#  Exact holographic tensor networks for the Motzkin spin chain

Rafael N. Alexander1, Glen Evenbly2, Israel Klich3 1 Center for Quantum
Information and Control, University of New Mexico, Albuquerque, New Mexico
87131-0001, USA
2 Département de Physique and Institut Quantique, Université de Sherbrooke,
Québec, Canada
3 Department of Physics, University of Virginia, Charlottesville, Virginia
22903, USA

###### Abstract

The study of low-dimensional quantum systems has proven to be a particularly
fertile field for discovering novel types of quantum matter. When studied
numerically, low-energy states of low-dimensional quantum systems are often
approximated via a tensor-network description. The tensor network’s utility in
studying short range correlated states in 1D have been thoroughly
investigated, with numerous examples where the treatment is essentially exact.
Yet, despite the large number of works investigating these networks and their
relations to physical models, examples of exact correspondence between the
ground state of a quantum critical system and an appropriate scale-invariant
tensor network have eluded us so far. Here we show that the features of the
quantum-critical Motzkin model can be faithfully captured by an analytic
tensor network that exactly represents the ground state of the physical
Hamiltonian. In particular, our network offers a two-dimensional
representation of this state by a correspondence between walks and a type of
tiling of a square lattice. We discuss connections to renormalization and
holography.

One of the hallmarks of critical behavior is the divergence of correlations,
and the emergence of scale invariance; the low-energy behavior of the system
seems to be of a similar nature on small and large scales. Beyond the physical
beauty of such states, their treatment has helped develop many important ideas
and tools, such as, conformal field theory (CFT) and the renormalization group
(RG) wilson1983renormalization ; fisher1998renormalization .

At temperatures approaching absolute zero, criticality is of a quantum nature
and is a focal point of interest for low-temperature many-body physics. At
such quantum critical points, long-range quantum fluctuations and correlations
allow the system to sustain emergent large-scale quantum behavior (see e.g.
Ref. sachdev2011quantum ). Such phenomena have been observed in many
experiments, including in magnetic systems where quantum critical behavior may
develop when a magnetic transition is driven by changes in chemical doping
lohneysen1994non , pressure julian1996normal , external fields
grigera2001magnetic and other system parameters.

In recent years, a lot of attention has been given to the detailed description
of scale-invariant ground states of quantum systems. This is a particularly
challenging problem since such systems are typically highly entangled, and
harder to study than gapped generic ground states with short-range
correlations.

A recent promising approach to simulating many-body states is via so-called
tensor networks. Tensor network notation Orus2014 ; bridgeman2017hand offers a
convenient graphical representation of the entanglement structure of many-body
quantum states. A particularly simple class of 1D tensor networks,
particularly useful for describing spin-chains with finite range correlations,
are known as matrix product states (MPS). These were introduced in Ref.
fannes1992finitely , and are the variational class of states used in White’s
density matrix renormalization group (DMRG) numerical procedure White1992 ,
arguably the most successful tool for numerical investigation of quantum
phases in 1D. Another class of tensor network states, known as the multi-scale
entanglement renormalization ansatz (MERA) Vidal2008MERA , was proposed by
Vidal Vidal2007ER to be specially tailored for describing quantum critical
points.

While there are a number of examples of physical spin systems for which an
exact matrix product state representation is known for the ground state—most
notably the celebrated AKLT state of Affleck, Lieb, Kennedy and Tasaki
affleck1987rigorous —the situation for critical systems is quite different. To
our knowledge, until now, there has been no exact example of a scale-invariant
tensor network that represents the ground state of a simple local Hamiltonian.
Such networks are typically computed numerically. The only prior case of an
analytical MERA we are aware of describes a sequence of approximate
descriptions for free fermions, shown in Ref. Evenbly2016 ;
haegeman2018rigorous .

Here we provide the first exact analytic tensor network for describing a
critical state: the ground state of the Motzkin spin chain. Our results
suggest that scale-invariant tensor networks may be useful beyond dealing with
CFTs. Indeed, it is well known that the Motzkin Hamiltonian does not have the
energy level scaling associated with a CFT Movassagh2016Power .

The Motzkin model, as well as the closely related Fredkin model, grew out of
the study of ”frustration-free” Hamiltonians. These are an important class of
Hamiltonians, where the ground state is also a simultaneous ground state for
the local interactions comprising the Hamiltonian. Frustration-free
Hamiltonians have been recently used for constructing novel quantum states of
matter and also as representations for a variety of quantum optimization
problems. Examples of frustration-free Hamiltonians include Kitaev’s toric
code kitaev2003fault , the AKLT Hamiltonian, and the Rokhsar-Kiveson model for
a quantum dimer gas rokhsar1988superconductivity . Moreover, any MPS state is
the ground state of a frustration-free Hamiltonian fannes1992finitely ;
perez2007peps ; schuch2010peps ; schuch2011classifying .

The Motzkin Hamiltonian, a spin-1 Hamiltonian introduced by Bravyi et al. in
Ref. bravyi2012criticality , represents an important new class of Hamiltonians
that are both frustration-free and critical. The model admits a
straightforward geometric interpretation in terms of random walks called
”Motzkin walks” and its ground state is exactly solvable. Moreover it is the
starting point for several generalizations that uncovered rich new
possibilities for how the entanglement in a ground state can scale. In
particular, Movassagh and Shor Movassagh2016Power showed how a higher spin
(”colored”) version of the model can feature much enhanced entanglement (going
from typical logarithmic behavior in critical spin chains to a power law).
Motivated by this model, Zhang, Ahmadain and Klich Zhang2017novel found a
parametric deformation of the model that yielded a continuous family of
frustration-free Hamiltonians featuring a new quantum phase transition
interpolating between an area-law phase and a ”rainbow” phase with volume
scaling of half-chain entanglement entropy. A spin 1/2 version of the model
has been proposed based on so-called Fredkin gates by Salberger and Korepin
salberger2017entangled ; dell2016violation , and its deformation was presented
in Ref. salberger2017deformed ; zhang2017entropy ; udagawa2017finite . An
interesting recent variation of this class of models can be found in, e.g.
sugino2018area using symmetric inverse semigroups.

Here we present two types of tensor network that faithfully capture both the
geometric properties and the entanglement structure of the colorless models:
The binary height network and the height renormalization network. As described
below, each path is mapped onto a tiling of a grid that defines the tensor
network. Thus, our networks have the rules that govern the random walker baked
into their building blocks. In particular, the second network we propose has a
MERA-like structure Bal2016 and defines a natural renormalization process of
Motzkin walk configurations. Moreover, this network is capable of tiling the
hyperbolic plane, potentially opening the door to connections with holography
swingle2012entanglement .

## Motzkin spin chains

As stated above, the spin-1 Motzkin model was introduced as an example of a
critical spin chain described by a frustration-free Hamiltonian with nearest-
neighbor interactions. These have been generalized to include a deformation
parameter $t$, where $t=1$ corresponds to a critical point.

The Motzkin Hamiltonian has a unique zero-energy frustration-free ground state
for any value of $t>0$. For our present purpose, we need the exact form of the
ground state as we describe below. (The interested reader may refer to the
detailed description of the Hamiltonian in the supplemental material and in
the references). This ground state is a superposition of walks called “Motzkin
Walks”. A Motzkin walk $w$ is a walk on the $\mathbb{Z}^{2}$ lattice using the
line segments $\\{\diagup,\text{{---}},\diagdown\\}$ that start at $(0,0)$, go
to $(2n,0)$, and never go below the $x$ axis. Each walk represents a spin
configuration via identifying the Motzkin line segments with the local spin
$\\{\ket{1},\ket{0},\ket{-1}\\}$ states, respectively. The ground state can be
written as Zhang2017novel :

$|\text{GS}\rangle=\frac{1}{\mathcal{N}}\sum_{\begin{subarray}{c}w\in\\{\text{Motzkin}\
\text{walks}\\}\end{subarray}}t^{\mathcal{A}(w)}|w\rangle.$ (1)

Here $\mathcal{A}(w)$ denotes the area below the Motzkin walk $w$, and
$\mathcal{N}$ is a normalization factor. A similar type of ground state occurs
in the Fredkin model, which is a half-integer spin model with essentially the
same structure; it only lacks the “flat” move.

The half-chain entanglement entropy—which is a measure of the degree of
quantum correlations in the system—is maximal for the $t=1$ case, where it
grows logarithmically in $n$. For $t<1$ and $t>1$, the ground state satisfies
an area law zhang2017entropy : the entanglement entropy is bounded by a
constant independent of system size. The deformed Motzkin and Fredkin walks
can naturally be viewed as constrained trajectories of a random walker in the
presence of drift; the “x” axis plays the role of time; and the $t$ parameter
measures the strength and direction of the drift. More details can be found in
references salberger2017entangled ; salberger2017deformed ; Zhang2017novel ;
zhang2017entropy . In this article, we will mainly be interested in tensor
network representations of the $t=1$ case. However, below we will show how the
first of our tensor networks can be generalized to the case of arbitrary
$t>0$.

The Motzkin model can also be studied with periodic boundary conditions
Movassagh2016Power . The ground space of the periodic model has a $4n+1$
degeneracy, and the ground states $\ket{\Psi_{k}}$ consist of equal
superpositions over all spin-$z$ configurations that have a total
magnetization $k$, where $-2n\leq k\leq 2n$ Movassagh2016Power . Recently,
such states were shown to act as approximate quantum error correcting codes
Brandao2018 .

## Binary height tensor network

The Motzkin model can be described as a ”height model”: the height of each
walk is encoded within a field value at each point. This is a natural starting
point for constructing a field theory description of the colorless Motzkin
state, as was done in Ref. Chen2017MotzkinLifshitz .

If we describe the state using a binary representation of the heights, we see
that—since at site $x$, the walk could have reached at most height $x$— we
need at most $\log_{2}(2x)$ bits to encode the height. The height encoded at
site $x+1$ results from the adding (or subtracting) the spin value at site
$x+1$. Thus, our first step is to generate a tensor network that implements
these additions. We find it convenient to encode both the walk and the binary
addition using a set of tiles, as explained below.

### Walks as tiles

Consider the square tiles represented below:

(2)

Top and bottom edges are labeled by $1$, $-1$, or $0$ if that edge touches a
“$\uparrow$”-line, a “$\downarrow$”-line, or neither, respectively. Left and
right edges are labeled by $1$ or $0$ depending on if that edge touches a
“$\rightarrow$”-line or not.

Given a tiling of a square grid, we say that a tiling is valid if all edges
match. To represent a $2n$-step Motzkin walk by a valid tiling, we consider a
square-gridded “step pyramid”, as shown in Fig. 1. The steps vary in length
because only a logarithmically growing number of bits
($\lfloor\text{log}_{2}2x\rfloor$) is required to store the height at the
$x^{\text{th}}$ column, for $1\leq x\leq n$. Also, the network is symmetric
about the halfway point. We impose boundary conditions on the exterior north,
east, and west edges such that they are all equal to zero. The values on the
south boundary correspond to the local spin values of the walk. As shown in
Fig. 1, the height of the walk is encoded in the binary strings between
columns of tiles. This defines an isomorphism between Motzkin walks and valid
tilings.

Figure 1: Tile representation (bottom) for a Motzkin walk (top). The tiles
from Eq. 2 are placed on a square-gridded step pyramid. The height of the walk
is encoded in binary between columns of tiles, where “1” and “0” values are
associated with the presence or absence of a horizontal arrow. Vertical arrows
act as “carry” bits in the binary addition process. Each valid tiling can be
described pictorially as follows: a value of 1 at the base of the pyramid
starts an arrowed line that travels through the bulk; arrowed paths travel
horizontally to the right; to move vertically, two arrowed paths must fuse;
similarly, an arrowed path bifurcates when moving downwards; all arrowed lines
terminate at the base of the pyramid.

### Tiles as tensors

Now we define a tensor network that sums over all configurations along the
bottom edge of a square-lattice step pyramid that yield a valid tiling. We
call this the binary-height tensor network. It is shown in Fig. 2(a).

Recall that the north, east, and west facing outer edges of each tiling must
be projected onto the value $0$. In the network, this is achieved by
contracting indices on these edges with the state $\ket{0}$ (see Fig. 2(b,
i)). The basic building block is a tensor $B$ (see Fig. 2(b, ii)), which is
the sum of rank-one operators in the spin-$z$ basis that are one-to-one with
tiles in Eq. 2.

$\displaystyle B=\sum^{6}_{l=1}A_{l},$ (3)

where each $A_{l}$ is a rank-one operator defined in Figs. 2(b, iii-viii)
using the four-index tensor

$\displaystyle\delta_{\vec{k}}(w,x,y,z)=\delta_{k_{1},w}\delta_{k_{2},x}\delta_{k_{3},y}\delta_{k_{4},z}.$
(4)

Figure 2: $\mathbf{(a)}$ 8-spin example of the binary height tensor network.
Physical indices for this network are located at the bottom. This network
requires $O(n\log n)$ square tensors. $\mathbf{(b)}$ $\mathbf{(i)}$ Tensor
network representation of the unit vector $\vec{v}=\ket{0}$. $\mathbf{(ii)}$
The four-index tensor $B$ (defined in Eq. 18). $\mathbf{(iii-viii)}$ Each tile
is represents a rank-one operator. The value of each index corresponds to type
of line: upwards and horizontal arrows represent $1$, downwards arrows
represent $-1$, and non-arrowed lines represent $0$.

Contracting this network gives a value of $1$ if the tiling is valid and $0$
otherwise. Therefore, it represents an equal-weight superposition of Motzkin
walks, which is exactly the $t=1$ Motzkin ground state.

The bond dimension between the two halves of the network grows as $O(n)$. The
total number of $B$ tensors required scales as $O(n\log{n})$ 111More
specifically, for $n=2^{y}-1$, where $y$ is a positive integer, exactly
$2(n+1)\log_{2}(n+1)-2n$ many $B$ tensors are required.. An immediate upper
bound on the entanglement entropy between the two sides—as estimated by the
number of cuts needed—is simply $\log{n}$.

The binary height network can be straightforwardly generalized to describe the
general $t\neq 0$ case, the spin $1/2$ Fredkin model, and these models with
periodic boundary conditions.

### Area weighted walks

To generalize to the $t\neq 0$ (area-weighted) case, we introduce a $t$\- and
height-dependent weighing of the $A_{j}$ tiles that contribute to $B$ (compare
with Eq. 3):

$\displaystyle B^{s}(t)=$ $\displaystyle
t^{2^{s-1}}A_{1}+t^{2^{s-1}}A_{2}+t^{2^{s}}A_{3}$
$\displaystyle+A_{4}+t^{2^{s-1}}A_{5}+t^{2^{s-1}}A_{6}$ (5)

where the $s$ index labels which row the tensor appears in, starting with
$s=1$ at the bottom and counting upwards. Each tile gets a factor of $t$
raised to the ${2^{s-1}-\text{th}}$ power for each horizontal arrow segment it
contains. This exponential scaling in $s$ is because a horizontal line
appearing in a given row is worth twice as much as one in the layer below, and
four times as much as one that is two layers below, etc.

### Spin 1/2 case

To re-purpose the binary-height network for the spin-1/2 Fredkin model, we
define an operator $P\in\text{Hom}(\mathbb{C}^{3},\mathbb{C}^{2})$ that
projects out the $0$ component of each spin and maps $1\mapsto\tfrac{1}{2}$
and $-1\mapsto-\tfrac{1}{2}$. It is given by

(6)

where the indices $k_{1}$ and $k_{2}$ are associated with the spin-1 and
spin-$\tfrac{1}{2}$ degrees of freedom, respectively. Appending $P^{\otimes
2n}$ to the bottom of the binary-height tensor network yields a tensor network
for the Fredkin model:

(7)

We note that Salberger and Korepin previously presented an MPS description of
the Fredkin state using the height representation of Dyck walks
salberger2017entangled . However, the elementary building blocks of such
networks have growing bond-dimension and lack the local structure of the
networks we achieve here.

### Boundary conditions

In order to generalize the binary-height tensor network to the periodic
Motzkin model, it will be useful to consider more generally how modifying the
boundaries of the network affects the state that it represents.

First, we consider modifying the shape of the network. If we extend the
width-$2n$ square-gridded step pyramid of the binary height-network to an
$m\times 2n$ square-gridded rectangle with $m\geq\lfloor\log_{2}2n\rfloor$,
then there is no increase in the number of valid tilings. In fact, tilings of
the rectangle are merely embedded tilings of the pyramid that have been padded
from above with blank $A_{4}$ tiles.

Next, consider generalizing the left and right boundary conditions of the
binary-height network (extended to a height $m$ rectangle) from
$\ket{0}^{\otimes\lfloor\log_{2}2n\rfloor}$ to a product of $\ket{0}$ and
$\ket{1}$ states

$\displaystyle\ket{\vec{b}_{\text{L(R)}}}=\bigotimes^{m}_{k=1}\ket{b_{k\text{L(R)}}}.$
(8)

where $b_{k\text{L(R)}}$ is the $k^{\text{th}}$ bit in the binary expansion of
the integer $p$ ($q$) between 0 and $2^{m}-1$.

Fig. 3 shows a tensor network that incorporates these changes. If
$2^{m}-1\geq\text{max}(p,q)+n$, this network represents the equally weighted
sum over all walk configurations with non-negative heights starting at height
$p$ and finishing at height $q$.

If the left and right height boundary conditions are increased so that
$\text{min}(p,q)\geq n$, then all length $2n$ walks with net height $q-p$ will
be generated by the network. This is because no walk is long enough to reach a
negative height value. This observation can be used to find a representation
of $\ket{\Psi_{k}}$. Choosing left and right boundary vectors to represent any
$p$ and $q$ such that $q-p=k$ and $\text{min}(p,q)\geq n$ will generate
$\ket{\Psi_{k}}$. One example is to set $\text{max}(p,q)=2n$ and
$\text{min}(p,q)=2n-\lvert k\rvert$.

Figure 3: Generalized binary-height tensor network for summing over all walks
that start at height $p$ and end at height $q$. The left and right boundaries
are contractions with various $\ket{b_{i}}$, where $b_{i}\in\\{0,1\\}$, and
$b_{m\text{L}}\dots b_{1\text{L}}$ and $b_{m\text{R}}\dots b_{1\text{R}}$ are
the binary expansions of $p$ and $q$, respectively. In the case where $p=q=0$,
this is equivalent to the original network shown in Fig. 2(a). The original
$B$ tensors (colored blue) have been padded from above with identical $B$
tensors (colored white).

## Zipping up: an exact network RG transformation

The binary-height tensor network discussed above offers a compact and
intuitive representation of the Motzkin ground state. Nevertheless, the
$O(n\log{n})$ number of $B$ tensors required is suboptimal. Here we show that
the ${t=1}$ Motzkin and Fredkin ground states, and their generalizations with
periodic boundary conditions, can be represented exactly using a tensor
network that only requires $O(n)$ many tensors.

We offer two equivalent constructions. In the main text we present a tile-
based approach consistent with the construction of the binary-height tensor
network presented above. In Appendix II we provide an independent construction
that is more closely related to existing tensor-network methods. In
particular, it leverages the framework of U(1)-invariant tensors, originally
described in Ref. singh2011tensor .

The key idea behind the tile-based construction is a network renormalization
group step associated to the ”zipper lemma”:

(9)

where the triangular tensor will be defined below, and the proof is given in
the appendix. Sequential application of the zipper lemma leads to a new tensor
network where each layer discards the highest frequency components of the
layer below. Thus, it naturally represents a renormalization process for
Motzkin walks, and resembles a MERA.

First we will treat the periodic case, which is more closely related to the
binary-height tensor network and involves a simpler tile set. After this, we
will generalize the construction to the original Motzkin model.

### The height renormalization network with periodic boundary conditions

We will require the following three index tensor:

(10)

where,

$\displaystyle t_{ijk}\coloneqq\delta_{i+j,k}$ (11)

and all indices $i,j$, and $k$ are restricted to values $\\{-1,0,1\\}$.
Consequently, the values of $i$ and $j$ cannot be both 1 or both -1.

Now we introduce the basic unit of renormalization. Consider the tensor
network shown below, which maps two spin values $s_{1}$ and $s_{2}$, to a
single spin value $s_{3}$

(12)

where

$\displaystyle s_{3}=\begin{cases}\lfloor(s_{1}+s_{2})/2\rfloor,&\text{if
}b_{\text{L}}=0\\\ \lceil(s_{1}+s_{2})/2\rceil,&\text{if
}b_{\text{R}}=1\end{cases}$ (13)

and

$\displaystyle b_{\text{R}}=b_{\text{L}}+s_{1}+s_{2}\quad(\text{mod}\,2).$
(14)

This map can be represented pictorially, as shown in Fig. 4(a).

Figure 4: $\mathbf{(a)}$ Table showing the map applied by the tensor network
in Eq. 12 between the top and bottom indices. The value of $s_{3}$ depends on
whether the left index value $b_{\text{L}}$ is $0$ or $1$. Equivalently, it
depends on if the segment corresponding to $s_{1}$ starts at an even or odd
height, respectively. Note that the values on the right two columns do not
depend on the height of the middle node in the first column. $\mathbf{(b)}$
Example of walk renormalization. The height achieved at each dashed line on
the top walk is equal to half the height (rounding down) at the same point in
the walk below. As was pointed out in (a), the height values at the thickly
dashed lines for the higher walks do not depend on the heights at the more
finely dashed lines for walks below. This is the information that is discarded
by the coarse-graining process.

Next, consider the action of a single layer of these 3-tensor renormalization
networks, as shown in Fig. 4(b). As a linear map acting on a walk
configuration on the $2n$ bottom indices, it outputs a “coarse-grained”
version of that walk on the $n$ top indices. More specifically, the height of
the walk between sites $l$ and $l+1$ on the top indices is half the height
(rounding down) of that between sites $2l$ and $2l+1$ at the bottom indices.

If this course-graining procedure were repeated $\lfloor\log_{2}(2n)\rfloor$
times, it could be represented by the network shown in Fig. 5(a). It requires
$O(n)$ many square and triangular tensors 222More specifically, for $2n=2^{y}$
for some integer $y>0$, exactly $4n(1-2^{-2n})$ many $B$ tensors are required,
and $2n(1-2^{-n})$ many $T$ tensors are required. . Note that we have left bit
values of the left and right boundary vectors $\ket{\vec{b}_{\text{L}}}$ and
$\ket{\vec{b}_{\text{R}}}$ unspecified. These encode two integers $p$ and $q$,
respectively.

Contracting this tensor network with a walk configuration gives the value 1 if
the walk reaches a net height of $p-q$, and gives zero otherwise. Therefore,
by choosing $\text{max}(p,q)=2n$ and $\text{min}(p,q)=2n-|k|$, this tensor
network represents the periodic ground state $\ket{\Psi_{k}}$.

One way to verify this claim would be to compute all valid tilings of this
network. The triangle tensors can be redefined in tile notation as follows

$\displaystyle T=\sum_{l=1}^{7}D_{l}$ (15)

where the triangular tiles $D_{l}$ are defined in Fig. 5(b), where

$\displaystyle\delta_{\vec{k}}(x,y,z)=\delta_{k_{1},x}\delta_{k_{2},y}\delta_{k_{3},z}.$
(16)

In Appendix III, we use the zipper lemma to show in detail how the height
renormalization tensor network is equivalent to the binary-height tensor
network representation of the periodic model (as shown in Fig. 3).

Figure 5: $\mathbf{(a)}$ The height renormalization tensor network. The square
tensors $B$ are identical to those defined in Fig. 2(b). $\mathbf{(b)}$ (i)
The triangular tensors $T$ are defined in Eq. 15. This definition involves the
triangular tiles (ii-viii), which are themselves defined using
$\delta_{\vec{k}}(x,y,z)$ (see Eq. 16).

Next, we show that by modifying the $B$ and $T$ tensors, the same type of
tensor network can represent the original Motzkin model (with open boundary
conditions).

### Original Motzkin model

The main modification we make to the network is that we increase the bond
dimension of all indices from three to four. In addition to the three spin
values $\\{-1,0,1\\}$, each index can also be assigned a non-physical value
labeled $\omega$. Each tensor will still be represented as the sum of tiles.
New tiles will indicate the $\omega$-value of an edge using a dotted line. In
order to still represent a spin-1 chain, physical indices of the network
(those that appear at the bottom) must take only spin-1 values. To ensure
this, we define the projector

$\displaystyle\Pi=1-\ket{\omega}\\!\\!\bra{\omega},$ (17)

which will be appended to all physical indices of the network.

We define the new tensor network in Fig. 6(a). We modify the square and
triangular building blocks of the tensor as follows.

The square tensors in Fig. 6(a) are denoted $C$ and are defined as

$\displaystyle C=B+A_{7}+A_{8}$ (18)

where $A_{7}$ and $A_{8}$ are defined in Fig. 6(b), and $B$ was defined in Eq.
3 . The triangle tensors in Fig. 6 (a) are denoted $S$, and are defined as

$\displaystyle S=\sum^{12}_{l=1}E_{l}$ (19)

where $E_{1},\dots E_{12}$ are defined in Fig. 6(c).

Figure 6: $\mathbf{(a)}$ Tensor network representation of the spin-1 Motzkin
model. The definitions of the $\Pi$ tensors, blue square $C$ tensors, and red
triangular $S$ tensors are given in Eqs. 17, 18, and 19. $\mathbf{(b)}$ Two
square tiles used in the $S$ tensor. The value of $\omega$ is represented by a
dotted line. $\mathbf{(c)}$ Twelve triangular tiles used in $S$.

Similar to the height renormalization tensor network for the periodic case, in
Appendix IV we verify that this network represents the Motzkin ground state by
proving its equivalence to the binary-height tensor network shown in Fig. 2.

We also note that the height renormalization tensor network can be generalized
to the spin-1/2 Fredkin model in the same way as the binary-height tensor
network. It merely involves appending $P^{\otimes 2n}$ (defined in Eq. 6) to
the physical indices of the network.

## Final Remarks

We presented exact holographic tensor networks for representing the ground
state of the spin-1 Motzkin spin-chain. These networks generate the sum over
tile configurations that are one-to-one with valid Motzkin walks. The networks
utilize the characterization of the states as height models, the ability to
encode the height efficiently in a binary way, and that binary addition can be
understood as a local operation between digits in an addition algorithm. The
tile-based two-dimensional representation of each walk provides a bulk
description of the spin chain: each valid bulk “picture” corresponds to a
particular boundary state in the ground state superposition. It is interesting
to note that bulk boundary correspondence in terms of tiling has played a
useful theme in tackling other hard problems, most recently in the context of
spin glasses klich2014glassiness .

At this point it is interesting to consider the relation between our scale-
invariant network, the renormalization group (RG), and the MERA. The height
renormalization network we describe is clearly similar in structure to a 1D
binary (scale-invariant) MERA evenbly2009algorithms , but where the
disentangling has been realized through the action of a matrix product
operator (MPO) as opposed to the tensor product of local unitary gates
utilized in a standard MERA. Indeed, the alternative construction of a network
for the Motzkin chain described in Appendix II proceeds from the perspective
of an RG transformation, building the network layer-by-layer in a manner
comparable to that constructions of MERA using entanglement renormalization
vidal2007entanglement , thus further illuminating the connection between these
ideas. However, a significant difference is that the tensors in our
holographic network lack the isometric constraints imposed on MERA tensors,
which are responsible for the finite-width “light cone” in MERA. Therefore our
network does not preserve locality when viewed as an RG transformation.

The results of this manuscript represent a significant achievement in relation
to tensor network descriptions of quantum states, as there are no previous
examples of holographic networks (MERA or otherwise) that exactly describe the
exact ground state of a gapless system in the thermodynamic limit, barring
possible exceptions based on network realizations of the Fourier transform
ferris2014fourier . While recent advances utilizing wavelets have allowed for
the analytic construction of MERA that approximate the ground states of
certain lattice CFTs with arbitrarily high precision Evenbly2016 ;
haegeman2018rigorous , these constructions only become exact in the limit of
infinite bond dimension. The inability of a finite bond dimension MERA to
achieve an exact representation of a CFT can be understood as a direct
consequence of the finite-width light cone in MERA, which, at finite bond
dimension, can only support a finite number of scaling operators
pfeifer2009entanglement . This is incompatible with achieving an exact
representation of a CFT, which typically possess an infinite set of scaling
operators. However, this suggests that one should consider holographic
networks of the form derived in this manuscript, which replace the (local)
unitary disentangling with a (non-local) disentangling implemented by an MPO,
more generally, as these do not have the limitation of only supporting a
finite number of scaling operators. Indeed, our present work opens the
exciting possibility that other systems, potentially including ground states
of lattice CFTs, could also have an exact representation as a (finite bond
dimension) network of this form. Notice also, that if the MPO disentangler was
required to be a unitary operator cirac2017matrix ; csahinouglu2017matrix ,
one could achieve a quasi-local RG transformation that may still be
computationally viable as a variational ansatz. It thus remains an interesting
avenue for future research to investigate whether such an ansatz, which
sacrifices the exact locality present in a standard MERA, could lead to
improved numeric simulation algorithms.

It has been noted that scale-invariant networks, and in particular MERA, have
a special connection to to holographic duals in the sense of the AdS/CFT
correspondence swingle2012entanglement . Here, the bulk of a MERA tensor
network can be understood as a discrete realization of 3d anti-de Sitter space
(AdS3), identifying the extra radial holographic dimension with the RG group
scale dimension in the MERA. While the MERA tensor network is used to
represent ground states of relativistic CFTs—which has the conformal group as
the symmetry group and where both time and space scale with dynamical exponent
equal to one—the tensor networks presented in this work perhaps more naturally
describe ground states of non-relativistic field theories that are invariant
under non-relativistic symmetry groups. These non-relativistic symmetry groups
are mainly characterized by anisotropic scale transformation of time and space
where the latter scales with a dynamical exponent greater than one. Lifshitz
and Schrodinger field theories are well-known examples of such field theories.

This relation may come to be valuable to the currently active attempts of
constructing holographic duals for non-relativistic field theories with
Lifshitz symmetry taylor2008non or Schrodinger symmetry
balasubramanian2008gravity ; son2008toward . For a recent thorough review of
Lifshitz holography, see taylor2016lifshitz and references therein.

It is natural to ask whether the tensor networks presented here can be
generalized to other spin models. In another work alexander2018rainbow , we
introduce another tile-based tensor network for the higher spin (colored)
generalizations of the Motzkin and Fredkin models, showing how an exact
holographic network can describe the rainbow phases these models exhibit.

Our results highlight the versatility of tensor networks and their potential
in describing complicated many body states, and in particular the power of
self-similar structures such as MERA in describing quantum critical phases.

_Acknowledgments._ We thank Amr Ahmadain, Zhao Zhang, Jacob Bridgeman,
Elizabeth Crosson, Joseph Avron, Ramis Movassagh, Vladimir Korepin, Diana
Vaman and Xiao-Liang Qi for discussions. The work of I.K. was supported in
part by the NSF grant DMR-1508245. The work of R.N.A. was partially supported
by National Science Foundation Grant No. PHY-1630114. The work of G.E. was
undertaken thanks in part to funding from the Canada First Research Excellence
Fund (CFREF).

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Supplementary Material

Rafael Alexander1, Glen Evenbly2 and Israel Klich3

[$1$] Center for Quantum Information and Control, University of New Mexico,
Albuquerque, New Mexico 87131-0001, USA
[$2$] Département de Physique and Institut Quantique, Université de
Sherbrooke, Québec, Canada
[$3$] Department of Physics, University of Virginia, Charlottesville, 22904,
VA, USA

##  I Definition of the Motzkin Hamiltonian

The Motzkin Hamiltonian can be defined by first identifying each local
spin-$z$ basis state $\\{\ket{1},\ket{0},\ket{-1}\\}$ with a line segment
$\\{\diagup,\text{{---}},\diagdown\\}$, respectively. This allows us to
represent states as a superposition of walks. The Hamiltonian, defined on a
spin-chain with $2n$ sites reads:

$H=\Pi_{\text{boundary}}+\sum_{j=1}^{2n-1}\Pi_{j}$ (S1)

where $\Pi_{j}$ acts on the pair of spins $j,j+1$ and

$\displaystyle\Pi_{j}$
$\displaystyle=|\Phi_{t}\rangle\langle\Phi_{t}|_{j,j+1}+|\Psi_{t}\rangle\langle\Psi_{t}|_{j,j+1}+|\Theta_{t}\rangle\langle\Theta_{t}|_{j,j+1},$
$\displaystyle\Pi_{\text{boundary}}$
$\displaystyle=|-1\rangle\langle-1|_{1}+|1\rangle\langle 1|_{2n}.$

where $\Phi,\Psi,\Theta$ are the following states on pairs of neighboring
spins

$\displaystyle|\Phi_{t}\rangle$
$\displaystyle\propto|1,0\rangle-t|0,1\rangle,$ (S2)
$\displaystyle|\Psi_{t}\rangle$
$\displaystyle\propto|0,-1\rangle-t|-1,0\rangle,$ (S3)
$\displaystyle|\Theta_{t}\rangle$
$\displaystyle\propto|1,-1\rangle-t|0,0\rangle.$ (S4)

To have periodic boundary conditions, the Hamiltonian in Eq. S1 can be simply
modified to include a $\Pi_{2n}$ term while omitting the boundary terms
$\Pi_{\text{boundary}}$.

##  II Construction of renormalization tensor network via U(1)-symmetric
tensors

In this appendix we provide an alternative derivation of an exact holographic
tensor network representation of the ground state of the spin-$1$ Motzkin spin
chain. Here, in order to better connect with established tensor network
methodology singh2011tensor , we formulate the solution in terms of U(1)
invariant tensors instead of the ‘flux’ preserving tiles discussed in the main
text; however both approaches are ultimately equivalent. The derivation
presented here is based on the dual notions of $\delta$-symmetric tensors, a
restricted subclass of U(1) tensors, and boundary-locked networks. We first
define these concepts before proceeding to demonstrate how they can be used to
construct a network for the Motzkin spin chain.

Let us recall the basics of U(1)-symmetric networks, as detailed in
Ref.singh2011tensor . A tensor network with U(1) symmetry is represented by an
oriented graph, where each index has an associated direction (depicted with an
arrow), such that the indices connected to a tensor can be regarded as either
incoming or outgoing with respect to that tensor. Each index
$i\in\\{1,2,\ldots,d\\}$ in the network is assigned a set of quantum numbers
$\vec{n}^{(i)}=[n^{(i)}_{1},n^{(i)}_{2},\dots,n^{(i)}_{d}]$ with charges
$n\in\mathbb{Z}$ associated to the $z$-component of the spin at that value of
the index. We say that a tensor is U(1) symmetric if it is left invariant
under transformation with a unitary representation of U(1) acting on each
index (where outgoing indices should be transformed by the dual representation
of U(1) with respect to incoming indices). It is known that U(1)-symmetric
tensors are those that conserve particle number, such that a tensor component
has zero weight unless the sum of the outgoing charges matches the sum of the
incoming charges. For the purposes of constructing a solution to the Motzkin
model, it is useful to restrict to a sub-class of symmetric tensors that we
call $\delta$-symmetric tensors, which we define as U(1)-symmetric tensors
where every (structurally non-zero) element is equal to unity. In other words,
these are U(1)-symmetric tensors with component equal to unity if the total
incoming charge $n_{\textrm{in}}$ matches the total outgoing charge
$n_{\textrm{out}}$ and zero otherwise, which can be understood as a tensor
version of the Kronecker-delta function.

The concept of a boundary-locked tensor network is now defined. Given a
lattice $\mathcal{L}$ of spin-1 sites, let $T$ be the tensor network built
from U(1)-symmetric tensors, representing a $S_{z}=0$ quantum state
$\ket{\psi}\in\mathcal{L}$, and let $\ket{\phi}$ be a U(1)-invariant product
state ($S_{z}=0$) on $\mathcal{L}$. We say that network $T$ if boundary-locked
if, by constraint of the U(1) symmetry, there is a single unique configuration
of the internal indices in $T$ that can give a non-zero contribution to the
scalar product $\braket{\phi}{\psi}$ for any input product state $\ket{\phi}$
with $S_{z}=0$. In other words, a network is boundary locked if, by fixing the
boundary indices in a configuration compatible the $S_{z}=0$ symmetry of the
network, the internal indices are then ‘locked’ in a unique configuration by
the constraint that the total incoming U(1) charges must match the total
outgoing charge in all tensors. It follows that a necessary (but not
sufficient) constraint for a network to be boundary locked is that the irrep
of charge $n$ on any tensor index is at most 1-fold degenerate (which allows
us to specify the value that an index takes by the charge $n$ that it
carries).

Figure S1: (a) Description of the U(1) charges on each index of the
$G^{ij}_{kl}$ tensors. (b) List of the non-zero components of $G$, which have
equal incoming and outgoing U(1) charge. (c) Description of the U(1) charges
on each index of the $W^{ij}_{k}$ tensors. (b) List of the non-zero components
of $W$. These have equal incoming and outgoing U(1) charge.

Given that the notions of $\delta$-symmetric tensors and boundary-locked
networks have been established, we are now able to construct the exact
holographic network for the ground state of the Motzkin chain. We begin by
focusing on a simpler task: given a finite chain of spin-$1$ sites
$\mathcal{L}$ we describe how to construct a network representing the equal
weight superposition U(1) states in the $S_{z}=0$ spin symmetry sector (or
equivalently, the superposition of all walks that start and end at zero
height). This solution can later be refined to exclude the paths that take
negative height values at any point, as described in the main text, such that
the ground state of the original Motzkin spin chain is recovered. Let us
consider a U(1)-invariant state $\ket{\psi}\in\mathcal{L}$ with $S_{z}=0$; if
$\ket{\psi}$ is described by a boundary locked network built from
$\delta$-symmetric tensors it automatically follows that $\ket{\psi}$ must be
the desired equal-weight superposition of all states in the $S_{z}=0$ sector.
This is true since the scalar product of $\ket{\psi}$ with any product state
of the $S_{z}=0$ spin sector must evaluate to unity, as only a single
configuration of indices can contribute and all configurations have a total
weight equal to unity, while the scalar product with any state outside of the
$S_{z}=0$ is trivially zero.

Figure S2: The MPO formed from $G$ tensors maps an input product state on
lattice $\mathcal{L}$ of local dim $d=3$ to a single product state on lattice
$\mathcal{L}^{\prime}$ of local dim $d=2$.

The remaining goal is thus is to build a boundary-locked network $T$of
$\delta$-invariant tensors that constitutes a proper holographic realization:
given a lattice $\mathcal{L}$ of $N$ sites, we want the network $T$ to be
organized into $O(\log N)$ self-similar layers, each with some finite bond
dimension that is independent of the system size $N$. Here we follow a similar
construction as used in the MERA, and build the network from a sequence of
coarse-graining (CG) transformations, each of which is comprised of a
disentangling step followed by a blocking step that reduces the number of
lattice sites by a factor of $2$. However, instead of using disentanglers with
finite local support, which prove inadequate for the exact construction, we
instead represent the disentangling operation as an MPO. We build this MPO
from copies of a four index $\delta$-invariant tensor $G^{ik}_{jl}$ as
depicted in Fig. S1(a-b). The virtual bond dimension of the MPO is set at
$d=2$, with U(1) charges on these indices $\vec{n}^{(i)}=\vec{n}^{(j)}=[0,1]$.
The incoming indices $k$ must match the spin-$1$ sites in $\mathcal{L}$, i.e.
such that $\vec{n}^{(k)}=[-1,0,1]$, while the output indices $l$ are set at
$d=2$ with U(1) charges $\vec{n}^{(l)}=[-1,1]$. Thus the action of this MPO is
to map states on the $N$-site lattice $\mathcal{L}$, which has local dimension
$d=3$, to states on an $N$-site lattice $\mathcal{L}^{\prime}$ of local
dimension $d=2$. If we assume $N$ to be even and that the virtual indices of
the edge MPO tensors are fixed in the $\ket{0}$ state, then it can be seen
that any $S_{z}=0$ product state on $\mathcal{L}$ is mapped to a unique
product state on $\mathcal{L}^{\prime}$, see Fig. S2 for example, consistent
with the desire for a boundary-locked network. This follows as, once the input
indices $i$ and $k$ have been set on a $G$ tensor, there is only a single
choice of the output indices $j$ and $l$ that satisfies U(1) charge
conservation. The blocking step of the CG transformation is now realized using
a 3-index $\delta$-symmetric tensor $W^{ij}_{k}$, whose input indices carry
U(1) charges $\vec{n}^{(i)}=\vec{n}^{(j)}=[-1,1]$ and whose output index
carries $\vec{n}^{(k)}=[-2,0,2]$, see also Fig. S1(c-d). This blocking step is
again consistent with realizing a boundary locked network, and clearly has
non-zero overlap with any state on $\mathcal{L}^{\prime}$.

Figure S3: (a) Depiction of the holographic tensor network. (b) The network is
boundary-locked; under the input product state (in the total spin $S_{z}=0$
sector) the internal indices are fixed in a single unique configuration in
order to satisfy the U(1) symmetry constraints on each tensor.  Figure S4: (a)
Tile representation of the components of the $G$ tensor from Fig.S1(b). (b)
Tile representation of the components of the $W$ tensor from Fig.S1(d). (c)
The product of $G$ and $W$ tensors is identical to the product of $B$ and $T$
tensors from Eq.12.

One may then repeat this coarse-graining transformation, consisting of
disentangling with the MPO formed from $G$ tensors and then blocking with the
$W$. Notice that the magnitude of U(1) individual charges carried on the
indices of the $G$ and $W$ tensors doubles with each coarse-graining step, but
the content of these tensors otherwise remains unchanged. The output index on
the final isometry, after taking $\log_{2}(N)$ coarse-graining steps where $N$
is assumed to be a power of 2, is fixed in the $n=0$ state to ensure that the
network is in the total spin $S_{z}=0$ sector, see Fig. S3. Thus our goal is
completed: we have an $S_{z}=0$ holographic network that (i) is constructed
entirely from $\delta$-symmetric tensors and (ii) is boundary-locked, which
implies that it represents an equal weighted superposition of all states in
the total spin $S_{z}=0$ sector.

In comparing the network derived in this appendix, Fig. S3(a), to that derived
in the main text, Fig. 5, one sees that they both have an equivalent
structure, with the $G$ and $W$ tensors derived in this appendix substituting
for the $B$ and $T$ tensors in Fig. 5. However, upon expressing the $G$ and
$W$ tensors in the ‘tiling’ representation used in the main text, see Fig.
S4(a-b), it is seen that they correspond to a different set of tiles than do
the $B$ tensors, Fig. 2, and the $T$ tensors, Fig. 5. Nevertheless, a
component-wise analysis of all permissible tilings reveals that the product of
two $G$ and a $W$ tensor is identical to the product of two $B$ and a $T$
tensor, as depicted in Fig. S4(c). Thus one indeed concludes that the network
constructed in this appendix is ultimately equivalent to that of the main
text.

##  III Equivalence between the renormalization and binary height networks for
the periodic model

Here we prove that the height renormalization tensor network defined in Fig. 5
represents the periodic Motzkin ground state. We do this by showing its
equivalence to the generalized binary-height tensor network defined in Fig. 3.

In order to represent the state $\ket{\Psi_{k}}$, the boundary vectors
$\ket{\vec{b}_{\text{L}}}$ and $\ket{\vec{b}_{\text{R}}}$ (see Eq. 8) are
chosen to encode integers $p$ and $q$ such that $q-p=k$ and
$\text{min}(p,q)\geq n$. We choose

$\displaystyle\text{max}(p,q)$
$\displaystyle=2^{\lfloor\log_{2}2n\rfloor+1}+2n$ (S5)
$\displaystyle\text{min}(p,q)$
$\displaystyle=2^{\lfloor\log_{2}2n\rfloor+1}+2n-|k|,$ (S6)

We use a rectangular binary-height tensor network with
$m=2^{\lfloor\log_{2}2n\rfloor+1}$ many layers. This choice guarantees that
$b_{m\text{L}}=b_{m\text{R}}=1$. See Fig. S5(a) for an example on 8 spins.

As usual, the upwards pointing indices on the $m^{\text{th}}$ layer have been
contracted with the state $\ket{0}$. In fact, all such indices must take the
value 0 regardless of this boundary condition. To see this, suppose some of
these indices took a value of 1 or -1, and let $\mu$ be the leftmost column
with a non-zero value at the top.

Then,

  * •

If the top index of $\mu$ takes value $-1$, the only valid tiling for $\mu$ is
to use tile $A_{6}$ for all square cells. Then, the height encoded at the left
boundary of $\mu$ is zero. However, from Eq. S6, $p>2n\geq\mu$, and since it
takes at least $y$ many columns to drop in height by $y$, there are no valid
tilings of the lattice to the left of $\mu$.

  * •

If the top index of $\mu$ takes value $1$, the only valid tiling for $\mu$ is
to use tile $A_{2}$ for all square cells. Then, the height encoded at the
right boundary of $\mu$ is zero. However, from Eq. S6, $q>2n\geq 2n-\mu$, and
since it takes at least $y$ many columns to climb in height by $y$, there are
no valid tilings of the lattice to the right of $\mu$.

Because non-zero values at the top indices of the network yield invalid
tilings, contractions of the network must evaluate to zero in such cases.

Therefore, there is some flexibility in choosing the upper boundary condition,
i.e., we are free to replace the contraction with $\ket{0}^{\otimes 2n}$ with
any other tensor that has equal support over this state. In particular, we can
use a binary tree of triangle $T$ tensors, as shown in Fig. S5(b). Tiling the
entire pyramid with $D_{5}$ tiles shows that it has support over the
$\ket{0}^{\otimes 2n}$ subspace.

Figure S5: $\mathbf{(a)}$ Rectangular binary-height tensor network with left
and right boundary conditions corresponding to Eqs. S5 and S6. $\mathbf{(b)}$
Equivalent tensor network with the top boundary contracted with a binary tree
of triangle $T$ tensors. $\mathbf{(c)}$ Tensor network after the zipper lemma
has been applied to all $T$ tensors.

Next, we require following lemma:

######  Lemma III.1 (Zipper Lemma).

The following tensor networks are equivalent:

(S7)

###### Proof.

By exhaustive search all valid tile configurations (see Fig. S6), the
following two tensor networks are equivalent

(S8)

The lemma follows by sequential application of this identity. ∎

Using this, we can pull a horizontal layer of triangular tensors in Fig. S5(b)
downwards through the tensor network. At each step, two $B$ tensors are merged
into one. Pulling down each triangular tensor is analogous to closing a
zipper. The end result is shown in Fig. S5(c).

The final step to prove equivalence with the height renormalization tensor
network from Fig. 5(a) is to note that the top $B$ tensor from Fig. S5(c) can
be removed, since

(S9)

Removing the top $B$ tensor modifies the left and right boundary heights so
that now $\text{max}(p,q)=2n$ and $\text{min}(p,q)=2n-|k|$.

Figure S6: Each tiling of the networks in Eq. S8 is uniquely specified by the
values on the left and bottom pointing indices. These can take values
$\\{0,1\\}$ and $\\{-1,0,1\\}$, respectively. This results in a total of 36
possible tilings. We list these for both networks and find an isomorphism
between configurations that share boundary conditions.

##  IV Equivalence between the renormalization and binary height networks for
the original model

Here we show that the height renormalization tensor network shown in Fig. 6 is
a valid representation of the Motzkin model. Our proof will follow a similar
trajectory to the case with periodic boundary conditions in Appendix III.

We begin with the step-pyramid of $B$ tensors from Fig. 2. We will embed the
width $2n$ pyramid within a $(\lfloor\log_{2}2n\rfloor+1)\times 2n$ rectangle
(recall that this does not change the represented state). We are also free to
append projectors $\Pi$ (see Eq. 17) to the base of the network because the
$B$ tensors contain no tiles with support on $\omega$ (they have no dotted
lines). The tensor network shown in Fig S7(a) incorporates both these changes.

Figure S7: $\mathbf{(a)}$ Binary height tensor network made up of $B$ tensors,
vectors $\ket{0}$, and projectors $\Pi$. This is equivalent to the step-
pyramid binary-height tensor network from Fig. 2. $\mathbf{(b)}$ The network
from (a) has been modified so that the $B$ tensors are replaced with $C$
tensors, and the top boundary of $\ket{0}^{\otimes 2n}$ has been replaced with
a tree of triangular $S$ tensors.

Next, we replace each of the $B$ tensors in Fig S7(a) with $C$ tensors defined
in Eq. 18. Though the $C$ tensors contain two additional tiles ($A_{7}$ and
$A_{8}$), the new network includes no additional tilings. To see this, note
that $A_{7}$ and $A_{8}$ contain vertical dotted lines. If these tiles appear
in some column, then any valid tiling of that column must connect the dotted
line to the bottom of the network. Contraction with $\Pi$ at the bottom of
that column will evaluate the network to zero. Next, we replace the top
boundary $\ket{0}^{\otimes 2n}$ vector with a tree of triangular $S$ tensors
(defined in Eq. 19). These changes are incorporated into the tensor network
shown in Fig. S7(b). This tensor network is equivalent to the network shown in
(a).

To prove this, we will show that the indices at the bottom of the pyramid
(equivalently, the indices at the top of the square lattice) in Fig. S7(b)
must take the value zero in order for the network contraction to give a non-
zero value. Suppose some of these indices took a value of 1, -1, or $\omega$,
and let $\mu$ be the leftmost column of the square lattice with a non-zero
value at the top.

Then,

  * •

If the top index of $\mu$ takes value $1$, the only valid tiling for $\mu$ is
to use tile $A_{2}$ for all square cells. Then, the height encoded at the left
boundary of $\mu$ is $2^{\lfloor\log_{2}2n\rfloor+1}$. Since it takes at least
$y$ many columns to climb in height by $y$, and the left boundary of the
network is set to height zero, there is no valid tiling that is compatible
with both the left boundary of the network and the right side of the column
$\mu$.

  * •

If the top index of $\mu$ takes value $-1$, then the tiling of the pyramid
network of $S$ tensors must include some $E_{7}$ tiles. Consider the topmost
$E_{7}$ tile(s). Any valid tiling of the pyramid must connect this dotted line
to the top of the pyramid. Then, contracting with the $\ket{0}$ vector at the
top index of the top $S$ tensor will evaluate the network to zero.

  * •

If the top index of $\mu$ takes value $\omega$, then $\mu$ can be tiled only
with $A_{8}$. Contraction with $\Pi$ at the base of $\mu$ evaluates the
network to zero.

Therefore, using a tree of $S$ tensors as the upper boundary of the square
lattice network is equivalent to contracting with $\ket{0}^{\otimes 2n}$.

Now, we reprove the zipper lemma (lemma S7) with tensors $C$ and $S$ replacing
tensors $B$ and $T$, respectively.

###### Proof.

First we prove the equivalence of the tensor networks shown in Eq. S8. We do
this by proving an equivalence of tilings. The tilings of each network are
specified by assigning values of $\\{0,1\\}$ to the left facing indices and
values of $\\{-1,0,1,\omega\\}$ to the indices on the bottom. This yields a
total of 64 distinct tilings. 34 of these were shown already in Fig. S6; only
numbers 9 and 10 cannot be included because $S$ does not contain the tile
$D_{7}$. The remaining 30 are shown in Fig. S8.

The zipper lemma follows from sequential application of the identity in Eq.
S8. ∎

Figure S8: A subset of the tilings of the networks in Eq. S8 using the $C$ and
$S$ tensors.

Using the zipper lemma, we can pull a horizontal layer of triangular tensors
in Fig. S7(b) downwards through the tensor network. At each step, two $C$
tensors are merged into one. Pulling down each triangular tensor is analogous
to closing a zipper. The end result is shown in Fig. S7(c).

The final step to proving equivalence between the original binary-height
tensor network (Fig. 2(a)) and the height renormalization tensor network (Fig.
6(a)) is to note that the top $C$ tensor from Fig. S7(c) can be removed, since

(S10)

where the square tile in the middle network is $A_{4}$.


# Predicting Digital Asset Prices
using Natural Language Processing: a survey

Trang Tran
Cornell University, USA
<EMAIL_ADDRESS>
###### Abstract

The introduction of blockchain technology has changed the way people think
about how they used to store and trade their assets, as it introduced us to a
whole new way to transact: using digital currencies. One of the major
innovations of blockchain technology is decentralization, meaning that
traditional financial intermediaries, such as asset-backed security issuers
and banks, are eliminated in the process. Even though the blockchain
technology has been utilized in a wide range of industries, its most prominent
application is still cryptocurrencies, with Bitcoin being the first one
proposed. At its peak in 2021, the market cap for Bitcoin once surpassed 1
trillion US dollars. The open nature of the crypto market poses various
challenges and concerns for both potential retail investors and institutional
investors, as the price of the investment is highly volatile and its
fluctuations are unpredictable. The rise of Machine Learning, and Natural
Language Processing, in particular, has shed some light on monitoring and
predicting the price behaviors of cryptocurrencies. This paper aims to review
and analyze the recent efforts in applying Machine Learning and Natural
Language Processing methods to predict the prices and analyze the behaviors of
digital assets such as Bitcoin and Ethereum.

Keywords: blockchain; cryptocurrency, natural language processing; price
prediction

##  1 Introduction

In recent years, blockchain technology and its desirable characteristics, such
as decentralization and anonymity, have motivated the industry and academia.
Even though the concept of using cryptographically secured chain of blocks was
first described by Haber and Stornetta (1990) in their paper ”How to Time-
Stamp a Digital Document”, the term blockchain started to attract significant
attention after the group of developers who go by the pseudonym Satoshi
Nakamoto described the blockchain model their white paper (Nakamoto, 2008) and
implemented the peer-to-peer electronic cash system in 2009. The open and
anonymous nature of the market has facilitated new opportunities while
introducing new risks never seen before with established financial
institutions, such as trust, unpredicted fluctuations, and fraudulent
crowdfunding schemes.

The digital age also sees the rise of social media platforms such as Twitter
and Reddit, where users can easily and conveniently express their opinions at
any time and from anywhere, in the form of a simple post or a ”tweet.”
Investment advice and opinions are among the most popular topics discussed
online because of its real-time nature. While Twitter has implemented some
forms of verifications to encourage people to use their real identities, such
as asking for phone numbers or emails, Reddit has done little to ensure the
users’ identities are traceable. As a result, the world has witnessed the
infamous case of Reddit’s subforum ”WallstreeBets” in January 2021, where the
explosive power of Internet users and their speculative investment research
can manipulate the investment prices, leading to the short squeeze of
Gamestop’s stock (GME) (Boylston et al., 2021). In an open space with less
regulation and more freedom like cryptocurrency, we see a clear opportunity to
utilize textual data from social media posts helping investors make informed
investment decisions.

The remainder of this paper is organized as follows. In Section 2, we briefly
covered the background and terminologies behind blockchain. In Section 3, we
similarly presented a brief literature review of the recent developments in
the Natural Language Processing space. We dived deep into the recent works
that use Natural Language Processing to predict the price and behaviours of
cryptocurrencies in Section 4, and concluded the paper in Section 5.

##  2 Background on Blockchain technology

At a high level, a blockchain is a distributed ledger where all transactions
are recorded in a chain of blocks. As new transactions are committed, new
blocks will be appended to the chain (Zheng et al., 2018). A distributed
ledger is a public database where all participants can contribute and
maintains a synchronized copy of the data.

###  2.1 Key characteristics

  * •

_Decentralization._ A key innovation of blockchain technology is the
elimination of traditional intermediaries (central banks). Cost and flow time
can be massively reduced since transactions can happen peer-to-peer (P2P).
There are no single points of failure, as every network participant has the
same copy of the data.

  * •

_Persistency._ blockchain is built to be an immutable ledger, meaning it is
impossible to alter or delete the data already recorded in the blocks.

  * •

_Anonymity._ In the blockchain network, users are not identified by real names
but by a randomly generated address, which will be liked to each user’s
transaction. Even though the transactions and the user’s address (or wallet
address) are public, the keys to prove ownership is private to each user.

  * •

_Auditability._ Since all participants have a copy of the data; it is possible
to verify and trace the transactions based on the public wallet address and
the digital timestamp.

###  2.2 Types of blockchain network

There are three major types of blockchain networks: Public, Private and
Hybrid.

  * •

_Public (Permissionless) blockchain._ In this network, users can join and
leave at any time without obtaining prior permission.

  * •

_Private (Permissioned) blockchain._ In this network, only a limited number of
users with permissions can read and write.

  * •

_Hybrid blockchain._ This is a combination of both public and private
blockchains. Transactions and private information are kept inside the network
but can still be verified.

###  2.3 Bitcoin (BTC) and Ether (ETC) and other crytocurrencies

Bitcoin, proposed by Nakamoto (2008), is a digital currency using blockchain
technology to facilitate transactions. It is still the most popular general-
purpose cryptocurrency and is considered the first cryptocurrency ever
created. There are 21 million bitcoins in total, and each bitcoin consists of
100 million smaller units (satoshis). Bitcoin is an example of public
blockchain.

Since Bitcoin launched, other alternative cryptocurrencies based on blockchain
technology have been introduced, including Ethereum - developed by Buterin
(2014). Its focus is providing a protocol for building decentralized
applications (dApps) deployed on Ethereum virtual machines (EVMs). Users can
create smart contracts to perform various tasks using a built-in Turing-
complete programming language to be executed on the blockchain. _Turing
complete_ (or computationally universal) refers to the idea that a system
needs to have the capacity to implement arbitrary computer algorithms. In
other words, it needs to be able to stimulate a Turing machine. Ether (ETH) is
the native cryptocurrency on the Ethereum network. Other popular
cryptocurrencies by market capitalization include Tether (USDT), USD Coin
(USDC), Binance Coin (BNB), etc. Both Tether, USD Coin, and Binance Coin were
initially built on the Ethereum network.

###  2.4 Predicting Prices and Trends of Cryptocurrency

As cryptocurrencies are considered valuable digital assets, various studies
have been done to analyze and protect cryptocurrencies, just like any other
type of investment. Past works have used Machine Learning methods extensively,
alongside classical time-series variables (i.e., closing prices), to predict
prices of cryptocurrencies such as Bitcoin, Ethereum, etc. Popular choices for
model architecture include Autoregressive Integrated Moving Average (ARIMA)
(Garg et al., 2018; Roy et al., 2018), Support Vector Machine (SVM) (Poongodi
et al., 2020).

In recent years, with the advancement in Machine Learning and Natural Language
Processing, we have seen broad adoption of language models to support past
works in predicting the Prices and Trends of an investment. Prior research
suggested that peers’ influence and public moods in the form of unstructured
texts have been found to play an important role in investment decisions and
market predictions (Farrell et al., 2022; Farimani et al., 2021; Vo et al.,
2019). As a result, it is common to see an architecture in recent literature
comprising two parts: a sentiment analyzer (to process the textual data such
as Tweets from Twitter) and a prediction model (to predict the prices).

Figure 1: An example system to predict the price of bitcoin by Pant et al.
(2018). The Recurrent Neural Networks (Section 3.2) take into account two
inputs: a sentiment score from a tweet, and the historical price of bitcoin to
predict the next price

##  3 Background on Natural Language Processing

To tackle a large amount of textual data, earlier works in the area have
focused on gauging the general sentiment or the polarity of the text and
quantifying it using manual labels (i.e., ”positive,” ”neutral,” and
”negative” that are associated with a numeric value). This is generally
considered the lexicon-based approach, in which a sentence will be transformed
into a vector via a simple method such as a token count matrix (Şaşmaz and
Tek, 2021). After that, a simple classification model such as Logistic
Regression, Support Vector Machine (SVM), or Naive Bayes classifier. Latest
developments in the Natural Language Processing field have pushed the frontier
in improving the performance of sentiment classification system by allowing
the vector representation of a word to have a context. Various recent state-
of-the-arts language models introduced recently include BERT (Devlin et al.,
2018), RoBERTa (Liu et al., 2019), XLNet (Yang et al., 2019), and ELECTRA
(Clark et al., 2020).

###  3.1 Artificial Neural Networks (ANNs)

An artificial neural network, or a neural network for short, is a mathematical
model primarily inspired by the structure and functionalities of the
biological neural system in the brain. Figure 1 showed the connection between
a biological neuron on the left and its mathematical model on the right. At a
high level, they work by taking input signals from their dendrites and output
signals along its axon. These axons eventually connect to the dendrites of
other neurons, creating the entire _network_. Another name for ANNs is Feed-
Forward Neural network because the input information will only be processed in
the forward direction, as seen in Figure 2.

Figure 2: A biological neuron (left) and its simplified mathematical model
(right)111https://cs231n.github.io/neural-networks-1/ Figure 3: Example
architecture of a simple 3-layer neural
network222https://cs231n.github.io/neural-networks-1/

###  3.2 Recurrent Neural Networks (RNNs)

Recurrent Neural Networks (RNNs) are models for data that involve times or
sequences, such as sound, video, or language. They addressed a few limitations
of Feed-Forward Neural networks, such as keeping track of or ”memorizing” some
of the inputs to make predictions in a situation where context and historical
information plays an important role.

###  3.3 Long short-term memory Networks (LSTMs)

Long Short-Term Memory networks or LSTMs are a type of RNN introduced by
Hochreiter and Schmidhuber (1997) to address the long-term dependency problem
in the original RNN architecture. The original RNN cannot make an accurate
prediction as the gap between the point containing the relevant information
and the point where the relevant information increases. As a result, LSTMs
have become the popular model architecture with significant use cases in Image
captioning, Language Translation, Algorithmic trading, and Text Generation.

##  4 Applications of NLPs in price prediction

We can categorize the works in this space into two major groups: Corpus-based,
and Advanced methods.

###  4.1 Corpus-based methods

Corpus-based methods are based on the actual, real, and authentic words as
they showed up in the document (Almutairi, 2016). By comparing and contrasting
raw words from a document with a dictionary like the Harvard-IV dictionary
(Dictionary with a list of positive and negative words), researchers can
quickly sense the overarching sentiment of a post by aggregating the number of
times that a positive or negative word shows up. Sul et al. (2017) utilized
this method to show that a trading strategy incorporating Twitter postings’
sentiments could generate an annual return between 10 and 15%.

###  4.2 Advanced NLP methods

Advanced NLP techniques attempt to go beyond relying purely on the count of
the words as they occur in a sentence. These methods gears towards language
understanding, as they allow the context to be incorporated into the vector
representation of a word, as discussed in Section 3.

Vo et al. (2019) built a price prediction system using both news data and
historical prices of Ethereum (ETH) to generate a prediction for future prices
(Figure 4). The authors created semantic vectors of the news document using a
classic $n$-gram language model and syntactic vectors utilizing a set of NLP
methods such as Dependency Parser, Coreference Resolution, and Named Entity
Recognition. These two vectors are combined to predict the sentiment score.
Eventually, the sentiment score and the historical cryptocurrency price are
used to predict the future price. The text data used in this work is crypto-
related, obtained from NewsNow333https://newsnow.co.uk, and manually labeled.
A label of 1 indicates a positive sentiment before a rise in the historical
price and vice versa. Otherwise, the news will be labeled as 0 (neutral) as
there is no change in the price. Figure 5 showed that the proposed
architecture achieved relatively better performance than the SVM baseline when
predicting ETH prices between July 2017 and October 2018.

Figure 4: Proposed system to predict the price of Ethereum (Vo et al., 2019)
Figure 5: Result of predicting Ethereum (ETH) prices using different
approaches (Vo et al., 2019)

To leverage the power of transfer learning, Cheuque Cerda (2021) used the
pretrained ULMFiT and BERT model in predicting the stance of a tweet from
crypto-influencers and used these insights in building price prediction models
using XGBoost and LSTM. The author showed that while there might be a weak
relationship between the asset price and crypto-influencers’ opinion, this
effect was much stronger and more causal in the short term.

Huang et al. (2021) proposed using news articles and social media platforms
such as Sina-Weibo, WeChat, and QQ groups to build a crypto-specific sentiment
analysis model based on the Long short-term memory (LSTM) Recurrent Network
Structure (Figure 6). The data used in the training process was manually
labeled as positive (1), neutral (0), and negative (-1). The social media
posts were tokenized using the manually-created crypto word vocabulary
specific for Chinese words and used as the input to the LSTM Sentiment
Analyzer.

Figure 6: Cryptocurrency sentiment analysis and price movement prediction
(Huang et al., 2021)

Ortu, Vacca, et al. (2022) showed via various empirical analyses that the
price of Bitcoin is highly correlated with the sentiment from the discussions
on social media forums such as Reddit. Using dynamic topic modeling and Hawkes
models, they were able to explain how public opinions on social media chatter
can influence each other and cause price fluctuations.

Ortu, Uras, et al. (2022) compared the performance of four different deep
learning models (Multi-Layer Perceptron, Multivariate Attention Long Short
Term Memory Fully Convolutional Network, Convolutional Neural Network, and
Long Short Term Memory in predicting the price movements of Ethereum and
Bitcoin between 2017 and 2020, utilizing BERT to extract the emotions from
social media posts. The competitive results showed a lot of promise in
detecting the price movements of Bitcoin and Ethereum.

##  5 Conclusion

The effort in using advanced machine methods to better understand
cryptocurrencies has proved that investing in digital currencies is considered
an alternative investment. As innovations are introduced in both the crypto
world and the natural language processing field every day, opportunities and
challenges exist in this niche research area. With cryptocurrency being a new
research topic, more work needed to be done in this area to create an even
more robust and accurate monitoring system to better utilize the massive
amount of data available, especially the textual data surrounding the topic.

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