# Emergent Area Laws from Entangled Matrices

Alexander Frenkel♯ and Sean A. Hartnoll♭ ♯Department of Physics, Stanford
University,
Stanford, CA 94305-4060, USA
♭Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge CB3 0WA, UK

###### Abstract

We consider a wavefunction of large $N$ matrices supported close to an
emergent classical fuzzy sphere geometry. The $SU(N)$ Gauss law of the theory
enforces correlations between the matrix degrees of freedom associated to a
geometric subregion and their complement. We call this ‘Gauss law
entanglement’. We show that the subregion degrees of freedom transform under a
single dominant, low rank representation of $SU(N)$. The corresponding Gauss
law entanglement entropy is given by the logarithm of the dimension of this
dominant representation. It is found that, after coarse-graining in momentum
space, the $SU(N)$ Gauss law entanglement entropy is proportional to the
geometric area bounding the subregion. The constant of proportionality goes
like the inverse of an emergent Maxwell coupling constant, reminiscent of
gravitational entropy.

###### Contents

  1. 1 Gauss law entanglement
  2. 2 Partitions of matrix systems
    1. 2.1 Projection onto subregions
    2. 2.2 Gauge charge in a subregion
    3. 2.3 Low rank representations
  3. 3 The fuzzy sphere state
    1. 3.1 Classical fuzzy sphere
    2. 3.2 Fluctuations and semiclassical state
    3. 3.3 The normal modes
  4. 4 Cap subregion
    1. 4.1 $SU(N)$ charge
    2. 4.2 UV/IR mixing and smoothing
    3. 4.3 Gauss law entanglement entropy
  5. 5 Discussion
    1. 5.1 Smearing and analogy with gravitational entropy
    2. 5.2 Other saddles and the holographic direction
    3. 5.3 Diffeomorphisms and permutations
  6. A Form of the matrix spherical harmonics
  7. B Details of the normal modes
  8. C Classicality of the quadratic Casimir

##  1 Gauss law entanglement

The quantum mechanics of large $N$ matrices is well-known to give rise to
emergent space. The simplest example is the emergence of two dimensional
string theory from the quantum mechanics of a single matrix [1], while the
richest examples involve the emergence of critical string theory or M-theory
from maximally supersymmetric Yang-Mills theories [2] or quantum mechanics [3,
4]. There are many models in between these two limits, with varying numbers of
matrices and degree of supersymmetry. A major open question is how, precisely,
quantum states of matrices encode emergent geometry. In this paper we will
make some general observations about the emergence of geometry from matrices,
with specific computations in a bosonic model of three matrices recently
discussed in [5].

We are able to make progress by restricting to technically tractable cases in
which the geometry emerges from semiclassical matrices. This means that the
matrix wavefunction is supported close to a classical noncommutative geometry,
where the noncommutativity is small at large $N$. This is, of course, a major
restriction. Such matrix states describe emergent $D$ brane worldvolumes [6]
rather than gravitating spacetimes. In these cases we will show that the
‘areas’ bounding subregions in the emergent space are directly encoded in what
we call the Gauss law entanglement of the matrices. This is an entanglement
due to the gauge-theoretic correlations in the matrix wavefunction, as we now
explain.

Consider the matrix quantum mechanics of several $N$ by $N$ bosonic Hermitian
matrices $X^{i}$ with conjugate momenta $\Pi^{i}$ and a Hamiltonian of the
general form

$H=\Tr[{\textstyle{\frac{1}{2}}}\vec{\Pi}\cdot\vec{\Pi}+V(\vec{X})]\,.$ (1)

The Hamiltonian is invariant under an $SU(N)$ symmetry generated by

$G=2i\sum_{i}[X^{i},\Pi^{i}]\,.$ (2)

In a string theory context this symmetry is usually gauged, so that physical
states $\ket{\Psi}$ obey:

$G\ket{\Psi}=0\,.$ (3)

Even if the microscopic gauge constraint is relaxed [7], certain holographic
states obey (3).

The Gauss law (3) expresses the global vanishing of $SU(N)$ charge. If one
restricts attention to a subset of ‘local’ degrees of freedom, these may carry
a net charge. This charge is, of course, cancelled by the remaining degrees of
freedom. Quantum mechanical fluctuations of the local charge will therefore
result in, via the Gauss law, correlations between local degrees of freedom
and their complement. This leads to what we call Gauss law entanglement. This
is not a new concept, but we wish to give the phenomenon a name that
emphasizes its physical origin. Gauss law entanglement in matrix theories was
first discussed in [8]. In more conventional local quantum systems it has been
widely studied in the guise of gauge-theoretic ‘edge modes’ [9, 10, 11, 12,
13]. These modes are typically gauge transformations that do not vanish on the
boundary of a geometric subregion and which therefore obstruct a geometric
factorization of the gauge-invariant Hilbert space. Because the edge modes
live on the boundary of subregions they produce ‘area’ (or, better,
‘boundary’) law entanglement.

It was shown in [14] that the $SU(N)$ Gauss law entanglement in the so-called
Quantum Hall matrix model — a matrix quantum mechanics with first order
kinetic terms in the Lagrangian [15, 16] — also exhibits a boundary law
behavior, but now in an emergent two dimensional space. The Gauss law
entanglement in the theory was shown to be in correspondence with that of
large gauge transformations in an emergent Chern-Simons theory. Therefore, in
that theory, matrix $SU(N)$ Gauss law correlations can be identified with
conventional emergent geometric edge modes. However, the Quantum Hall matrix
model leads to a topological gauge theory with no local dynamics in the
emergent bulk. It is therefore at once rather simple while also containing
subtleties specific to its topological nature. In this paper we will instead
compute the Gauss law entanglement in a model of the form (1), with a second
order kinetic term. The model is known to lead to an emergent $2+1$
dimensional Maxwell-scalar theory with genuine bulk dynamics on a fuzzy sphere
[5].

Our main result is equation (47) for the $SU(N)$ Gauss law entanglement
corresponding to a ‘cap’ subregion of the emergent fuzzy sphere. This
expression exhibits a geometric boundary law. Unlike in the Quantum Hall model
discussed in the previous paragraph, however, this entanglement is not simply
the edge mode entanglement of the emergent Maxwell field. The leading order
matrix entanglement obtained in (47) is enhanced by a factor of the inverse
dimensionless Maxwell coupling $\frac{\Lambda^{1/2}}{g_{M}}$. In this regard
it has some similarities with the well-known $\frac{1}{G_{N}}$ contribution to
gravitational entropy, as we discuss in §5.1. The matrix boundary law is
furthermore enhanced by an unconventional logarithimic factor. The logarithm
in (47) multiplies the boundary law and hence is distinct from the subleading
universal logarithm widely discussed in Maxwell entanglement, e.g. [17, 18,
19, 20]. We do not have a physical understanding of our logarithim;
technically it arises as the asymptotic behavior of a simple counting problem
in (20).

A further significant feature in our result (47) is the presence of a coarse-
graining scale $\Lambda$. To obtain a geometric entropy it is essential to
coarse-grain the reduced density matrix in order to smooth out the effects of
UV/IR mixing in the noncommutative theory. This fact has implications for the
general understanding of matrix entanglement. We initially define the
subregion degrees of freedom via a block partition of matrices, as suggested
by [21, 22]. However, the need for smearing to obtain a useful result shows
that the block decomposition is ‘too microscopic’ and is not, in and of
itself, sufficient to reveal the emergent locality of the system. We are able
to perform the smearing in our semi-classical state because we know a priori
what the low energy field-theoretic degrees of freedom will be. For more
strongly quantum mechanical states it may be difficult to make sense of the
entropy following from the matrix block decomposition. Coarse-graining is
discussed further in §5.1.

While our computation will involve a few subtle points, the essential reason
that a boundary law appears is simple and illustrated in Fig. 1.

Figure 1: Geometric partition of the sphere and corresponding partition of the
matrices. The ‘volume’ (area) $|A|$ of the cap is given by the size of the
upper left block of the matrix. The off-diagonal block of the matrices is low
rank, with a unique non-vanishing singular value proportional to the ‘area’
(perimeter) $|\partial A|$ of the cap. This singular value controls the
$SU(N)$ charge of the cap and hence the Gauss law entanglement.

As we will recall in §2.1 shortly, a geometric subregion can be associated to
a sub-block of the matrix degrees of freedom. In §2.2 we show, using the Gauss
law, that the off-diagonal block determines the $SU(N)$ charge of the
subregion. The off-diagonal block is found to be of low rank for states close
to a classical non-commutative geometry, with a unique nonvanishing singular
value given by the perimeter of the subregion — this fact that also
underpinned the emergence of an entanglement boundary law in the Quantum Hall
matrix model [14]. This singular value will be seen in §2.3 and §4 to
determine the number of $SU(N)$ ‘edge modes’ that are needed to ensure the
Gauss law is obeyed by the reduced density matrix of the subregion. The
$SU(N)$ edge modes will be shown to be maximally entangled and consequently
lead to a boundary-law entanglement.

##  2 Partitions of matrix systems

###  2.1 Projection onto subregions

We would like to partition the degrees of freedom in a matrix theory in a way
that corresponds to emergent locality. In theories with a single matrix it is
natural to partition the eigenvalues of the matrix. A ‘target space
entanglement’ [23] may be associated to this partition of eigenvalues, and has
been shown to match the geometric entanglement of an emergent scalar field in
a one dimensional space [24, 25, 26]. Once there is more than one matrix in
the theory, however, the various matrices cannot generically be simultaneously
diagonalized. On the other hand, recent works have noted that a given
geometric partition in fact selects a single preferred matrix [21, 22]. The
eigenvalues of this matrix may then be used to define a partition of all the
matrix degrees of freedom, as we now briefly review.

In string theoretic matrix quantum mechanics there is commonly an association
between the number of matrices and the number of emergent dimensions. That is,
each matrix $X^{i}$ corresponds to an emergent coordinate $x^{i}$. In string
theory the eigenvalues of the $X^{i}$ matrix give the location of $D$ branes
in the $x^{i}$ direction. This correspondence suggests that subregions defined
in terms of the coordinates can be pulled back to the matrix degrees of
freedom. Specifically, suppose that a region $\Sigma$ in the emergent space is
defined by $f_{\Sigma}(\vec{x})<0$. The function $f_{\Sigma}$ may be promoted
to a function of matrices $f_{\Sigma}(\vec{X})$. This step has an ordering
ambiguity because the matrices do not commute. We will eventually coarse-grain
the state over the noncommutativity scale, so this will not be a significant
issue for us. The ordering ambiguity, however, already suggests the need to
supplement the matrix block decomposition with a coarse-graining more
generally. Given $f_{\Sigma}(\vec{X})$, the crucial point is that
$f_{\Sigma}(\vec{X})$ is now a single matrix that may be diagonalized. For
example, $f_{\Sigma}(\vec{X})=X_{1}$ would correspond to a half-plane, while
$f_{\Sigma}(\vec{X})=\vec{X}\cdot\vec{X}-1$ corresponds to the interior of a
sphere. The eigenvalues of $f_{\Sigma}(\vec{X})$ can be partitioned according
to whether they are positive or negative, just as in the single-matrix
theories mentioned above.

The partition of eigenvalues of $f_{\Sigma}$ is described in terms of a matrix
and its complement:

$\Theta_{\Sigma}\equiv\theta(f_{\Sigma}(\vec{X})),\qquad\Theta_{\bar{\Sigma}}\equiv\theta(-f_{\Sigma}(\vec{X}))=\mathbb{1}-\Theta_{\Sigma}\,.$
(4)

Here $\theta$ is the Heaviside step function. The matrix $\Theta_{\Sigma}$ is
the matrix version of the characteristic function of the region $\Sigma$. The
eigenvalues of $\Theta_{\Sigma}$ are all 0 or 1 — it is a projection matrix
that obeys $\Theta_{\Sigma}^{2}=\Theta_{\Sigma}$. The rank of
$\Theta_{\Sigma}$ is defined to be $M$. With this projector at hand, all
matrices may individually be split into four parts,

$\left.\begin{array}[]{rcl}X^{i}_{AB}&\equiv&\Theta_{A}X^{i}\,\Theta_{B}\\\
X^{i}&=&\sum_{A,B}X^{i}_{AB}\end{array}\right\\}\qquad
A,B\in\\{\Sigma,\bar{\Sigma}\\}\,.$ (5)

The above construction is equivalent to going to a basis in which $f_{\Sigma}$
is diagonalized, with ordered eigenvalues. The projection $\Theta_{\Sigma}$ is
onto the subspace spanned by the lowest $M$ eigenvectors of $f_{\Sigma}$.
Equation (5) is block decomposition with respect to this subspace.

###  2.2 Gauge charge in a subregion

We may now explain the interplay of the matrix partition defined in (5) with
the Gauss law (3). Related considerations have appeared previously in [8, 14].
The most important conclusion of this subsection will be that the Gauss law
leads to maximally mixed density matrices in each nontrivial charge sector of
the ‘subregion’. This is equation (13) below.

The generator of the Gauss law (3) may itself be partitioned using
$\Theta_{\Sigma}$. This leads to the blocks $G_{\Sigma\Sigma}$,
$G_{\Sigma\bar{\Sigma}}$, $G_{\bar{\Sigma}{\Sigma}}$ and
$G_{\bar{\Sigma}\bar{\Sigma}}$, as in (5). It is clear that $G_{\Sigma\Sigma}$
are the $M^{2}$ generators of an $SU(M)$ subgroup of $SU(N)$, while
$G_{\bar{\Sigma}\bar{\Sigma}}$ generate $SU(N-M)$. It is useful to write down
these generators explicitly in terms of the partitioned matrix. In particular,

$G_{\Sigma\Sigma}=2i\sum_{i}([X^{i}_{\Sigma\Sigma},\Pi^{i}_{\Sigma\Sigma}]+X^{i}_{\Sigma\bar{\Sigma}}\Pi^{i}_{\bar{\Sigma}\Sigma}-\Pi^{i}_{\Sigma\bar{\Sigma}}X^{i}_{\bar{\Sigma}\Sigma})\,.$
(6)

The singlet constraint (3) then implies that

$\sum_{i}[X^{i}_{\Sigma\Sigma},\Pi^{i}_{\Sigma\Sigma}]\ket{\Psi}=\sum_{i}(\Pi^{i}_{\Sigma\bar{\Sigma}}X^{i}_{\bar{\Sigma}\Sigma}-X^{i}_{\Sigma\bar{\Sigma}}\Pi^{i}_{\bar{\Sigma}\Sigma})\ket{\Psi}\,.$
(7)

We may re-interpret this expression as a Gauss law for a different $SU(M)$
that is generated purely by degrees of freedom within the region $\Sigma$.
Namely, consider the $SU(M)$ generators

$\hat{G}_{\Sigma\Sigma}\equiv
2i\sum_{i}[X^{i}_{\Sigma\Sigma},\Pi^{i}_{\Sigma\Sigma}]\,.$ (8)

We may further define charges made from off-diagonal degrees of freedom of the
matrices

$Q_{\Sigma}\equiv
2i\sum_{i}\left(\Pi^{i}_{\Sigma\bar{\Sigma}}X^{i}_{\bar{\Sigma}\Sigma}-X^{i}_{\Sigma\bar{\Sigma}}\Pi^{i}_{\bar{\Sigma}\Sigma}\right)\,.$
(9)

The Gauss law (7) becomes the statement that the off-diagonal degrees of
freedom are charged under the $SU(M)$ generated by the
$\hat{G}_{\Sigma\Sigma}$:

$\hat{G}_{\Sigma\Sigma}\ket{\Psi}=Q_{\Sigma}\ket{\Psi}\,.$ (10)

Tracing out the $\bar{\Sigma}\bar{\Sigma}$, $\bar{\Sigma}\Sigma$, and
$\Sigma\bar{\Sigma}$ degrees of freedom leads to a reduced density matrix

$\rho_{\Sigma\Sigma}=\Tr_{\bar{\Sigma}\bar{\Sigma},\bar{\Sigma}\Sigma,\Sigma\bar{\Sigma}}\ket{\Psi}\bra{\Psi}\,.$
(11)

Crucially this trace is over states in a larger ‘extended’ Hilbert space in
which the Gauss law is not imposed [9, 10, 11, 12]. This is necessary because
some of the $SU(N)$ transformations mix up the different blocks of the
decomposition. Nonetheless, we may now use the fact that the state
$\ket{\Psi}$ obeys (10) to obtain

$[\hat{G}_{\Sigma\Sigma},\rho_{\Sigma\Sigma}]=\Tr_{\bar{\Sigma}\bar{\Sigma},\bar{\Sigma}\Sigma,\Sigma\bar{\Sigma}}\Big{(}Q_{\Sigma}\ket{\Psi}\bra{\Psi}-\ket{\Psi}\bra{\Psi}Q_{\Sigma}\Big{)}=0\,.$
(12)

The last step holds because $Q_{\Sigma}$ only acts on the degrees of freedom
that are being traced over. Using the cyclic property of the trace, the two
terms in (12) cancel. Because all of the $M^{2}$ generators of
$\hat{G}_{\Sigma\Sigma}$ commute with $\rho_{\Sigma\Sigma}$, the density
matrix must be proportional to the identify on each nontrivial irreducible
representation $R$ that appears in the Hilbert space. That it is to say, the
reduced density matrix for the gauge-theoretic ‘edge modes’ is maximally mixed
over each irreducible representation so that

$\rho_{\Sigma\Sigma}=\bigoplus_{R}\left(\frac{p_{R}}{d_{R}}\mathbb{1}_{d_{R}}\bigotimes\rho_{R}\right)\,.$
(13)

Here $d_{R}$ is the dimension of the representation, $p_{R}$ is the
probability of the representation in the state and the $\rho_{R}$ are
normalized density matrices in the unextended Hilbert space of the
subregion.111More explicitly, let $\psi(X^{i}_{\Sigma\Sigma})$ be a
wavefunction in the reduced Hilbert space of the region $\Sigma$. We may use
an $SU(M)$ transformation $U$ to fix a gauge, for example by setting
$X^{i}_{\Sigma\Sigma}=UX^{\text{gf}\,i}_{\Sigma\Sigma}U^{\dagger}$, with
$X^{\text{gf}\,3}_{\Sigma\Sigma}\equiv z$ diagonal and ordered. The
wavefunction may then be written as  $\ket{\psi}=\int
dUdzdX_{\Sigma\Sigma}^{\text{gf}\,1}dX^{\text{gf}\,2}_{\Sigma\Sigma}\,\Psi(U,z,X_{\Sigma\Sigma}^{\text{gf}\,1},X_{\Sigma\Sigma}^{\text{gf}\,2})\,\ket{U}\otimes\ket{z,X^{\text{gf}\,1}_{\Sigma\Sigma},X^{\text{gf}\,2}_{\Sigma\Sigma}}\,.$
(14) This defines a tensor factorization of the Hilbert space
$\mathcal{H}_{\Sigma\Sigma}=\mathcal{H}_{U}\otimes\mathcal{H}_{\text{gf}}\,,$
such that $SU(M)$ only acts on $\mathcal{H}_{U}$. From the Peter-Weyl theorem,
then,
$\mathcal{H}_{\Sigma\Sigma}=\left(\bigoplus_{R}\mathcal{H}_{R}\right)\otimes\mathcal{H}_{\text{gf}}$.

The entanglement entropy following from (13) is

$S(\rho_{\Sigma\Sigma})=\sum_{R}p_{R}\log d_{R}-\sum_{R}p_{R}\log
p_{R}+\sum_{R}p_{R}S(\rho_{R})\,.$ (15)

The first two terms here are what we are calling the Gauss law contribution to
the entanglement. We will find that in our semiclassical model there is a
single representation $R_{o}$ that dominates, so that $p_{R_{o}}\approx 1$ and
all other probabilities are suppressed in the large $N$ limit. In this case
the first term in (15) dominates and the Gauss law entanglement
$S^{\text{GL}}$ is

$S^{\text{GL}}(\rho_{\Sigma\Sigma})=\log d_{R_{o}}\,.$ (16)

Computing $S^{\text{GL}}$ in this case therefore reduces to evaluating the
dominant dimension $d_{R_{o}}$. The remaining final term in (15), from the
unextended Hilbert space, is then $S(\rho_{R_{o}})$. We will compare the Gauss
law and unextended contributions below.

The presence of Gauss law entanglement in matrix theories was first emphasized
in [8], who focused on a $U(1)^{M-1}$ subgroup of $SU(M)$. We will see below
that to obtain our leading order entanglement it is essential to consider the
full $SU(M)$ ‘edge modes’. We had previously argued in [14] that the edge mode
reduced density matrix should be maximally mixed over the dominant
representation of $SU(M)$, but we believe the proof we have now given in (12)
is more direct. It may be noted that there is some choice involved in
embedding the physical state into an ‘ungauged’ Hilbert space. Furthermore, as
in the more familiar case of gauge theories [9, 10, 11, 12, 13], the Gauss law
entanglement is due to $SU(M)$ edge modes that are introduced by the embedding
(cf. footnote 1). The embedding is, however, a well-defined procedure that
leads to a gauge-invariant Gauss law entanglement entropy that captures the
tension between the Gauss law and locality.

###  2.3 Low rank representations

In the remainder of this paper we will apply the formalism above to a state in
a particular matrix quantum mechanics. This state will describe Gaussian
fluctuations about a classical fuzzy sphere matrix configuration. We will see
that acting on this state in the large $N$ limit, $\hat{G}_{\Sigma\Sigma}$ is
dominated by a single representation $R_{o}$ and that in this representation
$\hat{G}_{\Sigma\Sigma}$ is a low rank matrix. In fact
$\hat{G}_{\Sigma\Sigma}$ will have rank two, which is the minimum possible
rank for a nonzero traceless Hermitian matrix. We will now explain that in
such cases there is a simple formula for the dimension $d_{R_{o}}$.

When $\hat{G}_{\Sigma\Sigma}$ is rank two it may be written in terms of a
complex vector $\xi$ and its conjugate momentum $\pi$, so that

$(Q_{\Sigma})_{ab}=i\left(\xi_{b}\pi_{a}-\xi^{\dagger}_{a}\pi^{\dagger}_{b}\right)\,.$
(17)

Here $\xi$ will be a function of the matrix components of
$X^{i}_{\Sigma\bar{\Sigma}}$ appearing in the charge (9). We have normal
ordered the operators and
$\pi_{a}=-i\partial/\partial\xi_{a}\equiv-i\partial_{a}$. The trace of the
constraint (7) implies that on physical states

$\xi_{a}\pi_{a}|\Psi\rangle=\xi^{\dagger}_{a}\pi^{\dagger}_{a}|\Psi\rangle\,.$
(18)

This constraint removes the $U(1)$ part of the generators (17). The generators
(17) and constraint (18) can be represented on polynomials in the $\xi_{a}$ of
some fixed degree $\ell$ multiplied by degree $\ell$ polynomials in the
complex conjugates $\bar{\xi}_{a}$. The dimension of the space of such
polynomials is given by

$d_{R}=\binom{M+\ell-1}{M-1}^{2}\,.$ (19)

The square is due to the two independent polynomials. The product of
polynomials does not immediately furnish an irreducible representation,
because of the presence of singlet factors $(\xi\cdot\bar{\xi})^{\mu}$ with
$\mu\leq\ell$. However, the reducible polynomials are subleading at large $M$
and hence we may use (19) as an approximation for the dimension of the
dominant irreducible subspace. See [27] for a more group theoretic discussion
of this point. We may further simplify (19) when $M,\ell\gg 1$. In this limit,
putting (19) into the Gauss law entanglement entropy (16) gives

$S^{\text{GL}}(\rho_{\Sigma\Sigma})=\log
d_{R}\approx\begin{cases}\displaystyle 2\ell\log\frac{\text{e}M}{\ell}&\ell\ll
M\\\\[10.0pt] \displaystyle
2M\log\frac{\text{e}\,\ell}{M}&M\ll\ell\end{cases}\,.$ (20)

In order to evaluate (20) it remains to determine the appropriate value of
$\ell$. In the remainder we do this in two steps. Firstly, we obtain the
semiclassical quantum state of fluctuations about the fuzzy sphere. Secondly,
we verify that upon restriction to a subregion the state transforms in a rank
two representation, and we fix the dimension of the representation by
computing the quadratic Casimir.

##  3 The fuzzy sphere state

###  3.1 Classical fuzzy sphere

We will focus on a model of three matrices $X^{i}$, $i=1,2,3$, with the
potential in (1) given by

$V(\vec{X})=\frac{1}{4}\left(\nu\epsilon^{ijk}X^{k}+i[X^{i},X^{j}]\right)^{2}\,.$
(21)

Here $\nu$ is a mass deformation parameter. This model is the bosonic sector
of a supersymmetric matrix quantum mechanics [28], which can be thought of as
a ‘mini-BMN’ theory [29]. The potential energy in (21) is a total square. The
classical ground states therefore obey the $SU(2)$ algebra

$[X_{\text{cl}}^{i},X_{\text{cl}}^{j}]=i\nu\epsilon^{ijk}X_{\text{cl}}^{k}\,.$
(22)

We will be concerned with the maximal ‘fuzzy sphere’ solution to (22) given by

$X_{\text{cl}}^{i}=\nu J^{i}\,,$ (23)

with $J^{i}$ being the $N$ dimensional irreducible representation of $SU(2)$.
Specifically, we take

$J^{3}=\sum_{n=0}^{N-1}\left({\textstyle{\frac{1}{2}}}(N-1)-n\right)|n)(n|\,,\quad
J^{+}=\sum_{n=1}^{N-1}\sqrt{n(N-n)}|n-1)(n|\,,\quad
J^{-}=(J^{+})^{\dagger}\,,$ (24)

where $J^{\pm}=J^{1}\pm iJ^{2}$.

The classical solution given by (23) and (24) is just one of many degenerate
minima of the potential, corresponding to different reducible representations
of the algebra (22). Domain walls between these different vacua were
constructed in [30]. Our objective here is to understand the emergence of
geometry in a simple setting, and for this we restrict attention to a quantum
state that is localized near the irreducible representation (24). We comment
on other representations in §5.2. The solution (23) obeys

$(X_{\text{cl}}^{1})^{2}+(X_{\text{cl}}^{2})^{2}+(X_{\text{cl}}^{3})^{2}=\frac{\nu^{2}}{4}\left(N^{2}-1\right)\,,$
(25)

which is suggestive of a spherical geometry with radius $\sim\nu N/2$. In
fact, the essential point about the fuzzy sphere solution is that fluctuations
about the background can be organized in terms of matrix spherical harmonics.
As we recall shortly, these are strongly analogous to the usual spherical
harmonics, but are cut off at angular momentum $j_{\text{max}}=N-1$. This
angular momentum cutoff in the fluctuations is the sense in which (24) defines
a fuzzy sphere. A smooth geometry is recovered in the limit $N\to\infty$.

###  3.2 Fluctuations and semiclassical state

We may now discuss quantum fluctuations about the classical solution. The
semiclassical limit in which these fluctuations remain close to the classical
geometry will be seen to be $\nu\to\infty$. It should be emphasized that this
is logically distinct from the $N\to\infty$ limit in which the classical
noncommutative geometry becomes smooth. We will be working in the limit in
which both $\nu$ and $N$ are large.

Fluctuations about the fuzzy sphere can be written as

$X^{i}=X_{\text{cl}}^{i}+\delta X^{i}=X_{\text{cl}}^{i}+\sum_{a}\delta
x_{a}Y^{i}_{a}\,.$ (26)

The normal modes $Y_{a}^{i}$ were obtained in [5], following [31, 32]. We will
review their construction in terms of matrix spherical harmonics shortly. The
$Y_{a}^{i}$ should be taken to be Hermitian matrices. They obey the
normalization condition
$\sum_{i=1}^{3}\tr\left[(Y^{i}_{a})^{\dagger}Y^{i}_{b}\right]=\delta_{ab}\,,$
leading to the perturbed quadratic Hamiltonian

$H^{(2)}={\textstyle{\frac{1}{2}}}\sum_{a}\left[\pi_{a}^{2}+\nu^{2}\omega_{a}^{2}(\delta
x_{a})^{2}\right]\,.$ (27)

Here $\pi_{a}$ is the momentum conjugate to $\delta x_{a}$. The normal mode
frequencies $\omega_{a}$ will be given shortly. The (unnormalized)
wavefunction close to the classical fuzzy sphere state is then

$\psi_{\text{fs}}(\delta
x)=\prod_{a}\exp\left(-{\textstyle{\frac{1}{2}}}\nu|\omega_{a}|\delta
x_{a}^{2}\right)\,.$ (28)

The Gaussian wavefunction (28) is the semiclassical state from which we will
obtain a Gauss law entanglement entropy.

It is apparent in (28) that at large $\nu$ the fluctuations about the
classical solution are small. However, there are many normal modes at large
$N$, so that their total zero point energy is large. This large zero point
energy is cancelled in supersymmetric extensions of the theory, but in the
purely bosonic theory it means that while (28) is a long-lived metastable
state in the quantum theory — and hence good enough as a tractable model of
emergent geometry — it is not the quantum ground state [5]. We return to this
point in §5.2.

###  3.3 The normal modes

The entanglement depends on the structure of the normal modes in (26). The
modes can be expressed in terms of matrix spherical harmonics, a basis of
matrices $\hat{Y}_{jm}$ that obey

$[J^{3},\hat{Y}_{jm}]=m\hat{Y}_{jm},\qquad\sum_{i}[J^{i},[J^{i},\hat{Y}_{jm}]]=j(j+1)\hat{Y}_{jm}\,.$
(29)

As with usual spherical harmonics, $-j\leq m\leq j$, but the angular momentum
is now upper bounded by $j\leq N-1$. We review the construction of these
matrices in Appendix A. In that Appendix we demonstrate that in the regime
$N\gg|m|$ these matrices are given, in the basis used in (24) and for $m\geq
0$, by

$\hat{Y}_{jm}=2\sqrt{\pi}\,c_{jm}\sum_{k=1}^{N-m}Y_{jm}{\textstyle\left(\theta_{k},0\right)}\,|k-1)(k+m-1|\,,$
(30)

where $Y_{jm}(\theta,\phi)$ are the usual spherical harmonics and the angles

$\cos\theta_{k}\equiv 1-\frac{2k+m}{N}\ \,.$ (31)

Negative $m$ then follows from the relation
$\hat{Y}_{j-m}=(-1)^{m}\hat{Y}_{jm}^{{\dagger}}$. Equation (30) shows that,
close to the commutative limit, the matrix spherical harmonics are essentially
discretized spherical harmonics.

The overall prefactor in (30) is determined by
$\frac{1}{N}\Tr[\hat{Y}_{jm}^{{\dagger}}\hat{Y}_{jm}]=\int_{S^{2}}d\Omega|Y_{jm}|^{2}=1$.
The coefficients $c_{jm}$ are given in Appendix A, the only property we shall
need is that $c_{jm}\to 1$ to leading order with $N\gg m$. We should note that
various $m/N$ corrections are included in the above formulae: in the limit of
the sum in (30), in the prefactor $c_{jm}$ and in the $m/N$ term in (31). It
will be physically illuminating to keep these corrections for the discussion
of UV/IR mixing below, even while other $m/N$ corrections have been dropped —
see the discussion in Appendix A. Ultimately we shall truncate in §4.2 to
$j,m\leq\Lambda\ll N$, whereupon (30) is rigorously correct and the $m/N$
corrections are negligible.

The normal modes in (26) may be expanded in terms of matrix spherical
harmonics. A given mode may be written

$Y^{i}=\sum_{jm}y^{i}_{jm}\hat{Y}_{jm}\,.$ (32)

It will be convenient to introduce $y^{\pm}_{jm}=y^{1}_{jm}\pm iy^{2}_{jm}$.
In Appendix B we briefly review the equations for the coefficients
$y^{i}_{jm}$, along with their explicit solution. The upshot is that the
normal modes may be labelled by $j$ and for each $j$ there are three possible
frequencies, each with a large degeneracy:

$\displaystyle\omega$ $\displaystyle=0$ $\displaystyle\quad\text{multiplicity
}N^{2}-1\,,$ $\displaystyle\omega$ $\displaystyle=-j$
$\displaystyle\quad\text{multiplicity }2(j-1)+1\,,$ (33) $\displaystyle\omega$
$\displaystyle=j+1$ $\displaystyle\quad\text{multiplicity }2(j+1)+1\,.$

Recall that $1\leq j\leq N-1$. There are then $3(N^{2}-1)$ normal modes in
total. The zero modes are $SU(N)$ gauge transformations. Because $\omega=0$
for the global pure gauge modes, these do not appear in the wavefunction (28)
and also will not contribute to any of our computations below. This confirms
that the observables we compute using the gauge-fixed wavefunction are in fact
gauge invariant. The remaining physical modes describe the coupled
fluctuations of a Maxwell field and a scalar field on the fuzzy sphere [5].
Each of the modes appearing in the above multiplets may be further labelled by
$m$, as we describe in Appendix B.

##  4 Cap subregion

###  4.1 $SU(N)$ charge

With the quantum state at hand, we now wish to obtain the charge $\ell$ of the
subregion to use in the expression (20) for the Gauss law entanglement. We
will do this by evaluating the expectation value of the quadratic Casimir
$\left\langle\Tr(\hat{G}_{\Sigma\Sigma}^{2})\right\rangle$ on the fuzzy sphere
state. We will furthermore establish that the Casimir is strongly peaked on
its expectation value and that the generators are low rank. That is to say,
the state of the subregion is in a dominant low rank representation and
fluctuations in the charge of the subregion are small.

On physical states the Gauss law (10) requires the generators of the reduced
$SU(M)$ to be given by $\hat{G}_{\Sigma\Sigma}=Q_{\Sigma}$. Inside the
expectation value, we may expand the $SU(M)$ charge $Q_{\Sigma}$ given in (9)
around the classical configuration using the normal modes (26) to obtain

$Q_{\Sigma}\approx
2i\nu\sum_{i}\sum_{a}\left(Y^{i}_{a\Sigma\bar{\Sigma}}J^{i}_{\bar{\Sigma}\Sigma}-J^{i}_{\Sigma\bar{\Sigma}}Y^{i}_{a\bar{\Sigma}\Sigma}\right)\delta\pi_{a}\,.$
(34)

Note that as an operator (34) no longer obeys the $SU(M)$ algebra, as it has
been linearized about its classical value. This linearization should be
understood to be performed within the expectation value, and will be important
later when we extract the value of $\ell$.

For simplicity we consider a ‘cap’ subregion around the north pole of the
fuzzy sphere. Such subregions are illustrated in Fig. 2.

Figure 2: The cap subregion is shaded grey and is defined by the angle
$\theta_{c}$ as shown.

The cap is the region $\theta<\theta_{\text{c}}$ of the polar angle $\theta$.
This region may equivalently be defined as a partition along the $X^{3}$ axis.
Per our discussion in §2.1, the $X^{3}$ direction is to be identified with the
eigenvalues of $X^{3}_{\text{cl}}$. In (23) and (24) we saw that
$X^{3}_{\text{cl}}$ has eigenvalues equally spaced between $\nu N/2$ and $-\nu
N/2$. We may project to the highest $M$ eigenvalues of $X^{3}_{\text{cl}}$
using

$\Theta_{\Sigma}=\sum_{n=0}^{M-1}|n)(n|\,.$ (35)

The projector has been written in the basis (24) in which $X^{3}_{\text{cl}}$
is diagonalized. This projection therefore corresponds to the region defined
by the angle

$\cos\theta_{c}=1-\frac{2M}{N}\,.$ (36)

It follows from (36) that $M$ is indeed the ‘volume’ (area) of the cap, as we
stated in Fig. 1. More precisely, in units where the sphere has unit radius,
the area of the cap is $|A|=4\pi M/N$.

Using the projector (35) we can see that the matrix
$Q_{\Sigma}=\hat{G}_{\Sigma\Sigma}$ in (34) is low rank. Using the explicit
form (24) of the $J^{i}$ gives

$\left(\hat{G}_{\Sigma\Sigma}\right)_{PQ}=i\nu\sqrt{M(N-M)}\sum_{a}\left(Y^{+}_{aPM}\delta_{(M-1)Q}-\delta_{P(M-1)}Y^{-}_{aMQ}\right)\delta\pi_{a}\,.$
(37)

Here $P$ and $Q$ run from $0$ to $M-1$. This matrix is manifestly of rank two
and remains so at a quantum level because all of the $\delta\pi_{a}$ operators
commute with each other. It may further be noted that the singular values of
the matrix (37) are proportional to

$\sqrt{M(N-M)}=\frac{N\sin\theta_{\text{c}}}{2}\,.$ (38)

This in turn is proportional to the length $|\partial A|$ of the boundary of
the cap, giving a direct connection between the geometry of the subregion and
gauge charge in the subregion. As in our previous work on matrix Chern-Simons
theories [14], this connection will be at the root of the emergent area law
entanglement. This singular value is directly inherited from the off-diagonal
block of the classical matrices in (24), as we illustrated in Fig. 1 above.

Using the explicit generators (37) we may compute the quadratic Casimir. This
will enable us to identify the representation in which the state transforms
and thereby obtain the entanglement entropy via (20),

$\displaystyle\left\langle\Tr\left(\hat{G}_{\Sigma\Sigma}^{2}\right)\right\rangle_{\psi_{\text{fs}}}=-{\textstyle{\frac{1}{2}}}\nu^{3}M(N-M)\sum_{a}|\omega_{a}|\left[Y^{+\,2}_{a(M-1)M}+Y^{-\,2}_{aM(M-1)}-2\sum_{P=0}^{M-1}|Y^{+}_{aPM}|^{2}\right]$
(39)

Here we used the wavefunction (28), which implies
$\left\langle\delta\pi_{a}\delta\pi_{b}\right\rangle_{\psi_{\text{fs}}}={\textstyle{\frac{1}{2}}}\nu|\omega_{a}|\delta_{ab}$,
and the classical background (24). We used $Y^{+}=(Y^{-})^{\dagger}$. The
first two terms in (39) may be dropped as they are subleading at large $N$.
The remaining term may be evaluated using the formula for $Y^{+}$ given in
Appendix B. As is explained in the Appendix, the different modes $a$ are
labelled by $\\{j,m\\}$, with two families of modes (having $\omega=-j$ and
$\omega=j+1$) for each $j$. The only fact we shall need from the Appendix is
that for each mode

$Y^{+}=\alpha_{jm\omega}\hat{Y}_{j(m+1)}+\beta_{jm\omega}\hat{Y}_{j(m-1)}^{\dagger}\,,\qquad|\alpha_{jm\omega}|^{2}+|\beta_{jm\omega}|^{2}=\frac{1}{N}\,.$
(40)

Furthermore $\beta_{j(-m)\omega}=(-1)^{m}\alpha_{jm\omega}$. We may then use
the explicit large $N$ form (30) of the matrix spherical harmonics to obtain,
being careful with the ranges of the various summations and to leading order
at $j,|m|\gg 1$,

$\left\langle\Tr\left(\hat{G}_{\Sigma\Sigma}^{2}\right)\right\rangle_{\psi_{\text{fs}}}=4\pi\nu^{3}\frac{M(N-M)}{N}\sum_{j=1}^{N}2j\sum_{m=1}^{\min(M,j)}c_{jm}^{2}|Y_{jm}(\theta_{M-m})|^{2}\,.$
(41)

The above expression suggests the scaling
$\Tr(\hat{G}_{\Sigma\Sigma}^{2})\sim\nu^{3}N^{4}$. Before developing (41)
further, it remains to verify that quantum fluctuations in
$\Tr(\hat{G}_{\Sigma\Sigma}^{2})$ are small. It is not uncommon for traces of
matrices to become classical at large $N$, and in Appendix C we verify that
the variance of $\Tr(\hat{G}_{\Sigma\Sigma}^{2})$ is indeed suppressed by a
power of $1/N$. It follows that the reduced state is strongly peaked on a
single low-rank representation of $SU(M)$.

###  4.2 UV/IR mixing and smoothing

We may make the geometric content of (41) clearer by re-expressing $M$ in
terms of the angle $\theta_{\text{c}}$ of the cap in (36). In particular,
using in addition (38) and (31), and changing the order of the summations, one
obtains

$\left\langle\Tr\left(\hat{G}_{\Sigma\Sigma}^{2}\right)\right\rangle_{\psi_{\text{fs}}}=\pi\nu^{3}N\sin^{2}\theta_{\text{c}}\sum_{m=1}^{M}\sum_{j=m}^{N}2jc_{jm}^{2}|Y_{jm}(\cos^{-1}\left[\cos\theta_{c}+m/N\right])|^{2}\,.$
(42)

There are two aspects of the expression (42) that look unnatural from a
continuum field theoretic perspective. The first is the $M$-dependent cutoff
on the $m$ summation and the second is the shift by $m/N$ inside the inverse
cosine. As we now explain, these features of (42) are indeed not innocuous.

The noncommutativity of the fuzzy sphere coordinates in (22) implies a (purely
classical) uncertainty relation between any pair of coordinates. In
particular, we may consider an excitation that is strongly localized to the
sphere (25) of radius $R=\frac{\nu}{2}N$ while localized to a range
$\Delta\theta$ about polar angle $\theta$ and a range $\Delta\phi$ about the
azimuthal angle $\phi=0$. Then

$\Delta[\cos\theta]\Delta\phi=\frac{\Delta Z}{R}\frac{\Delta
Y}{R\sin\theta}\geq\frac{\nu}{2}\frac{|\langle
X\rangle|}{R^{2}\sin\theta}=\frac{1}{N}\,.$ (43)

This equation explains both of the aforementioned unusual features in (42).
For an excitation with azimuthal angular momentum $m$ we may put
$\Delta\phi\sim 1/m$ in (43). It follows that $\Delta[\cos\theta]\gtrsim m/N$.
A given azimuthal angular momentum therefore necessitates a smearing over the
polar angle. This is precisely the effect seen in the argument of the
spherical harmonic in (42). Secondly, from (36) we have
$\Delta[\cos\theta]\sim\Delta M/N$. As we are considering modes that are
restricted to a subregion with matrix size $M$, we must have $\Delta M\leq M$.
The uncertainty relation therefore implies that $m\lesssim M$. This explains
the upper cutoff in (42) of the sum over $m$ by $M$, which is stronger than
the usual commutative bound of $m\leq j\leq N$.

The effects described in the previous paragraph are instances of UV/IR mixing,
a phenomenon that is familiar in noncommutative field theories [33]. In
particular, the bound $m\lesssim M$ is a short distance angular cutoff on
modes around the boundary of a region that depends on the volume ($\propto M$)
of the region enclosed. Given that (42) — and hence, shortly, the entanglement
entropy — is dominated by short distance modes, it is clear that this volume-
dependent cutoff will lead to violations of ‘boundary law’ behavior. Similar
effects have been noted in previous computations of the entanglement in
noncommutative theories [34, 35, 36, 37]. In fact, the expression is not even
symmetric under $\theta_{\text{c}}\leftrightarrow\pi-\theta_{\text{c}}$. This
is possible because we have traced out the off-diagonal modes in
$X^{i}_{\Sigma\bar{\Sigma}}$ and $X^{i}_{\bar{\Sigma}\Sigma}$ in addition to
the complementary diagonal block $X^{i}_{\bar{\Sigma}\bar{\Sigma}}$. However,
the asymmetry highlights the fact that (42) is non-geometric and does not
encode a boundary law.

Recovering an approximately smooth geometry in the large $N$ limit requires
smoothing over lengthscales that are afflicted by the UV/IR mixing that we
have just described. The UV/IR mixing is significant because high angular
momentum modes make a dominant contribution to (42). To reveal the approximate
geometric locality contained in the matrix wavefunction we must coarse-grain
the state.

The semiclassical matrix wavefunction (28) is a product of modes with
different angular momenta $j$. It is therefore straightforward to trace out
the high angular momentum modes in the full density matrix, by truncating the
angular momentum product over $j$ at some cutoff $\Lambda\ll M,N$. This leads
to a smoothed density matrix
$\rho^{\Lambda}\equiv|\psi^{\Lambda}_{\text{fs}}\rangle\langle\psi^{\Lambda}_{\text{fs}}|$.
This tracing out of modes in a product state, that is done before tracing out
the region $\bar{\Sigma}$, does not introduce any extra entanglement. In fact,
the Gauss law entanglement will be strictly reduced as the high angular
momentum modes are removed. Upon truncation (42) becomes

$\displaystyle\left\langle\Tr\left(\hat{G}_{\Sigma\Sigma}^{2}\right)\right\rangle_{\psi^{\Lambda}_{\text{fs}}}$
$\displaystyle=\pi\nu^{3}N\sin^{2}\theta_{\text{c}}\sum_{j=1}^{\Lambda}\sum_{m=1}^{j}2j|Y_{jm}(\theta_{c})|^{2}$
(44)
$\displaystyle\approx{\textstyle\frac{1}{6}}\nu^{3}\Lambda^{3}N\sin^{2}\theta_{\text{c}}\,.$
(45)

In the final expression, which is evaluated to leading order at large
$\Lambda$, we used the fact that
$\sum_{m=-j}^{j}|y_{jm}(\theta)|^{2}=\frac{2j+1}{4\pi}$. We have also set
$c_{jm}\to 1$ now that $m\ll N$.

In (45) we see that in the smoothed state
$\Tr(\hat{G}_{\Sigma\Sigma}^{2})\propto|\partial A|^{2}$, the boundary ‘area’
squared. We may now show that this implies a boundary-law entanglement
entropy.

###  4.3 Gauss law entanglement entropy

Having established that there is a dominant rank two representation of
$SU(M)$, the quantity $\ell$ in the formula for the Gauss law entanglement
entropy (20) may be obtained from the quadratic Casimir. For such
representations of $SU(M)$,
$C_{2}=2\left[\ell(M+\ell-1)-\frac{\ell^{2}}{M}\right]$ [38], with the overall
factor of $2$ because the edge mode states in §2.3 are the product of two
polynomials. However, the linearization performed in (34) means that we have
not computed the full Casimir. Specifically, linearizing the generators
directly rather than the Casimir itself means that we have effectively normal
ordered the Casimir. From the low rank $SU(M)$ generators (17) one may show
that
$\langle(Q_{\Sigma})_{ab}(Q_{\Sigma})_{ba}\rangle=\langle:(Q_{\Sigma})_{ab}(Q_{\Sigma})_{ba}:\rangle+2\ell
M$. That is, our linearized computation has missed the $2\ell M$ contribution
to the Casimir. The Casimir built from the linearized generators (37) is
simply a flat-space Laplacian type term, evaluating at large $\ell\gg 1$ to
$C_{2}^{\text{lin}}\approx 2\ell^{2}$. Therefore we have the identification

$\ell\approx\left(\frac{1}{2}\left\langle\Tr\left(\hat{G}_{\Sigma\Sigma}^{2}\right)\right\rangle_{\psi^{\Lambda}_{\text{fs}}}\right)^{1/2}\approx\frac{(\nu\Lambda)^{3/2}N^{1/2}}{2\cdot
3^{1/2}}\sin\theta_{\text{c}}\,.$ (46)

The entropy is obtained by using the value of $\ell$ from (46) in the formula
(20). In the limit that $\Lambda\ll\frac{1}{\nu}N^{1/3}$, which is weaker than
the initial requirement that $\Lambda\ll N$ so long as $\nu$ is not extremely
large, this leads to the coarse-grained Gauss law entanglement entropy
associated to the cap:

$S^{\text{GL}}(\rho_{\Sigma\Sigma}^{\Lambda})\approx\frac{1}{\sqrt{3\pi}}\frac{\Lambda^{1/2}}{g_{M}}\frac{|\partial
A|}{1/\Lambda}\times\log\frac{g_{M}N|A|}{\Lambda^{3/2}|\partial A|}\,.$ (47)

Here we have given the answer in terms of the length of the boundary of the
spherical cap

$|\partial A|\equiv 2\pi\sin\theta_{\text{c}}\,,$ (48)

in units where the sphere has unit radius. Recall also that the volume of the
cap $|A|=4\pi M/N$. We also introduced the coupling constant of the non-
commutative 2+1 dimensional Maxwell theory that arises on the emergent sphere
[5]

$g_{M}\equiv\sqrt{\frac{4\pi}{N\nu^{3}}}\,.$ (49)

Equation (47) is the promised boundary-law entanglement, up to a logarithmic
correction. It is interesting to interpret (47) from the perspective of the
low energy field theory on the emergent sphere. The term $\frac{|\partial
A|}{1/\Lambda}$ is the length of the boundary region in units of a momentum
cutoff $\Lambda$. This term is the natural magnitude for a conventional field-
theoretic contribution to the entanglement. Consistently with this
expectation, [5] obtained $S\approx 0.03\frac{|\partial A|}{1/\Lambda}$ for
the entanglement in the unextended Hilbert space of this model. The
contribution in [5] should correspond to the final term in the full
entanglement (15), albeit for a different microscopic partition of the degrees
of freedom. The prefactor in (47), however, shows an enhancement by
$\frac{\Lambda^{1/2}}{g_{M}}$. This is the inverse of the dimensionless
coupling constant of the emergent theory and is large in the regime where the
non-commutative Maxwell theory is weakly coupled. Such enhancement of the
gauge-theoretic entanglement due to microscopic degrees of freedom that are
building the space itself may possibly be analogous to the large entropy found
in string and gravitational theories [39, 40, 41, 42]. The relationship
between the microscopic and emergent field theoretic entanglement is discussed
further in §5.1.

A further point to make is that there is a phase transition in the
entanglement entropy if the cap becomes too small. At small angles
$|A|\sim\theta_{\text{c}}^{2}$ and $|\partial A|\sim\theta_{\text{c}}$. For

$\theta_{\text{c}}\lesssim\frac{\Lambda^{3/2}}{Ng_{M}}\,,$ (50)

then the entropy (20) is to be evaluated in the regime $M\ll\ell$, leading to
the ‘volume’ law

$S^{\text{GL}}(\rho_{\Sigma\Sigma}^{\Lambda})\approx\frac{N|A|}{2\pi}\times\log\frac{\Lambda^{3/2}|\partial
A|}{g_{M}N|A|}\,.$ (51)

It should be emphasized that this transition is not due to UV/IR mixing
effects, which have been smoothed out in our construction. Instead, it is due
to a change in the asymptotic regime of the counting problem that determines
the entropy. In this sense, the phase transition is similar in flavor to
‘deconfinement’ type transitions in free gauge theories [43]. It may be
interesting to elaborate on this connection in the future.

##  5 Discussion

We have obtained the boundary-law entropy (47) as the Gauss law entanglement
in a certain semiclassical state of a matrix quantum mechanics, associated to
the matrix partition (5) together with a coarse-graining of momenta. The Gauss
law entanglement captures the tension between the microscopic $SU(N)$ Gauss
law constraint and the emergent locality in the theory. In the remainder we
comment on some features of our construction that may warrant further thought,
and also note possible connections to other discussions of entanglement and
emergent geometry.

###  5.1 Smearing and analogy with gravitational entropy

To reveal the emergent locality it was necessary to supplement the matrix
partition with a coarse-graining that smears over regions of size $1/\Lambda$.
Without this smearing the dynamics is not local, due to the UV/IR mixing
effects of spatial noncommutativity. The introduction of a smearing parameter
$\Lambda$ means that the prefactor of the boundary-law entanglement (47) is
not given in terms of the microscopic parameters of the model alone. It is
plausible that this will be the case in any model with an emergent geometry:
if it were possible to partition the microscopic degrees of freedom in a way
that was in perfect correspondence with emergent low energy locality this
would imply that the geometry is already present in the microscopic theory and
not, in fact, emergent!

This point suggests that partitioning the matrix degrees of freedom as we have
done, following the proposal in [21, 22], is unlikely to be sufficient to
capture geometric partitions. Additional information about the nature of the
emergent geometry will be necessary. For semiclassical matrix states such as
the one we have considered, the presence of a classical non-commutative
geometry makes it possible to guess the correct smearing. However, in regimes
where the matrices are strongly quantum the necessary smearing may not be
obvious a priori. On the other hand, systems whose configuration space is just
one matrix may be sufficiently simple that the microscopic partition is
geometrically sensible [25, 24, 23, 14]. However, even in two-dimensional
string theory there is a nonlocality in the map from the collective field to
the string target space coordinates [44].

We may note that the Bekenstein-Hawking-Ryu-Takayanagi contribution [39, 40,
42] to entropy in gravitational theories depends on Newton’s constant, which
is an effective coupling in the low energy theory and also not microscopic.
The appearance of the dimensionless Maxwell coupling
$\frac{\Lambda^{1/2}}{g_{M}}$ in our entropy (47) is somewhat reminiscent of
that contribution and possibly should be thought of as an open-string analogue
of the gravitational entropy. A further point in support of this analogy is
that the entropy we have obtained can be thought of as counting strings that
cross the entanglement cut, cf. Fig. 1, in the spirit of the proposal of
Susskind and Uglum for gravitational entropy [41].

We may make two further remarks to emphasize that (47), while geometric,
should be thought of as having a matrix rather than field-theoretic origin.
The first is that the coupling $g_{M}$ is strictly a property of the non-
commutative Maxwell theory, appearing in front of the intrinsically non-
commutative term $[a_{i},a_{j}]_{\star}$ in the $U(1)$ field strength [5]. The
second is that, as we recall in §5.3 below, the emergent Maxwell gauge
transformations are associated to a $U(1)^{M-1}$ subgroup of $SU(M)$. These
are the diagonal components of the generators, in the basis where $X^{3}$ is
diagonalized. However, as we now explain, this subgroup does not contribute to
the result (47). From the generators (37) ones sees immediately that the only
nonzero diagonal component is
$\left(\hat{G}_{\Sigma\Sigma}\right)_{(M-1)(M-1)}$. This means the only
nontrivial edge modes transform under a single $U(1)\subset U(1)^{M-1}$.
However, $U(1)$ only has one dimensional representations and cannot contribute
to the Gauss law entanglement.

The field-theoretic entanglement due to the emergent Maxwell and scalar field
can instead to expected to arise at subleading order in the semiclassical
expansion. The Casimir at the first subleading order is proportional,
schematically, to $\langle(\delta x)^{2}(\delta\pi)^{2}\rangle$. Upon Wick
contraction this quantity is independent of $\nu$, and hence of the Maxwell
coupling $g_{M}$.

###  5.2 Other saddles and the holographic direction

We have restricted attention to an especially simple semiclassical state in
the theory. As noted above, the state describes the polarization of many $D$0
branes into a single $D$2 brane, in the spirit of [6], in a fixed background
(for example, AdS4 [28]). In the regime studied, there is no backreaction of
the branes onto the spacetime in the sense of AdS/CFT-type dualities [2]. In
particular, there does not seem to be an emergent ‘radial’ or ‘holographic’
dimension in the model. On the other hand, the emergent 2+1 dimensional
Maxwell theory, which is coupled to a scalar, does have area preserving
diffeomorphisms as a symmetry [5]. This symmetry is inherited, see the
following §5.3, from the original $SU(N)$ gauge invariance and the emergence
of space via noncommuative geometry [45, 46]. There may, therefore, be traces
of gravity-like physics even in this simple state.

A more direct way to probe the holographic dimension is to consider other
states of the theory, both semiclassical and beyond. Within the semiclassical
regime, reducible representations of the $SU(2)$ algebra (22) give classical
solutions describing multiple $D$2 branes. These can be coincident or
separated. Coincident $D$2 branes will carry non-abelian gauge fields that at
strong coupling will backreact into a holographic dimension. Distributions of
$D2$ branes can also extend into the radial direction. An explicit realization
of this connection arises in the maximally supersymmetric BMN matrix quantum
mechanics [4]. In that theory all of the classical fuzzy sphere solutions
survive as supersymmetric states in the quantum theory and are in
correspondence with dual spacetime geometries [32, 47, 48].

We have already noted that the classical fuzzy sphere solution is not the
ground state of the purely bosonic quantum theory. Even in mini-BMN theory,
the minimal supersymmetric completion of our model, with the same number of
bosonic matrices, it was shown in [5] that away from the semiclassical limit
the fuzzy sphere state we have considered eventually collapses into a strongly
quantum state that is not localized in the radial direction. That state is
also a candidate for a more strongly holographic regime of the theory.

###  5.3 Diffeomorphisms and permutations

Several ingredients of our analysis have appeared previously in our
computation of the entanglement in the Quantum Hall matrix model [14]. One
important difference, however, concerns the role of permutations. In [14] we
argued that states related by the symmetric group $S_{M}\subset SU(M)$ should
not be considered as distinct edge modes. This changed the counting problem,
as the edge modes were then given by symmetric polynomials rather than all
polymonials of a given degree. In that context, the symmetrization appeared
essential to recover the standard counting of chiral edge modes in Chern-
Simons theory and the corresponding area law entanglement. In the present
paper, however, we have found that an area law emerges without ‘re-gauging’
the $S_{M}$ symmetry. We feel that a better understanding of this ambiguity is
called for. In this subsection we explain that this is related to the role of
diffomorphism invariance.

In the large $N$ limit, the $SU(N)$ symmetry group approaches the group of
area preserving diffeomorphisms on the surface of the sphere [49, 50],
recently discussed in [51]. See [5] for an explicit discussion in our fuzzy
sphere model. One way to see the connection to diffeomorphisms is to start by
considering a particular diagonalized matrix to furnish a coordinate
parameterization of the sphere. For example, in the bulk of this work we
diagonalize $X^{3}=\sum_{n}z_{n}v_{n}v_{n}^{\dagger}$. The eigenvectors
$v_{n}$ define projectors $P_{n}\equiv v_{n}v_{n}^{\dagger}$ that have unit
trace and are mutually orthogonal, and are therefore naturally associated
(under the Moyal map) to non-overlapping unit area regions of the sphere.
These regions are labelled by the eigenvalues $z_{n}$ which, in string
theoretic realizations, are interpreted as the smeared locations of $D0$
branes. Under a unitary transformation each projector maps to a different
projector $UP_{n}U^{\dagger}=v_{n}^{\prime}v_{n}^{\prime\dagger}\equiv
P_{n}^{\prime}$. This transformation is therefore associated to a map from one
set of non-overlapping unit area regions to another. This is the area
preserving diffeomorphism.

Given the coordinate parameterization $X^{3}$ above, there are two subgroups
of $SU(N)$ that do not change the associated set of unit area regions. The
first are the $U(1)^{N-1}$ phases
$U=\sum_{n}e^{i\theta_{n}}v_{n}v_{n}^{\dagger}$, which leave the regions
invariant. As has been explained in [5, 14], these phase transformations are
the origin of the emergent $U(1)$ Maxwell field in this model. Strings
stretching between the $D$0 branes are charged under this transformation. The
second set of special transformations are the permutations $S_{N}$ which
exchange basis vectors and hence shuffle the regions among themselves. This
amounts to a relabelling of the regions, which is not evidently a physical
transformation. In [14] we found that quotienting the edge mode counting by
these permutations lead to a sensible result for the entanglement entropy in
the Quantum Hall matrix model. However, especially given the constraint (18),
this does not appear to be the case for the Gauss law entanglement in the
fuzzy sphere model.

## Acknowledgements

It is a pleasure to acknowledge helpful discussions with Jackson Fliss, Nabil
Iqbal, Zohar Komargodski, John McGreevy and Ronak Soni. We thank Juan
Maldacena for correcting a formula in an earlier version of this paper. The
work of SAH is partially supported by Simons Investigator award #620869 and by
STFC consolidated grant ST/T000694/1. AF is partially supported by the NSF
GRFP under grant number DGE-165-6518 and by the US Department of Energy Office
of Science under Award Number DE-SC0018134.

##  Appendix A Form of the matrix spherical harmonics

In this appendix we derive expression (30) in the main text for the matrix
spherical harmonics. We may start by recalling (see e.g. [5] for a recent
discussion) that these matrices are constructed, in analogy with the usual
spherical harmonics, via the generating relations

$\begin{split}&\hat{Y}_{j(-j)}=C(J^{-})^{j}\,,\\\
&\hat{Y}_{j(m+1)}=\frac{[J^{+},\hat{Y}_{jm}]}{\sqrt{(j-m)(j+1+m)}}\,.\end{split}$
(52)

The constant $C$ is to be fixed by the normalization
$\tr\left(\hat{Y}_{j(-j)}^{\dagger}\hat{Y}_{j(-j)}\right)=N$.

Recall from (24) that the only nonzero entries of $J^{-}$ are $|k)(k-1|$,
while $J^{+}$ has only $|k-1)(k|$ nonzero. One can then read off from (52)
that the matrix $\hat{Y}_{jm}$ only has nonzero entries for elements
$|k-1)(k+m-1|$. There is no approximation at this point. We now proceed to
obtain an equation for these nonzero entries. We set

$\hat{Y}_{jm}=\sum_{k=1}^{N-m}f(k)|k-1)(k+m-1|\,.$ (53)

The objective is to solve for $f(k)$ in the large $N$ limit.

The matrix spherical harmonics are defined by the equations (29). The first of
these is automatically obeyed given the form (53). The second equation in (29)
is then seen to imply

$[J^{+},[J^{-},\hat{Y}_{jm}]]=\Big{(}j(j+1)-m(m-1)\Big{)}\hat{Y}_{jm}\,.$ (54)

In the large $N$ limit, the left hand side of this equation becomes a
differential operator acting on $f(k)$. To see this, write

$J^{+}=\sum_{k=1}^{N-1}g(k)|k-1)(k|\,.$ (55)

Recall also that $J^{-}$ is just the transpose of $J^{+}$. Then we have

$\displaystyle[J^{+},[J^{-},\hat{Y}_{jm}]]$
$\displaystyle=\Big{(}g(k+m-1)[g(k+m-1)f(k)-g(k-1)f(k-1)]$
$\displaystyle\qquad+g(k)[g(k)f(k)-g(k+m)f(k+1)]\Big{)}|k-1)(k+m-1|$ (56)
$\displaystyle\approx\Big{(}-g(k)^{2}f^{\prime\prime}(k)-2g(k)g^{\prime}(k)f^{\prime}(k)$
$\displaystyle\qquad+[m(m-1)g^{\prime}(k)^{2}-mg(k)g^{\prime\prime}(k)]f(k)\Big{)}|k-1)(k+m-1|\,.$
(57)

In the second line we have assumed that $k\gg\\{m,1\\}$ in order to Taylor
expand the functions. If we use (57) and (53) in equation (54) we may equate
the coefficients to obtain a differential equation for $f(k)$. Recall from
(24) that $g(k)=\sqrt{k(N-k)}$. The differential equation acquires a familiar
form if we perform the change of variables $k\equiv\frac{N}{2}(x+1)$, whereby

$(1-x^{2})f^{\prime\prime}(x)-2xf^{\prime}(x)+\frac{m^{2}}{x^{2}-1}f(x)=-j(j+1)f(x)\,.$
(58)

The regular solutions to this differential equation are the associated
Legendre polynomials

$f(x)=c_{1}P_{j}^{m}(x)=c_{2}Y_{jm}(\cos^{-1}(x),0)\,.$ (59)

Here $c_{1}$ and $c_{2}$ are constants.

Expression (59) gives a direct connection between matrix spherical harmonics
and regular spherical harmonics. This has required $k\sim N\gg m$. There are
good reasons for this restriction. As we explain in §4.2 in the main text,
outside of this limit there are intrinsically matrix-like effects of spatial
noncommutativity. However, an improvement to (59) is possible that captures
some of these effects. The full difference equation obeyed by $f(k)$, obtained
from (56) and the explicit form for $g(k)$, is symmetric under the reflection
$k\leftrightarrow N-m-k+1$. Solutions for $f(k)$ can then be taken to be even
or odd under this symmetry. The expression (59) can be minimally promoted to
one that manifests this symmetry by shifting the argument of the Legendre
polynomial by $k\to k+(m-1)/2$ to give

$f(k)=c_{2}Y_{jm}\left(\cos^{-1}\left(1-\frac{2k+m-1}{N}\right),0\right)\,.$
(60)

This is the form we have quoted in the main text (30). The overall
normalization $c_{2}$ is discussed in the following paragraph below. In (31)
in the main text we have dropped the final $-1/N$ term in (60) that is
subleading at large $N$. One may verify that while the shifted expression (60)
does remove some of the leading order $m/N$ corrections to the expanded
equation of motion (57), it does not remove all of them and therefore does not
capture $m/N$ corrections in controlled way. Our final results in the main
text, with the cutoff $\Lambda$ introduced, are only sensitive to the regime
$m\ll N$ and do not depend on these corrections. The improved expression (60)
gives a sense of the effects of matrix noncommutativity — see the discussion
in §4.2 — and furthermore turns out to be in very good agreement with exact
numerical answers even in the noncommutative regime, as illustrated in Figs. 3
and 4.

Figure 3: Comparison of the exact $f(k)$ in the matrix spherical harmonics
(solid dots) with the expression (60), given by the solid line, with $N=39$,
$j=5$ and $m=4$. The dashed line shows the expression (60) without the shift
by $(m-1)/N$. Even while small in this regime, the shift significantly
improves the comparison. Figure 4: Same comparison as in Fig. 3 but now with
$N=13$, $j=7$ and $m=6$. The agreement remains good despite noncommutative
corrections being significant.

The normalization constant $c_{2}$ in (60) is fixed by the normalization of
the matrix spherical harmonics
$\frac{1}{N}\Tr[\hat{Y}_{jm}^{{\dagger}}\hat{Y}_{jm}]=1$ as well as the fact
that $\int_{-1}^{1}Y_{jm}(\cos^{-1}(x),0)^{2}\,dx=(2\pi)^{-1}$. In the large
$N$ limit the trace may be turned into an integral so that

$c_{2}^{2}=4\pi
c_{jm}^{2}\,,\qquad\frac{1}{c_{jm}^{2}}=2\pi\int_{-1+\frac{m}{N}}^{1-\frac{m}{N}}Y_{jm}(\cos^{-1}(x),0)^{2}\,dx$
(61)

The sums over $j$ and $m$ that give the entanglement and other observables are
dominated by large $j$ and $m$. Although we will not actually need to do so in
this paper, for possible future reference we note that in this regime one may
use the WKB form of the spherical harmonics

$\displaystyle
Y_{jm}(\theta)=\frac{1}{\pi}\frac{\Theta(\sin^{2}\theta-\hat{m}^{2})}{\left(\sin^{2}\theta-\hat{m}^{2}\right)^{1/4}}$
$\displaystyle\times$ (62)

where $\hat{m}=m/(j+\frac{1}{2})$ and $\Theta$ is the heaviside step function.
This form may be derived from the differential equation obeyed by the
$Y_{jm}(\theta)$ and may be explicitly verified using numerics to match the
full expressions very well at large $j$ and $m$.

##  Appendix B Details of the normal modes

The expansion (32) for the normal modes (26) leads to equations for the
coefficients [5]:

$\displaystyle
y_{jm}^{3}+\frac{1}{2}\sqrt{(j+m+1)(j-m)}y_{j(m+1)}^{+}-\frac{1}{2}\sqrt{(j-m-1)(j+m)}y_{j(m-1)}^{-}=\omega
y_{jm}^{3}\,,$ (63) $\displaystyle(\omega\pm m)y_{j(m\pm
1)}^{\pm}=\pm\sqrt{(j\pm m+1)(j\mp m)}y_{jm}^{3}\,.$ (64)

Here we see immediately that to find a mode with a given frequency $\omega$ we
may pick a single nonzero $y_{jm}^{3}$, corresponding to a given value of $j$
and $m$, and then the only other coefficients that are forced to be nonzero
from (64) are $y^{+}_{j(m+1)}$ and $y^{-}_{j(m-1)}$. For the perturbations
$Y^{i}$ to be hermitian we must have

$\bar{y}^{3}_{jm}=(-1)^{m}y^{3}_{j(-m)}\,,\qquad\bar{y}_{jm}^{+}=(-1)^{m}y_{j(-m)}^{-}\,,$
(65)

These conditions further require that we have a nonzero $y_{j(-m)}^{3}$,
$y^{+}_{j(-m+1)}$ and $y^{-}_{j(-m-1)}$.

In order to remain in the gauge slice that $X^{3}$ is diagonal, we must impose
$Y^{3}=0$. To achieve this we may add a pure gauge mode, with $\omega=0$, to
the physical mode, which has $\omega\neq 0$. We may choose
$y_{jm}^{3}(\omega)+y_{jm}^{3}(0)=0$, so that the full mode physical mode has

$\displaystyle Y^{3}$
$\displaystyle=\left(y_{jm}^{3}(\omega)+y_{jm}^{3}(0)\right)\hat{Y}_{jm}+\text{h.c.}=0\,,$
(66) $\displaystyle Y^{+}$
$\displaystyle=\left(y_{j(m+1)}^{+}(\omega)+y_{j(m+1)}^{+}(0)\right)\hat{Y}_{j(m+1)}+\left(\bar{y}_{j(m-1)}^{-}(\omega)+\bar{y}_{j(m-1)}^{-}(0)\right)\hat{Y}_{j(m-1)}^{\dagger}\,.$
(67)

And recall that $Y^{+}=(Y^{-})^{\dagger}$. The normalization condition, given
below (26), implies that

$1=\tr\left((Y^{+})^{\dagger}Y^{+}\right)=N\left(\left|y_{j(m+1)}^{+}(\omega)+y_{j(m+1)}^{+}(0)\right|^{2}+\left|y_{j(m-1)}^{-}(\omega)+y_{j(m-1)}^{-}(0)\right|^{2}\right)\,.$
(68)

Here we used the normalization of the matrix spherical harmonics, given below
(31), as well as the orthogonality of these harmonics at different values of
$m$.

The equations (64), gauge condition (66) and normalization (68) may be solved
explicitly to give the coefficients of the mode. For example,

$\left|y_{j(m-1)}^{-}(\omega)+y_{j(m-1)}^{-}(0)\right|^{2}=\frac{1}{2N}\frac{(1+j-m)(j+m)(m+\omega)^{2}}{j^{2}(m^{2}+\omega^{2})+j(m^{2}+\omega^{2})-m^{2}(m^{2}+\omega(\omega-2))}\,.$
(69)

This result determines the modulus of the first coefficient in (67). The phase
of the coefficient can be chosen freely. Purely real and purely imaginary
coefficients correspond to orthogonal Hermitian combinations of the $m$ and
$-m$ modes.

##  Appendix C Classicality of the quadratic Casimir

In this appendix we compute the variance of the quadratic Casimir
$\Tr(\hat{G}_{\Sigma\Sigma}^{2})$. The variance may be obtained using the
usual Wick contraction rules for the Gaussian wavefunction (28). In particular

$\left\langle\delta\pi_{a}\delta\pi_{b}\delta\pi_{c}\delta\pi_{d}\right\rangle_{\psi_{\text{fs}}}=\frac{\nu^{2}}{4}\left(|\omega_{a}||\omega_{c}|\delta_{ab}\delta_{cd}+|\omega_{a}||\omega_{b}|\delta_{ac}\delta_{bd}+|\omega_{a}||\omega_{b}|\delta_{ad}\delta_{bc}\right)\,.$
(70)

Therefore, using the expression (37) for $\hat{G}_{\Sigma\Sigma}$ we obtain,
to leading order at large $M,N$,

$\displaystyle\Big{\langle}$^2 ⟩_ψ_fs \- ⟨Tr(^G_ΣΣ^2)⟩_ψ_fs^2
$\displaystyle=\nu^{6}M^{2}(N-M)^{2}\sum_{ab}|\omega_{a}||\omega_{b}|\sum_{PQ}Y^{-}_{bMQ}Y^{+}_{aQM}\left(Y^{-}_{bMP}Y^{+}_{aPM}+Y^{-}_{aMP}Y^{+}_{bPM}\right)$
(71)
$\displaystyle=32\pi^{2}\nu^{6}\frac{M^{2}(N-M)^{2}}{N^{2}}\sum_{j,j^{\prime}=1}^{N}4jj^{\prime}\sum_{m=1}^{\min(M,j)}|Y_{jm}(\theta_{M-m})|^{2}|Y_{j^{\prime}m}(\theta_{M-m})|^{2}\,.$
(72)

The final expression follows from similar manipulations to those leading to
(41) previously. We further used the fact that $Y_{jm}(\theta)$ is real. The
variance (72) scales likes $\nu^{6}N^{7}$. This is smaller than
$\langle\Tr(\hat{G}_{\Sigma\Sigma}^{2})\rangle^{2}\sim\nu^{6}N^{8}$ from (41).
It follows that the large $N$ Casimir is classical.

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