aainstitutetext: Stanford Institute for Theoretical Physics, Stanford
University,
382 Via Pueblo, Stanford CA 94305

# APD-Invariant Tensor Networks from Matrix Quantum Mechanics

Alexander Frenkel<EMAIL_ADDRESS>
###### Abstract

We propose a simple connection between matrix quantum mechanics and tensor
networks. This allows us to imbue tensor networks with some interesting
additional structure. The geometry of the graph describing the tensor network
state is determined dynamically, giving a notion of background independence.
The tensor network states have a $U(N)$ invariance, which (a) allows us to
consider continuous families of entanglement cuts even with a finite number of
tensors and (b) includes a notion of bulk coordinate reparameterization and
area-preserving diffeomorphism invariance in the large $N$ limit. These tensor
networks also have a natural scale of nonlocality that behaves similarly to a
string scale, suggesting a potential toy model for sub-AdS physics. Emergent
$U(p)$ gauge fields naturally appear on the tensor network links.

##  1 Introduction

Following the second superstring revolution schwarz1987review ;
polchinski1996tasi  a family of holographic dualities was proposed between
large $N$ quantum theories and various corners of string theory banks1999m ;
Maldacena:1997re ; itzhaki1998supergravity ; berenstein2002strings ;
gubser1998gauge ; witten1998anti . These dualities share a heart – their
quantum mechanical sides are generically worldvolume theories of some stack of
$N$ D$p$-branes in the decoupling limit. The most well-studied example is the
gravitational theory emergent from dimensionally reduced 9+1d $\mathcal{N}=4$
Super Yang-Mills (SYM) Maldacena:1997re ; itzhaki1998supergravity . The
worldvolume theory of D3-branes in particular is conformal, and the emergent
spacetime is string theory in AdS${}_{5}\crossproduct S^{5}$ Maldacena:1997re
, so the case $p=3$ is a particular instantiation of the more general AdS/CFT
correspondence. However, following the general arguments of t1993planar , even
non-conformal large-$N$ theories in the appropriate limit often define some
emergent string theory.

In this paper we focus on matrix quantum mechanics (MQM), in which the
‘boundary’ theory is neither conformal, nor a QFT, but a quantum mechanical
system with a finite number of degrees of freedom living on a 0+1 dimensional
base space. The BFSS banks1999m  and BMN berenstein2002strings  models are
particular examples of type IIA string theory emergent from interacting
D0-branes. The degrees of freedom consist of $N\times N$ Hermitian matrices
$X^{i}$ and fermionic matrices $\lambda^{\alpha}$. The entries of these
matrices are denoted $x^{i}_{nn^{\prime}}$ and
$\lambda_{nn^{\prime}}^{\alpha}$, with the color indices $n$ running from 1 to
$N$. When considered as the worldvolume theory of $N$ D0-branes, the
eigenvalues of the matrix $X^{i}$ are interpreted as the positions of the
branes along the $i$th spatial dimension and the off-diagonal elements
$x^{i}_{nn^{\prime}}$ as modes of strings stretching between the $n$th and
$n^{\prime}$th branes polchinski1996tasi . See Fig. 1(a) for a diagram. As
before, MQM Lagrangians with stringy duals have the schematic form of
dimensionally reduced SYM:

$L=\Tr[\sum_{i}\dot{X}^{i2}+\sum_{ij}[X^{i},X^{j}]^{2}+\text{susy
terms}+\ldots].$ (1)

The commutator squared term may be recognized as the zero-mode piece of the
covariant SYM derivative. It is precisely this term that is responsible for
imbuing strings with a potential energy linear in their length (see
taylor2001m  for a review).

(a)

(b)

Figure 1: On the left, in Fig. 1(a), we have depicted a collection of
D0-branes with their interactions mediated by strings. The positions of the
D0-branes are associated to eigenvalues of Hermitian matrices $X^{i}$, denoted
$x^{i}_{n}$. The off-diagonal entries of $X^{i}$, denoted
$x^{i}_{nn^{\prime}}$, capture the state of string modes stretched between the
$n$th and $n^{\prime}$th branes. On the right, in Fig. 1(b), we have drawn a
tensor network state with pentagonal tensors (as in e.g.
pastawski2015holographic ). Contracted legs are drawn in purple, the free
external legs associated to a holographic boundary are black, and bulk
external legs are depicted by the maroon dots on the tensors. The two cartoons
bear a resemblance, and the purpose of this paper is to begin building the
technical relationship between them. Similar figures have been drawn in
Hampapura:2020hfg .

Since the discovery of holographic duality, there has been a concerted effort
to develop tractable toy models that capture important aspects of its physics.
Tensor networks pastawski2015holographic ; Hayden:2016cfa ;
swingle2012entanglement  have been one of the most celebrated and fruitful of
these models, and particularly excel in capturing the emergence of bulk
geometry from quantum correlation in the boundary theory. Among other
phenomena they capture holographic error correction, bulk reconstruction, the
Ryu-Takayanagi formula for boundary entropy ryu2006holographic , and a notion
of sub-AdS locality yang2016bidirectional . Suprisingly, they have even
recently been shown to recover bulk gravitational forces that appear to
directly arise from the entanglement pattern Sahay:2024vfw . They are
essentially lattice-regularized holography.

The key building block of of a tensor network is a generic state on $k$ qudits
with Hilbert space dimension $d$. The most general such state is described in
terms of a multilinear operator (i.e. tensor) $t^{\mu_{1}\ldots\mu_{k}}$:

$\ket{t}=\sum_{\\{\mu_{i}\\}=0}^{d-1}t^{\mu_{1}\ldots\mu_{k}}\ket{\mu_{1}}\otimes\ldots\otimes\ket{\mu_{k}}.$
(2)

These states are pictorially represented as a geometric shape with $k$
external legs, with one qudit associated to each leg (see Fig. 1(b) for an
example). General tensor network states are built by entangling qudits on the
external legs of different tensors. This is done by taking inner products of
the tensors $t^{\mu_{1}\ldots\mu_{k}}$. To represent entangled qudits, we
connect the legs of the tensors to form an internal edge. As such, tensor
network states are graphs (again, see Fig. 1(b)).

While they successfully capture many aspects of AdS/CFT, tensor networks
suffer from deficiencies. As they are defined on a lattice determined by some
graph, they lose the diffeomorphism invariance and background independence
that constitute general covariance of the bulk geometry (see Akers:2024wab
for an alternative recent approach for introducing diffeomorphism invariance
using the Chern-Simons description of 3d gravity). It is further unclear how
tensor networks properly describe the backreaction of RT surfaces on the
geometry Dong:2023kyr ; Akers:2024wab . While capable of probing some sub-AdS
lengthscale physics, they do not capture $\alpha^{\prime}$ corrections and the
UV/IR mixing we expect from physics at the string scale.

In this paper we argue that MQM states are naturally thought of as a
generalization of tensor network states. The construction is built on a simple
observation – the Hilbert space of MQM is identical in structure to the
Hilbert space of tensor networks. Moreover, the matrices $X^{i}$ defining
compact noncommutative manifolds have a natural interpretation as graph
adjacency matrices. Mapping tensor network states into $U(N)$ invariant MQM
models allows us to define tensor networks on noncommutative manifolds. The
$U(N)$ symmetry in the large $N$ limit contains area-preserving
diffeomorphisms (APDs) hoppe1982quantum  (see swain2004topology  for
discussion of some subtleties on this point), and a judicious choice of
Hamiltonian allows the background geometry to be determined dynamically, so we
may think of this construction as a step toward introducing general covariance
and background independence into tensor network models. This construction may
be viewed as a particular instantiation of the ideas in Cheng:2022ori .

###  1.1 Overview of the Construction

The construction in this paper relies on results and intuition from tensor
networks, noncommutative geometry, and noncommutative field theory. For the
convenience of a reader that may unfamiliar with one or more of these concepts
we give a brief sketch of the core idea, where we give the essential results
but leave their technical justification for §3.

As we review in §2, a noncommutative manifold $\mathcal{M}_{\theta}$ is
defined via a set of $N\times N$ matrices $X^{i}$. The $X^{i}$ are the
generators of the algebra of functions on $\mathcal{M}_{\theta}$, and are
themselves interpreted as the coordinate functions. One of the observations of
§2 is that $X^{i}$ carry a natural interpretation as the adjacency matrices of
a graph. For similar observations, see Hampapura:2020hfg ; Frenkel:2021yql .
An example of such a graph in the case of $N=7$ is drawn in Fig. 2.

Figure 2: A sketch of the graph interpretation of a noncommutative geometry
defined by three coordinate matrices $X$, $Y$, and $Z$, in the case $N=7$. We
have chosen to diagonalize $Z$. Each node represents the location of one of
the eigenvalues $z_{n}$ of $Z$. To each nonzero entry of $X$ and $Y$ we assign
an edge. $X$ edges are drawn above the nodes and in purple, $Y$ edges are
drawn below the nodes and in green. In this example entry $x_{1,3}$ is nonzero
and the entry $x_{1,7}$ is zero.

We wish to put a tensor network on this graph in a manner that respects the
$U(N)$ coordinate reparameterization symmetry of noncommutative manifolds. To
do so, we introduce $N\times N$ matrices of qubits $\lambda^{\alpha}$.
$\alpha$ is a flavor index that runs from $1$ to $N_{F}$. Our total Hilbert
space therefore consists of the $X^{i}$ and $\lambda^{\alpha}$ degrees of
freedom, and there is a natural adjoint action of the symmetry group $U(N)$ on
this Hilbert space. The explicit generators of this action are given in (21).
In view of Fig. 1(b), maximally entangling the
$\lambda_{nn^{\prime}}^{\alpha}$ qubits with the
$\lambda_{n^{\prime}n}^{\alpha}$ qubits creates a link between the $n$th and
$n^{\prime}$th nodes.

Figure 3: A sketch of the tensor network defined on a noncommutative geometry.
This graph should be interpreted the same way as Fig. 1(b). The black nodes
and edges represent the geometry as in Fig. 2. On this geometry only nearest
neighbor nodes are connected, such as in the fuzzy sphere (14). The colored
edges represent entanglement between qubits on the corresponding nodes – for
example, the edge $E_{2,4}$ represents a maximally entangled state between
$\lambda^{\alpha}_{2,4}$ and $\lambda^{\alpha}_{4,2}$. These colored edges are
the same objects as the edges of the graph and Fig. 1(b). The vertical dashed
lines are two different choices of entanglement cut. The entanglement across
these cuts is determined by the number of colored edges they intersect. The
entanglement across the leftmost cut is $3N_{f}\log 2$, and the entanglement
across the rightmost cut is $N_{f}\log 2$. For a more geometric depiction of
this network see Fig. 4. It may seem that we may only partition this network
along the $z$ coordinate, but it turns out the action of $U(N)$
transformations on the $X^{i}$ and $\lambda^{\alpha}$ matrices correspond to
area-preserving diffeomorphisms that allow us to consider any possible
subregion of the geometry (again see Fig. 4).

The key to the construction is how we determine the state of the tensor
network. We write $U(N)$-invariant MQM Hamiltonians on the $X^{i}$ and
$\lambda^{\alpha}$ degrees of freedom and consider their low energy states.
While we expect the relationship between MQM and tensor networks to be quite
general, for simplicity the specific Hamiltonians we consider in this paper
take the form

$\displaystyle H=H_{geom.}+\kappa\left(H_{1}+H_{2}\right),$ (3) $\displaystyle
H_{geom.}=\Tr[\sum_{i}{\Pi^{i}}^{2}+V(X^{i})],$ (4) $\displaystyle
H_{1}=i\sum_{\alpha}\Tr[(\lambda^{{\dagger}\alpha})^{2}-(\lambda^{\alpha})^{2}]$
(5) $\displaystyle
H_{2}=\sum_{\alpha}\Tr[\lambda^{\alpha{\dagger}}(g_{1}[X^{i},[X^{i},\lambda^{\alpha}]]+g_{2}[X^{i},[X^{i},[X^{j},[X^{j},\lambda^{\alpha}]]]]+\ldots)].$
(6)

$H_{geom.}$ is the Hamiltonian that sets the geometry of the noncommutative
manifold. It consists of a kinetic term (the $\Pi^{i}$ are the conjugate
momenta of the $X^{i}$) and a potential term $V(X^{i})$. This could be
something like mini-BMN Anous:2017mwr ; Han:2019wue , which has the fuzzy
sphere as a ground state. $H_{1}$ is designed to maximally entangle all
$\lambda_{nn^{\prime}}^{\alpha}$ qubits with their
$\lambda_{n^{\prime}n}^{\alpha}$ counterparts. $\kappa$ controls the strength
of the backreaction of the fermion modes $\lambda^{\alpha}$ on the geometry.
For the purposes of this paper we work in the regime of $\kappa\ll 1$, so as
not to worry about this backreaction.

The ground state of $H_{1}$ is an all-to-all connected tensor network graph.
$H_{2}$ is the coupling between the geometry and the tensor network qubits. On
its own, the ground state of $H_{2}$ is the completely unentangled state of
all the $\lambda^{\alpha}$ qubits – without $H_{1}$, this would define a
completely disconnected network. The commutator structure has a nice
interpretation in the noncommutative geometry – the commutator
$[X^{i},[X^{i},\cdot]]$ is the noncommutative analog of the Laplacian
$\Delta$. We should therefore read $H_{2}$ as

$H_{2}\sim\int_{\mathcal{M}_{\theta}}\sqrt{g}d^{D}\sigma\,\lambda^{{\dagger}}\left(g_{1}\Delta\lambda+g_{2}\Delta^{2}\lambda+\ldots\right),$
(7)

where the couplings $g_{i}$ set a derivative expansion in the Hamiltonian of
some spinor fields $\lambda^{\alpha}(\sigma)$ propagating on
$\mathcal{M}_{\theta}$. The structure of $H_{2}$ suppresses high momentum
modes of $\lambda^{\alpha}$, and by tuning $g_{i}$ we may set a momentum scale
for modes that can propagate in the low-energy subspace of (3). The nature of
noncommutative geometry is that momentum scales are related to length-scales –
high momentum modes spread out in the transverse directions to the direction
of propagation McGreevy:2000cw . By setting a momentum scale, we also set a
length-scale across which internal edges of the tensor network can stretch.
This is the scale of nonlocality of the network. The competition between
$H_{1}$ and $H_{2}$ is what sets the scale of nonlocality and creates the
local entanglement strucure of the states we consider.

It is crucial that the Hamiltonian of (3) is $U(N)$ invariant, so that we may
restrict ourselves to the subspace of $U(N)$-invariant states (i.e. we may
gauge this symmetry111In quantum mechanics, as opposed to quantum field
theory, we often have a choice about whether or not to take a symmetry to be
global symmetry or a gauge redundancy. It is only a question of whether we
restrict to the subspace of the Hilbert space annihilated by the symmetry
generators.). So far, as depicted in Fig. 3, we have described tensor networks
where each node has a fixed $z$ coordinate. It may seem that this only allows
us to partition our network along $z$. However, by acting with a unitary
transformation on the $X^{i}$ and $\lambda^{\alpha}$ matrices, we may
transform the state of our tensor network to a new basis that admits a Hilbert
space factorization along any curvilinear coordinate222There is a UV cutoff on
the curvature of coordinates we may consider given by $1/N$, the scale of
noncommutativity. However, it is important to note that unlike a standard
tensor network, we may consider continuous families of coordinates (and
therefore a continuous families of entanglement cuts). $f(x)$ of our choosing
(see Fig. 4).

A punchline here is that in the large $N$ limit, area-preserving
diffeomorphisms are a gauge symmetry of these matrix tensor networks.
Different $U(N)$ gauge fixing conditions behave as different coordinate
systems on $\mathcal{M}_{\theta}$, and each coordinate system presents us with
a different coordinate along which we may partition the network and compute an
entanglement entropy (again see Fig. 4). It is quite nontrivial that the
ground state of (3) has area-law entanglement for any choice of entanglement
cut333The cut should be weakly curved compared to the scale of nonlocality we
introduce in §§1.2.. This is the main technical result of this paper and what
is shown in §§3.3.2.

Figure 4: In this figure we depict how the tensor network drawn in Fig. 3
lives on a noncomutative geometry. We have chosen the fuzzy sphere for
simplicity. In (a) we have chosen to diagonalize the $Z$ coordinate, so each
node of the network is associated to a region of area $1/N$ localized to some
$z=z_{n}$. The entries of the other matrix entries $X^{i}$ that are not
diagonal are closely related by the lengths of the borders between these
regions in a manner described in §2. To go from (a) to (b), we have performed
a change of basis implemented by some unitary $U\in U(N)$. In the new basis a
different curvilinear coordinate $F=f(X,Y,Z)$ is diagonalized, so now each
area $1/N$ region is localized to some $f(x,y,z)=f_{n}$. Again, the entries of
$X^{i}$ are related to the lengths of the borders between these regions. This
is the noncommutative geometry analog of an area-preserving diffeomorphism
(APD) (see the discussions in hoppe1982quantum ; Frenkel:2021yql ;
swain2004topology ). The same unitary $U$ that implements this APD also
transforms the state of the fermions $\lambda^{\alpha}$, so the tensor network
state transforms covariantly with this coordinate reparameterization. In both
(a) and (b) the total number of tensor network edges (i.e. sets of entangled
$\lambda^{\alpha}_{nn^{\prime}}$ qubits) crossing each boundary between
regions is equal to the length of this boundary, so the area law entanglement
properties of the tensor network are preserved in any choice of coordinate
system.

###  1.2 A Planck Scale and a String Scale

The holographic bulk of D$p$-brane systems has three essential length-
scales444More precisely, two independent unitless ratios of length-scales. –
the scale of the geometry (i.e. the AdS radius) $r_{\Lambda}$, the string
scale $l_{s}\sim\sqrt{\alpha^{\prime}}$, and the Planck length $l_{p}$. The
string scale in particular is the scale of nonlocality – the Ryu-Takayanagi
formula picks up $\alpha^{\prime}$ corrections as extremal surfaces become
more and more curved Dong:2013qoa ; Wall:2015raa , and we expect that the area
law contribution will cease to dominate below the string scale. In
Maldacena:1997re ; itzhaki1998supergravity , it is emphasized that the ratios
of these length-scales are given by

$\frac{l_{s}}{r_{\Lambda}}=\lambda^{-1/3}=\left(g_{\text{YM}}^{2}N\right)^{-1/3},\quad\frac{l_{p}}{r_{\Lambda}}=N^{-1/4}.$
(8)

$\lambda$ is the ‘t Hooft coupling, and sets the string scale in the bulk. In
the holographic dualities considered in Maldacena:1997re ;
itzhaki1998supergravity  and related works, the ‘t Hooft coupling (and
therefore the ratio between the string scale and the AdS length-scale) is
$O(1)$ in $N$ scaling. The Planck length $l_{p}$, meanwhile, is separated from
the AdS lenghtscale and the string scale by nontrivial factors of $N$.

The tensor networks we consider naturally impose on us three length-scales
that have a behavior strikingly similar to those in (8). Consider Figs. 3 and
4, and call the size of the geometry $\tilde{r}_{\Lambda}$. One natural
length-scale to consider is the typical separation between nodes of the
network, or equivalently between D0-branes in Fig. 1(a). This is analogous to
$l_{p}$ – it is the shortest possible distance scale, and sets the total
number of degrees of freedom. The other length-scale is the typical lengths of
the tensor network edges – the purple $\lambda_{nn^{\prime}}$ edges in the two
figures. They need not only connect nearest-neighbor nodes, and may in
principle stretch long distances along the geometry. This is the scale of
nonlocality of the network, and behaves similarly to the string scale $l_{s}$.
We refer to these scales as $\tilde{l}_{p}$ and $\tilde{l}_{s}$ respectively.
The ratio of these two length-scales is set by the structure of $H_{2}$ in
(3), and in particular by the strength and structure of the couplings $g_{i}$
in the derivative expansion.

Note that in light of the D0-brane origin of holographic MQM (as in Fig.
1(a)), the interpretation of this scale of nonlocality as a string scale might
be taken quite literally. Below this scale, the off-diagonal matrix elements
$x^{i}_{nn^{\prime}}$ and $\lambda_{nn^{\prime}}^{\alpha}$ modes connecting
branes arehighly excited and entangled with nearby degrees of freedom. Above
this scale, $x^{i}_{nn^{\prime}}$ and $\lambda_{nn^{\prime}}^{\alpha}$ are
suppressed to their ground states. This exactly matches the behavior of off-
diagonal matrix elements (i.e. string modes connecting branes) in D-brane
matrix quantum mechanics taylor2001m ; Polchinski:1999br .

To find the nice property that area law entanglement is simply given by the
number of tensor network legs crossing a given entanglement cut, we find in §3
that we must consider the regime

$N^{-1/2}\ll\frac{\tilde{l}_{s}}{\tilde{r}_{\Lambda}}\ll 1.$ (9)

In particular, the ‘string scale’ must be much greater than the ‘Planck
scale’. It is perhaps an interesting coincidence that this regime includes the
analog of the ‘t Hooft limit. The string scale of the matrix tensor networks
we define depends on the Hamiltonian we choose, and we will find area law
entanglement only above the string scale we set.

###  1.3 Layout of the Paper

In §2 we review the relevant main ideas of noncommutative geometry. We also
argue that the matrices representing the noncommutative coordinate functions
have a natural interpretation as the adjacency matrices of a graph.

§3 contains the main results of the paper. We embed the structure of tensor
networks into the MQM Hilbert space. We find that these noncommutative tensor
networks have a natural scale of nonlocality, mimicking a string scale. This
string scale may be tuned arbitarily small in comparison to the size of the
geometry in the large $N$ limit, similar to the limit of strong ‘t Hooft
coupling. We show that the low-energy states of reasonable $U(N)$-invariant
MQM Hamiltonians produce states with area-law entanglement above this string
scale, demonstrating that local geometry may emerge dynamically for these
models.

In §4 we discuss the results and comment on future directions.

##  2 Fuzzy Coordinate Functions as Adjacency Matrices

Noncommutative geometry has a storied history within string theory in general
seiberg1999string ; douglas2001noncommutative ; szabo2003quantum ;
witten1986non ; Ardalan:1998ce  and matrix quantum mechanics in particular
taylor2001m ; Steinacker_2010 ; Steinacker:2011ix ; Han:2019wue . The bulk of
holographic systems, especially the compact dimensions, exhibit phenomena
similar to physics on noncommutative backrounds McGreevy:2000cw ; ho2000large
; ho2001fuzzy . In this section we briefly review how noncommutative manifolds
are defined and how they naturally emerge in MQM systems.

Noncommutative manifolds (which we denote $\mathcal{M}_{\theta}$), much like
their commutative counterparts, are defined through their algebra of functions
and metric structure. The essential aspect is that the coordinate functions
$x^{i}$ parameterizing $\mathcal{M}_{\theta}$ do not commute, and instead obey
some relations

$[x^{i},x^{j}]=\theta^{ij}(x).$ (10)

The algebra of functions on $\mathcal{M}_{\theta}$ is then generated by all
sums and products of the $x^{i}$. Perhaps surprisingly, from this basic
starting point we may introduce a differential and integral structure.
Consider some function $f(x)$ on $\mathcal{M}_{\theta}$. Its derivatives are
given by

$[x^{i},f(x)]=\theta^{ij}\partial_{j}f(x).$ (11)

This definition is particularly natural because commutators obey the Leibniz
rule. The Laplacian, which in turn determines the metric structure, is defined
as

$\Delta f(x)=\sum_{i}[x^{i},[x^{i},f(x)]].$ (12)

Integrals are given by traces –

$\int_{\mathcal{M}_{\theta}}\sqrt{g}d^{d}x\,f(x^{i})=\Tr[f].$ (13)

The simplest way to work with noncommutative manifolds is by finding a
representation of their algebra. The prototypical example is the fuzzy sphere,
which is defined by the relations

$X^{2}+Y^{2}+Z^{2}=R^{2}\mathbb{1},\quad[X^{i},X^{j}]=i\epsilon^{ijk}X^{k}.$
(14)

We recognize these expressions as the defining relations of $\mathfrak{su}(2)$
representations, and may find an explicit realization of the fuzzy sphere by
choosing an irrep of $\mathfrak{su}(2)$ –

$Z=\sum_{m=-N/2}^{N/2}|m)(m|,\quad
X+iY=\sum_{m=-N/2}^{N/2-1}\sqrt{(N/2+m)(N/2-m+1)}|m)(m+1|.$ (15)

This is the basis in which $Z$ is diagonal and its eigenvalues are ordered. We
interpret the spectrum of $Z$ as the distribution of $D0$-brane positions over
the $z$ axis. That the eigenvalues are evenly spaced is a signature of the
commutative sphere measure being $\sin\theta d\theta d\phi=d\phi dz$.

With an explicit representation given by $N\times N$ matrices, there is a
close relationship between derivatives and length scales. In particular,
consider some function on a noncommutative manifold represented by a matrix
$F$ with entries $F_{nn^{\prime}}$. Its commutator with a diagonal coordinate
matrix (say $Z$) is

$[Z,[Z,F]]_{nn^{\prime}}=(z_{n}-z_{n}^{\prime})^{2}F_{nn^{\prime}}.$ (16)

The ratio $\Tr[F^{{\dagger}}[Z,[Z,F]]]/\Tr[F^{{\dagger}}F]$ therefore measures
the typical distance between $z$ coordinates that elements of $F$ stretch
between, weighted by the size of the elements $F_{nn^{\prime}}$. By extension,
the noncommutative gradient $\Tr[F[X,[X,F]]+F[Y,[Y,F]]+F[Z,[Z,F]]]$ measures
the average length squared
$(x_{n}-x_{n^{\prime}})^{2}+(y_{n}-y_{n^{\prime}})^{2}+(z_{n}-z_{n^{\prime}})^{2}$
of the matrix elements excited in $F$, and therefore sets a length-scale for
$F$.

In the basis where $Z$ is diagonal, $X$ and $Y$ are sparse matrices with
support only on the diagonals adjacent to the main diagonal. This means that
e.g. in Fig. 5, $X_{nn^{\prime}}$ is only nonzero in a particular $U(N)$ basis
if $n$ and $n^{\prime}$ are adjacent D0 branes in the coordinate system
corresponding to this basis. In this sense, $X$ and $Y$ precisely behave as
adjacency matrices of a graph. Digging a bit deeper, $|x_{nn^{\prime}}|$ is
precisely the length of the perimeter between the $n$th and $n^{\prime}$th
D0-branes. The reason for this is that projection matrices $\Theta_{\Sigma}$
are the noncommutative analog of step functions, and their derivatives
$[X^{i},\Theta_{\Sigma}]$ (the natural noncommutative analog of a delta
function) pick out the off-diagonal blocks of the coordinate matrices $X^{i}$.

Figure 5: The $Z$ basis, pictured on the left, is one in which each $D0$-brane
on the fuzzy sphere has a well-defined $Z$ coordinate. We can transform to
another basis by diagonalizing a different Hermitian matrix $F$. We interpret
$F$ as a different curvilinear coordinate $f(x^{i})$, and the basis
transformation as taking us to a new coordinate system where each $D0$ brane
has a well defined $f$ coordinate. This coordinate change is interpreted as
the fuzzy regularization of an area-preserving diffeomorphism (as in
hoppe1982quantum ; swain2004topology ). In each basis, the coordinate matrices
$X^{i}$ only have nonzero elements (to leading order in $1/N$) if the $n$th
and $n^{\prime}$th $D0$-branes have adjacent $f$ coordinates (i.e. share a
boundary on the right hand figure).

So far we have taken the coordinate functions $X^{i}$ to be fixed. One way in
which noncommutative geometry naturally arises in MQM is by interpreting the
dynamical Hermitian matrices in (1) as the coordinate matrices of a
noncommutative manifold $\mathcal{M}_{\theta}$. Because the metric structure
is determined by the $X^{i}$, if we make the fuzzy coordinates dynamical we
also make the metric dynamical. The fuzzy sphere, for example, arises as one
possible vacuum state in the BMN (or mini-BMN Han:2019wue ; Anous:2017mwr )
matrix model. (1) is precisely designed to energetically suppress string modes
propagating between distant D-branes, and we can see this effect explicitly
from (14) in that only the matrix elements between adjacent D-branes are
populated with nonzero entries. In this way, it is natural for low-energy
states of MQM models to behave as dynamical adjacency matrices of local graphs
that only have connections between nearby nodes. Because the geometry and
topology of these graphs are determined by the dynamics of the theory, they
come with a manifest sense of background independence.

Intuition tells us that in low energy states of quantum systems degrees of
freedom which are more strongly coupled are more entangled. Moreover, it is
often tempting to interpret strings stretching across an entanglement cut as
extended degrees of freedom which should be cut to produce entanglement edge
modes Susskind:1994sm ; levin2006detecting ; ghosh2015entanglement ;
hampapura2021target ; Frenkel:2021yql ; frenkel2023emergent . The way in which
we make this intuition precise is by interpreting the graph arising from the
adjacency matrix picture of $X^{i}$ as the graph of a tensor network. This is
the task we take up in §3.

###  2.1 Subregions of Noncommutative Manifolds

Considering subalgebras and entanglement on the tensor networks we are
defining is closely related to the idea of considering geometric subregions of
noncommutative geometries. On a typical commutative manifold, a subregion
$\Sigma$ is defined via a characteristic function $\chi_{\Sigma}(x^{i})$.
$\chi_{\Sigma}(x^{i})$ evaluates to 1 inside $\Sigma$ and evaluates to 0 on
the complement $\bar{\Sigma}$. Any function $f(x^{i})$ may be decomposed into
two parts:

$f_{\Sigma}(x^{i}):=\chi_{\Sigma}(x^{i})f(x^{i}),\quad
f_{\bar{\Sigma}}(x^{i}):=(1-\chi_{\Sigma}(x^{i}))f(x^{i}).$ (17)

Figure 6: A subregion $\Sigma$ of a compact commutative manifold. The
characteristic function $\chi_{\Sigma}(x^{i})$ evaluates to 1 inside $\Sigma$
and evaluates to 0 outside of $\Sigma$. When we consider noncommutative
manifolds, $\chi_{\Sigma}(x^{i})$ will be promoted to a projection matrix
$\Theta_{\Sigma}$ and all functions $F$ will decompose into four distinct
pieces (as in Fig. 7.) For an analogous decomposition for the graph structure
we described in the previous section, see Fig. 8.

On a noncommutative manifold characteristic functions are promoted to
projection matrices $\Theta_{\Sigma}$ satisfying
$\Theta_{\Sigma}^{2}=\Theta_{\Sigma}$. Unlike the commutative case, a
noncommutative function $F$ has four distinct pieces to consider:

$\begin{split}&F_{\Sigma\Sigma}:=\Theta_{\Sigma}F\Theta_{\Sigma},\quad
F_{\Sigma\bar{\Sigma}}:=\Theta_{\Sigma}F(1-\Theta_{\Sigma}),\\\
&F_{\bar{\Sigma}\Sigma}:=(1-\Theta_{\Sigma})F\Theta_{\Sigma},\quad
F_{\bar{\Sigma}\bar{\Sigma}}:=(1-\Theta_{\Sigma})F(1-\Theta_{\Sigma}).\end{split}$
(18)

If we choose a basis where $\Theta_{\Sigma}$ is diagonal with ordered
eigenvalues, this is nothing more than a block decomposition of the $N\times
N$ matrix $F$ (see Fig 7). The blocks $F_{\Sigma\bar{\Sigma}}$ and
$F_{\bar{\Sigma}\Sigma}$ are associated to the boundary of the noncommutative
manifold, and in particular should be thought of as modes that cross the
boundary (see Fig. 8).

Figure 7: A depiction of the decomposition of a function on a noncommutative
manifold. As depicted in Fig. 8, $F_{\Sigma\Sigma}$ is the piece of the
function completely contained in the interior of $\Sigma$,
$F_{\bar{\Sigma}\bar{\Sigma}}$ is the piece in the interior of $\bar{\Sigma}$,
and the $F_{\bar{\Sigma}\Sigma}$, $F_{\Sigma\bar{\Sigma}}$ pieces are those
that are in contact or cross the boundary of $\Sigma$ and may interact with
modes in both regions. Figure 8: A depiction, in terms of the graph language
of Figs. 2-4, of the structure of subregions of a noncommutative manifold.
This is one way to picture the noncommutative analog of Fig. 6. The block
decomposition of Fig. 7 should be kept in mind. The vertical dashed line is
again the entanglement cut separating subregion $\Sigma$ on the left and
subregion $\bar{\Sigma}$ on the right. $X^{i}_{\Sigma\Sigma}$, as an example,
contains all the black geometric graph edges entirely contained within the
region $\Sigma$. Similarly, $\lambda^{\alpha}_{\bar{\Sigma}\bar{\Sigma}}$
consists of all of the purple tensor network edges that are entirely contained
in $\bar{\Sigma}$. Any matrix subblock with mixed indices such as
$X^{i}_{\sigma\bar{\Sigma}}$ or $\lambda^{\alpha}_{\Sigma\bar{\Sigma}}$
consists of edges that cross the entanglement cut.

To consider RT surfaces, instead of minimizing over a discrete set of
entanglement cuts we must minimize over the continuous set of projection
matrices. This type of minimization will appear shortly in ffhs:2024xx , and
would be interesting to directly apply to entropy calculations and bulk
reconstruction in the tensor networks we are considering.

##  3 Embedding Tensor Networks into MQM

###  3.1 The Embedding Map

We interpret the color indices $n$ of the matrices as nodes in the network. To
construct quantum states, we introduce dynamical degrees of freedom in the
form of $N_{f}$ matrices of spinors $\lambda^{a}$ and $N_{f}$ vectors of
spinors $\psi^{a}$. For simplicity, instead of treating these spins as
fermions we take their (anti-)commutation relations to be

$\begin{split}&\\{\lambda_{nn^{\prime}},(\lambda_{nn^{\prime}})^{{\dagger}}\\}=\\{\psi_{n},\psi_{n}^{{\dagger}}\\}=1\\\
&[\lambda_{ab},\lambda_{cd}]=0\text{ if $a\neq c$ or $b\neq
d$},\quad[\psi_{a},\psi_{b}]=0\text{ if $a\neq b$}.\end{split}$ (19)

We take this unusual choice to simplify the representations of the operators,
so that we may take e.g.

$\lambda_{nn^{\prime}}=\begin{bmatrix}0&1\\\
0&0\end{bmatrix}\otimes\bigotimes_{(n_{1},n_{2})\neq(n,n^{\prime})}\mathbb{1}_{nn^{\prime}}\quad\lambda_{nn^{\prime}}^{{\dagger}}=\begin{bmatrix}0&0\\\
1&0\end{bmatrix}\otimes\bigotimes_{(n_{1},n_{2})\neq(n,n^{\prime})}\mathbb{1}_{nn^{\prime}}.$
(20)

With these conventions, the generators of $U(N)$ transformations on the $X$,
$\lambda$, $\psi$ variables take the form

$\begin{split}&G^{X}_{nn^{\prime}}=2i\sum_{n^{\prime\prime}}\left(X_{nn^{\prime}}\Pi_{n^{\prime}n^{\prime\prime}}-X_{n^{\prime\prime}n^{\prime}}\Pi_{nn^{\prime}}\right),\\\
&G^{\lambda}_{nn^{\prime}}=2i\sum_{n^{\prime\prime}}\left(\lambda_{nn^{\prime\prime}}\lambda^{{\dagger}}_{n^{\prime\prime}n^{\prime}}-\lambda^{{\dagger}}_{nn^{\prime\prime}}\lambda_{n^{\prime\prime}n^{\prime}}\right)-4i\delta_{n,n^{\prime}}\lambda_{nn}^{{\dagger}}\lambda_{nn},\\\
&G^{\psi}_{nn^{\prime}}=\psi^{{\dagger}}_{n^{\prime}}\psi_{n}.\end{split}$
(21)

The $N_{f}$ creation operators $\lambda_{nn^{\prime}}^{a{\dagger}}$ act on the
$D:=2^{N_{f}}$ dimensional Hilbert space $\mathbb{H}_{nn^{\prime}}$ that lives
on the link connecting nodes $n$ and $n^{\prime}$. On each factor of the
Hilbert space, we label as $\ket{0}$ the states annihilated by
$\lambda_{nn^{\prime}}$ and $\ket{1}$ the states annihilated by
$\lambda^{{\dagger}}_{nn^{\prime}}$. We interpret the dynamical variables
$\psi_{n}$ as living on ‘external’ legs of the tensor network with
corresponding spaces $\mathbb{H}_{n}$. For holographic models these can be
interpreted as either boundary or bulk external legs, depending on where the
node $n$ is located in the fuzzy geometry. For notational convenience we
introduce a basis of states $\ket{\mu}_{nn^{\prime}}$ or $\ket{\mu}_{n}$ on
the corresponding Hilbert spaces, with the index $\mu$ running from 0 to
$D-1$.

Figure 9: A depiction of how the MQM degrees of freedom in (19) map onto a
tensor network as in pastawski2015holographic  and Fig. 1(b).
$\lambda^{\alpha}$ set the state of internal legs, and $\psi^{\alpha}$ the
state of external legs (either bulk or boundary).

We have now introduced all the degrees of freedom necessary to construct any
tensor network state depending on what state $\ket{\chi}_{nn^{\prime}}$ we fix
on the products $\mathbb{H}_{nn^{\prime}}\otimes\mathbb{H}_{n^{\prime}n}$. For
the most straightforward type of network, we can create a link (or a lack of a
link) between the nodes $n$ and $n^{\prime}$ by choosing one of two states on
these spaces:

$\begin{split}\ket{\chi}_{nn^{\prime}}&=\frac{1}{\sqrt{D}}\sum_{\mu}\ket{\mu}_{nn^{\prime}}\otimes\ket{\mu}_{n^{\prime}n}\quad\Longleftrightarrow\quad\text{(there
{is} a link between nodes $n$ and $n^{\prime}$)},\\\
\ket{\chi}_{nn^{\prime}}&=\ket{0}_{nn^{\prime}}\otimes\ket{0}_{n^{\prime}n}\quad\Longleftrightarrow\quad\text{(there
{is not} a link between nodes $n$ and $n^{\prime}$)}.\end{split}$ (22)

Of course, as Hayden:2016cfa ; Cheng:2022ori  point out, it is fruitful to
consider more general types of connections between nodes besides maximally
entangled states. With this in mind, we generally take a state in
$\mathbb{H}_{\lambda}$ as defining the graph the tensor network lives on, so
we denote elements of $\mathbb{H}_{\lambda}$ as $\ket{\Gamma}$.

To introduce random tensor networks, random tensors are defined on the states

$\ket{V_{n}}\in\tilde{\mathbb{H}}_{n}:=\mathbb{H}_{n}\otimes\bigotimes_{n^{\prime}}\mathbb{H}_{nn^{\prime}},\quad\dim\tilde{\mathbb{H}}_{n}=D^{N+1}=2^{(N+1)N_{f}}.$
(23)

Correspondingly, given a ‘graph state’ $\ket{\Gamma}$ the state on the
external links is given by

$\ket{\Psi}=\bra{\Gamma}\bigotimes_{n}\ket{V_{n}}.$ (24)

###  3.2 Partitions and Subalgebras

Now that we have described the Hilbert space of the theory (22), we explain
how to consider subregions of MQM systems and matrix tensor networks in
particular. This story is essentially the same as that of target space
entanglement in MQM das2020bulk ; Das:2020jhy ; karczmarek2014entanglement ;
hampapura2021target ; Han:2019wue ; frenkel2023emergent ; Frenkel:2021yql ,
but contains a crucial unique ingredient in how we treat the off-diagonal
matrix elements.

Remember that each matrix entry $nn^{\prime}$ has a Hilbert space associated
to it. This is true of both the $X^{i}$ and the $\lambda^{\alpha}$. The
essential idea, then, is to choose an appropriate basis in which to write our
vectors $\psi^{\alpha}$ and matrices555This should be thought of as a gauge
fixing condition for our $U(N)$ invariant wavefunctions. and split the system
along the row index (see Fig. 10). More precisely, we write

$\mathbb{H}=\mathbb{H}_{\Sigma}\otimes\mathbb{H}_{\bar{\Sigma}},\quad\mathbb{H}_{\Sigma}=\bigotimes_{n\leq
M}\mathbb{H}_{n}\otimes\bigotimes_{n^{\prime}}\mathbb{H}_{nn^{\prime}},\quad\mathbb{H}_{\bar{\Sigma}}=\bigotimes_{n>M}\mathbb{H}_{n}\otimes\bigotimes_{n^{\prime}}\mathbb{H}_{nn^{\prime}}.$
(25)

The most convenient way to split a matrix is to introduce a projection matrix
$\Theta_{\Sigma}$ on color space. The label $\Sigma$ may be associated to a
subregion of the noncommutative manfiold of area $M:=\Tr[\Theta_{\Sigma}]$
(see §§2.1). $\Theta_{\Sigma}$ is the noncommutative analog of a
characteristic function, which evaluates to 1 on some subset $\Sigma$ of a
manifold and evaluates to 0 on the complement $\bar{\Sigma}$ of $\Sigma$.

Figure 10: A schematic of how to partition the degrees of freedom of a matrix
tensor network. We write the state in a particular $U(N)$ frame, then take the
top $M$ rows of all matrix degrees of freedom $X^{i}$ and $\lambda^{\alpha}$.
The entanglement cut is represented by the dashed line across the figure. The
dominant contribution to the entanglement entropy is the entanglement of
qubits in the $\Sigma\bar{\Sigma}$ block with those in the
$\bar{\Sigma}\Sigma$ block. To emphasize this, the purple lines connecting the
$\Sigma\bar{\Sigma}$ and $\bar{\Sigma}\Sigma$ blocks are the same as the edges
of the tensor network diagram in Figs. 3 and 8 that are cut by the
entanglement cut.

###  3.3 Dynamically Introducing Locality

Thus we have introduced the tools to in principle describe tensor networks on
potentially all-to-all connected (or otherwise nonlocal) graphs. To make
contact with the typical notion of tensor network, however, we want there to
be a sense in which only nearby nodes are connected. We could by hand choose a
graph state $\ket{\Gamma}\in\mathbb{H}_{\lambda}$ that reproduces a local
network, but this simply reduces us to the construction explored in
Hayden:2016cfa . In this section consider some simple MQM-like Hamiltonians
and argue that their low-energy states have the properties of tensor networks
with only geometrically nearby nodes connected.

We consider two classes of MQM models. The first are based on Matrix Quantum
Hall models Susskind:2001fb ; Hellerman:2001rj ; Polychronakos:2001mi ;
Tong:2015xaa  where there are two coordinate matrices $X$ and $Y$ that commute
at a classical level but not as quantum operators. This is the usual way by
which noncommutative geometry arises in quantum hall models, as $x$ and $y$
become conjugate phase space variables in the limit of a strong magnetic
field. We refer to this setup as classically commutative, quantum
noncommutative, and explore it in §3.3.1. This is the notion of noncommutative
geometry considered in e.g. Zhu:2022gjc .

The second class consists of field theories on honest noncommutative
geometries with classically noncommuting coordinates. We refer to this as the
classically noncommutative setup and discuss it in §3.3.2. This setup has a
much richer structure and is closer in spirit to MQM with multiple matrices.

####  3.3.1 Classically commutative, Quantum noncommutative geometries

As in Polychronakos:2001mi ; Tong:2015xaa  we consider a complex matrix
$Z=X+iY$. We make the assumption that $[Z,Z^{{\dagger}}]=0$, so that $Z$ is
unitarily diagonalizable. We first consider the case where the matrix $Z$ is
fixed and has eigenvalues $z_{i}$. For simplicity, we drop the $\alpha$ index
on the $\lambda$ and $\psi$ spinors for now. The Hamiltonian we consider is

$H=\Tr[i(\lambda^{{\dagger}2}-\lambda^{2})+\lambda^{{\dagger}}[Z,[Z^{{\dagger}},\lambda]]].$
(26)

We choose to work in a basis where $Z$ is diagonal. In components, this
Hamiltonian is written as

$\begin{split}&H=\sum_{nn^{\prime}}H_{nn^{\prime}},\quad
H_{nn^{\prime}}:=\left(H_{1,nn^{\prime}}+|z_{n}-z_{n^{\prime}}|^{2}H_{2,nn^{\prime}}\right)\\\
&H_{1,nn^{\prime}}:=2i\sum_{nn^{\prime}}\left(\lambda_{nn^{\prime}}^{{\dagger}}\lambda^{{\dagger}}_{n^{\prime}n}-\lambda_{nn^{\prime}}\lambda_{n^{\prime}n}\right),\quad
H_{2,nn^{\prime}}:=2\sum_{nn^{\prime}}\lambda_{nn^{\prime}}^{{\dagger}}\lambda_{nn^{\prime}}\end{split}$
(27)

This Hamiltonian decouples on each Hilbert space factor
$\mathbb{H}_{nn^{\prime}}\otimes\mathbb{H}_{n^{\prime}n}$, where each
$\mathbb{H}_{n^{\prime}n}$ is two dimensional. The ground state of
$H_{1,nn^{\prime}}$ is the maximally entangled state
$\frac{1}{\sqrt{2}}\left(\ket{11}+\ket{00}\right)$. The ground state of
$H_{2,nn^{\prime}}$ is $\ket{00}$. The entanglement $S$ of the ground state of
$H_{1,nn^{\prime}}+gH_{2,nn^{\prime}}$ as a function of the coupling $g$ is

$S(g)=-\frac{1}{4}\left|\sqrt{4+g^{2}}-g\right|^{2}\log(\frac{1}{4}\left|\sqrt{4+g^{2}}-g\right|^{2}).$
(28)

Its behavior is shown in Fig. 11, but the only important point about its
functional form is that it is $O(1)$ for $g<1$ and falls off as
$\log(eg^{2})/g^{2}$ for $g\gg 1$. The overall ground state wavefunction will
simply be a tensor product of the ground states of each $H_{nn^{\prime}}$.

Figure 11: A plot of (28) – the entanglement of the ground state of the
Hamiltonian $H_{1}+gH_{2}$ as a function of $g$ as in (27). For large $g$,
this function falls off as $S(g)\approx\log(eg^{2})/g^{2}$.

So far we have not moved beyond familiar tensor network graphs. To do so, we
introduce dynamics on $Z$. We take the Hamiltonian studied in
Polychronakos:2001mi ; Tong:2015xaa , which we refer to as the matrix quantum
hall (MQH) model.

$H=\Tr[Z^{{\dagger}}Z+2i\left(\lambda^{{\dagger}2}-\lambda^{2}\right)+\lambda^{{\dagger}}[Z^{{\dagger}},[Z,\lambda]]].$
(29)

The matrix elements of $Z$ obey the commutation relations

$[Z_{ab},(Z^{{\dagger}})_{cd}]=\delta_{ad}\delta_{bc},$ (30)

so without the coupling to $\lambda$ the system decomposes into $N^{2}$
variables obeying the harmonic oscillator algebra.

We treat the system by passing to a collective field description, where we
define the eigenvalue density

$\rho(z)=\sum_{n}\delta(z-z_{n}).$ (31)

In terms of $\rho$ the Hamiltonian takes the form666I would like to thank
Laurens Lootens for pointing out the similarity to continuous tensor network
models as in Tilloy:2018gvo . One may view this construction as a fuzzy-
regulated version of a continuous tensor network.

$\begin{split}H=&\int\rho(z)d^{2}z\,|z|^{2}+\\\
&+2i\kappa\int\rho(z)\rho(z^{\prime})d^{2}zd^{2}z^{\prime}\,\left[\lambda^{{\dagger}}(z,z^{\prime})\lambda^{{\dagger}}(z_{2},z_{1})-\lambda(z_{1},z_{2})\lambda(z,z^{\prime})\right]+\\\
&+\kappa\int\rho(z_{1})\rho(z_{2})d^{2}zd^{2}z^{\prime}\,|z-z^{\prime}|^{2}\lambda^{{\dagger}}(z,z^{\prime})\lambda(z,z^{\prime}).\end{split}$
(32)

We have introduced an additional coupling constant $\kappa$, which we may take
to be small to ensure the bilocal fields $\lambda(z,z^{\prime})$ do not
strongly backreact on the geometry. Each $\lambda(z,z^{\prime})$ mode now
decouples, and the entanglement of a subregion $\Sigma$ with its complement
$\bar{\Sigma}$ is now calculated to leading order in $N$ as

$S_{\Sigma}=\int_{\Sigma}d^{2}z\int_{\bar{\Sigma}}d^{2}z^{\prime}\rho(z)\rho(z^{\prime})s(|z-z^{\prime}|).$
(33)

Noting that we may plug into (28) $g=|z-z^{\prime}|^{2}$, $s(|z-z^{\prime}|)$
falls off as

$s(|z-z^{\prime}|)\approx\frac{\log(e|z-z^{\prime}|^{4})}{|z-z^{\prime}|^{4}}\,,\quad|z-z^{\prime}|\gg
1.$ (34)

With this in mind, the integral in (33) is dominated by the UV contribution
localized to the entanglement cut, so we find an area law entanglement.

####  3.3.2 Classically noncommutative models

We turn to a case where the background geometry on which the tensor network
lives is honestly noncommutative at a classical level, as in §2. For clarity
and concreteness we focus on the fuzzy sphere as in (14) Madore:1991bw , but
the results of this section are readily extended to any weakly curved
noncommutative geometry with small noncommutativity parameter.

We first consider the case where the background geometry is fixed, so the
$X^{i}$ don’t fluctuate. The Hamiltonian on the $\lambda$ degrees of freedom
takes the form as in (3)

$\begin{split}&H=H_{1}+H_{2}\\\
&H_{1}=i\sum_{\alpha}\Tr[(\lambda^{{\dagger}})^{2}-\lambda^{2}]\\\
&H_{2}=\sum_{\alpha}\Tr[\lambda^{{\dagger}}(g_{1}[X^{i},[X^{i},\lambda]]+g_{2}[X^{i},[X^{i},[X^{j},[X^{j},\lambda]]]]+\ldots)].\end{split}$
(35)

Sums over the repeated $i,j$ indices are implicit. As in §§3.3.1, we have
dropped the $\alpha$ index as it just comes along for the ride. The ground
state of (35) is again easy to determine, this time by decomposing $\lambda$
into matrix spherical harmonics Taylor:1999qk ; Han:2019wue :

$\begin{split}&\lambda=\sum_{jm}\lambda_{jm}\hat{Y}_{jm},\\\
&H=\sum_{j,m>0}\lambda^{{\dagger}}_{j,m}\lambda^{{\dagger}}_{j,-m}-\lambda_{j,m}\lambda_{j,-m}+\sum_{jm}\epsilon(j)\lambda_{jm}^{{\dagger}}\lambda_{jm},\\\
&\epsilon(j)=\sum_{k}g_{k}j^{k}(j+1)^{k}.\end{split}$ (36)

The entanglement between the $\hat{Y}_{jm}$ mode and the $\hat{Y}_{j,-m}$ mode
(for $m\geq 1$) is obtained by plugging $\epsilon(j)$ in the place of $g$ in
(28).

It is most instructive to analyze this system in the diagonal $Z$ basis, where
$\hat{Y}_{jm}$ is only supported on the $m$th diagonal (see Fig. 12). It is
clear from the structure of (36) that the $\hat{Y}_{jm}$ fermion mode will
only be entangled with the $\hat{Y}_{j,-m}$ mode. Because the matrix entries
of matrix spherical harmonics are a discretization of the usual commutative
spherical harmonics $y_{jm}(\theta,\phi)$, we may evaluate traces of products
of the $\hat{Y}_{jm}$ in terms of integrals of $y_{jm}(\theta,\phi)$ up to
corrections suppressed by $O(1/(N-j))$.

Figure 12: We demonstrate the structure of matrix spherical harmonics. In the
diagonal $Z$ basis, the matrix spherical harmonic $\hat{Y}_{jm}$ is supported
only on the $m$th diagonal (as depicted in (a)). In (b) we depict the
structure of the matrix entries themselves along this diagonal. They form a
discretized version of the continuum spherical harmonics
$y_{jm}(\theta,\phi)$, evaluted at $\phi=0$ and the appropriate points
$\theta_{k}$. There are $N-m$ entries along the $m$th diagonal, so the length-
scale of discretization of the spherical harmonic is $1/(N-m)$.

The entanglement structure of the ground state depends on the structure of
$\epsilon(j)$. Modes $\hat{Y}_{jm}$ for which $\epsilon(j)$ is small will have
large entanglement, and vice versa for modes for which $\epsilon(j)$ is large.
We take $\epsilon(j)$ to have a form depicted in Fig. 13: it is a growing
function of $j$ that sharply transitions from much less than 1 to much greater
than 1 at some cutoff $j=\Lambda$. As is standard on noncommutative
geometries, a momentum scale $\Lambda$ is also a length-scale (see the
discussion below (16)). Physically, $\Lambda$ is the length-scale of extended
objects which propagate along the fuzzy sphere geometry. As such, it is the
scale of nonlocality of the theory, and we refer to it as the string scale. To
ensure a local entanglement structure for our network we take $\Lambda\ll N$.

Figure 13: The structure we take for the function $\epsilon(j)$ appearing in
(36). Specifying $\epsilon(j)$ is equivalent to specifying the couplings
$g_{i}$ appearing in the derivative expansion of (35). We assume
$\epsilon(j)\ll 1$ up to some cutoff scale $j=\Lambda$, after which it sharply
transitions to $\epsilon(j)\gg 1$. From the point of view of the emergent
noncommutative field theory, $\epsilon(j)$ is the spectrum of the kinetic
term.

Unlike in §§3.3.1, there are two distinct contributions to the entanglement
structure of the ground state. $H_{1}$ couples (and therefore entangles) the
$j,m$ and $j,-m$ matrix spherical harmonic modes, whereas $H_{2}$ couples
entries of $\lambda$ modes along the diagonals pictured in Fig. 12. To make
contact with standard tensor networks, we wish for the former contribution to
dominate. We do this by estimating the scaling of these two contributions in
$N$ and $\Lambda$.

The first issue is that the $\hat{Y}_{jm}$ modes are delocalized across the
sphere. In view of Fig. 12, we look for a fourier-like transform that takes
the delocalized form of spherical Harmonics along the matrix diagonal into a
more local form. For a particular cutoff $\Lambda\ll N$, the $m$th diagonal
will have $\Lambda-m$ activated modes entangled with the opposite diagonal.
There are $N-m$ lattice sites along the $m$th diagonal, so we may squeeze each
qudit to occupy $(N-m)/(\Lambda-m)$ lattice sites. The order of magnitude of
the number of entangled qudits $N_{q}$ within the off-diagonal block (i.e. the
number of cut tensor network edges contributing to the entanglement as in Fig.
14) therefore scales as

$N_{q}\sim O(\Lambda^{3}/N).$ (37)

On the other hand, the number of diagonals with excited degrees of freedom
that cross the entanglement cut simply scales as $\Lambda$. This contribution
to the entanglement is precisely what is computed in
karczmarek2014entanglement , and scales as $\Lambda\log\Lambda$. As we take
$\Lambda\ll N$, note that we are within the area law regime found
karczmarek2014entanglement . We therefore wish to take the limit
$\Lambda^{3}/N\gg\Lambda$. In order to for the scale of nonlocality to be much
smaller than the scale of the background geometry, we must take $\Lambda\ll
N$. We therefore consider the parameter regime

$N^{1/2}\ll\Lambda\ll N.$ (38)

Figure 14: In light of Figs. 10 and 12, if we suppress all modes
$\hat{Y}_{jm}$ with $j>\Lambda$ to their ground state then all excited qubits
contributing to tensor network edges that cross the entanglement cut will be
localized to the shaded corners of the $\Sigma\bar{\Sigma}$ and
$\bar{\Sigma}\Sigma$ subblocks. In the parameter regime we are considering
with $\Lambda\gg N^{1/2}$, this is the dominant contribution to the
entanglement between the $\Sigma$ and $\bar{\Sigma}$ subregions of the
network.

To evaluate the entanglement, we simply need to rotate to a basis where
$\Theta_{\Sigma}$ induces a block decomposition of the matrix as in Fig. 10
and count the number of qubits in the $\Sigma\bar{\Sigma}$ block that are
entangled with their partners in the $\bar{\Sigma}\Sigma$ block. This is given
by the sum

$N_{q}=N_{f}\sum_{j=1}^{\Lambda}\sum_{m=-j}^{j}\left(\Tr[(\hat{Y}_{jm})^{{\dagger}}_{\Sigma\bar{\Sigma}}(\hat{Y}_{jm})_{\bar{\Sigma}\Sigma}]+\Tr[(\hat{Y}_{jm})^{{\dagger}}_{\bar{\Sigma}\Sigma}(\hat{Y}_{jm})_{\Sigma\bar{\Sigma}}]\right).$
(39)

As an instructive example we explicitly evaluate this sum in a simple the
simple case where $\Sigma$ is a cap subregion on the sphere (see Fig. 15) so
that $\Theta_{\Sigma}$ is diagonal in the same basis as $Z$. The spherical
harmonics then take the form drawn in Fig. 12. The sum may be rewritten as

$N_{q}=N_{f}\sum_{j=1}^{\Lambda}\sum_{m=1}^{j}\left(\Tr[(\hat{Y}_{jm})^{{\dagger}}[\hat{Y}_{jm},\Theta_{\Sigma}]]-\Tr[(\hat{Y}_{j,-m})^{{\dagger}}[\hat{Y}_{j,-m},\Theta_{\Sigma}]]\right).$
(40)

As the $\hat{Y}_{jm}$ form a complete basis for $N\times N$ matrices, we may
expand the matrix $\Theta_{\Sigma}$ itself in terms of spherical harmonics:

$\Theta_{\Sigma}=\sum_{jm}c_{\Sigma,jm}\hat{Y}_{jm},\quad
c_{\Sigma,jm}:=\Tr[\hat{Y}_{jm}^{{\dagger}}\Theta_{\Sigma}].$ (41)

Figure 15: A cap subregion of the fuzzy sphere, of the sort considered in
karczmarek2014entanglement ; frenkel2023emergent . It is rotationally
symmetric around the $z$ axis, and its boundary $\partial\Sigma$ (i.e. the
entanglement cut) is simply a line of latitude at some constant $z$ value
$z_{c}$. If the fuzzy sphere has radius 1, we may consider $z_{c}=2M/N-1$ for
any integer $M$ such that $1\leq M\leq N-1$. To evaluate the entanglement, we
simply need to count the number of tensor network lines as pictured in Fig.
4(a) that cross $\partial\Sigma$.

Via the Moyal map Moyal:1949sk ; Susskind:2001fb ; Steinacker:2011ix  (and as
strongly suggested by Fig. 12) we may approximate $c_{\Sigma,jm}$ up to
$O(1/N)$ corrections as

$c_{\Sigma,jm}=\int_{0}^{\pi}d\theta\int_{s}^{2\pi}d\phi\sin\theta d\theta
d\phi\,\chi_{\Sigma}(\theta,\phi)y_{jm}(\theta,\phi)+O(1/N).$ (42)

The coefficients $c_{\Sigma,jm}$ are therefore simply determined by the
properties of the Fourier decomposition of step functions. In particular, for
large $j$ they have the structure

$\sqrt{\sum_{m=-j}^{j}|c_{\Sigma,jm}|^{2}}\approx\frac{|\partial\Sigma|}{\pi
j^{2}}+O(C/j^{3}).$ (43)

$C$ is the lengthscale of the radius of curvature of $\partial\Sigma$. Put
simply, the projection of a characteristic function onto high-momentum
(compared to the radius of curvature) fourier mode subspaces has magnitude
proportional to the surface area (i.e. perimeter for the $2d$ case). For the
case of the cap subregion symmetry demands that only $c_{\Sigma,j0}$ is
nonzero, so we simply have

$|c_{\Sigma,j0}|\approx\frac{|\partial\Sigma|}{\pi j^{2}}+O(1/j^{3}).$ (44)

Plugging this back into (39) we find

$N_{q}\approx
2N_{f}|\partial\Sigma|\underbrace{\sum_{j=1}^{\Lambda}\sum_{m=1}^{j}\sum_{j^{\prime}=1}^{N}\frac{1}{\pi{j^{\prime}}^{2}}\Tr[\hat{Y}_{jm}^{{\dagger}}[\hat{Y}_{jm},\hat{Y}_{j^{\prime}0}]]}_{\text{A
numerical coefficient independent of $\Sigma$}}+\,O(1/\Lambda).$ (45)

Taking $\Lambda$ large, we find that the number of tensor network legs
partitioned by the entanglement cut is proportional to the perimeter length
$|\partial\Sigma|$. As the entanglement is proportional to $N_{q}$, we find
area law entanglement in this parameter regime.

We may extend this argument to more general subregions using the techniques of
Frenkel:2023yuw . Instead of $Z$ we diagonalize some other matrix $F$ (as in
Fig. 4), and take the radius of curvature $C$ to be given by
$C\sim\sqrt{\Tr[F[X^{i},[X^{i},F]]]/\Tr[F^{2}]}$. It is important to take
$C\ll\Lambda$. This matrix corresponds to the curvilinear coordinate
$f(\theta,\phi)$ on the sphere, along with an orthogonal curvilinear
coordinate $\phi_{f}$ which satisfies
$[F,\cdot]\leftrightarrow-i\partial_{\phi_{f}}$. The computation was easy in
the case of the cap subregion because spherical harmonics are eigenstates of
$[Z,\cdot]\leftrightarrow-i\partial_{\phi}$. Because high-momentum spherical
harmonics have the structure of plane waves on the surface of the sphere, we
may find local rotations of the spherical harmonics that satisfy

$\begin{split}\tilde{y}_{jm}(f,\phi_{f}):=\sum_{m}c^{f}_{j,m,m^{\prime}}(\theta,\phi)y_{jm^{\prime}}(\theta,\phi),\quad\sum_{m^{\prime\prime}}c^{f}_{j,m,m^{\prime\prime}}(\theta,\phi)\bar{c}^{f}_{j,m^{\prime},m^{\prime\prime}}(\theta,\phi)=\delta_{m,m^{\prime}},\end{split}$
(46)

such that $\tilde{y}_{jm}(f,\phi_{f})$ is an eigenfunction of
$\partial_{\phi_{f}}$. This definition may be promoted to the fuzzy sphere by
promoting the $c^{f}_{j,m,m^{\prime}}(\theta,\phi)$ to be functions of the
$X^{i}$. The $\tilde{y}_{jm}(f,\phi_{f})$ then get promoted to matrices
$\tilde{Y}_{jm}$ that satisfy

$\partial_{\phi_{f}}\tilde{y}_{jm}(f,\phi_{f})=im\tilde{y}_{jm}(f,\phi_{f})\leftrightarrow[F,\tilde{Y}_{jm}]=m\tilde{Y}_{jm}.$
(47)

This implies that $\tilde{Y}_{jm}$ now have the exact structure depicted in
Fig. 12.

We now recall that the matrix commutator becomes the Poisson bracket
$\\{\cdot,\cdot\\}$ on the fuzzy sphere surface. We therefore have

$\begin{split}&\\{\tilde{y}_{j_{1},m_{1}}(f,\phi_{f}),\tilde{y}_{j_{2},m_{2}}(f,\phi_{f})\\}=\\\
&=\sum_{m_{1}^{\prime},m_{2}^{\prime}}c_{j_{1},m_{1},m_{1}^{\prime}}^{f}(\theta,\phi)c_{j_{2},m_{2},m_{2}^{\prime}}^{f}(\theta,\phi)\\{y_{j_{1},m_{1}^{\prime}}(\theta,\phi),y_{j_{2}m^{\prime}_{2}}(\theta,\phi)\\}+O(C/\min(j,j^{\prime})),\end{split}$
(48)

as the terms that dominate the Poisson bracket are those where the derivatives
hit the $\tilde{y}_{jm}$ instead of the $c_{j,m,m^{\prime}}$. Pulling this
back to matix expression in turn implies

$[\tilde{Y}_{jm},\tilde{Y}_{j^{\prime}m^{\prime}}]=\sum_{m_{1}^{\prime},m_{2}^{\prime}}c^{f}_{j_{1},m_{1},m_{1}^{\prime}}(X^{i})c^{f}_{j_{2},m_{2},m_{2}^{\prime}}(X^{i})[\hat{Y}_{jm},\hat{Y}_{j^{\prime}m^{\prime}}]+O(C/\min(j,j^{\prime})).$
(49)

Using this expansion we may now run the exact same computation as before,
finding again the result (45) with the same $O(1)$ overall numerical
coefficient:

$N_{q}\approx
2N_{f}|\partial\Sigma|\sum_{j=1}^{\Lambda}\sum_{m=1}^{j}\sum_{j^{\prime}=1}^{N}\frac{1}{\pi{j^{\prime}}^{2}}\Tr[\hat{Y}_{jm}^{{\dagger}}[\hat{Y}_{jm},\hat{Y}_{j^{\prime}0}]]+\,O(C/\Lambda).$
(50)

To help the reader trust these quick noncommutative function manipulations, we
present numerical results. See Fig. 16 for a direct numerical evaluation of
(39) for the cap subregion that confirms the area law behavior. See Figs. 17
and 18 for numerics for another family of entanglement cuts that includes
topologically nontrivial subregions.

Figure 16: We numerically compute the sum that calculates $N_{q}$ in (39) for
the cap subregion (see Fig. 15) for various choices of $z$ at which we
partition the sphere (blue dots). We compare the result to the functional form
of $\sin(\arccos(z))$ – the expression for the perimeter length
$|\partial\Sigma|$ at each value of $z$ (green dots). Note that for small
subregions the entanglement is smaller than what we would expect from the
perimeter law. This occurs because the region area decreases much faster than
the perimeter length, and when $\Sigma$ is of roughly the length-scale of
nonlocality $\Lambda/N$ the entanglement saturates at a volume law.

(a)

(b)

Figure 17: We consider a family of partitions parameterized by level sets of
the function $f(\vec{x})=z+x^{2}$. At $f(\vec{x})=1$, the subregion on the
fuzzy sphere splits from one component (as in Fig. 17(a)) to two disconnected
components (as in Fig. 17(b)). The matrix tensor network entanglement for this
family of subregions is plotted against the perimeter in Fig. 18. Figure 18:
The numerical evaluation of (39) for the family of entanglement cuts depicted
in Fig. 17. The cusp at $z=1$ is precisely the point at which the subregion
separates into two (i.e. the transition from the regime in Fig. 17(a) to the
regime in Fig. 17(b). The overall coefficient in the match between
entanglement and perimeter is identical to that in Fig. 16. Note that after
the region separates into two (i.e. to the right of the cusp) the entanglement
is subleading compared to the perimeter law. This is attributed to two
effects. Immediately after the region splits into two, the two disconnected
subregions are very close to each other and there will be many nonlocal tensor
network legs stretching between them (note that at $\Lambda=25$, the scale of
nonlocality is $25\%$ of the sphere radius.) As the two regions shrink, we
enter the regime where the area law saturates into a volume law, and the
volume to area ratio increases with the number of disconnected components of
the subregion.

###  3.4 Emergent Non-Abelian Gauge Fields on the Links

The observation that tensor networks may incorporate emergent bulk gauge
fields living on their links is nothing new Dong:2023kyr ; Akers:2024ixq ;
Akers:2024wab ; Qi:2022lbd . In this section, we simply demonstrate that this
construction naturally arises from the structure of MQM.

The $U(N)$ symmetry of the network acts in the adjoint representation on
$\lambda_{nn^{\prime}}$. Consider acting with a diagonal $U\in U(1)^{N}\subset
U(N)$. This will rotate the matrix elements as

$\lambda^{\alpha}_{nn^{\prime}}\rightarrow\lambda^{\alpha}_{nn^{\prime}}e^{i(\theta_{n}-\theta_{n}^{\prime})}.$
(51)

Note that $\lambda^{\alpha}_{nn^{\prime}}$ transforms precisely with the
opposite phase as with its partner $\lambda^{\alpha}_{n^{\prime}n}$. There is
therefore a natural $U(1)$ emergent bulk gauge field that lives on the tensor
network links. This is the same $U(1)$ of the emergent Maxwell (or Chern-
Simons) theory that lives on the surface of the fuzzy sphere Iso:2001mg . To
make contact with Dong:2023kyr ; Akers:2024ixq , we promote the emergent
$U(1)$ gauge theory to a $U(p)$ gauge theory in the standard way Iso:2001mg ;
Susskind:2001fb ; Dorey:2016mxm  – by stacking $p$ fuzzy spheres on top of one
another. Specifically, we take our coordinate matrices to be

$\tilde{X}^{i}:=X^{i}\otimes\mathbb{1}_{p},$ (52)

where the $X^{i}$ are an irreducible representation of the $\mathfrak{su}(2)$
algebra as in (14). We take $X^{i}$ to be $N\times N$ and $\tilde{X}^{i}$ to
be $pN\times pN$. The structure of $\tilde{X}^{i}$ encourages us to introduce
a block decomposition for the $\lambda_{nn^{\prime}}$. We introduce the symbol
$\mathfrak{l}_{kk^{\prime}}$ to denote the $p\times p$ block

$\mathfrak{l}_{kk^{\prime}}=\begin{bmatrix}\lambda_{pk+1,pk^{\prime}+1}&\ldots&\lambda_{pk+1,pk^{\prime}+p}\\\
\vdots&\ddots&\vdots\\\
\lambda_{pk+p,pk^{\prime}+1}&\ldots&\lambda_{pk+p,pk^{\prime}+p}\end{bmatrix}.$
(53)

We have introduced an index $k,k^{\prime}$ which runs from $1$ to $N$, whereas
$n,n^{\prime}$ runs from $1$ to $pN$.

We plug $\tilde{X}^{i}$ into the Hamiltonian (35). Note that $\tilde{X}^{i}$
satisfy

$[\tilde{X^{i}},U]=0\quad\text{for}\quad U\in U(p)^{N}.$ (54)

This Weyl-like subgroup of $U(pN)$ acts on the $p\times p$
$\mathfrak{l}_{kk^{\prime}}$ blocks as

$\mathfrak{l}_{kk^{\prime}}\rightarrow
U_{k}\mathfrak{l}_{kk^{\prime}}U_{k^{\prime}}^{{\dagger}},\quad
U_{k},U_{k^{\prime}}\in U(p).$ (55)

We have therefore promoted our tensor network to one where the links are
$U(p)$ bifundamentals as opposed to $U(1)$ bifundamentals, and have found an
emergent $U(p)$ gauge theory. The structure of (35) ensures that
$\mathfrak{l}_{kk^{\prime}}$ only appear in the Hamiltonian in the combination
$\Tr[\mathfrak{l}_{kk^{\prime}}^{{\dagger}}\mathfrak{l}_{k^{\prime}k}]$,
$\Tr[\mathfrak{l}_{kk^{\prime}}\mathfrak{l}_{k^{\prime}k}]$, or
$\Tr[\mathfrak{l}_{kk^{\prime}}^{{\dagger}}\mathfrak{l}_{k^{\prime}k}^{{\dagger}}]$.
The ground state is therefore invariant under the larger group $U(p)^{N^{2}}$
that acts as

$\mathfrak{l}_{kk^{\prime}},\mathfrak{l}_{k^{\prime}k}\rightarrow
U_{kk^{\prime}}\mathfrak{l}_{kk^{\prime}}U_{kk^{\prime}}^{{\dagger}},U_{kk^{\prime}}\mathfrak{l}_{k^{\prime}k}U_{kk^{\prime}}^{{\dagger}},\quad
U_{kk^{\prime}}\in U(p).$ (56)

The Hilbert space of $\mathfrak{l}_{kk^{\prime}}$ decomposes into $U(p)$
irreps. By standard arguments ghosh2015entanglement , in the ground state the
irreps of $\mathfrak{l}_{kk^{\prime}}$ are maximally entangled with the
corresponding irreps of $\mathfrak{l}_{k^{\prime}k}$.

##  4 Discussion and Outlook

We have introduced a new family of tensor network states inspired by MQM and
noncommutative geometry. They appear to naturally include notions of
background independence, area-preserving diffeomorphism invariance, emergent
gauge theory, and a string scale. We have demonstrated that despite the
nonlocality UV/IR mixing endemic to noncommutative field theories, the low-
energy tensor network states exhibit area law entanglement.

It would be interested to explicitly check the RT surface and bulk
reconstruction properties of these networks, especially on fuzzy AdS spaces,
perhaps using the methods of Cheng:2022ori ; ffhs:2024xx . In particular, it
would be interesting to make sense of tensor network links that might stretch
far across the geometry. These modes suggest a natural way to encode mutual
information between distant subregions, perhaps inducing backreaction between
RT surfaces (see Fig. 19). One could also ask whether the dynamical background
independence we consider in this work is identical to the notion of background
independence explored in Akers:2024ixq .

Figure 19: Consider a tensor network on a fuzzy AdS spacetime – perhaps a
noncommutative analog of the HaPPY code pastawski2015holographic . The tensor
network states we consider naturally have very nonlocal links contributing to
the entanglement structure. This nonlocal entanglement serves to pull RT
surfaces closer to one another. It would be interesting to check if the ground
state of Hamiltonians like (35) may induce the appropriate backreaction
between RT surfaces to match the behavior in the bulk of AdS/CFT.

It would also be interesting to explore whether such matrix tensor networks
are the appropriate language in which to capture $\alpha^{\prime}$ (or general
higher-derivative) corrections to the entanglement entropy (as in Dong:2013qoa
; Wall:2015raa ). More generally, it is an interesting question whether a
matrix tensor network on a fuzzy $S^{5}$ background (perhaps using the results
of Fiore:2022twy ) is the correct way to capture how the compact directions of
AdS/CFT may be incorporated into tensor networks. Similarly, it is interesting
to check whether these networks accurately capture the ground-state
entanglement behavior of BFSS.

Lastly, although we have proposed a manner if incorporating spatial APDs into
tensor networks, it is interesting whether time reparameterizations may also
be incorporated into such a context, e.g. by using constructions such as
Banados:2001xw . Alternatively, it is possible that the dynamics generated by
(35) or similar Hamiltonians might be related in some way to bulk time
evolution. ffhs:2024xx

## Acknowledgements

I am partially supported by the NSF GRFP under grant no. DGE-165-651. I am
indebted especially to Xiao-Liang Qi and Aditya Cowsik for extensive
insightful discussions. I am also grateful to Ronak M. Soni, Watse Sybesma,
Laurens Lootens, Chris Akers, Jason Pollack, and Annie Y. Wei for extensive
comments on an early draft. I would like to thank Annie Y. Wei in particular
for discussion that inspired the addition of §§3.4. I would also like to thank
DAMTP and the University of Cambridge for hospitality during a significant
portion of this work.

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11institutetext: Fermi National Accelerator Laboratory, Batavia, IL 60510, USA

# Particle Flow and PUPPI in the Level-1 Trigger at CMS for the HL-LHC

Benjamin Kreis for the CMS Collaboration 11<EMAIL_ADDRESS>
###### Abstract

With the planned addition of tracking information to the Compact Muon Solenoid
(CMS) Level-1 trigger for the High-Luminosity Large Hadron Collider (HL-LHC),
the trigger algorithms can be completely reconceptualized. We explore the
feasibility of using particle flow-like reconstruction and pileup per particle
identification (PUPPI) pileup mitigation at the hardware trigger level. This
represents a new type of multi-subdetector pattern recognition challenge for
the HL-LHC. We present proof-of-principle studies on both physics and
hardware-resource performance of a prototype algorithm for use by CMS in the
HL-LHC era.

##  1 Introduction

In the mid-2020s, the High-Luminosity Large Hadron Collider (HL-LHC) at CERN
will begin colliding protons with an instantaneous luminosity leveled to
$5\times 10^{34}\ \mathrm{cm}^{-2}\mathrm{s}^{-1}$ with the goal of delivering
approximately $300\ \mathrm{fb}^{-1}$ per year for ten years hllhc . This
increase in luminosity will enable the general-purpose experiments at the HL-
LHC, CMS and ATLAS, to maintain progress in their rich physics programs of
precision measurements of the standard model and searches for physics beyond
the standard model, such as supersymmetry and dark matter.

The challenge for the HL-LHC experiments will be to maintain detection
efficiency for interesting physics processes at the electroweak energy scale
occurring in the same bunch crossings as additional inelastic interactions,
known as pileup. Pileup produces extra hits in the tracking subdetectors and
extra energy depositions in the calorimeter subdetectors that must be
separated from the process of interest. In addition, the subdetectors must be
able to withstand the radiation damage caused by the particles produced by
these many proton interactions. At the HL-LHC, the average number of pileup
interactions is expected to reach 200.

The CMS experiment cms  will undergo a comprehensive upgrade to maintain
performance in the high-luminosity conditions of the HL-LHC. In this paper, we
focus on upgrades to the Level-1 trigger, the first stage of data processing
that has a coarse, global view of the detector and determines which collision
events to fully read out for further analysis. We explore the feasibility of
using particle-level reconstruction and pileup mitigation, the best performing
algorithms in current offline processing, in the upgraded trigger. In Section
2, we describe the inputs to the upgraded Level-1 trigger and its
architecture. The inputs will include tracks for the first time, and the
trigger will have a correlator that forms the global view of each collision
event. Particle-level reconstruction and pileup mitigation algorithms for the
correlator are described in Section 3. In Section 4, results on physics
performance and hardware resource requirements based on a proof-of-principle
study are presented.

##  2 Level-1 Trigger for HL-LHC

Upgrades to the CMS detector for HL-LHC, described in detail elsewhere
cmsupgrade ; interim , will bring new and improved inputs to the Level-1
trigger. The inputs, referred to as trigger primitives, are formed by trigger
primitive generators (TPGs). For the first time at CMS, the Level-1 trigger
will receive trigger primitives representing tracks with transverse momentum
$p_{T}$ $>$ 2-3 GeV from a new outer tracking subdetector. The TPGs of a new
endcap calorimeter will provide energy deposition information at improved
granularity. New TPGs in the barrel electromagetic calorimeter will also
improve granularity, while the TPGs of the hadron calorimeter will be upgraded
without changing granularity. Finally, the TPGs of the muon subdetectors will
upgraded and will include trigger primitives from new muon detectors added to
the endcap region.

The data flow from the CMS subdetectors to the TPGs and through the Level-1
trigger is shown in Figure 1. The barrel calorimeter trigger processes trigger
primitives to form high-resolution clusters. The barrel and endcap muon track
finders identify muons. The output of the barrel calorimeter trigger, muon
track finders, and the trigger primitives of the tracking subdetector, endcap
calorimeter, and forward hadronic calorimeter reach the correlator at a
bandwidth of $\mathcal{O}(10)$ Tbps, where the information is correlated to
form a global view of the event. This is passed on to the global trigger,
where the trigger decision is made.

Figure 1: Schematic of the CMS Level-1 Trigger for HL-LHC. A number of
possible, additional direct links are not pictured (see interim ).

The Level-1 trigger systems will use Field Programmable Gate Arrays (FPGAs)
for all data processing. Modern high performance FPGAs have high speed
input/output able to handle the large data bandwidth and hundreds of thousands
of logic cells to do pipelined and parallelized data processing. The
correlator has a latency budget of approximately 2.5 $\mu s$ and must be
pipelined to the LHC collision rate of 40 MHz.

##  3 Particle-Level Algorithms for Level-1 Trigger

The availability of tracking information in the Level-1 trigger correlator
opens up the possibility of performing reconstruction and pileup mitigation at
the particle level, the best performing approach in CMS’s software-level
trigger and offline reconstruction pf ; puppicms . To perform these algorithms
in the correlator, they must be implemented in FPGAs and meet the pipelining
and latency constraints.

###  3.1 Particle-Flow Reconstruction in Level-1 Trigger

Particle-flow reconstruction correlates tracks from the tracking and muon
subdetectors and clusters from the calorimeter subdetectors to identify each
final-state particle and combines the subdetector measurements to reconstruct
the identified particles’ properties pf . Here we describe a proof-of-
principle algorithm for performing particle-flow reconstruction in the FPGAs
of the Level-1 trigger correlator. A schematic of the algorithm is shown in
Figure 2.

Figure 2: Schematic of particle flow algorithm for CMS Level-1 trigger
correlator.

The algorithm inputs are muon candidates from the muon track finders, tracks
from the tracking subdetector TPGs, and electromagnetic and hadronic
calorimeter energy deposits from the barrel calorimeter trigger and endcap
calorimeter TPGs. The algorithm reconstructs particles centered within a
restricted region in pseudorapidity ($\eta$) and azimuthal angle ($\phi$)
based on inputs from an enlarged region to correlate across the $\eta$-$\phi$
boundaries. Multiple copies of the algorithm will run in parallel to cover the
whole detector.

Muon candidates and tracks closest in $p_{T}$ within a $\Delta
R\equiv\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}$ requirement are linked to form
particle-flow muon candidates. The linked tracks are removed from further
consideration. Electromagnetic calorimeter clusters and tracks closest in
$\Delta R$ within a $\Delta R$ requirement are linked. Linked electromagnetic
energy clusters and tracks form particle-flow electrons, and the linked tracks
are removed from further consideration. If the electromagnetic energy
cluster’s $p_{T}$ is significantly larger than that of the track, the excess
$p_{T}$ forms a particle-flow photon. Particle-flow photons are also formed
from unlinked electromagnetic energy clusters. The electromagnetic and
hadronic calorimeter energy is clustered and calibrated, and the energy of the
particle-flow electrons and photons is removed. The resulting calorimeter
clusters are linked with tracks based on $\Delta p_{T}$ and $\Delta R$. Linked
calorimeter clusters and tracks and unlinked low-$p_{T}$ tracks form particle-
flow charged hadrons. If the calorimeter cluster’s $p_{T}$ is significantly
larger than that of the track, the excess $p_{T}$ forms a particle-flow
neutral hadron. Particle-flow neutral hadrons are also formed from unlinked
calorimeter clusters.

###  3.2 PUPPI Pileup Mitigation in Level-1 Trigger

The best performing pileup mitigation technique used in CMS offline
reconstruction is PileUp Per Particle Identification (PUPPI), an algorithm
that removes charged particles with tracks not originating at the primary
vertex and downweights neutral particles based on the probability that they
originate from pileup puppi ; puppicms . The weight is calculated from the
neighboring charged particles not originating from pileup. Here we describe a
proof-of-principle implementation of PUPPI for the FPGAs of the Level-1
trigger correlator. A schematic is shown in Figure 3.

Figure 3: Schematic of PUPPI pileup mitigation algorithm for CMS Level-1
trigger correlator.

The position along the beam axis ($z$) of the primary vertex is computed from
all tracks in the event. One algorithm for finding the primary vertex is to
find the $z$ position of the peak in a histogram of track $z$ weighted by
track $p_{T}$. Charged particles with tracks not originating near the primary
vertex are removed. For each neutral particle $i$, we compute the sum of
$p_{T}/\Delta R_{ij}$ of the remaining charged particles $j$ within a $\Delta
R$ requirement. The sum addresses a precomputed lookup table of weights.

##  4 Results

###  4.1 Physics Performance

We test the physics performance of particle-flow reconstruction and PUPPI
pileup mitigation in the Level-1 trigger using simulated samples of $t\bar{t}$
events decaying semileptonically as signal and minimum-bias events as
background. The average number of pileup interactions in the simulation is
140, corresponding to a baseline HL-LHC scenario.

Figure 4 (left) shows the rate of a trigger on missing transverse energy (MET)
computed with particle-flow candidates versus signal efficiency for events
with MET>100 GeV at the generator level. We compare with two alternative MET
triggers. The first computes MET from calorimeter energy deposits. The second
computes MET from tracks with a small $\Delta z$ with respect to the primary
vertex. Figure 4 (right) compares trigger efficiency at a fixed rate of 20 kHz
versus the scalar sum of jet $p_{T}$ ($H_{T}$) at the generator level. The
jets are clustered using the anti-$k_{T}$ algorithm with a radius of 0.4
antikt  and are required to have $p_{T}>30$ GeV and $\eta<2.4$.

Figure 4: MET trigger rate versus signal efficiency (left) and $H_{T}$ trigger
turn-on curve (right) for the particle-flow and PUPPI based trigger (blue), a
calorimeter-only trigger (purple), and a trigger based on tracks consistent
with the primary vertex (red) interim .

###  4.2 Hardware Implementation

We divide the detector into $\eta\times\phi\approx 0.55\times 0.55$ regions
accepting up to 25 tracks and 20 calorimeter clusters each. Implmented in
Xilinx Vivado High Level Synthesis version 2016.4 for an Ultrascale+ VU9P
FPGA, four regions can be processed with approximately $40\%$ of the FPGA
resources. The latencies for particle flow and PUPPI are approximately 500 ns
and 100 ns, respectively.

##  5 Conclusions and Outlook

We explore the feasibility of using particle-flow reconstruction and PUPPI
pileup mitigation, the best performing algorithms in current offline
processing, in the CMS Level-1 trigger correlator to maintain performance in
the high pileup conditions of the HL-LHC. We present proof-of-principle
implementations of these algorithms for FPGAs and find that they significantly
improve the physics performance of the Level-1 trigger while being feasible in
terms of FPGA resource usage and latency.

The concept of adapting algorithms used in offline processing to the FPGAs of
the Level-1 trigger may give rise to additional opportunities to improve
physics performance. We note in particular that machine learning techniques,
which are being explored for offline processing, are also a promising
direction for the Level-1 trigger hls4ml .

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