collaborative Innovation Center of Advanced Microstructures, Nanjing University
5 Department of Physics and Center for Field Theory and Particle Physics, Fudan University
6 560012 Bangalore , India
In this paper we study the recently proposed tensor networks/AdS correspondence. We found that the Coxeter group is a useful tool to describe tensor networks in a negatively curved space. Studying generic tensor network populated by perfect tensors, we nd that the physical wave function generically do not admit any connected correlation functions of local operators. To remedy the problem, we assume that wavefunctions admitting such semi-classical gravitational interpretation are composed of tensors close to, but not exactly perfect tensors. Computing corrections to the connected two point correlation functions, we nd that the leading contribution is given by structures related to geodesics connecting the operators inserted at the boundary physical dofs. Such considerations admit generalizations at least to three point functions. This is highly suggestive of the emergence of the analogues of Witten diagrams in the tensor network. The perturbations alone however do not give the right entanglement spectrum. Using the Coxeter construction, we also constructed the tensor network counterpart of the BTZ black hole, by orbifolding the discrete lattice on which the network resides. We found that the construction naturally reproduces some of the salient features of the BTZ black hole, such as the appearance of RT surfaces that could wrap the horizon, depending on the size of the entanglement region A.
AdS-CFT Correspondence; Black Holes
1 Introduction 2
Coxeter group and tensor network
An overview of Coxeter group
Towards constructing tensor network
Some recursion relations
3
Perfect tensors
2.1
2.2
2.3
3.1
3.2
3.3
4.1
4.2
4.3
4.4
5.1
5.2
5.3
5.4
5.5
5.6
legs
{ 1 {
Reduced density matrix and modular Hamiltonian
3.1.1
Renyi entropy
Stabilizers and symmetries
Correlation functions
3.3.1
3.3.2
3.3.3
3.3.4
Operator pushing and stabilizers
Some examples- the hexagonal code
Proof of factorization of correlation functions
A remark on correlation functions and multipartite entanglement
4
Perturbations away from perfection 5 Simulating BTZ geometry in the Tensor Network
Corrections to entanglement entropy
4.1.1
Quadratic order correction
Correction to correlation functions
Three point correlation function and fusion coe cients
Constraints
Orbifolding and BTZ
BTZ via Coxeter group
Entanglement entropy and RT surface in the BTZ background
Black hole | A region of high bond dimensions?
A comparison with the random tensor construction
Computing the thermal spectrum
5.6.1
An extra correction in the Renyi entropy
6
Conclusion
A Curvature and bulk locality B Inspecting a remedy to the at entanglement spectrum | Weighting the
B.1 The maximal number of weights when pushing toward the inner most layer
Introduction
Recently there is a wave of new understanding of the AdS/CFT correspondence driven
by developments in the understanding of quantum entanglement and its manifestation in
a gravity theory via the AdS/CFT correspondence. Most notably, the entanglement
entropy of some region in con guration space in some states in a CFT corresponds to the
area of a minimal surface in the bulk gravity theory [1, 2]. It is by now clear that this
minimal surface, often called the Ryu-Takayanagi (RT) surface, is profoundly connected to
the Bekenstein-Hawking entropy formula. Till now, however, in the absence of a quantum
gravitational theory, it is not clear how such a surface could emerge to encode the
entanthe space of geodesics in AdS space, termed the kinematic space [4{12].1 More recently, [31]
proposes a concrete construction of a tensor network such that the entanglement entropy
of the physical degrees of freedom is given precisely by the size of a minimal surface that
cuts through the bulk of the tensor network. The important ingredient that goes into the
proposal is perfect tensors, which have the magic property of being a unitary map between
any choice of a set of half of the indices to the other half. This proposal is further developed
in [32, 33] so that there is a map between every state in the physical theory and a bulk
tensor network, relaxing the restriction to the \code-subspace" in [31]. Perfect tensors
were shown [33] to emerge rather typically for tensors with large bond dimension. Given
this success however, there are obvious aws to the proposal. To start with, it is known
that the entanglement spectrum is always at, which follows unfortunately also from the
perfect-ness of the tensors. We would like to systematically study these tensor networks. A
natural framework to discuss these tensor networks is the hyperbolic Coxeter group, which
describes all tessellations in a negatively curved space. In section 2, we will begin with a
brief review of the Coxeter group, and how it is related to a tensor network. In section 3,
we study properties of the perfect tensor network, and
nd that any tensor networks built
from a collection of perfect tensors in a negatively curved lattice has no connected
correlation functions between su ciently local operators. To remedy the problem, we explore the
1Interested readers are referred to [13, 14] for a comprehensive review on tensor networks and its diverse
applications in various branches of theoretical physics. Also we list out some references [15{30] where the
connection between reconstruction of bulk spacetime and entanglement in the context of holography have
been explored. This list is by no means complete and the readers are encouraged to check the citations of
these papers and the references within.
{ 2 {
possibility that the wave-functions are built from tensors that depart from exact perfect
tensors, albeit only by a \small amount", parameterized by a small number . As we will
describe in section 4, the leading order correction to the correlation function comes from a
set of perturbations at nodes forming a connected geodesic connecting the operators acting
at the boundary. This situation generalizes to three point functions. It is tempting to
interpret these as the analogy of Witten diagrams in the large mass limit, which emerges
naturally as we perturb away from the perfect tensor background. Then in section 5, we
discuss how the BTZ black hole for example admits a tensor network construction by
orbifolding the lattices, exactly as it is constructed by orbifolding in the continuous AdS space.
We will end with some concluding remarks in section 6. Various subtleties connected to a
at geometry and a discussion of a cure to the at entanglement spectrum problem using
\weights" are discussed in the appendix.
2
Coxeter group and tensor network
To understand tensor networks beyond the speci c lattice structures suggested in [31]
and [33], we would like to make better contact between the isometries of the actual AdS
space and the symmetries of the tensor network, and study systematically tensor networks
that can t into a negatively curved space. To do so, we will need to use the Coxeter group.
2.1
An overview of Coxeter group
The main idea we would like to explore is how one could systematically \tessellate"
spacetime. That is achieved by exploiting re ections across various planes which in turn is
connected to various properties of Coxeter group. First one can consider any Riemannian
space M n with constant curvature, i.e it can either be at, positively curved or negatively
curved. Then consider polyhedra made up of locally
nite number of intersections of
halfspaces. We call them \chambers" or Coxeter polyhedra. We consider re ection across each
of these faces of the polyhedra which are (n
1) dimensions. Now consider another such
polyhedra and re ections generated by their edges. Combining all these re ections we can
\tile" the whole space. To be explicit, let us consider generating a picture like gure 2 in
a 2d space. One rst determines the angles of a single triangle as will be described in the
next section and
gure 1. Then all the other triangles are related to the seed triangle by
subsequent re ections across each of the sides of the seed triangle, and the process
continues inde nitely. Re ection across each of the edge of the collection of all the triangles
form the generators of the Coxeter group corresponding to this speci c triangulation. i.e.
The isometries generated by all these re ections form one particular \Coxeter Group". We
note that all isometries of the AdS space can be understood as combinations of re
ections across codimension 1 planes. Therefore a speci c triangulation based on the Coxeter
group basically preserves a discrete group of the full isometries of the AdS space. This
will be very crucial to our construction, as we will demonstrate that combination of these
re ections would generate rotation (boost) and we will exploit that in the next sections.
The algebraic representation of these re ections across an edge on a Poincare disk will be
described in detail in section 5.2. The group multiplication of these generators however can
{ 3 {
be readily recovered by simply looking at a picture of the triangulation, and combine series
of re ections across edges of triangles. Suppose F1;
; Fn are the faces of the Coxeter
polyhedra characterized by the re ections s1;
; sn satisfying
si2 = 1:
(si sj )mij = 1:
We also de ne the dihedral angle between the two face as mij for all mij
2 so they are always acute. Then
Now these mij are the elements of the Coxeter matrix. We note that all the isometries
of the hyperbolic space can be achieved by re ections in the
at embedding space. For
detailed review of Coxeter group interested readers are referred to [34{38].
2.2
Towards constructing tensor network
For our case we will consider H2 which will be the constant time slices of AdS3. Now we
want to tessellate this space time using the Coxeter re ections. To tile a space by polytopes
one important condition is that theses polytopes satisfy the Gauss-Bonnet theorem. In
general because of this we can actually list all such polytopes when one tries to tessellate
two sphere or at (Euclidean) space. Now to tessellate H2 we can start with regular p-gons
in hyperbolic 2-space with angles 2q . Any such polygon will tessellate H2 with q copies
of these p-gons meeting at each vertex. Then we rst divide these p-gons into p number
of triangles meeting at the center with the three angles q
; q ; 2p . Each of the angle of the
to tile the H2 must satisfy the Gauss-Bonnet theorem implies that
polyhedra will be bisected by the sides of these triangles. Now we divide further these
triangles through the angle 2p into two right angled triangle there by bisecting one edge of
the polyhedra also. Now these triangles will have angles q
; 2 ; p : All these polyhedra used
1
p
+
+
1
q
1
2
< 1:
{ 4 {
2
3
4
labels of the 6 legs of the tensor. These tensors are generally not symmetric under permutation of
the legs.
fundamental building blocks for constructing the lattice. In the following we will describe
a simple map between these tessellations and architecture of tensor networks.
2.3
Some recursion relations
As demonstrated in [31], there is a recursion relation that can be derived, relating the
number of tensors at each layer to the next. To de ne what we mean by a layer in the
Coxeter tessellation of the hyperbolic space speci ed for example by the angles [p; q; 2] (if
any of the entries takes value 2 it is usually omitted in standard notation which we will
follow in all the subsequent sections.), where p would be specifying the p-gon that holds
a perfect tensor in its center, formed from 2p triangles, as already described. Each edge
is shared by two p-gons , which should be interpreted as a tensor contraction between two
tensors. One could see that the hexagon code in [31] would correspond to a [6; 4] tessellation
in this language. This is illustrated in gure 2.2 We note that conversely, providing only the
data about the number of legs p in each tensor and the number q of nodes being adjacent
to each other does not uniquely specify the tessellation. For example, the code in [33] could
have been equally well described by [5,5], although the map between vertices, edges and
tensors and their contraction would have to follow some di erent rules. This is connected
to the subtlety of what isometries are in fact preserved in the speci c tessellation. We will
in this paper base our discussion on the prescription as described above.
To assign a layer number to the nodes sitting at the center of a p-gon, we rst pick
a reference node in the bulk, and assign it a layer number 1. Then the layer number of
2Many of the Coxeter tessellation diagrams in this paper are based on the diagrams obtained from
Wikipedia [38].
{ 5 {
HJEP08(216)
4
3
2
4
3
1
2
5
hexagons apart from 2 are truncated and we are only illustrating here the di erent tensor types
we discussed in the text. It can be seen that at layer 2 there are 3 tensors of type 2(2) and at
layer 3 there are 3 tensors of type 2(3) and 4 tensors of type 1(3). Finally there are two type g(4)
tensors, and 8 type 1(4) tensors.
each neighbouring nodes sitting at the center of a p-gon sharing an edge with the reference
p-gon increases by 1. Then the next layer is obtained by collecting all the p-gons sharing
one edge with the layer -2 p-gons apart from the layer 1 reference p-gon. This process can
go on, giving a consistent assignment of layer number that increases as we move towards
the boundary, as long as q is even. (When q is odd, there is a con ict in the assignment
using the above simple rules. )
Using such an assignment, we will again come to the conclusion that no tensor is
connected to other tensors in the same layer. Also, there would again be two types of
tensors as in [31], one type where it has two legs connected to the previous layers, and the
other type where there is only 1 leg connected to the previous layer.
At layer n, we denote type I tensors with two legs connected to the previous layer by
g(n). Then, we can subdivide type II tensors in each layer with only 1 leg connecting to
the previous layer into subsets i, denoted i(n), where i denotes the shortest separation in
layers they are from the next type I tensor. Clearly 1
i
q=2, since q p-gons meet at a
vertex. The case in which [p = 6; q = 4] is illustrated in gure 3.
One can see that the following recursion relation is satis ed:
k = 1:
4) (n + 1) + g(n + 1)
(ii) (n) ! g(n + 1) + (p
3) (n + 1)
{ 6 {
Perfect tensors
One can check that this recovers the result in the appendix of [31] where p = 5; q = 4
The initial values are N ( k(1)) = p and 0 for other kinds of tensors. And if we stop at
Nbounday = N (g(n)) + X N ( i(n))
Nbulk = 1 + X(N (g(n)) + N ( i(n)));
n;i
where N (X) denotes the number of tensors of type X. Some numerical results:
i
n
i
(3.1)
In [31], it is demonstrated that a tensor network description of a wave-function such that
it is de ned on a negatively curved space and that each tensor is chosen to be a perfect
tensor leads to a natural emergence of the RT surface when one computes the entanglement
entropy of some connected group of chosen physical spins. Perfect tensors Ta1 a2n are
tensors with an even number of indices 2n, such that an arbitrary separation of the indices
into two groups TI , I = fb1;
bng and
= fbn+1
b2ng, TI is a unitary map satisfying
TI T yI0 = II0 = Y
bib0i :
{ 7 {
When the number of indices featuring in I is less than that in the set , T becomes a
norm-preserving projector from
to I, thus satisfying
TI T yI0 = II0 Ds;
(3.2)
where s is the di erence in the number of extra indices in the collection , and D is the
bond dimension of each index. The appearance of these delta functions play a signi cant
role in the following discussions. It is behind the emergence of a bulk that appears local
in many respect.
3.1
Reduced density matrix and modular Hamiltonian
To begin with, we revisit the entanglement between physical degrees of freedom in some
region A described by this family of tensor network wavefunctions. As demonstrated in [31],
a RT surface emerges, which speci es how a Schmidt-decomposition between region A and
its complement. Under the decomposition of the bulk wavefunction along the RT surface,
the reduced density matrix is basically given by the isometry V and it's complex conjugate,
A = Vxa11;;x2;ajAxjjRT j ja1;
ajAjiha01;
a0jAjjVxy1;ax02;; ax0jAjRjT j :
1
(3.3)
The modular Hamiltonian is thus a projector with eigenvalue log D, where D is the
bond dimension, for the linear combo of states that mapped to the e ective spins xi, and
in nity for any other linear combo. The spectrum is thus completely at.
3.1.1
Renyi entropy
The fact that the entanglement Hamiltonian has a complete at spectra also suggests that
all the Renyi entropies for any Renyi index n has exactly the same value. This is also
observed in [33]. We will explore the corrections to this issue.
3.2
Stabilizers and symmetries
Stabilizers correspond to symmetries. Any operator that do not commute with the
stabilizers correspond to an operator charged under the symmetry.
Examples of perfect tensors were constructed from stabilizer code. However, purely
from the fact that these perfect tensors satisfy
HJEP08(216)
(3.4)
(3.5)
for arbitrary O0, it means that
ie one can construct a set of operators that keep T invariant. In the case of spin 1/2
systems, where each index takes only 2 values, it is clear that for T having L- indices, we
can construct a set of 2L independent and commuting stabilizers, where O can be chosen
to be say
z acting on each site and O0 obtained by conjugating O by T . This generates
the complete set of stabilizers that specify the L-index tensor uniquely. Therefore, every
perfect tensor has a corresponding complete set of stabilizers.
T:O = O0T
O
0 1:T:O = T;
{ 8 {
This construction can be generalized to any perfect tensor whose index takes D values.
We simply replace Pauli matrices by the generalized Pauli matrices. These are constructed
by rst de ning
D
X
k
D
X !
k
X =
jk >< k + 1(modD)j;
Z =
k 1jk >< kj;
(3.6)
where ! = exp(2 i=D). The rest of the D2 basis, can be generated by products of X and Z.
Given these symmetries at each node, which can be treated as global symmetries of
the boundary theory, one can expect that any correlation function is non-zero only if the
operators have trivial charge. This is highly non-trivial given the amount of symmetry.
We then turn our attention to correlation functions, which is a crucial aspect of the
AdS/CFT correspondence [39{41]. As we will see, a tensor network built from perfect
tensors generically does not lead to any connected correlation function between local
operators. In a discretized system, a local operator is one that acts on some number P of
spins, such that P does not scale with the system size. The number of spins in the physical
system grows exponentially, and the tensor network is a self-repeating structure.
Therefore, we could presumably take our boundary links as an e ectively coarse grained set of
degrees of freedom, and consider operators that act strictly locally on individual spins in
the boundary layer. This consideration can be relaxed as we generalize.
3.3.1
Operator pushing and stabilizers
Action of an operator at the boundary can be readily pushed to action on the interior, i.e
one needs to work out O0 using O. The perfect tensor at each node is constructed as a
code stabilized by a set of stabilizers. Taking for example the hexagonal code in [31] it is
clear from (3.5) that O
0 1 and O together form another stabilizer. Therefore we can work
backwards using the list of stabilizers and obtain O0 from any given O.
3.3.2
Some examples- the hexagonal code
In this section we will demonstrate how correlation functions can be computed explicitly
in the hexagon code. We will mainly consider the hexagonal holographic code consisting of
six legs perfect tensors (T
) constructed from 5 qubit stabilizer code discussed in [31].
They are essentially characterized by the [6,4] triangulation which is already shown in
gure 2.
As we want to compute correlation functions of some operators Oi we exploit the
operator pushing property of the prefect tensor as mentioned in (3.5). The operators Oi
acts on the free legs of perfect tensors at the outer layer of the network. Oi's are typically
made up from Pauli operators so we need to know only the e ect of pushing Pauli operators
through three legs of the perfect tensors. For example consider the six index perfect states
j
>= T
j ; ; ; ; ; >. j ; ; ; ; ; > is the orthonormal basis. The stabilizers
{ 9 {
for the tensor is given by [31, 42]
S1 = XZZXII;
S5 = XXXXXX;
S2 = IXZZXI;
S6 = ZZZZZZ;
where X; Z are Pauli matrices x;z respectively. Interested readers are referred to [42] for
more details about the stabilizer codes. One can show that by taking products of these
stabilizers, it is possible to generate a new stabilizer such that one can make X acts on
any of the one spin, and that there are two I's acting on another 2 arbitrary sites. For
example, there is a stabilizer
where,
~
S
j
>= j
> :
S~ = x
I
I
x
y
S3 = IIXZZX;
S4 = XIIXZZ;
(3.7)
(3.8)
(3.9)
I the
I
I
(3.10)
(3.11)
(3.12)
of other Pauli matrices in the interior leg in the hexagon code.
j
~ >= xc4c04 yc5c05 yc6c06 jc4; c5; c6 >
jc04c05c06
>;
y:
y
So from that it is evident that if we act on the free legs of the tensor by x
I
e ect is to act on the inner three legs by the operator x
y: Similarly for y
one has to act with y
z
z on the inner three legs. This is illustrated in gure 4.
Now we will exploit this property to calculate the correlation functions. When the
operator acts on the free legs after pushing them through a node it will act on the inner
legs and hence will act on the bulk wavefunction formed by contractions of those inner legs.
First we write down that bulk wavefunction for this tensor network. For now we focus on
a single tensor.
j
>= Tb1b2b3c4c5c6 U c4c5c6 jb1; b2; b3
>;
like to compute j ~ >=
b1
x j
>. Using (3.9) we immediately obtain
where U denotes all the other tensors T and the boundary physical degrees of freedom are
contracted to. Tb1b2b3b4b5;b6 is the perfect tensor at the central node. Suppose we would
j
~ >= Tb1b2b3c4c5c6 xc4c04 yc5c05 yc6c06 U c04c05c06 jb1; b2; b3
>
Note that T can be viewed as a unitary transformation from b1; b2; b3 to c4; c5; c6.
Therefore, what we have done is e ectively obtaining a state
where U is also an isometry that de nes a mapping from the boundary links to the links
which the x;y are contracted with.
Now when we compute
<
x >=<
~ > :
(3.13)
This will give products of Tr x and Tr y which are zero.
Using this trick we can also check the factorization of the correlator as described in
the following section very easily. Most of the cases we will get zero connected correlations
unless we consider correlations of highly non local operators | namely those operators
that act on close to more than half of the total number of boundary sites.
3.3.3
Proof of factorization of correlation functions
When exactly do we get more non-trivial correlation functions?
One can see that the tricks of operator pushing would start to fail when we have too
many operators acting on the boundary. The question is how many is deemed \many"?
We can do the computation in a slightly di erent way. Decompose the bulk
wavefunction into a product of two big tensors U and V , each forming an isometry from some region
A (or its complement Ac respectively) to a RT cut. Suppose the choice of region is one
that contains the action of all the boundary operators.
Then the correlation function is
hOi = Uxa11;; xajjRATj j Oa1;a01 Uyy1a;01 yjaRjTAjj Vx1;
ba; xcjjRATcjj Vyy1;ba;yjRcTjAjcj
where O denotes a group of operators acting on sites in region A. The product of V 's and
it's Hermitian conjugate generate a set of delta's on x and y. This is guaranteed when we
place the tensors in a negatively curved lattice, in which the number of ingoing legs from
the boundary to the interior layers keep decreasing exponentially, and that the tensors
being perfect, are isometries from a layer to the next. We are thus left with the product
of U 's with all the indices contracted with each other except for a few contracted to O,
assuming that they are sets of local operators.
Therefore, we are immediately left with
Moreover, since O can be a set of local operators acting on multiple separated sites, we have
hOi = trO:
h
Y Oii = Y trOi:
i
i
(3.14)
(3.15)
(3.16)
We have a factorization of the operators.
3.3.4
A remark on correlation functions and multipartite entanglement
In [31] there is a discussion on multi-partite entanglement. For a division of the boundary
into 4 regions A; B; C; D, each of them is surrounded by an RT surface and the volume
enclosed between the RT surface and the boundary region A for example is denoted by the
causal region of A = C(A). It is noted that the union of C(A); C(B); C(C); C(D) does not
O(D)
inserted. The dotted lines correspond to geodesics forming the boundaries of the causal wedges of
each boundary region. The regions colored red denote residual regions outside of the union of the
four causal wedges. Connected correlation functions could feature when the residual region itself is
connected.
cover the entirety of the hyperbolic space and there are some residual regions. The size of
the residual regions has a size that can be estimated from the Gauss-Bonnet theorem. The
gure describing these in [31] is reproduced here in gure 5 to illustrate its implications
when we inspect correlation functions.
Suppose we consider correlation of operators with support in region B and D. The
evaluation of the correlation functions hO(B)O(D)i involves tracing out, A and C, which
would lead to a set of delta functions pushed all the way to the geodesic which bounds
C(A) and C(C). When jAjjCj
jBjjDj, it is clear that the residual region, denoted by the
red dots in the gure, are disconnected. As a result, the correlation functions are actually
disconnected since the delta functions would pinch o
the throat. Only until jAjjDj
jBjjDj; there is a connected region of residual regions, in which the correlation functions
could be connected. But this immediately suggests that the support of the operator has
to scale as system size before one obtains non-trivial connected correlation functions.
4
Perturbations away from perfection
Since perfect tensors admit a lot of symmetries, as we demonstrated above, most of the
correlation functions of local operators vanish. To obtain non-trivial correlation functions one
has to move beyond perfect tensors. However, we would like to stay close to perfect tensors
since it poses properties that can be identi ed with classical gravity, such as saturating the
RT formula when computing entanglement entropy. We will therefore consider departure
away from the perfect tensors as a small perturbation, and inspect their contributions to
entanglement entropy and correlation functions.
The tensors at each node is thus given by
X = T + t;
(4.1)
where T is a perfect tensor, and is a small number. For example, in [33] it is demonstrated
that random tensors weighted by the Haar measure approach a perfect tensor in the large
tensor dimension (D) limit. Therefore
1=D.
The wavefunction is then given by
j i = (Y Xv) 1 Lj 1;
Li
v
(4.2)
where Qv Xv corresponds to the appropriate tensor contraction speci ed in a tensor
network diagram, Xv denote tensors at each node and i are boundary degrees of freedom.
Corrections to entanglement entropy
One could compute the change to the entanglement entropy in the presence of perturbations
t. To leading order in t, the correction to the reduced density matrix involves only one t
located at some vertex in the tensor network.
Consider those cases in which this t is located away from the RT surface. In that case,
in the computation of the reduced density matrix, one can consider splitting the calculation
into three bits. Consider for simplicity the case of a single connected region A. Suppose t
is located \outside" of the RT surface. When computing the reduced density matrix, one
rst construct an isometry from the complement of A, denoted by Ac, to the boundary
plus a hole surrounding the node at which t is located. For a negatively curved graph,
this generically means we can draw a ow diagram from incoming boundary links in Ac
that ow out to the hole and the RT surface, de ning an isometry from the boundary to
the hole and the RT surface. Then when the links in Ac are contracted, one obtains the
identity operator for the out-going legs, which are legs at the hole and the RT surface. The
identity operator is of course factorizable and acts independently on the hole and the RT
surface!. As a result, the spectrum of the reduced density matrix, which depends on the
schimdt decomposition between Ac and A at the RT surface still has a
at spectrum as
before dictated by the identity matrix there, except for an overall change of normalization
(1 + tr(T:ty + t:T y)) that came from the hole. As a result, they do not contribute to any
change in the entanglement entropy. This is illustrated in gure 6. We note here, however
that negative curvature again plays a crucial role. Consider for example the four leg tensor
considered in [33]. The tensor structure there is describable by the Coxeter group [4,5],
if we inspect the arrangement of the self-repeating units. However, within each unit, the
tensors are arranged in a way that is essentially
at. In that case, when a hole is dug
within that unit, it is not necessarily possible to construct a ow diagram without internal
loops such that one de nes an isometry from the incoming legs near the boundary to the
legs connecting to the next layer and to the hole. We will discuss this issue in more detail
in the appendix.
A similar calculation can be done when t is located \within" the RT surface. In that
case, the trace over region Ac leads again to a simple delta function on the links of the RT
surface, and that the reduced density matrix is basically
A = Tx1;x2;x3;:::;xLU 11 A
N x1;x2;x3;:::xL 1
A0jU y 1
N x01;x02;x03;:::;x0L 1tyx01x02x03:::x0L 1
+(T $ t)) + h.c.;
(4.3)
show that they only change the normalization of the reduced density matrix, but do not contribute
to changes in the entanglement spectrum.
where now U is an isometry constructed from the boundary to the hole and the RT surface,
and we are working with an L-leg tensor T . For precisely the same reason as the previous
scenario, the presence of t in the interior do not lead to any change of the entanglement
spectrum, and so the entanglement entropy is not altered to this order of the perturbation.
In fact, for that reason, any disconnected appearance of t, small number of them
relative to bulk size, away from the RT surface, could only contribute to change in the
overall normalization, and no change to the entanglement spectrum, and thus preserve the
entanglement entropy.
The leading order change has to come from t that falls exactly on top of the RT surface,
which then actually changes the spectra of Schmidt coe cients along the RT surface. In
fact the change to the reduced density matrix due to perturbation t at site i on the RT
surface is given by
i
A0j + (T $ t) + h.c.:
A(i) =
X U 11 i 1 x1x2xL 1 i+1
A
N Tx1;x2;x3; i U y0 1 i 1 x1x2:::xL 1 i+1
0
1
A
N tyx1;x2;xL 1; i
This is highly reminiscence of the bulk calculation of changes to the entanglement
entropy, in which the entanglement entropy is only sensitive to the changes in the metric
on top of the RT surface.
To linear order, the correction to the entanglement entropy thus comes from
S + S =
tr(( 0A +
A) log( 0A +
A)):
The form of A is a projector, in which it has equal eigenvalues along some jDj
dimensional sub-space H , and vanishing eigenvalues in the orthogonal DjAj
D
dimensional
(4.4)
(4.5)
Diagonalizing 0A +
eigenvalues of the form
where
subspace H . On the other hand,
A generically has o -diagonal elements
and also
elements purely in the orthogonal subspace
A
.
A, to linear order in the small perturbation, it is forced to have
U ( 0A +
A)U y =
1
D
diagf1 + 1
;
; 1 + D ; D +1;
DjAjg;
DjAj
X
i
i = 0:
The leading contribution to
A contains only t very close to the RT surface. In practice
therefore the number of eigenvalues i in the orthogonal subspace is actually less than
DL 1, where L is the number of legs of the tensor.
HJEP08(216)
Therefore, we have 1
D
D
X
i
S =
i(ln D
1)
DjAj D
X
i
D
D +i ln
D
D +i :
There are two issues of note. First, there is necessarily an ln contribution, which is
non-perturbative in the small parameter, whenever
A leads to extra eigenvectors in the
orthogonal subspace. If the holographic code is generically an isometry from boundary
degrees of freedom to bulk degrees of freedom, even with more interesting entanglement
spectrum than the at spectrum that follows from perfect tensors, one would expect that
any perturbation that involves the orthogonal subspace could lead to such non-perturbative
terms. It would be a very strong test of the proposal to understand if such a term could
arise in the AdS/CFT correspondence. We note that for generic perturbations of couplings
these would not arise. The only possibility is perhaps when a coupling perturbation is
mixed with the change of cut-o scales.
A second issue of note is that if j = 0 for j > D , then
S = 0 following from
tr
A = 0. This is expected because within the subspace V
entanglement is maximal. To
obtain non-trivial perturbation to the entanglement entropy it is necessary to move on to
a second order perturbation.
4.1.1
Quadratic order correction
To complete our discussion, it is instructive to inspect also second order contribution. To
second order in t, the density matrix takes the following form
= N 2(j 0ih 0j + j 0ih 1j + h.c. + j 0ih 2j + h.c. + j 1ih 1j
)
(4.9)
where j ni denotes corrections to the state corresponding to the sum of all con gurations
in which n of the nodes is replaced by t, and that N is the normalization, which itself
admits an expansion in t. For later convenience, let us denote
Pn = trA
j iih j j:
X
i+j=n
(4.6)
(4.7)
(4.8)
(4.10)
Precisely for the same reason as already discussed, when one replaces two nodes by t
away from the RT surface, their contributions to the Renyi entropy are canceled by the
normalization. Let us focus on the case in which
A projects onto the same sub-space as
A, so that non-perturbative logarithmic terms do not appear. In that case the leading
quadratic contribution to the Renyi entropy comes from j 1ih 1j. In particular, since t
aligns in the same subspace that the perfect tensors T projects into, j 0ih 2j + h.c. do
not contribute to the quadratic order in entanglement entropy.
The systematics of N in this case is as follows:
N
X tr(Tiy:ti + tiy:Ti);
=
X tr(Tiy:ti + tiy:Ti):
i2=
i2
where the tr means that all four indices between T and t are contracted with each other.
where
We then have
tr( nA) =
1
Dn (1
i2
+
i;j2=
i2
=D)n trIn + ntr(In 1(P1 + P2 +
)) +
n(n 1)
2
trIn 2P12 +
Focussing on the quadratic contribution, we have
trP12 =
X tr[(Tiy:ti + tiy:Ti)2] +
tr(Tiy:ti + tiy:Ti)tr(Tjy:tj + tjy:Tj )
X
i;j2 ; i6=j
X tr(Tiy:ti + tiy:Ti)tr(Tjy:tj + tjy:Tj ):
The last two terms are canceled out by the normalization. We nally have
Sn =
1
1
n
ln tr( nA) =
ln D
2 n
2D
i2
X tr[(Tiy:ti + tiy:Ti)2]:
In the following, we will discuss how t enters into the computation of the correlation
function. Looking ahead, comparing with (4.20), (4.21) the above quadratic correction to
the Renyi entropy can be related to the conformal dimension of operators as
2 X tr[(Tiy:ti + tiy:Ti)2] =
X 2 2ab =
X exp( 2 ab):
ab
ab
4.2
Correction to correlation functions
Given (4.1)
h jO1(i)O2(j)j i =
Y Xvy
v
!
1 i j
O1(i) i 0i O2(j) j 0j
Y Xp
p
!
1 0i 0j
where Om(i) is some operator located at site i.
(4.11)
(4.12)
:
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
of perturbations t along nodes along a geodesic connecting the operators inserted at the end points.
The legs of a tensor is labeled according to the contraction de ned in the transfer matrix (4.20).
Now consider the correction to linear order in . This means that in (4.17) exactly
one node features some tensor tv, while all other nodes are occupied by perfect tensor Tv.
Given the property of the perfect tensors, the contraction of s where s 6= i and s 6= j
means that most of these tensors Xv; Xp contract to give delta functions. Exactly as in
the previous section, the correlation function would factorize into
h jO1(i)O2(j)j i = trO1(i)trO2(j)tr(T:t)
(4.18)
To obtain a connected correlation function, it is thus clear that we need a string of
nodes replaced by a departure from perfect tensors. To obtain the leading term in the
expansion, it is thus given by a string of nodes that connects O1(i) and O2(j) along a
geodesic, the shortest path through the tensor network that connects the two boundary
sites i and j as illustrated in
gure 7.
In fact the leading contribution would take the form
h jO1(i)O2(j)j i = j jtr
Y TvO1(i)O2(j) Y tyv +
= j jtr O1(i)O2(j) Y
j j
v
j j
p
!
v
Tv
!
(4.19)
where j j is the number of nodes along the geodesic, and
denote the terms where the T 's
and ty at arbitrary node v along the geodesic are interchanged. This is in turn rewritten
in terms of a transfer matrix T . Let us illustrate this with the hexagon code which is
characterized by the [6,4] lattice. This is illustrated in gure 7.
For the orientation of the tensor as indicated in the diagram,
(To1 )aa0bb0 = [(T:ty + t:T y)v]aab0b0 = Tas1s2bs3s4 tya0s1s2b0s3s4 + h.c.:
(4.20)
Here, the T:ty denotes contraction of only 4 of the 6 indices of each of these tensors, and the
sub-script o1 denote the transfer matrix for a speci c orientation of the geodesic. Suppose
T and t are same at every node for a geometry that is homogenous, then the correlation
function would be controlled by the eigenvalues of the transfer matrix T .
In fact, suppose in the 6-leg network constructed in [31] the transfer matrix is
diagonalized to
for some appropriate unitary matrices U , then the 2 point correlation functions are given by
(U 1T U )aa0bb0 = ab aa0 bb0 ;
hOab(i)Oab(j)i = j j jabj 12;
where Oab is assumed to be operators in the appropriate basis that diagonalizes T . These
operators would correspond to conformal primaries, as pointed out in [33]. The conformal
dimension are given by
ab =
log(
ab);
for a holographic code such that j j = log
x, where
x is the distance between two sites
i; j at the boundary connected by the geodesic . That
is small for the approximation
to be valid means that we are naturally working with an operator with large conformal
dimension. This coincides with the AdS intuition that the Green's function of a very
massive particle is approximated by geodesics.
We note that this approach also naturally recovers the counting in [33] in which
correlation functions are found to scale also with 1=Dj j, since
plays the role of 1=D when
one averages over random matrices of large dimensions D.
Before we move on, let us also comment on the calculation here and that of the
entanglement entropy in the previous section. It is well known that in a 1+1 d CFT, the
Renyi entropy can be computed by considering essentially correlation functions of local
twist operators, whose dimension depends on the central charge and Renyi index n by
n =
n
c
6
1
n
:
(4.21)
(4.22)
(4.23)
(4.24)
HJEP08(216)
A computation of the correlation function would require that a non-trivial result appears
at order
, where
is the length of the geodesic, leading to an entanglement entropy
SEE
0
log . The lion's share of the entanglement entropy however starts with order
. This entanglement follows from correlation between highly non-local operators. The
computation of the reduced density matrix in the previous section recovers small corrections
of this large amount of entanglement, starting with log . This puts in tension between
the relationship of the tensor network and the CFT. We note that indeed as observed
in the previous section, such perturbations away from perfection falls short of recovering
the n-dependence of the Renyi entropy of a CFT. It is expected that other xes, such as
the introduction of weights, is necessary to correct this problem. It is also possible that
there is a non-trivial interplay between computing the n ! 1 limit in which the conformal
dimension necessarily becomes small, and the small expansion.
As a constraint of preserving isometry, one would expect that if the geodesic is oriented
in a di erent direction, the spectra of operators remain the same. If such is the case, we
should require that another transfer matrix corresponding to a di erent orientation in the
six leg tensor network
(To2 )aa0bb0 = Ts1 a s2s3 b s4 tys1a0s2s3b0s4 + h.c.
to have the same spectra as To1 .
The symmetry of this tensor naturally allows for one more orientation of the geodesic
(To2 )aa0bb0 = Ts1s2as3s4btys1s2a0s3s4b0 + h.c.
Now geodesics can be orientated in very di erent ways. For an [L,M] lattice, geodesics
are most naturally running across the L-gon in L=2 di erent ways. (see for example gure 8,
in which the three geodesics meeting at a point happen to have three di erent orientations.
) It appears that for an L leg tensor embedded in a [L,M] lattice it is natural to have L=2
di erent orientations of the geodesic that respect lattice symmetry, and that we should
require that they preserve the same spectrum to allow for the same spectra of operators
independently of the relative orientation of the inserted operators.
Three point correlation function and fusion coe cients
We can continue our quest and move on to three point correlation functions. Following the
same logic, we can see that leading connected three point correlation functions are given
by products of perturbations t away from perfect tensors T along three paths that connect
the three points xi; i = f1; 2; 3g where operators are inserted at the boundary, and that
these paths are joined at a point z. This con guration must be such that the number of
t's involved is minimal, and thus the paths connecting z and xi should be a boundary to
bulk geodesic. Moreover, the point z should be chosen such that the sum of these three
geodesics have a minimal length among all these geodesics.
In a discrete graph, it is not immediately obvious where the point z is located. Let us
resort to the knowledge of continuous AdS space, which would be a reasonable
approximation when the lattice representation of the AdS space is su ciently
ne-grained.
It is already observed in [43] that the three point correlation functions of some massive
spin states can be approximated in the bulk by three geodesics joining at a point. The
correct central point that minimizes the length of this path was also found there, using
AdS isometry. Explicitly, it is found that the 3-point correlation function takes the following form { 19 { 1
X
i
(4.25)
(4.26)
hO 1 (x1)O 2 (x2)O 3 (x3)i = exp(W [1; 2; 3]);
W =
i ln z2 + (x
xi)2
; (4.27)
hO 1 (x1)O 2 (x2)O 3 (x3)i
where is the radial z cuto surface, and (x; z) is the position of the bulk point at which
the three boundary-bulk geodesics join.
It is then found that a suitable choice of x and z gives the right lengths of these
geodesics such that
precisely the correct universal form of the three point correlation function of a CFT.
jx12j 1+ 2
3 jx13j 2+ 3
1 jx23j 2+ 3
1
;
(4.28)
(a1,b1)
s
3
geodesics meeting at a point. The tensor at which they meet de nes a fusion matrix, and the legs
of the tensor are labeled according to (4.29).
Putting this result in our context, we have obtained a set of \primary operators"
in the previous subsection, by diagonalizing the transfer matrices.
Therefore working
in that basis, the three boundary to bulk geodesics would automatically generate the
expression (4.27), from which the knowledge of the existence of an appropriate bulk junction
leads to a correct form of the three point function.
This then gives us a handle of how the fusion coe cients between operators are
computed.
At precisely the junction, the transfer matrices along the three boundary-to-bulk
geodesics would be contracted with a tensor at the junction. Taking again the 6-leg perfect
tensor construction, the three point function takes the form
hO1(1)O2(2)O3(3)i = O 1 a1;b1 ((T1)1 )aa011bb011 O 2 a2;b2 ((T2) 2 )aa022bb022 O 3 a3;b3 ((T3) 3 )aa033bb033
Ta01s1a02s2a03s3 tby01s1b02s2b03s3 + ta01s1a02s2a03s3 Tby01s1b02s2b03s3
:
(4.29)
This is illustrated in the following gure 8. One can immediately see that the fusion matrix is given by
F
(a1;b1);(a2;b2);(a3;b3) =
Ta1s1a2s2a3s3 tby1s1b2s2b3s3 + ta1s1a2s2a3s3 Tby1s1b2s2b3s3 :
(4.30)
One important requirement following from isometry, is thus that the fusion coe cients
should have at least the same eigenvalues when any of the four indices of T is contracted
with the corresponding index in t, since depending on the speci c diagram, the junction
can have di erent orientation and it is expected that the fusion coe cient should stay
\unchanged" (possibly up to a change of basis at each boundary site).
4.4
From rotational invariance of two point function which also follows from the homogeneity
and isotropy of the tensor network, it is natural to require that
and so on.
naively, we should set
A generic complex perturbation t have DL number of components for an L leg tensor at
each node. The condition (4.31) gives 2(L=2
1)
D4 constraint equations. Now from the
three point function analysis of the previous section, we come to know there will be CL
3
di erent orientations of Fi: e.g. for the six leg tensor, Fi are constructed by picking any
three of the indices out of the 6, and contract them between T and ty:
If we demand rotational invariance coming from homogeneity and isotropy, most
The number of equations is equal to 2(C3L
1)D6
It is evident that with L < 6, it becomes di cult to satisfy all the constraints barring
isolated solutions that may or may not appear. We tested these conditions with the 4-leg
tensor and the 6-leg tensor detailed in [31], and found that in the case of the 4-leg tensor,
it is possible to solve the 2-pt function constraints together with 2 of the 3 constraints
following from the 3-pt correlation. In the case of the hexagon code we could solve all
the 2-pt constraints together with 9 of the 19 constraints in the 3 point function, using
only real matrices t, which appears already to be doing far better than the naive counting
would suggest.
Another possibility is that the constraints that we are imposing is too stringent. After
all, the basis operators acting on di erent sites need not coincide precisely, and that perhaps
it is reasonable to require only that the eigenvalues of the transfer matrices match.
Toi = Uij Toj Vij 1;
(4.34)
where U; V also need not be identical, since our transfer matrices are not Hermitian upon
complex conjugating, which exchanges the pair (a; b) and (a0; b0). Rather, it is \Hermitian"
upon exchange of (a; a0) $ (b; b0). Matching only eigenvalues (or strictly speaking singular
values for asymmetric matrices) leads to (L=2
1)D2 equations. Once these Uij ; Vij are
determined, they have to be recycled in the computation of the 3-point function. This
reduction of constraints therefore is not signi cant enough to avoid anisotropy of 3-pt
fusion. It is perhaps more natural that a generic tensor network with a proper gravitational
interpretation should admit some fairly large number of legs at each node.
Toi = Toj :
Fi = Fj :
(4.31)
(4.32)
(4.33)
Having a concrete understanding of the subset of isometries of the AdS space preserved
in a given lattice using the Coxeter group, we are now in a position to make better use
of these symmetries and study the BTZ black hole, which is obtained by orbifolding the
AdS3 space.
In the previous sections we have demonstrated that using the tensor network we can
compute 2 pt and 3pt correlation functions which give us the necessary information about
conformal dimensions and fusion matrices. It is also shown that they have the correct
scaling properties as expected from CFT. So at least these tensor networks reproduce
some basic features of AdS/CFT although we are still far from giving it a proper gravity
description. But nonetheless all these observation suggest that one can implement these
tensor networks to understand dynamics of the gravity and possibly how Einstein equation
emerges. In this section we will take another important step towards understanding these
concept by constructing a BTZ black hole with this network purely using the symmetry
considerations. We will see that our construction gives all the information which has been
known from these tensor networks in the previous studies [31] but our construction is more
physical and it backs up our idea of constructing tensor network using the Coxeter group
constructions.
We rst review the construction of BTZ black hole by orbifolidng Poincare AdS
following [44, 45]. Below we quote the Euclidean BTZ metric,
ds2 =
r2 ) dr2 + d +
r+(ir )
r2
d
:
(5.1)
r+ and r denote the location of the two horizons.
is the Euclidean time. We know that
Riemann curvature tensor takes the same form for the AdS3 and BTZ black hole. We now
de ne the following coordinate transformations,
w =
2
r
r2
r
r
+2 1=2
2
exp
r+ + r
l
( + i ) ; z =
r
2
r2
r
r
+
2
2 1=2
exp
r+
l
+
i r
l
Using this we get,
ds2 =
l
2
z2 (dwdw + dz2):
w
e
w; z
e
and z > 0: So this restrict the space time in the upper half plane and can be thought of
quotienting the three hyperbola. Now the periodicity condition on
+ 2 implies,
:
(5.2)
(5.3)
(5.4)
where in the case at hand we have restricted our attention to non-rotating black holes,
and thus real temperature . The orbifolding transformation thus correspond to an overall
re-scaling of ! and z. Now for our network construction we generally consider constant
slice of H3 and so we will have w = w, and that there is no angular momentum, so
that the temperature
is real, and r
= 0. We obtain an orbifold of the AdS3 geometry
by identifying points in Poincare AdS related by the symmetry transformation to obtain
the BTZ geometry. This is the crucial ingredient in constructing the corresponding tensor
network.
As demonstrated earlier we know that using the generators of Coxeter group which are
basically some re ections about a chosen plane one can construct all the Lorentz generators.
So combination of a nite number of re ections can generate any Lorentz transformations.
The mirrors appropriate for the speci c re ection can be found by constructing planes
in the at embedding space and then obtain their intersection with the hyperbolic space.
Intersections of planes (that pass through the origin) with an H2 centered also at the
origin are geodesics. Boosts are therefore generated by re ections across geodesics in H2.
Orbifolding an H2 tessellation respecting these re ection symmetries determine the BTZ
geometry on a constant time slice. To demonstrate this we write the equation of the H2
space in embedding coordinate [46],
R is the radius of the hyperbola. To go to the Poincare coordinate we will use,
As described already in the introduction of the Coxeter group, every isometry of the
hyperbolic space can be recovered by a re ection across planes also embedded in at space.
Therefore, the orbifolding transformation of a given point in the hyperbolic space, which
corresponds to a boost in embedding space, can be recovered by specifying a plane across
which the point re ects. Under re ection across a plane de ned by X~ :~n = 0, with ~n
specifying the normal vector of the plane, a point transforms as
=
X~~ = X~
2 X~ :~n ~n:
boost in the X0
X2 plane.
Writing, one immediately see that
In the following, we will restrict ourselves to the t = 0 slice, so that X3 = 0, and that the
slice, an AdS2 space, is invariant under the orbifolding transformation (5.4). Meanwhile,
X0 and X2 transforms and the other two remains unchanged. From the perspectives of
these at space embedding coordinates, the orbifold transformation is in fact a Lorentz
X~0 = X0 cosh( ) + X2 sinh( ) ;
X~2 = X2 cosh( ) + X0 sinh( ):
X0 =
X2 =
z
2
z
2
1 +
1
R2 + ww
(R2
ww)
z2
z2
;
;
X1 =
X3 =
R
z
R
2 i z
(w + w) ;
(w
w):
(5.6)
(5.5)
(5.7)
(5.8)
(5.9)
Comparing (5.7) and (5.9), we get a consistency condition on the normal vector ~n =
(n0; n1; n2; n3)
Now without loss of generality, we would rst start with a reference mirror, speci ed by
n0 = 0 and thus n2 = 1. Points acquiring a boost (5.9) under re ection across this plane
lie on a plane satisfying
Starting with this plane and reference mirror, we would like to obtain subsequent mirrors
so that the images acquire further boosts again given by (5.7) . It is not hard to see that
the next mirror which would produce the correct boost on the image produced by the
reference mirror is related to the reference mirror by precisely the same boost (5.7). Since
n0 = 0 for the reference mirror, it is basically located at X2 = 0. Under the boost (5.7) it
means that the next mirror satis es
n
2
0
n22 =
X2 =
X0
tanh :
1
2
X0
X2 = tanh :
In fact all subsequent set of mirrors that generate boosts by re ection on the previous
images are all generated by the reference mirror by sequentially boosting it by (5.7). We
can plot these planes on the Poincare disk. This is done by solving for the intersection of
these planes with the hyperbolic space:
X~ :n~i = 0 ;
where ~ni is the normal of one of these in nite sequence of planes. These intersections
are geodesics on the hyperbolic space. One can plot them on the Poincare unit disk
(Y1; Y2); Y12 + Y22
1 via the map
Y1 =
X2
1 + X0
;
Y2 =
X1
1 + X0
:
The reference mirror we have chosen is thus a vertical line at Y1 = 0 at the center of the
Poincare disk. Subsequent curves can be obtained readily. For example, the second mirror
on the hyperbolic plane corresponding a curve satisfying both (5.5) and (5.12) is given by
the parametric equation:
Y1 =
X0 tanh
1 + X0
;
Y2 =
q
X02
cosh2
cosh (X0 + 1)
:
The collection of these mirrors in the Poincare disk looks like a set of \parallel curves" that
do not meet. To obtain an orbifolded tensor network, it now corresponds to rst picking
a tessellation that contains the set of mirrors generating the orbifolding boosts, and then
identify all the nodes related to each other by re ection by any of these mirrors. This
would lead to periodic contractions of legs. The tensor network would acquire the topology
of a cylinder. We take the [6,4] lattice as an example and illustrate it in
gure 9. In the
next section we will compute explicitly the entanglement entropy that follows from this
tensor network.
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)
(5.15)
T
3
T
4
T
2
T
1
T5
r=r
h
T6
The black curves are the mirrors generating the desired boosts. The fundamental region we have
chosen is indicated by the red arrow. The dashed line is the apparent horizon, across which degrees
of freedom are entangled. The nodes corresponding to di erent tensors are marked. The tensor
network that follows is illustrated on the right panel. Also we have marked the two RT surfaces
corresponding to tracing out node 7 and node 2 and 3 in red.
Entanglement entropy and RT surface in the BTZ background
In this section we will compute the entanglement entropy for the particular tensor network
and we will demonstrate that the result matches with BTZ result perfectly. To do so, we
rst brie y review the AdS3 calculation of the entanglement entropy for BTZ using the
standard RT formula which can be found in [2] details. We will just point out some key
points which we will use in this section. As for the BTZ as the starting point is not a pure
state so entropy for the region A and its complement Ac are not equal.
The main conclusion is that we will have generally two di erent geodesics - 1.) One
geodesic corresponding to A do not cross the region and 2.)
Other one corresponding
to Ac which wraps the black hole one time. Next we will determine these two geodesics
from the tensor network. This is illustrated in the gure 9. Now we write down a speci c
wavefunction for this fundamental domain below as illustrated in gure 9.
j
> =
Ta7bc1c2c3c4 Te2ac5c6c7c8 Te31bc9c10c11c12 Te1t1e1t1f1g1
Tf41xc13c14c15c16 Tg61x1c17c18c19c20 Tx5x1c21c22c23c24 jc1c2
c23c24 > :
(5.16)
T 1 lies at the horizon. T 4 , T 5 and T 6 are in the region which is on the other side of
the horizon as explained in the picture and always traced. Now to compute the reduce
matrix for computing entanglement entropy when we trace out the tensor at 2 and 3 and
we immediately see that the RT surface doesn't cross the horizon. The contribution only
comes form cutting the link between 2, 3 and 7. Also when we trace out the tensor at 7
and we will see that the geodesic wraps the horizon once. The RT surfaces are shown in
the gure 9.
5.4
In [31] and [33], it is observed that a region of high bond dimensions behaves in very similar
ways to a black hole, in which the entangling surface for example, would naturally avoid
the boundary of this region of high bond dimensions, mimicking the behaviour of the RT
surface near a black hole horizon. In the current construction, the RT surface clearly never
cuts through the horizon separating the two halves of the lattice. It is a curiosity how this
construction of the BTZ black hole is related to the intuition of the high bond dimensions.
On closer inspection, we reckon there is an e ective region of high bond dimensions
realized by the arrangement of the bulk lattice. The key lies in the relation (5.2). The
exponential maps means what appears to be equally spaced in
is highly uneven in !
coordinates, on which the tensor network is based. The bottom line is that comparing the
e ective length of the rst half of the cycle as
to the full length of the cycle
+ 2 , one sees that
(! =2
(! =
! =0)
! =0)
1;
x =
:
(5.17)
This means that traversing half of the
cycle still corresponds to an uncannily tiny region
in the ! coordinates, particularly so in the limit of high temperature. The relative number
of physical degree of freedom within the region residing in the rst half of the space would
in fact constitutes a small fraction of the total number of degree of freedom available. There
is thus an e ective region of high bond dimensions outside of the region, which it has no
way of entangling with and the RT surface would thus look very far away from the horizon,
which mimics an avoidance of the horizon. We interpret this as a geometrical realization
of a region of high bond dimensions.
5.5
A comparison with the random tensor construction
One can readily obtain a general picture of generic wave-functions constructed on these
networks with large bond dimension. As discussed in [33], we can obtain a generic picture
when bond dimensions are high, in which case the typical behaviour can be deduced by
averaging the tensors weighted by Haar measure.
Let us consider in particular the case where we are computing the Renyi entropy at
n = 2. It is discussed in detail in [33] that the computation of the Renyi entropy can be
mapped to that of computing the partition function of an Ising model.
In particular, since upon average of the random tensor located at each node gives
jVxihVxj jVxihVxj =
Ix + Fx ;
Dx2 + Dx
where Ix corresponds to the identity , where Fx corresponds to swapping the two copies of
the tensors at Vx, and Dx the dimension of the Hilbert space at vertex x, which is equal
to the product of dimensions of all the bonds that meet at x.
The Renyi entropy of some boundary region A is then given by
S2
log
Z1
Z0
Z1 =
X exp( A[fsxg]);
sx
(5.18)
(5.19)
a large region, such that the domain wall covers the \horizon" as well as its complement. The right
panel is the corresponding con guration of the Renyi entropy of a small region, with only a single
domain wall.
with
X 1
xy 2
X 1
2
A =
log Dxy(sxsy
1)
1):
(5.20)
Here sx 2 f1; 1g are the e ective \spin" degree of freedom which denotes whether the
bond at vertex x is connected to the next replica, or with itself. hx dictates whether a
bond dangling at the boundary should be connected to the same replica for spins outside
of region A (in which hx = 1) or the other replica for boundary spins inside region A, in
which hx =
1. (Note that in our case we have chosen a wave-function that is a pure state
in the bulk and thus there is no further contribution from a bulk density matrix as in [33]).
We are tracing out everything across the horizon by default to simulate a thermal
mixed state. In this case therefore, the saddle point contribution in the large Dx limit is
such that there are domain walls separating regions with opposite spins that are adjacent
to region A at the boundary and its complement, whose e ective boundary spin degree of
freedom are subjected to opposite directions of the e ective magnetic eld hx.
In this picture, it is then clear that for su ciently small region A, the domain wall
again encloses the small boundary region A. This is illustrated in gure 10. This should be
contrasted with the results following from perfect tensors, where we recover also two RT
surfaces, one that wraps the horizon and another that does not, depending on the size of
the region A.
5.6
Computing the thermal spectrum
In this section we investigate the thermal spectrum. We will use the wavefunction as
mentioned in (5.16). It can be easily seen the tensor T 1 that connects the nodes between
the two sides of the horizon plays the most important role. For simplicity we consider a
wavefunction only made up of T 1 as follows,
>= T 1
j
> :
(5.21)
We now construct the reduce density matrix using this wavefunction and trace out the
region inside the horizon. Then we compute the eigenvalues of these reduce density matrix
( thermal), that in turn gives the thermal spectrum.
thermal = Tr
0 0 T 1
>< 0 0 0 0jT 10 0 0 0 0 0 :
(5.22)
For six index perfect tensor, all the eigenvalues of this density matrix are equal, in fact
they are maximally entangled. So the thermal spectra we get is at like the pure AdS case.
Next we consider a bit more non trivial example where there are two nodes on the horizon.
The wavefunction is,
>= Ta1cdbef Tbc1d1ae1f1 jcdef c1d1e1f1 > :
The corresponding reduced density matrix is,
thermal = Trefe1f1e2f2e3f3 Ta1cdbef Tbc1d1ae1f1 jcdef c1d1e1f1 >
(5.23)
< c2d2e2f2c3d3e3f3jTa1c2d2be2f2 Tbc3d3ae3f3 :
(5.24)
Now the tensor contractions in (5.23) is not unique. We observe that there are two cases.
For some speci c contractions we again get maximal entanglement between the four bits.
But there exist a particular con guration for which we get maximal entanglement but
only two bits contribute. We then checked for higher number of nodes on the horizon and
we reach the same conclusions. There exists always some con gurations when all the bits
contribute and they became maximally entangled and there exist some con gurations when
not all the bits contribute but yet the contributing bits are maximally entangled with each
other. Also as we increase the number of such nodes at the horizon it becomes more and
more di cult to nd these con gurations. Almost all the cases all the bits participate and
they are maximally entangled.
5.6.1
An extra correction in the Renyi entropy
The calculation can be inspected in closer detail at arbitrary Renyi index n, which is given
by [33]
Sn(A) =
1
1
log 1
Z(n)
0
Z(n) ' 1
1
n
Z(n)
log 1 ;
Z(n)
0
where the expression is again applicable only in the large D limit. When the density matrix
corresponding to n-replica is averaged over the Haar measure at each vertex, it gives
with gx the element of the permutation group of order n, and Cn;x the normalization
constant. Then
jVxihVxj n =
1
Cn;x gx2Symn
X
gx
Z1n = tr[ P CA
n n Y
jVxihVxj n]
n n O gx
=
=
X
fgxg
fgxg
1
Cn;x
x
tr
X e A(n)[fgxg]
(5.25)
(5.26)
(5.27)
and
A
(n)[fgxg] =
X logDxy( (gx 1gy)
n)
X logDx@ (gx 1hx) + X logCn;x
(5.28)
hxyi
where (gx 1gy) is the number of permutation cycles of the element gx 1gy, and
hx =
( n
Cx
Ix
x 2 A
x 2 A
where Cxn is the cyclic permutation of all the replicas at vertex x. The P is the density
matrix of bulk states. In the case that has a direct analogue with the tensor network, it
is simply given by the expression, P =
introduced in the \bulk" corresponding to a maximally entangled state living on the bond
that connects the vertices x and y. It is maximally entangled to recover simple tensor
contraction across a link. In the form of (5.27) and (5.28), [33] suggested that calculating
the Renyi entropy is the same as calculating the partition function of the spin con guration
on the same lattice where spin state was one of the elements of the permutation group, hx
the boundary pinning eld, and bond dimension logD the inverse of the temperature. So
in the large D limit, the leading contribution of the Renyi entropy should come from the
ground state con guration of the corresponding spin model with boundary condition fhxg.
The energy of two adjacent spin was
Qhxyi jxyihxyj where jxyi is an auxiliary state
logDxy( (gx 1gy)
n)
So it is a ferromagnetic model, and then it was concluded in [33] that the ground state
is given by a con guration in which all the spins are parallel to each other, except at the
geodesic, where a domain wall is formed between the C
leading term in Renyi entropy is
n area and I area. The resulting
Sn(A) ' 1
1
j AjlogD (Cn) n =
1
1
(1
n)j AjlogD
(5.29)
= j AjlogD
which is the RT formula. However, there is some subtleties in the calculation, which could
contribute to some extra non-vanishing terms even in the large D limit.
Consider the following: that there are not one but two domain walls, that are very
close to each other, both lying very close to the geodesic. The length of each domain
wall would almost be j Aj, and in the intermediate area, the \spins" are chosen to take
value g, such that g = ((12)(3) : : : (n)), so that (Ig 1) = n
1 and (Cng 1) = 2. Such
con gurations are illustrated in gure 11.
The contribution to Renyi entropy is S0 = 1 1
n
n) + ( 1))j AjlogD = j AjlogD
which is exactly the same as the term in (5.29). Meanwhile, viewing the cyclic nature of
the element C and I, there are many other possible g that leads to exactly the same action:
g = ((1)(23)(4) : : : (n)); ((1)(2)(34) : : : (n)) : : : :
(5.30)
g
Cn
known contribution with only one domain wall, if the lattice is su ciently ne-grained.
Summing all these saddles with equal weight would contribute to a l1ognn to the Renyi
entropy. It is suppressed compared to the leading term only by a factor of log[D].
It appears that there are more saddles that could contribute, and the Renyi entropy
has more structure less suppressed perhaps than previously thought.
6
In this paper, we have attempted a remedy of some obvious issues that plague the
perfecttensor-network proposal that attempts to re-enact the AdS/CFT correspondence. These
obvious issues include a necessarily
at entanglement spectrum, and the absence of
connected correlation functions between local operators. We make a naive proposal that the
tensors involved in the tensor network are perturbed away from exact perfection.
We
demonstrate that such perturbations naturally lead to emergence of some analogues of
Witten diagrams, when we consider 2 pt and 3 pt correlation functions of local operators.
We note that these perturbations would also make contributions to the Renyi entropies.
It appears that such perturbations naturally lead to non-analytic contribution behaving
like log , if the perturbations away from perfection is orthogonal to the subspace of the
boundary Hilbert space the background perfect tensors projects to. To quadratic order
they would give rise to non-trivial Renyi entropy, although they are insu cient to
x the
issue with the at spectrum. We explored some other x of the entanglement spectrum,
which appears to work even though they might be rather contrived.
We have also constructed the BTZ black hole under the tensor network framework,
by orbifolding the discrete lattice characterized by some Coxeter group, analogous to how
the BTZ black hole is constructed in the continuous limit. We recover a tensor network
that exhibits some salient features of the BTZ black hole, including a horizon which no
entangling surface penetrates, and that the entanglement entropy of a region A is di erent
from that of its complement, given by two entangling surfaces one wrapping the horizon
while the other does not. The black hole entropy in this case is precisely an entanglement
entropy between degrees of freedom across the black hole horizon.
The tensor network proposal has led to quite a few surprises and excitement. The
basic feature underlying all of that is the emergence of the notion of a local bulk, which
manifests itself as a product of delta functions occurring at each edge whenever we trace
out some parts of the boundary of the tensor. It is, as in any other developments, perhaps
raising more questions than answers.
One important issue is to x the entanglement spectrum. More importantly, it is necessary to begin discussing time evolution of these networks, in order to actually make contact with other extremely important aspects of the
AdS/CFT correspondence. Some systematics such as the handling of isometries is now at
hand, using the Coxeter group, and more generally, via these discussions of the PSL(2; Qp)
groups [47, 48]. It appears that time-evolution based on a triangulation of the AdSd+1
space and then introducing tensors where some indices are now taken as unitary maps
from one time slice to the next is a natural generalization of the current construction of
tensor networks on a given spatial slice. We comment that it is observed in [49] that perfect
tensors are naturally describing quantum chaotic evolution. Given that the locality of the
network and covariance of space-time begs for the use of perfect tensors in the complete
space-time network, a coherent picture of gravity, and perhaps black holes in particular
being fast scramblers might emerge for free via the tensor network construction. Such
construction is currently underway, and we will report them elsewhere [50].
Acknowledgments
We thank Zheng-Cheng Gu for telling us about the Coxeter group. We also thank Wei
Song for very helpful discussions, and for probing us about the BTZ black hole in the
tensor network construction which motivated part of this work. AB and LYH would like
to acknowledge support by the Thousand Young Talents Program, and Fudan University.
ZSG and SNL would like to acknowledge support by Fudan University. AB and LYH would
like to thank Sudipta Sarkar and Bin Chen for discussing various relevant issues. Authors
would also like to thank Chris Lau for his tremendous help with the gures.
A
Curvature and bulk locality
In this paper, the main focus is the absence of connected correlation function between local
operators. It is found that departure from perfection in isolated nodes in the tensor network
lead only to a change of the overall normalization of operators, but do not contribute to
connected correlation functions.
That this works actually depend heavily also on the
negative curvature of the tessellation. For example, the discussion in [33] is based on a
4-leg tensor forming a square lattice. The lattice arranges itself into self-repeating units
which can be described by the tessellation [4,5]. The self-repeating units are 4-gons, as
indicated in gure 12.3
A crucial point here is that while the network is built up from 4-leg tensors, the actual
unit that de nes the geodesics and RT surfaces would be determined by these blocks of
tensors, rather than individual 4-leg tensors. To visualize that, let us focus on one block,
depicted in gure 13.
3We adopted the gure from [33].
4-leg tensors cannot be treated as a single unit. The leading contribution in connected correlation
functions in this case correspond to a collection of these blocks each with one node replaced by t
forming a connected set of blocks.
Within this block, we could view the left and bottom set of dangling legs as input
legs, and the top and right set of dangling legs as output legs in a unitary map de ned by
the block of 4-leg tensors. Being on a at geometry means that on average the number of
ingoing legs connecting from a layer closer to the boundary dangling legs to out-going legs
connecting to an inner layer roughly balances. Suppose we now introduce a hole somewhere
inside this block. It is immediately clear from our ow diagram that it is impossible to
construct a conserved
ow from the input legs that ows out to the output legs and the
hole, without either introducing loops in the ow, or having unbalanced in/out legs in a
given node. This means that the unitary map of the block is destroyed by introducing only
one hole in this block that could contain many 4-leg unit tensor. This follows from the fact
that the square lattice has the structure such that a contraction of the outer layers have
less than half of the total number of legs, ingoing and outgoing included, unlike the case of
negatively curved lattices. As a result, if we were computing correlation functions in a at
lattice, our previous argument would break down: namely, that a single node perturbed
away from perfection would immediately contribute to connected correlation functions. To
make connection with our discussion in the tessellation of a negatively curved space, it is
thus necessary to view the block of tensors as a single unit, in which case the averaged
number of incoming legs from the boundary still vastly exceeds the out-going legs toward
the bulk, and our de nition of the transfer matrices and fusion matrices would work equally
well then.
B
Inspecting a remedy to the
at entanglement spectrum |
Weighting the legs
In the main text, we have seen that the entanglement spectrum is completely at, which
follows from the nature of the perfect tensors. We have also seen that a small deviation
from perfection alone does not lead to a satisfactory spectrum.
middle of the network would destroy the unitary map from the incoming links surrounded by the
blue curve to the out-going links surrounded by a blue curve. The red dots denote nodes where the
number of incoming arrows and outgoing arrows are not balanced.
In [33] it is suggested that perhaps one could introduce non-maximally entangled
auxiliary states along the edges connecting vertices. In our tensor network construction, this
is equivalent to introducing weights when tensors are contracted. i.e.
Ta1 Ka1a01 Ta01 ;
(B.1)
for some matrix K satisfying tr(K:Ky) = 1. In [33], it is observed that the averaged tensors
with large bond dimension is not very sensitive to such weights. Things are quite di erent
in a given tensor network with speci c perfect tensors sitting at each node. While T:T y = ,
T:K:Ky:T y generically do not recover delta functions. As a result, the RT surface would
fail to emerge in the presence of these weights, unless they are also randomly chosen.
However, given the negative curvature of the tensor network, one could ask whether it
is still possible to change the entanglement spectrum while preserving the RT surface. One
observation is that if the weights are not too densely populated | weights K are sparsely
populating the edges, then it is possible that their presence do not destroy the RT surface.
The question is: what is the maximum density of weighted edges such that the RT surface
remains intact?
We note immediately that in the 4-leg code, locally the geometry is at. For the same
reasons as discussed in section A, the addition of weights quickly delocalizes the RT surface
(i.e. Schmidt-decomposition) into blocks of the self-repeating units.
It's more instructive to work with the 2L-leg tensor in which the geometry is manifestly
negatively curved.
A weight on an internal leg prevents the indices of perfect tensor and its adjoint from
being contracted directly. Therefore, a weight works as a \barrier" obstructing the ow
of contraction that propagate from tensor to tensor into the interior of the tensor network
as some parts of the physical degrees of freedom is traced out. The e ect of a weight
disappears if both nodes connected by this weighted link have at least half of their legs
already contracted, so that the weighted link would itself be contracted with a delta function
and sum to 1. Making use of this constraint, we can draw some conclusions as follows.
B.1
The maximal number of weights when pushing toward the inner most
layer
In this section, we study the maximal number of weights without spoiling the propagation
of delta function in the next layer k as one contracts some legs connecting to the k + 1 layer
such that the process eventually hits the center of the network, i.e. layer 1. Suppose there
are two tensors, tensor A at Layer k and tensor B at k + 1, and that they are connected
via an internal leg AB. Consider two cases.
Case 1: AB is an internal leg with weight. As explained above, we need delta-functions
to remove the e ect of the weight. That is to say, among the other legs of A and B, there
should be at least L (out of the total number of tensor legs 2L) that have no weights.
Case 2: AB is an internal leg without weight. We need L legs, not including AB,
without weight to give a delta-function on leg AB. As for tensor A, since we have already
had an index contracted, we only need another L
1 legs without weight, in order to propagate the delta functions from layer k to layer k 1. It is thus necessary that each tensor has less than L weighted legs.
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