aainstitutetext: Department of Physics, University of California, Santa
Barbara, CA 93106, USAbbinstitutetext: Department of Physics, Cornell
University, Ithaca, NY 14853, USA

# Holographic Tensor Networks with Bulk Gauge Symmetries

Xi Dong a Sean McBride a, b and Wayne W. Weng<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Tensor networks are useful toy models for understanding the structure of
entanglement in holographic states and reconstruction of bulk operators within
the entanglement wedge. They are, however, constrained to only prepare so-
called “fixed-area states” with flat entanglement spectra, limiting their
utility in understanding general features of holographic entanglement. Here,
we overcome this limitation by constructing a variant of random tensor
networks that enjoys bulk gauge symmetries. Our model includes a gauge theory
on a general graph, whose gauge-invariant states are fed into a random tensor
network. We show that the model satisfies the quantum-corrected Ryu-Takayanagi
formula with a nontrivial area operator living in the center of a gauge-
invariant algebra. We also demonstrate nontrivial, $n$-dependent contributions
to the Rényi entropy and Rényi mutual information from this area operator, a
feature shared by general holographic states.

##  1 Introduction

The ultimate goal of the AdS/CFT correspondence is to understand, concretely,
the relationship between a bulk gravitational theory and its dual boundary
conformal field theory. Holographic duality posits that the partition
functions of the two theories are equal and that there exists an isomorphism
between the Hilbert space of states of a theory of quantum gravity
$\mathcal{H}_{\textrm{bulk}}$ and the Hilbert space of a seemingly unrelated
quantum mechanical system $\mathcal{H}_{\textrm{{boundary}}}$. If we were to
understand the precise relation between these Hilbert spaces, we would have a
tractable handle with which to study quantum gravity, in whatever form it may
ultimately arise.

In practice, the UV degrees of freedom in the bulk are not well-understood, so
one must often be satisfied with studying a subspace of states given by small
fluctuations around a fixed semiclassical saddle. These states span a code
subspace of the quantum gravity Hilbert space, and are thus embedded in the
larger Hilbert space of the dual boundary theory, in the same way as the
logical qubits of a quantum error correcting code (QECC) are embedded in a
larger Hilbert space of physical qubits Almheiri:2014lwa .

In the last decade, a useful tool for developing intuition about the bulk-to-
boundary map has been tensor networks. Tensor networks, specifically projected
entangled pair states (PEPS) and PEPS-inspired tensor networks, originally
arose in many-body physics as a generalization of matrix product states, which
allowed one to efficiently prepare spin chain states with area law
entanglement Verstraete:2004cf .

As a toy model for holography, tensor networks found their niche due to the
fact that they obey the Ryu-Takayanagi (RT) formula Ryu:2006bv  and its
refinements Faulkner:2013ana ; Swingle:2009bg ; Lewkowycz:2013nqa ;
Dong:2017xht . In particular, random tensor networks (RTNs) Hayden:2016cfa
reproduce several desirable properties of a holographic QECC, namely
satisfying a quantum-corrected RT formula and the Petz recontruction of local
operators Jia:2020etj .

We now give a short overview of holographic RTNs and their entanglement
properties, as well as their issues. A rank-$k$ tensor can be represented by
its components $T_{\mu_{1}\cdots\mu_{k}}$, with $\mu_{i}=1,\dots,D_{i}$ (the
bond dimension). We can associate to each leg a $D_{i}$-dimensional Hilbert
space $\mathcal{H}_{i}$ spanned by an orthonormal basis of states
$\\{|\mu_{i}\rangle,\,\,\mu_{i}=1,\cdots,D_{i}\\}$. The tensor $T$ can then be
thought of as a state on the tensor product Hilbert space
$\bigotimes_{i=1}^{k}\mathcal{H}_{i}$:

$|T\rangle=\sum_{\mu_{1},\cdots,\mu_{k}}T_{\mu_{1}\cdots\mu_{k}}|\mu_{1}\rangle\otimes\cdots\otimes|\mu_{k}\rangle.$
(1)

To construct a tensor network, we consider a set of vertices and links which
form a network. To each vertex $x$ we associate a state $|T_{x}\rangle$, such
that the collection of all tensors defines a product state
$\otimes_{x}|T_{x}\rangle$. Adjacent tensors are those connected by a link;
their corresponding legs are contracted by projecting onto a maximally
entangled state. For simplicity, we assume that all contracted legs have the
same bond dimension $D$. Denoting the tensor product Hilbert space on the two
legs connecting the tensors at vertices $x$ and $y$ as
$\mathcal{H}_{xy}\otimes\mathcal{H}_{yx}$, this means that we project onto the
state $\ket{xy}=D^{-1/2}\sum_{\mu=1}^{D}\ket{\mu_{xy}}\otimes\ket{\mu_{yx}}$.
Uncontracted legs are called “dangling” and come in two types: bulk legs
(viewed as input) and boundary legs (viewed as output). We write the boundary
state in the following way:111Here, we have chosen a pure state as the bulk
input, but generalizing to mixed states is straightforward.

$\ket{\Psi_{\partial}}=\left(\bra{\Phi_{b}}\otimes\bigotimes_{\expectationvalue{xy}}\bra{xy}\right)\left(\bigotimes_{x}\ket{T_{x}}\right),$
(2)

where we project the bulk input legs onto a bulk state $\ket{\Phi_{b}}$. In an
RTN, we choose $T_{x}$ to be independent random tensors and take $D$ to be
large. We will not go into details on how one computes Rényi entropy in the
RTN here; the important point is that, for a boundary subregion $R$, one finds
the following answer for the Rényi entropy $S_{n}(R)$:

$S_{n}(R)=\absolutevalue{\gamma_{R}}\log D+S_{n}(\rho_{r}),$ (3)

where $\absolutevalue{\gamma_{R}}$ is the number of links cut by the minimal
surface $\gamma_{R}$ homologous to $R$ and $S_{n}(\rho_{r})$ is the Rényi
entropy of the bulk subregion $r$ bounded by $R\cup\gamma_{R}$ (we will call
$r$ the entanglement wedge). Analytically continuing to $n=1$ recovers the
Faulkner-Lewkowycz-Maldacena (FLM) formula

$S_{\textrm{vN}}(R)=\frac{\big{\langle}\hat{A}\big{\rangle}}{4G_{N}}+S_{\textrm{vN}}(\rho_{r}),$
(4)

with $\absolutevalue{\gamma_{R}}\log D$ identified with the expectation value
of the area operator $\langle\hat{A}\rangle/4G_{N}$.

In a state with vanishing bulk Rényi entropy (such as a product state), the
boundary Rényi entropy (3) is consequently independent of $n$. The RTN thus
exhibits a flat entanglement spectrum due to the projection of contracted legs
onto maximally mixed states.222The HaPPY code Pastawski:2015qua  also features
a flat Rényi spectrum for similar reasons. This differs sharply from what we
expect from generic situations in AdS/CFT. For example, the Rényi entropy for
an interval $R$ of length $\ell$ in the vacuum state of a two-dimensional CFT
takes the form

$S_{n}(R)=\frac{c}{6}\left(1+\frac{1}{n}\right)\log\left(\frac{\ell}{\epsilon}\right),$
(5)

which is manifestly $n$-dependent. One possible solution is to instead project
contracted legs onto a non-maximally entangled link state Dong:2021clv ;
Cheng:2022ori . By tuning the entanglement spectrum appropriately, this allows
one to reproduce the correct single-interval CFT vacuum Rényi entropy (5), but
does not work in more general cases such as that of multiple disjoint
intervals. To see this, consider two disjoint intervals $R_{1}$ and $R_{2}$
(see Figure 1), and for simplicity consider the case where the mutual
information between the intervals is small in the sense that the RT surfaces
are always in a disconnected phase. The boundary Rényi entropy can be obtained
by inserting appropriate cosmic branes into the bulk Dong:2016fnf . The
tension of the cosmic branes is proportional to $1-1/n$. In a fully
gravitating system, the two cosmic branes homologous to $R_{1}$, $R_{2}$ will
backreact and affect each other in an $n$-dependent way. This results in a
nonzero Rényi mutual information between the two intervals that cannot be
reproduced in RTNs by simply adding non-maximally entangled links, because
they would not allow the minimal surfaces to affect each other.

From the gravity point of view, the RTN prepares a so-called fixed-area state
Akers:2018fow ; Dong:2018seb , which is an eigenstate of the area operator
$\hat{A}$ in (4). Such eigenstates form a complete basis for semiclassical
states prepared via the gravitational path integral, so in principle any
semiclassical state can be represented as a superposition over fixed-area
basis states $\ket{\alpha}$, where $\alpha$ labels the eigenvalues of the area
operator. As the area operator lives on the RT surface dividing the
entanglement wedge $r$ and its complement $\overline{r}$, it naturally belongs
to the center of the algebra of bulk operators in $r$. This view was espoused
in Harlow:2016vwg , where it was shown that the FLM formula (4) can be derived
from a quantum error correcting code with complementary recovery. In that
language, the area operator is a specific element of the center of the bulk
von Neumann algebra on $r$. The usual RTN implements a special case of this
where the algebra has a trivial center, i.e., the center consists of
$c$-numbers only and is therefore isomorphic to C. In particular, this means
that the area operator must be a $c$-number, which, as previously discussed,
is incongruous with what one observes in gravitational holography.

Figure 1: The cosmic branes that arise in computing the Rényi entropy for
disjoint subregions. These branes have nonzero, $n$-dependent tension, and so
would backreact in a realistic holographic system.

The goal of this paper is to construct a model where the algebra on $r$ has a
nontrivial center and to identify a nontrivial area operator living in the
center.333Having a nontrivial center does not guarantee that the area operator
is not a $c$-number; for example, see Cao:2023mzo . An ad hoc way of getting a
nontrivial center is to “stack” multiple layers of tensor networks by hand to
form superpositions of fixed-area states. We will not do this but will instead
pursue a more physically motivated approach. In particular, one would like to
incorporate something akin to “edge states”, degrees of freedom which live on
the minimal surface, in order to go beyond fixed-area states and produce a
nontrivial area operator.444Initial work in this direction was taken in
Donnelly:2016qqt  by generalizing the HaPPY code. Our goal in this work is to
give a model which provides a physical origin for these edge states. Inspired
by similar operators found in gauge theory Donnelly:2011hn , we will add a
second layer on top of the standard RTN which imposes gauge invariance. This
alters the algebra of operators in the bulk, and as we will show, it
introduces a nontrivial contribution to the area operator of the following
form:

$\Delta\widetilde{A}=\bigoplus_{\alpha}\widetilde{P}^{\alpha}\log d_{\alpha},$
(6)

where roughly speaking $\alpha$ denotes a superselection sector in the gauge-
invariant Hilbert space, $\widetilde{P}^{\alpha}$ is the projection onto that
superselection sector, and $d_{\alpha}$ is the dimension of $\alpha$ viewed as
an irreducible representation. The important thing to note at the moment is
that this operator is not a $c$-number and is therefore nontrivial.

The structure of this paper is as follows. In Section 2 we will set up our
model – a two-layer gauged random tensor network – and introduce the formalism
for gauge theory on a graph. In Section 3 we will analyze the Hilbert space of
gauge-invariant states and the algebras of gauge-invariant operators for a
subregion. In Section 4 we will compute entanglement and Rényi entropies in
both the pre-gauged and gauge-invariant algebras, which we will use to derive
the new area operator for our model. We conclude with some discussion and
future directions.

##  2 The Gauged Random Tensor Network

We now construct our model. It has two layers: a top layer consisting of a
gauge theory on a graph, and a bottom layer made of a standard random tensor
network. We illustrate some examples of this two-layer model in Figure 2. The
top layer produces a gauge-invariant state which is then fed into the bottom
layer as input. The final output of the model is the boundary state produced
by the bottom RTN. We can then analyze properties of the boundary state (such
as its entropy) using the usual techniques for the random tensor network.

This construction has some nice properties. In particular, one might be
worried that if the structure of the RTN is altered, Petz reconstruction of
local operators might no longer hold. Here we avoid this potential issue by
keeping the tensor network the same, but changing the space of states that can
be fed into the network.

Figure 2: Some examples of our two-layer model, with a gauge theory on a
directed graph on the top layer and a random tensor network with dangling
boundary legs on the bottom. In these examples, we choose each tensor in the
bottom layer to have a bulk input leg which is either a vertex or edge on the
graph. The light gray planes in the right example are included for visual
clarity.

Given this construction, we would like to understand what set of gauge-
invariant states we will be feeding into the bottom layer. The following is
based on a non-dynamical version of the standard Kogut-Susskind construction
in lattice gauge theory Kogut:1974ag .555See also related discussion in
Dolev:2021ofc . As we do not require our graph to be a lattice, i.e. there is
not necessarily a regular tiling, we will refrain from calling our top layer a
lattice gauge theory.

Our starting point is an arbitrary directed graph $\Lambda=(V,E)$ consisting
of vertices $V=\\{v\\}$ and edges $E=\\{e\\}$. We require the graph to be
directed so we have a well-defined orientation on each edge, though we
emphasize that the choice of orientation is arbitrary. We impose no additional
conditions on the graph. In particular, the graph could have loops, adjacent
vertices could be connected by multiple edges, and the graph does not need to
be planar.

We start with a gauge group, which we choose to be a compact Lie group $G$. It
does not have to be connected, and in particular, we could consider finite
groups such as ${\mathbb{Z}}_{2}$ if we wish. We assign a (pre-gauged) Hilbert
space to each vertex and edge of the graph $\Lambda$. The Hilbert space
$\mathcal{H}_{e}$ on each edge $e$ is taken to be $L^{2}(G)$, the space of
square-integrable functions on $G$. A state $\ket{\psi}_{e}$ in this
$\mathcal{H}_{e}=L^{2}(G)$ can be written as an integral666In cases where $G$
is finite, the integral is understood as a sum: $\ket{\psi}_{e}=\sum_{g\in
G}\frac{1}{\absolutevalue{G}}\psi(g)\ket{g}_{e}$, where $|G|$ is the order of
$G$. over orthonormal basis elements $\ket{g}$ labeled by $g\in G$:

$\ket{\psi}_{e}=\int dg\psi(g)\ket{g}_{e},$ (7)

where $dg$ is the Haar measure777The Haar measure on $G$ is invariant under
left and right group multiplication ($g\rightarrow g^{\prime}g$ and
$g\rightarrow gg^{\prime}$) and is normalized such that $\int dg=1$. on $G$.
For our purposes, it will be useful to work with another orthonormal basis

$\ket{\alpha ij}_{e},\quad i,j=1,2,\cdots,d_{\alpha}$ (8)

for the same $\mathcal{H}_{e}=L^{2}(G)$, where $\alpha$ labels irreducible
representations (irreps) of $G$ and $d_{\alpha}$ is the dimension of the
representation $\alpha$. This representation basis is orthonormal:

${}_{e}\langle\alpha ij|\beta
k\ell\rangle_{e}=\delta_{\alpha\beta}\delta_{ik}\delta_{j\ell},$ (9)

and can be written in terms of the previously defined group basis
$\ket{g}_{e}$:

$\ket{\alpha ij}_{e}=\sqrt{d_{\alpha}}\int dg\,D^{\alpha}_{ij}(g)\ket{g}_{e},$
(10)

where $D^{\alpha}_{ij}(g)$ are elements of a unitary matrix $D^{\alpha}(g)$
representing $g$ in $\alpha$. This can be viewed as a “Fourier transform”
between the representation basis and the group basis.

The group action induces a set of unitaries $L_{e}(g)$ and $R_{e}(g)$ which
act as left and right group multiplications on the group basis:

$L_{e}(g)\ket{h}_{e}=\ket{gh}_{e},\quad
R_{e}(g^{-1})\ket{h}_{e}=\ket{hg^{-1}}_{e}.$ (11)

In the representation basis, the group unitaries instead act as unitary matrix
multiplication on one of the two indices $i$, $j$:

$L_{e}(g)\ket{\alpha ij}_{e}=\sum_{k}D^{\overline{\alpha}}_{ki}(g)\ket{\alpha
kj}_{e},\quad R_{e}(g^{-1})\ket{\alpha
ij}_{e}=\sum_{k}D^{\alpha}_{kj}(g)\ket{\alpha ik}_{e},$ (12)

where $\overline{\alpha}$ denotes the complex conjugate representation of
$\alpha$ defined by $D^{\overline{\alpha}}_{ij}(g)=D^{\alpha*}_{ij}(g)$. Thus
there are two copies of $G$ acting on $\mathcal{H}_{e}$: $L_{e}(g)$ gives the
group action of the first copy under which $\ket{\alpha ij}_{e}$ transforms in
the representation $\overline{\alpha}$, and $R_{e}(g^{-1})$ gives the action
of the second copy under which $\ket{\alpha ij}_{e}$ transforms in the
representation $\alpha$. Altogether, $\ket{\alpha ij}_{e}$ transforms in the
external tensor product888For representations $\alpha_{1}$, $\alpha_{2}$ of
$G$, their external tensor product $\alpha_{1}\boxtimes\alpha_{2}$ is a
representation of $G\times G$ with an underlying vector space
${\mathcal{H}}^{\alpha_{1}}\otimes{\mathcal{H}}^{\alpha_{2}}$, where
${\mathcal{H}}^{\alpha_{1}}$ transforms under the first $G$ in the
$\alpha_{1}$ representation and ${\mathcal{H}}^{\alpha_{2}}$ transforms under
the second $G$ in the $\alpha_{2}$ representation. Note that this is different
from the (usual) tensor product $\alpha_{1}\otimes\alpha_{2}$ which is a
representation of $G$ (not $G\times G$), with an underlying vector space
${\mathcal{H}}^{\alpha_{1}}\otimes{\mathcal{H}}^{\alpha_{2}}$ where
${\mathcal{H}}^{\alpha_{1}}$ and ${\mathcal{H}}^{\alpha_{2}}$ transform under
the same $G$. representation $\overline{\alpha}\boxtimes\alpha$ of $G\times
G$. Using this, we decompose $\mathcal{H}_{e}$ as

$\mathcal{H}_{e}\cong\bigoplus_{\alpha}\mathcal{H}^{\overline{\alpha}}\otimes\mathcal{H}^{\alpha}\cong\bigoplus_{\alpha}\left(\mathcal{H}^{\alpha}\right)^{\oplus
d_{\alpha}},$ (13)

where the sum runs over all irreducible representations $\alpha$ of $G$ and
$\mathcal{H}^{\alpha}$ is a Hilbert space of dimension $d_{\alpha}$
transforming in the $\alpha$ representation. It will be convenient to use the
representation basis for the remainder of the paper.

Now we turn to the (pre-gauged) Hilbert space ${\mathcal{H}}_{v}$ on a vertex
$v$. In general, ${\mathcal{H}}_{v}$ may be chosen quite arbitrarily
(corresponding to specifying any number of matter degrees of freedom including
the case of no matter), but it needs to furnish some representation under the
group action of $G$. This representation could be reducible or trivial, but it
can always be decomposed into a direct sum of irreducible representations of
$G$. Using this, we may decompose a general ${\mathcal{H}}_{v}$ as

$\mathcal{H}_{v}=\bigoplus_{\alpha}\left(\mathcal{H}_{v}^{\alpha}\right)^{\oplus
n_{\alpha}}.$ (14)

Here the sum again runs over all distinct irreducible representations $\alpha$
of $G$ and $n_{\alpha}$ is the multiplicity of the representation $\alpha$ in
${\mathcal{H}}_{v}$. Note that $n_{\alpha}$ could be any nonnegative integer,
and in particular, it could be zero (representing the absence of a given
representation $\alpha$ in ${\mathcal{H}}_{v}$). Thus, the simplest choice of
${\mathcal{H}}_{v}$ is a trivial Hilbert space with no matter (corresponding
to $n_{\alpha}=0$ for all $\alpha$), but in the discussion below we will
consider the general case (14) with arbitrary $n_{\alpha}$. Furthermore, we
will allow ${\mathcal{H}}_{v}$ to vary from one vertex to another. An
orthonormal basis of states for the Hilbert space ${\mathcal{H}}_{v}$ can be
written as

$\ket{\alpha ij}_{v},\quad i=1,\cdots,n_{\alpha},\quad j=1,\cdots,d_{\alpha},$
(15)

where the first index $i$ runs over the multiplicity $n_{\alpha}$ and the
second runs over the dimension $d_{\alpha}$. The group action of $G$ on
${\mathcal{H}}_{v}$ is given by unitary operators $U_{v}(g)$, which act on the
$\ket{\alpha ij}_{v}$ basis as

$U_{v}(g)\ket{\alpha ij}_{v}=\sum_{k}D^{\alpha}_{kj}(g)\ket{\alpha ik}_{v}.$
(16)

Note that $U_{v}(g)$ only acts on the second index $j$ and is analogous to the
action of $R_{e}(g^{-1})$ in (11). Thus, we find an important distinction
between the vertex Hilbert space ${\mathcal{H}}_{v}$ and the edge Hilbert
space ${\mathcal{H}}_{e}$. To see this, first note that the two Hilbert spaces
share some similarities. In particular, ${\mathcal{H}}_{e}$ is a direct sum of
irreducible representations $\alpha$ with multiplicity $d_{\alpha}$ as shown
on the right-hand side of (13), and this is the analogue of (14) for
${\mathcal{H}}_{v}$. The representation basis (8) of ${\mathcal{H}}_{e}$ is
similar to the basis (15) of ${\mathcal{H}}_{v}$. However, the difference is
that an edge has the additional structure of allowing another group action
$L_{e}(g)$ that acts on the first index $i$ of $\ket{\alpha ij}_{e}$, whereas
at a vertex the first index $i$ of $\ket{\alpha ij}_{v}$ is a multiplicity
index that does not admit a natural group action.

The pre-gauged Hilbert space for the entire graph is then

$\mathcal{H}=\left(\bigotimes_{v\in
V}\mathcal{H}_{v}\right)\otimes\left(\bigotimes_{e\in
E}\mathcal{H}_{e}\right).$ (17)

We refer to the algebra of all bounded operators on ${\mathcal{H}}$ as
$\mathcal{A}=\mathcal{B}\left(\mathcal{H}\right)$. As ${\mathcal{H}}$
completely factorizes over the vertices and edges, so too does the algebra of
operators

$\mathcal{A}=\left(\bigotimes_{v\in
V}\mathcal{A}_{v}\right)\otimes\left(\bigotimes_{e\in
E}\mathcal{A}_{e}\right).$ (18)

Using the representation basis (8) of ${\mathcal{H}}_{e}$, $\mathcal{A}_{e}$
can be written as

$\mathcal{A}_{e}=\textrm{span}\\{\ket{\alpha ij}_{e}\bra{\beta k\ell}\\},$
(19)

where the indices $i,j,k,\ell$ run over the irrep dimension. Similarly, using
(15) we write

$\mathcal{A}_{v}=\textrm{span}\\{\ket{\alpha ij}_{v}\bra{\beta k\ell}\\},$
(20)

where $i,k$ run over the irrep multiplicity and $j,\ell$ run over the irrep
dimension.

For each vertex $v$, we now define a gauge transformation $A_{v}(g)$ as the
following unitary operator acting on $v$ and all its associated edges:

$A_{v}(g)\equiv U_{v}(g)\prod_{e\in E^{-}(v)}L_{e}(g)\prod_{e\in
E^{+}(v)}R_{e}(g^{-1}),$ (21)

where $E^{-}(v)$ consists of edges associated to $v$ oriented away from the
vertex and $E^{+}(v)$ consists of edges oriented into the vertex. Physical
states are defined to be those invariant under gauge transformations
$A_{v}(g)$ for all $g$ and $v$. The easiest way of generating a gauge-
invariant state is to average over all gauge transformations acting on a state
in ${\mathcal{H}}$. The operator that implements this averaging on a vertex
$v$ is the following projector:

$\Pi_{v}=\int dgA_{v}(g).$ (22)

$\Pi_{v}$ obeys the usual properties of a projector such that
$\Pi_{v}^{2}=\Pi_{v}$ and $\Pi_{v}=\Pi_{v}^{\dagger}$. The gauge-invariant
projector on the entire graph is simply the product of individual projectors
on all vertices:

$\Pi_{\text{GI}}=\prod_{v\in V}\Pi_{v}.$ (23)

It is easy to verify that $[A_{v}(g),A_{v^{\prime}}(g^{\prime})]=0$ for all
$v$, $v^{\prime}$, $g$, $g^{\prime}$, and therefore
$[\Pi_{v},\Pi_{v^{\prime}}]=0$.

Throughout the paper, we will denote fully gauge-invariant spaces, states, and
operators with a tilde; for instance, the gauge-invariant states
$\ket{\widetilde{\psi}}$ are elements of $\widetilde{\mathcal{H}}$ defined via

$\widetilde{\mathcal{H}}\equiv\Pi_{\text{GI}}\mathcal{H}.$ (24)

The gauge-invariant algebra $\widetilde{\mathcal{A}}$ is defined as the space
of bounded operators on $\widetilde{\mathcal{H}}$. $\widetilde{\mathcal{A}}$
can alternatively be represented by conjugation of the pre-gauged algebra
${\mathcal{A}}$ with the projector $\Pi_{\text{GI}}$:

$\widetilde{\mathcal{A}}=\Pi_{\text{GI}}\mathcal{A}\Pi_{\text{GI}}.$ (25)

We should comment on the interpretation of the operators in this gauge-
invariant algebra. Every operator
$\widetilde{\mathcal{O}}\in\widetilde{\mathcal{A}}$ can be extended to a pre-
gauged operator $\mathcal{O}\in\mathcal{A}$ which acts identically on gauge-
invariant states. There is generally more than one extension to $\mathcal{A}$,
and to choose a unique extension one must specify the action of the pre-gauged
operator on the orthogonal complement of $\widetilde{\mathcal{H}}$. We make
the natural choice that the extension ${\mathcal{O}}$ should annihilate the
orthogonal complement. Moreover, for notational simplicity, we identify every
$\widetilde{\mathcal{O}}\in\widetilde{\mathcal{A}}$ with its natural extension
${\mathcal{O}}\in{\mathcal{A}}$ (which annihilates the orthogonal complement),
as we have done in (25). The reason for this natural extension will become
clearer in later sections.

We now feed any gauge-invariant state $\ket{\widetilde{\psi}}$ as the bulk
input into the RTN on the the bottom layer, in a manner illustrated by Figure
2. In particular, the bulk dangling legs of the RTN should match and connect
to the edges and vertices of the graph $G$ on the top layer, for
$\ket{\widetilde{\psi}}$ lives on these edges and vertices. In other words,
each edge or vertex of $G$ is fed into a bulk dangling leg of the RTN.999In
principle, the RTN could also take any pre-gauged state as the bulk input, but
we choose to feed only gauge-invariant states because as we will see, this
restriction leads to a nontrivial area operator.

In order to utilize the full machinery of the original RTN, we would like the
Hilbert spaces associated with the tensors on the bottom layer to be finite-
dimensional (as is the case for the original RTN). When $G$ is an infinite
group, ${\mathcal{H}}_{e}=L^{2}(G)$ is infinite-dimensional and there are an
infinite number of irreducible representations to sum over, so in order to
avoid a tensor in the bottom layer having an infinite-dimensional leg, we
impose a cutoff on our edge and vertex Hilbert spaces. This can take the form
of, e.g., a cutoff in the sums in (13) and (14). Therefore, we are only
feeding in states that live in a finite-dimensional subspace of
$\widetilde{\mathcal{H}}$. This does not affect the discussion in the next
section of the gauge-invariant algebra; the cutoff is only relevant when we
compute entanglement measures in Section 4.

##  3 Deriving the Gauge-Invariant Algebra

Now that we have defined our gauge-invariant states, we would like to
understand the structure of the algebra of gauge-invariant operators. Our
overarching goal is to write down the gauge-invariant subalgebra for a
subregion $r$ of the top layer which we will later use to derive an FLM
formula for the gauged RTN.

###  3.1 The structure of the gauge-invariant Hilbert space

We now study the decomposition of $\widetilde{\mathcal{H}}$ when our graph
$\Lambda$ is divided into a subregion and its complement. We define a
subregion $r$ of $\Lambda$ to be an arbitrary subset of vertices and edges
(without further restrictions). We call the complement subregion
$\overline{r}$.

In order to work out a useful basis for gauge-invariant states, it is
convenient to divide the set $V$ of all vertices into three types: those
strictly in $r$ (meaning that the vertex and its associated edges are all in
$r$), those strictly in $\overline{r}$, and vertices “on the cut” (meaning
that the vertex and its associated edges are partly in $r$ and partly in
$\overline{r}$). We call these sets $V_{r}$, $V_{\overline{r}}$, and
$V_{c}\equiv V/\left(V_{r}\cup V_{\overline{r}}\right)$, respectively.
Consequently, the gauge-invariant projector can be decomposed in the following
way:

$\Pi_{\text{GI}}=\Pi_{V_{r}}\Pi_{V_{c}}\Pi_{V_{\overline{r}}},$ (26)

where $\Pi_{V_{i}}$ is defined as the product of individual projections
$\Pi_{v}$ over all vertices $v\in V_{i}$, for $i=r,c,\overline{r}$. First, let
us discuss a partial gauging of the pre-gauged Hilbert space. Using the tensor
decomposition of
$\mathcal{H}=\mathcal{H}_{r}\otimes\mathcal{H}_{\overline{r}}$, we can write
$\widetilde{\mathcal{H}}=\Pi_{\text{GI}}{\mathcal{H}}$ as

$\displaystyle\widetilde{\mathcal{H}}$
$\displaystyle=\Pi_{V_{r}}\Pi_{V_{c}}\Pi_{V_{\overline{r}}}\left(\mathcal{H}_{r}\otimes\mathcal{H}_{\overline{r}}\right)$
$\displaystyle=\Pi_{V_{c}}\left(\left(\Pi_{V_{r}}\mathcal{H}_{r}\right)\otimes\left(\Pi_{V_{\overline{r}}}\mathcal{H}_{\overline{r}}\right)\right).$
(27)

We define the two terms in the parentheses as

$\hat{\mathcal{H}}_{r}\equiv\Pi_{V_{r}}\mathcal{H}_{r},\quad\hat{\mathcal{H}}_{\overline{r}}\equiv\Pi_{V_{\overline{r}}}\mathcal{H}_{\overline{r}}.$
(28)

These are “partially gauged” Hilbert spaces, in the sense that states in
$\hat{\mathcal{H}}_{r}$ ($\hat{\mathcal{H}}_{\overline{r}}$) are invariant
under gauge transformations associated to vertices in $V_{r}$
($V_{\overline{r}}$), but not so under gauge transformations on the cut. We
denote the partially gauged Hilbert space on the full graph as

$\hat{\mathcal{H}}=\hat{\mathcal{H}}_{r}\otimes\hat{\mathcal{H}}_{\overline{r}}.$
(29)

As $\hat{\mathcal{H}}$ tensor factorizes, the algebra of operators on
$\hat{\mathcal{H}}$ also factorizes as

$\hat{\mathcal{A}}=\hat{\mathcal{A}}_{r}\otimes\hat{\mathcal{A}}_{\overline{r}}.$
(30)

Now that we have a partially gauged Hilbert space $\hat{{\mathcal{H}}}$, it
remains to impose gauge invariance “on the cut” and obtain the fully gauged
Hilbert space $\widetilde{\mathcal{H}}=\Pi_{V_{c}}\hat{{\mathcal{H}}}$. The
gauge transformation (21) associated to each vertex $v_{i}\in V_{c}$ can be
decomposed into unitary operators in $r$ and $\overline{r}$:

$A_{v_{i}}(g_{i})=A_{v_{i},r}(g_{i})A_{v_{i},\overline{r}}(g_{i}).$ (31)

Let $n\equiv\absolutevalue{V_{c}}$ be the number of vertices on the cut. The
gauge-invariant projector on the cut $\Pi_{V_{c}}$ acts by integrating over
the gauge transformations associated to the $n$ vertices in $V_{c}$:

$\displaystyle\Pi_{V_{c}}$ $\displaystyle=\int dg_{1}\cdots
dg_{n}A_{v_{1}}(g_{1})\cdots A_{v_{n}}(g_{n})$ $\displaystyle=\int
dg_{1}\cdots dg_{n}A_{v_{1},r}(g_{1})\cdots
A_{v_{n},r}(g_{n})A_{v_{1},\overline{r}}(g_{1})\cdots
A_{v_{n},\overline{r}}(g_{n})$ $\displaystyle\equiv\int
dgA_{r}(g)A_{\overline{r}}(g),$ (32)

where we have defined $A_{r}(g)=\prod_{i=1}^{n}A_{v_{i},r}(g_{i})$ (and
similarly for $A_{\overline{r}}(g)$), $g=(g_{1},\cdots,g_{n})$ is a element of
$G^{n}$ (the direct product of $n$ copies of $G$ on the cut), and $dg$ is the
Haar measure on $G^{n}$. Thus $A_{r}(g)$ is a $G^{n}$ action on
$\hat{\mathcal{H}}_{r}$, and $\hat{\mathcal{H}}_{r}$ can be decomposed into
irreps of $G^{n}$. We decompose $\hat{\mathcal{H}_{r}}$ into the following
way:

$\hat{\mathcal{H}}_{r}\cong\bigoplus_{\alpha,i}\hat{\mathcal{H}}^{\alpha
i}_{r},$ (33)

where $\alpha$ as an irreducible representation of $G^{n}$ can also be thought
of as the external tensor product of $n$ irreps of $G$, i.e., $\alpha$ denotes
the external tensor product
$\alpha_{1}\boxtimes\alpha_{2}\boxtimes\cdots\boxtimes\alpha_{n}$. Thus, we
will sometimes write $\alpha$ as a tuple of $G$ irreps
$(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})$. The index $i=1,\cdots,n_{\alpha}$
denotes the multiplicity of the $\alpha$ irrep. The sum ranges over all
$G^{n}$ irreps but some irreps may appear with zero multiplicity, as in the
single vertex Hilbert space (14).

From the decomposition (33), we write an orthonormal basis for
$\hat{\mathcal{H}}_{r}$ as $\\{\ket{\alpha ik}_{r}\\}$, where again the first
index $i=1,\cdots,n_{\alpha}$ runs over the irrep multiplicity and the second
index $k=1,\cdots,d_{\alpha}$ labels an orthonormal basis for each
$\hat{\mathcal{H}}_{r}^{\alpha i}$. Similarly, we write an orthonormal basis
for $\hat{\mathcal{H}}_{\overline{r}}$ as
$\\{\ket{\overline{\beta}j\ell}_{\overline{r}}\\}$, where
$j=1,\cdots,{\overline{n}}_{\overline{\beta}}$, and
${\overline{n}}_{\overline{\beta}}$ is the multiplicity of the
$\overline{\beta}$ irrep on $\overline{r}$. Explicitly, $A_{r}(g)$
($A_{\overline{r}}(g)$) acts on the basis states of $\hat{\mathcal{H}}_{r}$
($\hat{\mathcal{H}}_{\overline{r}}$) via

$\displaystyle A_{r}(g)\ket{\alpha ik}_{r}$
$\displaystyle=\sum_{k^{\prime}}D^{\alpha}_{k^{\prime}k}(g)\ket{\alpha
ik^{\prime}}_{r},$ $\displaystyle
A_{\overline{r}}(g)\ket{\overline{\beta}j\ell}_{\overline{r}}$
$\displaystyle=\sum_{\ell^{\prime}}D^{\overline{\beta}}_{\ell^{\prime}\ell}(g)\ket{\overline{\beta}j\ell^{\prime}}_{\overline{r}}.$
(34)

Combining the basis for $\hat{\mathcal{H}}_{r}$ and for
$\hat{\mathcal{H}}_{\overline{r}}$, we write an orthonormal basis for
$\hat{\mathcal{H}}$ as $\\{\ket{\alpha
ik}_{r}\ket{\overline{\beta}j\ell}_{\overline{r}}\\}$. It is worth noting that
the multiplicities ${\overline{n}}_{\overline{\alpha}}$ on $\overline{r}$ are
generally independent from the multiplicities $n_{\overline{\alpha}}$ on $r$;
in particular, ${\overline{n}}_{\overline{\alpha}}$ could vanish while
$n_{\overline{\alpha}}$ is nonzero, and vice versa.

In a sense, we have done as much gauging as we can while keeping the
factorization of the Hilbert space between $r$ and $\overline{r}$.
$\hat{\mathcal{H}}$ is similar to what is often called the extended Hilbert
space Donnelly:2011hn ; Donnelly:2014gva ; Aoki:2015bsa ; Ghosh:2015iwa ,
which is a choice of Hilbert space into which one can embed gauge-invariant
states such that the extended Hilbert space factorizes across the cut. Here we
arrive at a similar prescription by restricting from a larger Hilbert space
${\mathcal{H}}$.

Now we will write a basis of states for the fully gauge-invariant Hilbert
space $\widetilde{\mathcal{H}}$.

######  Lemma 1.

The fully gauge-invariant Hilbert space
$\widetilde{\mathcal{H}}=\Pi_{V_{c}}\left(\hat{\mathcal{H}}_{r}\otimes\hat{\mathcal{H}}_{\overline{r}}\right)$
is given by

$\widetilde{\mathcal{H}}=\left\\{\sum_{\alpha ijk}\widetilde{\psi}_{\alpha
ij}\ket{\alpha
ik}_{r}\ket{\overline{\alpha}jk}_{\overline{r}}:\widetilde{\psi}_{\alpha
ij}\in\text{C}\right\\}.$ (35)

###### Proof.

Since we already have a basis for the partially gauged Hilbert space, it
suffices to demonstrate the action of $\Pi_{V_{c}}$ on these basis states,
which is given by

$\Pi_{V_{c}}\ket{\alpha
ik}_{r}\ket{\overline{\beta}j\ell}_{\overline{r}}=\sum_{k^{\prime}\ell^{\prime}}\int
dgD^{\alpha}_{k^{\prime}k}(g)D^{\overline{\beta}}_{\ell^{\prime}\ell}(g)\ket{\alpha
ik^{\prime}}_{r}\ket{\overline{\beta}j\ell^{\prime}}_{\overline{r}}.$ (36)

We recall the Schur orthogonality relation for compact groups:

$\int
dgD^{\alpha}_{k^{\prime}k}(g)D^{\overline{\beta}}_{\ell^{\prime}\ell}(g)=\frac{\delta_{\alpha\beta}\delta_{k^{\prime}\ell^{\prime}}\delta_{k\ell}}{d_{\alpha}},$
(37)

so that the fully gauge-invariant basis states are

$\Pi_{V_{c}}\ket{\alpha
ik}_{r}\ket{\overline{\beta}j\ell}_{\overline{r}}=\frac{1}{d_{\alpha}}\delta_{\alpha\beta}\delta_{k\ell}\sum_{k^{\prime}\ell^{\prime}}\delta_{k^{\prime}\ell^{\prime}}\ket{\alpha
ik^{\prime}}_{r}\ket{\overline{\beta}j\ell^{\prime}}_{\overline{r}}=\frac{1}{d_{\alpha}}\delta_{\alpha\beta}\delta_{k\ell}\sum_{k^{\prime}}\ket{\alpha
ik^{\prime}}_{r}\ket{\overline{\beta}jk^{\prime}}_{\overline{r}}.$ (38)

Choosing $\alpha=\beta$ and $k=\ell$ gives the desired form (35). ∎

######  Remark 1.

(35) immediately implies a natural Hilbert space isomorphism

$\widetilde{\mathcal{H}}\cong\bigoplus_{\alpha}\widetilde{\mathcal{H}}_{r}^{\alpha}\otimes\widetilde{\mathcal{H}}_{\overline{r}}^{\overline{\alpha}}.$
(39)

Here $\widetilde{\mathcal{H}}_{r}^{\alpha}$ denotes a Hilbert space of
dimension $n_{\alpha}$ with orthonormal basis states $\ket{\alpha i}_{r}$
transforming in the $\alpha$ representation of $G^{n}$, and
$\widetilde{\mathcal{H}}_{\overline{r}}^{\overline{\alpha}}$ similarly denotes
a Hilbert space of dimension $\overline{n}_{\overline{\alpha}}$ with
orthonormal basis states $\ket{\overline{\alpha}j}_{\overline{r}}$
transforming in the $\overline{\alpha}$ representation. Note that although
irrep labels such as $\alpha$ appear in the basis states, they are fixed
within each Hilbert space $\widetilde{\mathcal{H}}_{r}^{\alpha}$ or
$\widetilde{\mathcal{H}}_{\overline{r}}^{\overline{\alpha}}$.

More explicitly, the natural isomorphism (39) maps an arbitrary state of (35)
in the following way:

$\ket{\widetilde{\psi}}=\sum_{\alpha ijk}\widetilde{\psi}_{\alpha
ij}\ket{\alpha
ik}_{r}\ket{\overline{\alpha}jk}_{\overline{r}}\quad\rightarrow\quad\sum_{\alpha
ij}\sqrt{d_{\alpha}}\widetilde{\psi}_{\alpha ij}\ket{\alpha
i}_{r}\ket{\overline{\alpha}j}_{\overline{r}}.$ (40)

The $\sqrt{d_{\alpha}}$ is a crucial factor which ensures that the isomorphism
preserves the inner product.

Given this decomposition, our next goal will be to define an algebra of gauge-
invariant operators on $r$, which we will call $\widetilde{\mathcal{A}}_{r}$.
Given the lack of factorization of $\widetilde{\mathcal{H}}$ as indicated by
(39), we cannot easily write $\widetilde{\mathcal{A}}_{r}$ as
$\mathcal{B}(\widetilde{\mathcal{H}}_{r})$ for some putative Hilbert space
$\widetilde{\mathcal{H}}_{r}$. Rather, we will use the known algebra of
operators on $\mathcal{H}_{r}$ and $\hat{\mathcal{H}}_{r}$ to define
$\widetilde{\mathcal{A}}_{r}$.

###  3.2 The gauge-invariant subregion algebra

It is tempting to define the algebra of gauge-invariant operators in a
subregion $r$ via restriction of the pre-gauged algebra in that region

$\widetilde{\mathcal{A}}_{r}=\Pi_{\text{GI}}\mathcal{A}_{r}\Pi_{\text{GI}},$
(41)

similar to (25). There is a second possible description of the gauge-invariant
algebra, which is that $\widetilde{\mathcal{A}_{r}}$ consists of the set of
operators $\\{\widetilde{\mathcal{O}}_{r}=\mathcal{O}_{r}\Pi_{\text{GI}}\\}$
for all operators $\mathcal{O}_{r}\in\mathcal{A}_{r}$ which commute with the
gauge-invariant projector: $[\mathcal{O}_{r},\Pi_{\text{GI}}]=0$. We will call
this algebra $\widetilde{\mathcal{A}}_{r}^{(1)}$, and the algebra (41) defined
by conjugation by the gauge-invariant projector
$\widetilde{\mathcal{A}}_{r}^{(2)}$. At first blush it is only obvious that
$\widetilde{\mathcal{A}}_{r}^{(1)}$ is a subset of
$\widetilde{\mathcal{A}}_{r}^{(2)}$, as

$\mathcal{O}_{r}\Pi_{\text{GI}}=\mathcal{O}_{r}\Pi_{\text{GI}}^{2}=\Pi_{\text{GI}}\mathcal{O}_{r}\Pi_{\text{GI}}\Rightarrow\widetilde{\mathcal{A}}_{r}^{(1)}\subseteq\widetilde{\mathcal{A}}_{r}^{(2)},$
(42)

but it is not obvious the two definitions are equivalent. Here we aim to show
that.

######  Lemma 2.

$\widetilde{\mathcal{A}}_{r}^{(1)}=\widetilde{\mathcal{A}}_{r}^{(2)}$.

###### Proof.

We again use the group action on the cut $A(g)=A_{r}(g)A_{\overline{r}}(g)$
and the gauge-invariant projector on the cut $\Pi_{V_{c}}$ which integrates
over the group action:

$\Pi_{V_{c}}=\int dgA(g).$ (43)

We define an element of $\widetilde{\mathcal{A}}_{r}^{(2)}$ by acting on an
arbitrary pre-gauged operator $\mathcal{O}_{r}\in{\mathcal{A}}_{r}$ via

$\displaystyle\Pi_{\text{GI}}\mathcal{O}_{r}\Pi_{\text{GI}}$
$\displaystyle=\Pi_{V_{\overline{r}}}\left(\Pi_{V_{c}}\Pi_{V_{r}}\mathcal{O}_{r}\Pi_{V_{r}}\Pi_{V_{c}}\right)\Pi_{V_{\overline{r}}}$
$\displaystyle=\left(\Pi_{V_{c}}\Pi_{V_{r}}\mathcal{O}_{r}\Pi_{V_{r}}\Pi_{V_{c}}\right)\Pi_{V_{\overline{r}}}$
$\displaystyle\equiv\left(\Pi_{V_{c}}\hat{\mathcal{O}}_{r}\Pi_{V_{c}}\right)\Pi_{V_{\overline{r}}}$
(44)

where
$\hat{\mathcal{O}}_{r}\equiv\Pi_{V_{r}}\mathcal{O}_{r}\Pi_{V_{r}}\in\hat{\mathcal{A}}_{r}$
is an operator on the partially gauged Hilbert space $\hat{\mathcal{H}}_{r}$.
Conjugation via the gauge-invariant projector on the cut yields

$\Pi_{V_{c}}\hat{\mathcal{O}}_{r}\Pi_{V_{c}}=\int
dgdg^{\prime}A(g)\hat{\mathcal{O}}_{r}A(g^{\prime}).$ (45)

Using the right-invariance of the Haar measure, we can shift $g\rightarrow
g(g^{\prime})^{-1}$ to obtain

$\displaystyle\Pi_{V_{c}}\hat{\mathcal{O}}_{r}\Pi_{V_{c}}$ $\displaystyle=\int
dgA(g)\int dg^{\prime}A((g^{\prime})^{-1})\hat{\mathcal{O}}_{r}A(g^{\prime})$
(46) $\displaystyle=\Pi_{V_{c}}\int
dg^{\prime}A((g^{\prime})^{-1})\hat{\mathcal{O}}_{r}A(g^{\prime})\equiv\Pi_{V_{c}}\hat{\mathcal{O}}_{r}^{\prime},$
(47)

where $\hat{\mathcal{O}}_{r}^{\prime}$ is defined by the integral over
$g^{\prime}$. We could equivalently send $g^{\prime}\rightarrow
g^{-1}g^{\prime}$ to obtain

$\displaystyle\Pi_{V_{c}}\hat{\mathcal{O}}_{r}\Pi_{V_{c}}$ $\displaystyle=\int
dgA(g)\hat{\mathcal{O}}_{r}A(g^{-1})\Pi_{V_{c}}$ $\displaystyle=\int
dgA(g^{-1})\hat{\mathcal{O}}_{r}A(g)\Pi_{V_{c}}=\hat{\mathcal{O}}_{r}^{\prime}\Pi_{V_{c}}$
(48)

where we use the fact that the Haar measure is invariant under inversion
$dg\rightarrow d(g^{-1})$. This shows
$\hat{\mathcal{O}}_{r}^{\prime}\Pi_{V_{c}}=\Pi_{V_{c}}\hat{\mathcal{O}}_{r}^{\prime}$,
so $\hat{\mathcal{O}}_{r}^{\prime}$ commutes with the gauge-invariant
projector on the cut. By construction, $\hat{\mathcal{O}}_{r}^{\prime}$ also
commutes with $\Pi_{V_{r}}$ and $\Pi_{V_{\overline{r}}}$, so it commutes with
$\Pi_{\text{GI}}$.

Now we show that $\hat{\mathcal{O}}_{r}^{\prime}$ is an element of
$\mathcal{A}_{r}$, which is not obvious as $A(g)$ on the cut acts on both $r$
and $\overline{r}$. However, we can write

$\displaystyle\hat{\mathcal{O}}_{r}^{\prime}$ $\displaystyle=\int
dgA(g^{-1})\hat{\mathcal{O}}_{r}A(g)=\int
dgA_{r}(g^{-1})A_{\overline{r}}(g^{-1})\hat{\mathcal{O}}_{r}A_{\overline{r}}(g)A_{r}(g)$
$\displaystyle=\int dgA_{r}(g^{-1})\hat{\mathcal{O}}_{r}A_{r}(g),$ (49)

as $\hat{\mathcal{O}}_{r}$ commutes with operators in $\overline{r}$. Thus
$\hat{\mathcal{O}}_{r}^{\prime}$ is in $\mathcal{A}_{r}$.

Combining the above, we can write any element of
$\widetilde{\mathcal{A}}_{r}^{(2)}$ as

$\Pi_{\text{GI}}\mathcal{O}_{r}\Pi_{\text{GI}}=\hat{\mathcal{O}}_{r}^{\prime}\Pi_{V_{c}}\Pi_{V_{\overline{r}}}=\hat{\mathcal{O}}_{r}^{\prime}\Pi_{V_{r}}\Pi_{V_{c}}\Pi_{V_{\overline{r}}}=\hat{\mathcal{O}}_{r}^{\prime}\Pi_{\text{GI}},$
(50)

which belong to $\widetilde{\mathcal{A}}_{r}^{(1)}$ as
$\hat{\mathcal{O}}_{r}^{\prime}$ is an operator in $\mathcal{A}_{r}$ that
commutes with $\Pi_{\text{GI}}$. Therefore,
$\widetilde{\mathcal{A}}_{r}^{(2)}\subseteq\widetilde{\mathcal{A}}_{r}^{(1)}$.
Morevoer, as we argued earlier, we have
$\widetilde{\mathcal{A}}_{r}^{(1)}\subseteq\widetilde{\mathcal{A}}_{r}^{(2)}$.
Thus, we have shown
$\widetilde{\mathcal{A}}_{r}^{(1)}=\widetilde{\mathcal{A}}_{r}^{(2)}$. ∎

######  Remark 2.

It will be more convenient to use $\widetilde{\mathcal{A}}_{r}^{(1)}$ as our
definition of $\widetilde{\mathcal{A}}_{r}$ in later discussions. We now
rewrite it by introducing the following notation. For the rest of the paper,
we will denote the subset of operators in an algebra that commute with the
gauge-invariant projector with a superscript $\Pi$; for example, the algebra
$\mathcal{A}^{\Pi}$ is defined by

$\mathcal{A}^{\Pi}\equiv\\{\mathcal{O}\in\mathcal{A}:[\mathcal{O},\Pi_{\text{GI}}]=0\\}.$
(51)

It is clear that this subset is itself a von Neumann algebra, as it contains
the identity, which necessarily commutes with any projector, and is closed
under addition, multiplication, and involution101010Closure of
$\mathcal{A}^{\Pi}$ under addition and multiplication is obvious, and closure
under involution follows from the projector being Hermitian.. Similarly, we
define the subalgebra $\mathcal{A}_{r}^{\Pi}$ as

$\mathcal{A}_{r}^{\Pi}=\\{\mathcal{O}_{r}\in\mathcal{A}_{r}:[\mathcal{O}_{r},\Pi_{\text{GI}}]=0\\},$
(52)

and define $\hat{\mathcal{A}}_{r}^{\Pi}$ as

$\hat{\mathcal{A}}_{r}^{\Pi}=\\{\hat{\mathcal{O}}_{r}\in\hat{\mathcal{A}}_{r}:[\hat{\mathcal{O}}_{r},\Pi_{\text{GI}}]=0\\}.$
(53)

So far, we have shown

$\widetilde{\mathcal{A}}_{r}=\mathcal{A}_{r}^{\Pi}\Pi_{\text{GI}}.$ (54)

######  Lemma 3.

$\mathcal{A}_{r}^{\Pi}\Pi_{\text{GI}}=\hat{\mathcal{A}}_{r}^{\Pi}\Pi_{\text{GI}}$.

###### Proof.

It is clear that
$\hat{\mathcal{A}}_{r}^{\Pi}\Pi_{\text{GI}}\subseteq\mathcal{A}_{r}^{\Pi}\Pi_{\text{GI}}$,
so we only need to show the opposite inclusion. Consider any operator
$\mathcal{O}_{r}\in\mathcal{A}_{r}^{\Pi}$. As this operator commutes with
$\Pi_{\text{GI}}$, we can use the decomposition of the gauge-invariant
projector to write $\mathcal{O}_{r}\Pi_{\text{GI}}$ as

$\mathcal{O}_{r}\Pi_{\text{GI}}=\Pi_{\text{GI}}\mathcal{O}_{r}\Pi_{\text{GI}}=\Pi_{\text{GI}}\left(\Pi_{V_{r}}\mathcal{O}_{r}\Pi_{V_{r}}\right)\Pi_{\text{GI}}=\left(\Pi_{V_{r}}\mathcal{O}_{r}\Pi_{V_{r}}\right)\Pi_{\text{GI}}.$
(55)

Note that $\Pi_{V_{r}}\mathcal{O}_{r}\Pi_{V_{r}}$ is an operator on
$\hat{\mathcal{H}}_{r}$ that commutes with $\Pi_{\text{GI}}$, so it belongs to
$\hat{\mathcal{A}}_{r}^{\Pi}$. Thus, every element of
$\mathcal{A}_{r}^{\Pi}\Pi_{\text{GI}}$ is an element of
$\hat{\mathcal{A}}_{r}^{\Pi}\Pi_{\text{GI}}$. This shows the inclusion
$\mathcal{A}_{r}^{\Pi}\Pi_{\text{GI}}\subseteq\hat{\mathcal{A}}_{r}^{\Pi}\Pi_{\text{GI}}$,
from which we conclude
$\mathcal{A}_{r}^{\Pi}\Pi_{\text{GI}}=\hat{\mathcal{A}}_{r}^{\Pi}\Pi_{\text{GI}}$.
∎

######  Corollary 4.

$\widetilde{\mathcal{A}}_{r}=\hat{\mathcal{A}}_{r}^{\Pi}\Pi_{\text{GI}}$.

Using the corollary above, we will now construct a generic operator in
$\widetilde{\mathcal{A}}_{r}$.

######  Lemma 5.

$\widetilde{\mathcal{A}}_{r}$ can be written in the following two forms:

$\displaystyle\widetilde{\mathcal{A}}_{r}$
$\displaystyle=\left\\{\hat{\mathcal{O}}_{r}\Pi_{\text{GI}}:\hat{\mathcal{O}}_{r}=\sum_{\alpha
ijk}\hat{\mathcal{O}}_{\alpha ij}\ket{\alpha ik}_{r}\bra{\alpha
jk}\otimes\text{1}_{\overline{r}},\,\hat{\mathcal{O}}_{\alpha
ij}\in\text{C}\right\\}$ (56)
$\displaystyle=\left\\{\widetilde{\mathcal{O}}_{r}=\sum_{\alpha
ii^{\prime}jk\ell}\widetilde{\mathcal{O}}_{\alpha ij}\ket{\alpha
ik}_{r}\bra{\alpha
j\ell}\otimes\ket{\overline{\alpha}i^{\prime}k}_{\overline{r}}\bra{\overline{\alpha}i^{\prime}\ell}:\widetilde{\mathcal{O}}_{\alpha
ij}\in\text{C}\right\\},$ (57)

with $\widetilde{\mathcal{O}}_{r}$ in (57) identified with
$\hat{\mathcal{O}}_{r}\Pi_{\text{GI}}$ in (56) under
$\widetilde{\mathcal{O}}_{\alpha ij}=\hat{\mathcal{O}}_{\alpha
ij}/d_{\alpha}$.111111In a slight abuse of notation, we have referred to the
matrix elements of an operator with the same symbol as the operator itself,
but with irrep labels and indices such as $\alpha$, $i$, and $j$. We could
have referred to $\hat{{\mathcal{O}}}_{\alpha ij}$ as
$\hat{{\mathcal{O}}}_{r,\alpha ij}$ in (56), but for simplicity, we will use
the former.

###### Proof.

We show this by noting
$\widetilde{\mathcal{A}}_{r}=\hat{\mathcal{A}}_{r}^{\Pi}\Pi_{\text{GI}}$ and
constructing a generic operator therein. Recall that $\\{\ket{\alpha
ik}_{r}\\}$ is a basis for $\hat{\mathcal{H}}_{r}$, so an operator
$\hat{\mathcal{O}}_{r}\in\hat{\mathcal{A}}_{r}$ (not necessarily gauge-
invariant) can be written as

$\hat{\mathcal{O}}_{r}=\sum_{\alpha\beta
ijk\ell}\hat{\mathcal{O}}_{\alpha\beta ijk\ell}\ket{\alpha ik}_{r}\bra{\beta
j\ell}\otimes\text{1}_{\overline{r}}$ (58)

with some $\hat{\mathcal{O}}_{\alpha\beta ijk\ell}\in\text{C}$. Now we require
$\hat{\mathcal{O}}_{r}\in\hat{\mathcal{A}}_{r}^{\Pi}$, so we will try to
impose
$\hat{\mathcal{O}}_{r}\Pi_{\text{GI}}=\Pi_{\text{GI}}\hat{\mathcal{O}}_{r}$.
We find

$\displaystyle\hat{\mathcal{O}}_{r}\Pi_{\text{GI}}$
$\displaystyle=\hat{\mathcal{O}}_{r}\Pi_{V_{r}}\Pi_{V_{\overline{r}}}\Pi_{V_{c}}$
$\displaystyle=\left(\sum_{\alpha\beta ijk\ell}\hat{\mathcal{O}}_{\alpha\beta
ijk\ell}\ket{\alpha ik}_{r}\bra{\beta
j\ell}\otimes\text{1}_{\overline{r}}\right)\Pi_{V_{\overline{r}}}\Pi_{V_{c}}$
$\displaystyle=\left(\sum_{\alpha\beta\gamma ijk\ell
i^{\prime}k^{\prime}}\hat{\mathcal{O}}_{\alpha\beta ijk\ell}\ket{\alpha
ik}_{r}\bra{\beta
j\ell}\otimes\ket{\overline{\gamma}i^{\prime}k^{\prime}}_{\overline{r}}\bra{\overline{\gamma}i^{\prime}k^{\prime}}\right)\Pi_{V_{c}}$
$\displaystyle=\sum_{\alpha\beta\gamma ijk\ell
i^{\prime}k^{\prime}\ell^{\prime}}\hat{\mathcal{O}}_{\alpha\beta
ijk\ell}\frac{1}{d_{\beta}}\delta_{\beta\gamma}\delta_{\ell
k^{\prime}}\ket{\alpha ik}_{r}\bra{\beta
j\ell^{\prime}}\otimes\ket{\overline{\gamma}i^{\prime}k^{\prime}}_{\overline{r}}\bra{\overline{\gamma}i^{\prime}\ell^{\prime}}$
$\displaystyle=\sum_{\alpha\beta ijk\ell
i^{\prime}\ell^{\prime}}\hat{\mathcal{O}}_{\alpha\beta
ijk\ell}\frac{1}{d_{\beta}}\ket{\alpha ik}_{r}\bra{\beta
j\ell^{\prime}}\otimes\ket{\overline{\beta}i^{\prime}\ell}_{\overline{r}}\bra{\overline{\beta}i^{\prime}\ell^{\prime}},$
(59)

where we have used the basis of $\hat{\mathcal{H}}_{\overline{r}}$ in going to
the third line and used (38) in going to the fourth line. We can apply the
same procedure to write $\Pi_{\text{GI}}\hat{\mathcal{O}}_{r}$ as

$\displaystyle\Pi_{\text{GI}}\hat{\mathcal{O}}_{r}$
$\displaystyle=\Pi_{V_{c}}\sum_{\alpha\beta\gamma ijk\ell
i^{\prime}k^{\prime}}\hat{\mathcal{O}}_{\alpha\beta ijk\ell}\ket{\alpha
ik}_{r}\bra{\beta
j\ell}\otimes\ket{\overline{\gamma}i^{\prime}k^{\prime}}_{\overline{r}}\bra{\overline{\gamma}i^{\prime}k^{\prime}}$
$\displaystyle=\sum_{\alpha\beta\gamma ijk\ell
i^{\prime}k^{\prime}\ell^{\prime}}\hat{\mathcal{O}}_{\alpha\beta
ijk\ell}\frac{1}{d_{\alpha}}\delta_{\alpha\gamma}\delta_{kk^{\prime}}\ket{\alpha
i\ell^{\prime}}_{r}\bra{\beta
j\ell}\otimes\ket{\overline{\gamma}i^{\prime}\ell^{\prime}}_{\overline{r}}\bra{\overline{\gamma}i^{\prime}k^{\prime}}$
$\displaystyle=\sum_{\alpha\beta ijk\ell
i^{\prime}\ell^{\prime}}\hat{\mathcal{O}}_{\alpha\beta
ijk\ell}\frac{1}{d_{\alpha}}\ket{\alpha i\ell^{\prime}}_{r}\bra{\beta
j\ell}\otimes\ket{\overline{\alpha}i^{\prime}\ell^{\prime}}_{\overline{r}}\bra{\overline{\alpha}i^{\prime}k}.$
(60)

One way to proceed is to find conditions on $\hat{\mathcal{O}}_{\alpha\beta
aijk\ell}$ such that the two expressions (59), (60) are equal, but doing this
explicitly turns out to be slightly complicated (in cases where the
multiplicities $\overline{n}_{\overline{\alpha}}$,
$\overline{n}_{\overline{\beta}}$ vanish but $n_{\alpha}$, $n_{\beta}$ do
not). Instead, we will use the equality of (59) and (60) to directly show that
$\hat{\mathcal{A}}_{r}^{\Pi}\Pi_{\text{GI}}$ contains and is contained in the
right-hand side of (56), which we now define as

$\widetilde{{\mathcal{A}}}_{r}^{(3)}\equiv\left\\{\hat{\mathcal{O}}_{r}\Pi_{\text{GI}}:\hat{\mathcal{O}}_{r}=\sum_{\alpha
ijk}\hat{\mathcal{O}}_{\alpha ij}\ket{\alpha ik}_{r}\bra{\alpha
jk}\otimes\text{1}_{\overline{r}},\,\hat{\mathcal{O}}_{\alpha
ij}\in\text{C}\right\\}.$ (61)

First, we show that $\widetilde{{\mathcal{A}}}_{r}^{(3)}$ defined by (61) is
equal to (57) as claimed. To see this, we simply apply (59) to the special
case of $\hat{\mathcal{O}}_{r}=\sum_{\alpha ijk}\hat{\mathcal{O}}_{\alpha
ij}\ket{\alpha ik}_{r}\bra{\alpha jk}\otimes\text{1}_{\overline{r}}$ and find
$\hat{\mathcal{O}}_{r}\Pi_{V_{c}}$ to be identical to
$\widetilde{\mathcal{O}}_{r}$ in (57) under $\widetilde{\mathcal{O}}_{\alpha
ij}=\hat{\mathcal{O}}_{\alpha ij}/d_{\alpha}$. Moreover, applying (60) to this
case yields the same operator, so we find that this special
$\hat{\mathcal{O}}_{r}$ commutes with $\Pi_{\text{GI}}$. Thus,
$\widetilde{{\mathcal{A}}}_{r}^{(3)}$ is contained in
$\hat{\mathcal{A}}_{r}^{\Pi}\Pi_{\text{GI}}$.

Finally, we will show that $\hat{\mathcal{A}}_{r}^{\Pi}\Pi_{\text{GI}}$ is
contained in $\widetilde{{\mathcal{A}}}_{r}^{(3)}$. Any
$\hat{\mathcal{O}}_{r}\Pi_{\text{GI}}\in\hat{\mathcal{A}}_{r}^{\Pi}\Pi_{\text{GI}}$
can be written explicitly as

$\displaystyle\hat{\mathcal{O}}_{r}\Pi_{\text{GI}}$
$\displaystyle=\Pi_{\text{GI}}\hat{\mathcal{O}}_{r}\Pi_{\text{GI}}=\Pi_{V_{c}}\sum_{\alpha\beta
ijk\ell i^{\prime}\ell^{\prime}}\hat{\mathcal{O}}_{\alpha\beta
ijk\ell}\frac{1}{d_{\beta}}\ket{\alpha ik}_{r}\bra{\beta
j\ell^{\prime}}\otimes\ket{\overline{\beta}i^{\prime}\ell}_{\overline{r}}\bra{\overline{\beta}i^{\prime}\ell^{\prime}}$
$\displaystyle=\sum_{\alpha\beta ijk\ell
i^{\prime}k^{\prime}\ell^{\prime}}\hat{\mathcal{O}}_{\alpha\beta
ijk\ell}\frac{1}{d_{\alpha}d_{\beta}}\delta_{\alpha\beta}\delta_{k\ell}\ket{\alpha
ik^{\prime}}_{r}\bra{\beta
j\ell^{\prime}}\otimes\ket{\overline{\beta}i^{\prime}k^{\prime}}_{\overline{r}}\bra{\overline{\beta}i^{\prime}\ell^{\prime}}$
$\displaystyle=\sum_{\alpha ijk\ell
i^{\prime}k^{\prime}}\hat{\mathcal{O}}_{\alpha\alpha
ijk^{\prime}k^{\prime}}\frac{1}{d_{\alpha}^{2}}\ket{\alpha ik}_{r}\bra{\alpha
j\ell}\otimes\ket{\overline{\alpha}i^{\prime}k}_{\overline{r}}\bra{\overline{\alpha}i^{\prime}\ell},$
(62)

which is identical to $\widetilde{\mathcal{O}}_{r}$ in (57) under
$\widetilde{\mathcal{O}}_{\alpha
ij}=\sum_{k^{\prime}}\hat{\mathcal{O}}_{\alpha\alpha
ijk^{\prime}k^{\prime}}/d_{\alpha}^{2}$, and thus belongs to
$\widetilde{{\mathcal{A}}}_{r}^{(3)}$. Combining the above results, we
conclude $\widetilde{\mathcal{A}}_{r}=\widetilde{\mathcal{A}}_{r}^{(3)}$. ∎

After all of this machinery, it is clear that one is justified in writing the
algebra $\widetilde{\mathcal{A}}_{r}$ of gauge-invariant operators on a
subregion $r$ as a restriction via $\Pi_{\text{GI}}$ of the pre-gauged algebra
${\mathcal{A}}_{r}$ on $r$. Crucially, however, $\widetilde{\mathcal{A}}_{r}$
is not a subalgebra of $\mathcal{A}_{r}$, as is obvious from the nontrivial
action of (57) on $\hat{\mathcal{H}}_{\overline{r}}$. This is manifest from
the fact that $\Pi_{\text{GI}}$ is an element of $\mathcal{A}$, not of
$\mathcal{A}_{r}$, and so the projection takes one out of the pre-gauged
subregion algebra $\mathcal{A}_{r}$.

###  3.3 The center of the algebra

For spatial subregions the following inclusion is obvious:

$\mathcal{A}_{\overline{r}}\subseteq\left(\mathcal{A}_{r}\right)^{\prime},$
(63)

as causally disconnected operators must commute. Here
$\left(\mathcal{A}_{r}\right)^{\prime}$ denotes the commutant of
${\mathcal{A}}_{r}$. Haag duality is the saturation of the above bound:

$\mathcal{A}_{\overline{r}}=\left(\mathcal{A}_{r}\right)^{\prime},$ (64)

that is, the commutant of the algebra of operators in a subregion is equal to
the algebra of operators in the complement region.121212There are
counterexamples to Haag duality in quantum field theories with global or gauge
symmetries; see for example Casini:2020rgj .

In our model, Haag duality certainly holds for the pre-gauged algebras, but
does it also hold for the gauge-invariant algebras? We will now show that it
does, i.e.,

$\widetilde{\mathcal{A}}_{\overline{r}}=\left(\widetilde{\mathcal{A}}_{r}\right)^{\prime}.$
(65)

######  Proposition 6.

The Hilbert space isomorphism (39) induces the following isomorphisms between
algebras:

$\widetilde{\mathcal{A}}_{r}\cong\bigoplus_{\alpha}\widetilde{\mathcal{A}}_{r}^{\alpha}\otimes\text{1}^{\overline{\alpha}}_{\overline{r}},\qquad\widetilde{\mathcal{A}}_{\overline{r}}\cong\bigoplus_{\alpha}\text{1}_{r}^{\alpha}\otimes\widetilde{\mathcal{A}}^{\overline{\alpha}}_{\overline{r}},$
(66)

where
$\widetilde{\mathcal{A}}_{r}^{\alpha}\equiv\mathcal{B}(\widetilde{\mathcal{H}}_{r}^{\alpha})$,
the algebra of bounded operators on $\widetilde{\mathcal{H}}_{r}^{\alpha}$,
and similarly we define
$\widetilde{\mathcal{A}}_{\overline{r}}^{\overline{\alpha}}\equiv\mathcal{B}(\widetilde{\mathcal{H}}_{\overline{r}}^{\overline{\alpha}})$.
Moreover, $\text{1}_{r}^{\alpha}$,
$\text{1}^{\overline{\alpha}}_{\overline{r}}$ denote the identity operators on
$\widetilde{\mathcal{H}}_{r}^{\alpha}$,
$\widetilde{\mathcal{H}}_{\overline{r}}^{\overline{\alpha}}$, respectively.

###### Proof.

Recall from (39) that $\widetilde{\mathcal{H}}$ is isomorphic to a direct sum
of factorizing Hilbert spaces:

$\widetilde{\mathcal{H}}\cong\bigoplus_{\alpha}\widetilde{\mathcal{H}}_{r}^{\alpha}\otimes\widetilde{\mathcal{H}}_{\overline{r}}^{\overline{\alpha}};$
(67)

where the two sides are identified under the natural isomorphism (40), which
we reproduce here:

$\ket{\widetilde{\psi}}=\sum_{\alpha ijk}\widetilde{\psi}_{\alpha
ij}\ket{\alpha
ik}_{r}\ket{\overline{\alpha}jk}_{\overline{r}}\quad\rightarrow\quad\sum_{\alpha
ij}\sqrt{d_{\alpha}}\widetilde{\psi}_{\alpha ij}\ket{\alpha
i}_{r}\ket{\overline{\alpha}j}_{\overline{r}}.$ (68)

We now apply this isomorphism to our algebra $\widetilde{\mathcal{A}}_{r}$.
Consider a general element of $\widetilde{\mathcal{A}}_{r}$ defined via (57).
Under (68), this element becomes

$\displaystyle\sum_{\alpha ii^{\prime}jk\ell}\widetilde{\mathcal{O}}_{\alpha
ij}\ket{\alpha ik}_{r}\bra{\alpha
j\ell}\otimes\ket{\overline{\alpha}i^{\prime}k}_{\overline{r}}\bra{\overline{\alpha}i^{\prime}\ell}$
$\displaystyle\rightarrow\sum_{\alpha
ii^{\prime}j}d_{\alpha}\widetilde{\mathcal{O}}_{\alpha ij}\ket{\alpha
i}_{r}\bra{\alpha
j}\otimes\ket{\overline{\alpha}i^{\prime}}_{\overline{r}}\bra{\overline{\alpha}i^{\prime}}$
$\displaystyle=\sum_{\alpha ij}d_{\alpha}\widetilde{\mathcal{O}}_{\alpha
ij}\ket{\alpha i}_{r}\bra{\alpha
j}\otimes\text{1}_{\overline{r}}^{\overline{\alpha}},$ (69)

which is an element of
$\widetilde{\mathcal{A}}_{r}^{\alpha}\otimes\text{1}^{\overline{\alpha}}_{\overline{r}}$.
Thus, we have demonstrated the isomorphism for $\widetilde{\mathcal{A}}_{r}$
in (66). The isomorphism for $\widetilde{\mathcal{A}}_{\overline{r}}$ follows
from a similar argument. ∎

######  Corollary 7.

$\widetilde{\mathcal{A}}_{r}$ obeys Haag duality, such that
$\left(\widetilde{\mathcal{A}}_{r}\right)^{\prime}=\widetilde{\mathcal{A}}_{\overline{r}}$,
where the commutant is defined with respect to the full gauge-invariant
algebra $\widetilde{\mathcal{A}}$.

###### Proof.

This immediately follows from the algebra isomorphisms (66) and

$\left(\bigoplus_{\alpha}\widetilde{\mathcal{A}}_{r}^{\alpha}\otimes\text{1}^{\overline{\alpha}}_{\overline{r}}\right)^{\prime}=\bigoplus_{\alpha}\left(\widetilde{\mathcal{A}}_{r}^{\alpha}\otimes\text{1}^{\overline{\alpha}}_{\overline{r}}\right)^{\prime}=\bigoplus_{\alpha}\text{1}^{\alpha}_{r}\otimes\widetilde{\mathcal{A}}_{\overline{r}}^{\overline{\alpha}}.$
(70)

∎

The center of an algebra is defined to be the intersection of the algebra with
its commutant. As our gauge-invariant subalgebra $\widetilde{\mathcal{A}}_{r}$
obeys Haag duality, the center is

$\widetilde{\mathcal{Z}}_{r}=\widetilde{\mathcal{A}}_{r}\cap\widetilde{\mathcal{A}}_{r}^{\prime}=\widetilde{\mathcal{A}}_{r}\cap\widetilde{\mathcal{A}}_{\overline{r}}.$
(71)

######  Lemma 8.

The center $\widetilde{\mathcal{Z}}_{r}$ is

$\widetilde{\mathcal{Z}}_{r}=\left\\{z_{\alpha}\widetilde{P}^{\alpha}:z_{\alpha}\in\text{C}\right\\},$
(72)

where $\widetilde{P}^{\alpha}$ are mutually orthogonal projections defined via

$\widetilde{P}^{\alpha}=\frac{1}{d_{\alpha}}\sum_{ijk\ell}\ket{\alpha
ik}_{r}\bra{\alpha
i\ell}\otimes\ket{\overline{\alpha}jk}_{\overline{r}}\bra{\overline{\alpha}j\ell}.$
(73)

###### Proof.

Under the algebra isomorphisms (66) for $\widetilde{\mathcal{A}}_{r}$ and
$\widetilde{\mathcal{A}}_{\overline{r}}$, we can immediately identify the
center as

$\widetilde{\mathcal{Z}}_{r}\cong\bigoplus_{\alpha}\text{C}\left(\text{1}^{\alpha}_{r}\otimes\text{1}^{\overline{\alpha}}_{\overline{r}}\right).$
(74)

That is, the center $\widetilde{\mathcal{Z}}_{r}$ is the direct sum of complex
multiples of the identity within each superselection sector $\alpha$. We can
write the identity in a superselection sector as

$\text{1}^{\alpha}_{r}\otimes\text{1}^{\overline{\alpha}}_{\overline{r}}=\sum_{ij}\ket{\alpha
i}_{r}\bra{\alpha
i}\otimes\ket{\overline{\alpha}j}_{\overline{r}}\bra{\overline{\alpha}j},$
(75)

and examine the pullback of these operators under the natural isomorphism (68)
to find the corresponding operators in $\widetilde{\mathcal{A}}$. We obtain

$\text{1}^{\alpha}_{r}\otimes\text{1}^{\overline{\alpha}}_{\overline{r}}\quad\Rightarrow\quad\frac{1}{d_{\alpha}}\sum_{ijk\ell}\ket{\alpha
ik}_{r}\bra{\alpha
i\ell}\otimes\ket{\overline{\alpha}jk}_{\overline{r}}\bra{\overline{\alpha}j\ell}=\widetilde{P}^{\alpha}.$
(76)

We identify these operators $\widetilde{P}^{\alpha}$ as the (properly
normalized) projections onto the $\alpha$ superselection sector, where we
remind the reader that $\alpha$ is an irreducible representation of $G^{n}$.
These operators can alternatively be written as

$\widetilde{P}^{\alpha}=\left(\hat{P}^{\alpha}_{r}\otimes\text{1}_{\overline{r}}\right)\Pi_{\text{GI}}=\left(\text{1}_{r}\otimes\hat{P}^{\overline{\alpha}}_{\overline{r}}\right)\Pi_{\text{GI}},$
(77)

where $\hat{P}^{\alpha}_{r}$ and $\hat{P}^{\overline{\alpha}}_{\overline{r}}$
are orthogonal projections in $\hat{\mathcal{H}}_{r}$ and
$\hat{\mathcal{H}}_{\overline{r}}$, respectively:

$\hat{P}^{\alpha}_{r}=\sum_{ik}\ket{\alpha ik}_{r}\bra{\alpha
ik},\quad\hat{P}^{\overline{\alpha}}_{\overline{r}}=\sum_{ik}\ket{\overline{\alpha}ik}_{\overline{r}}\bra{\overline{\alpha}ik}.$
(78)

One can show the $\widetilde{P}^{\alpha}$ are orthogonal and idempotent such
that
$\widetilde{P}^{\alpha}\widetilde{P}^{\beta}=\delta_{\alpha\beta}\widetilde{P}^{\alpha}$.
∎

###  3.4 Traces in $\mathcal{A}_{r}$ and $\widetilde{\mathcal{A}}_{r}$

We now define traces in our von Neumann algebras. When an algebra is
$\mathcal{B}(\mathcal{H})$ for some Hilbert space ${\mathcal{H}}$, we can
simply identify the minimal projections as projections onto a pure state in
$\mathcal{H}$, and the trace is the usual trace of a square matrix. Our
algebras are not always of this form; an example is
$\widetilde{\mathcal{A}}_{r}$. Therefore, we will first identify the minimal
projections, which are then used to define a normalized trace on the algebra.
In particular, for $\widetilde{\mathcal{A}}_{r}$ our task is to find the
minimal projections $\widetilde{P}_{r}$ in $\widetilde{\mathcal{A}}_{r}$ and
use them to define a “rescaled” trace $\widetilde{\Tr}_{r}$ which satisfies

$\widetilde{\Tr}_{r}\widetilde{P}_{r}=1.$ (79)

Let us first discuss the case that we understand well: that of
$\mathcal{A}_{r}$, and by extension $\hat{\mathcal{A}}_{r}$. As
$\mathcal{A}_{r}=\mathcal{B}(\mathcal{H}_{r})$, the minimal projections are
projections onto a pure state in $\mathcal{H}_{r}$, and we define the trace
$\Tr_{r}$ in $\mathcal{A}_{r}$ such that the minimal projections have trace
$1$. As $\hat{\mathcal{A}}_{r}=\mathcal{B}(\hat{\mathcal{H}}_{r})$, we proceed
similarly. Recall that the basis states of $\hat{\mathcal{H}}_{r}$ are
$\\{\ket{\alpha ik}_{r}\\}$, and so we define the trace $\hat{\Tr}_{r}$ in
$\hat{\mathcal{A}}_{r}$ via

$\hat{\Tr}_{r}\ket{\alpha ik}_{r}\bra{\alpha ik}=1.$ (80)

As the minimal projections in $\hat{\mathcal{A}}_{r}$ are also minimal
projections in $\mathcal{A}_{r}$, the two traces agree (on
$\hat{\mathcal{A}}_{r}$):

$\Tr_{r}=\hat{\Tr}_{r},$ (81)

so we will use only $\Tr_{r}$ (not $\hat{\Tr}_{r}$) moving forward.

Now consider $\widetilde{\mathcal{A}}_{r}$. Although
$\widetilde{\mathcal{A}}_{r}$ is not the algebra of all bounded operators on a
Hilbert space, the algebra isomorphism (66) shows that we can write it as a
direct sum of algebras for which we can easily identify minimal projections.
In particular, the pullback of minimal projections onto pure states
$\ket{\alpha i}_{r}\in\widetilde{\mathcal{H}}_{r}^{\alpha}$ under the natural
isomorphism (68) gives minimal projections in $\widetilde{\mathcal{A}}_{r}$.
Thus, we write these minimal projections $\widetilde{P}^{\alpha
i}_{r}\in\widetilde{\mathcal{A}}_{r}$ as

$\widetilde{P}^{\alpha i}_{r}=\frac{1}{d_{\alpha}}\sum_{jk\ell}\ket{\alpha
ik}_{r}\bra{\alpha
i\ell}\otimes\ket{\overline{\alpha}jk}_{\overline{r}}\bra{\overline{\alpha}j\ell}$
(82)

for all non-empty sectors $\alpha$, defined as those with nonzero
$n_{\alpha}$, $\overline{n}_{\overline{\alpha}}$. If $n_{\alpha}$ vanishes,
the index $i$ above has an empty range, and if
$\overline{n}_{\overline{\alpha}}$ vanishes, $\widetilde{P}^{\alpha i}_{r}$
vanishes due to the empty sum over $j$ in (82).

We can alternatively write $\widetilde{P}^{\alpha i}_{r}$ as

$\widetilde{P}^{\alpha i}_{r}=\hat{P}^{\alpha i}_{r}\Pi_{\text{GI}},$ (83)

where the projections $\hat{P}^{\alpha i}_{r}$ are defined similarly to (78):

$\hat{P}^{\alpha i}_{r}\equiv\sum_{k}\ket{\alpha ik}\bra{\alpha
ik}_{r}\otimes\text{1}_{\overline{r}}.$ (84)

Although we already argued that $\widetilde{P}^{\alpha i}_{r}$ are minimal
projections using the natural isomorphism, we now show it more directly.

######  Lemma 9.

The projections $\widetilde{P}_{r}^{\alpha i}$ (for non-empty sectors
$\alpha$) are minimal projections in $\widetilde{\mathcal{A}}_{r}$.

###### Proof.

We recall that minimal projections are nonzero and have the property that any
subprojection $\widetilde{Q}_{r}$ of $\widetilde{P}^{\alpha i}_{r}$ is either
zero or $\widetilde{P}^{\alpha i}_{r}$. As an element of
$\widetilde{\mathcal{A}}_{r}$, $\widetilde{Q}_{r}$ must be of the form

$\widetilde{Q}_{r}=\sum_{\beta jj^{\prime}k}\hat{Q}_{\beta
jj^{\prime}}\left(\ket{\beta jk}_{r}\bra{\beta
j^{\prime}k}\otimes\text{1}_{\overline{r}}\right)\Pi_{\text{GI}}$ (85)

with complex coefficients $\hat{Q}_{\beta jj^{\prime}}$. The subprojection
$\widetilde{Q}_{r}$ is left fixed under conjugation via
$\widetilde{P}_{r}^{\alpha i}$, so we have

$\widetilde{Q}_{r}=\widetilde{P}_{r}^{\alpha
i}\widetilde{Q}_{r}\widetilde{P}_{r}^{\alpha i}=\sum_{k}\hat{Q}_{\alpha
ii}\left(\ket{\alpha ik}_{r}\bra{\alpha
ik}\otimes\text{1}_{\overline{r}}\right)\Pi_{\text{GI}}=\hat{Q}_{\alpha
ii}\widetilde{P}^{\alpha i}_{r}.$ (86)

Additionally imposing $\widetilde{Q}_{r}^{2}=\widetilde{Q}_{r}$, we find

$\widetilde{Q}_{r}^{2}=\hat{Q}_{\alpha ii}^{2}\widetilde{P}^{\alpha
i}_{r}=\hat{Q}_{\alpha ii}\widetilde{P}^{\alpha i}_{r}.$ (87)

Unless $\widetilde{Q}_{r}$ is zero, we obtain $\hat{Q}_{\alpha ii}=1$ and thus
$\widetilde{Q}_{r}=\widetilde{P}_{r}^{\alpha i}$. So
$\widetilde{P}_{r}^{\alpha i}$ (for a non-empty sector $\alpha$) is indeed a
minimal projection. ∎

Therefore, we define the trace $\widetilde{\Tr}_{r}$ in
$\widetilde{\mathcal{A}}_{r}$ by imposing

$\widetilde{\Tr}_{r}\widetilde{P}_{r}^{\alpha i}=1$ (88)

for every non-empty sector $\alpha$ and every $i=1,\cdots,n_{\alpha}$.

How do we understand this trace acting on a general operator in
$\widetilde{\mathcal{A}}_{r}$? Such an operator can always be written in the
form (56):

$\widetilde{\mathcal{O}}_{r}=\hat{\mathcal{O}}_{r}\Pi_{\text{GI}},\quad\hat{\mathcal{O}}_{r}=\sum_{\alpha
ijk}\hat{\mathcal{O}}_{\alpha ij}\ket{\alpha ik}_{r}\bra{\alpha
jk}\otimes\text{1}_{\overline{r}}.$ (89)

Taking the trace using $\widetilde{\Tr}_{r}$, we find

$\widetilde{\Tr}_{r}\widetilde{\mathcal{O}}_{r}=\widetilde{\Tr}_{r}\sum_{\alpha
i}\hat{\mathcal{O}}_{\alpha ii}\widetilde{P}_{r}^{\alpha i}=\sum_{\alpha
i}\hat{\mathcal{O}}_{\alpha ii}.$ (90)

If we were to take the trace $\Tr_{r}$ of the corresponding
$\hat{\mathcal{O}}_{r}$, we would instead find

$\Tr_{r}\hat{\mathcal{O}}_{r}=\Tr\sum_{\alpha i}\hat{\mathcal{O}}_{\alpha
ii}\hat{P}_{r}^{\alpha i}=\sum_{\alpha i}d_{\alpha}\hat{\mathcal{O}}_{\alpha
ii}.$ (91)

Thus, it is tempting to relate the trace $\widetilde{\Tr}_{r}$ to $\Tr_{r}$
using an appropriate rescaling by $1/d_{\alpha}$ in each sector. A more
precise version of this statement is the following: for any operator
$\widetilde{\mathcal{O}}_{r}^{\alpha}\in\widetilde{\mathcal{A}}_{r}$ that acts
only in the $\alpha$ sector such that it can be written as

$\widetilde{\mathcal{O}}_{r}^{\alpha}=\hat{\mathcal{O}}_{r}^{\alpha}\Pi_{\text{GI}},\quad\hat{\mathcal{O}}_{r}^{\alpha}=\sum_{ijk}\hat{\mathcal{O}}_{\alpha
ij}\ket{\alpha ik}_{r}\bra{\alpha jk}\otimes\text{1}_{\overline{r}},$ (92)

i.e., with no sum over $\alpha$, the two traces are related by

$\widetilde{\Tr}_{r}\widetilde{\mathcal{O}}_{r}^{\alpha}=\frac{1}{d_{\alpha}}\Tr_{r}\hat{\mathcal{O}}_{r}^{\alpha}.$
(93)

Summing both sides over $\alpha$ recovers (90) and (91).

###  3.5 Reduced states

Our ultimate goal is to relate the von Neumann entropies for the same gauge-
invariant state $\widetilde{\rho}$ in $\widetilde{\mathcal{H}}$ on two
different subalgebras: $\mathcal{A}_{r}$ and $\widetilde{\mathcal{A}}_{r}$.

The first thing to note is that, when we consider the full graph (instead of
restricting to a subregion $r$), we have
${\mathcal{A}}={\mathcal{B}}({\mathcal{H}})$ and
$\widetilde{\mathcal{A}}={\mathcal{B}}(\widetilde{\mathcal{H}})$ where
$\widetilde{\mathcal{H}}$ is a subspace of ${\mathcal{H}}$, so minimal
projections in $\widetilde{\mathcal{A}}$ are also minimal projections in
${\mathcal{A}}$, and the trace $\widetilde{\Tr}$ in $\widetilde{\mathcal{A}}$
therefore agrees with the trace $\Tr$ in ${\mathcal{A}}$ when acting on gauge-
invariant states. Hence, a gauge-invariant state $\widetilde{\rho}$ on the
full graph that is properly normalized under the $\widetilde{\Tr}$ trace is
also properly normalized under the $\Tr$ trace, and can therefore be viewed as
a properly normalized state $\rho=\widetilde{\rho}$ in ${\mathcal{H}}$ (albeit
a special one). Thus, we will use only $\rho$ (not $\widetilde{\rho}$) for
notational simplicity in the following discussions. We should still remember
that $\rho$ is a special state that belongs to $\widetilde{\mathcal{A}}$.

The above statements do not hold for reduced states on subregions. In
particular, we need to distinguish a properly normalized state $\rho_{r}$ in
${\mathcal{A}}_{r}$ from a properly normalized $\widetilde{\rho}_{r}$ in
$\widetilde{\mathcal{A}}_{r}$. Now we derive the relation between these two
states.

Recall that to find $S(\rho,\mathcal{A}_{r})$ for a general subalgebra
$\mathcal{A}_{r}\subset\mathcal{A}$, we need to find a reduced state
$\rho_{r}\in\mathcal{A}_{r}$ satisfying

$\Tr_{r}(\rho_{r}\mathcal{O}_{r})=\Tr(\rho\mathcal{O}_{r})$ (94)

for all $\mathcal{O}_{r}\in\mathcal{A}_{r}$. For our particular
$\mathcal{A}_{r}$ (the pre-gauged algebra on $r$), the answer is, of course,
$\rho_{r}=\Tr_{\overline{r}}\rho$.

Now we work out the reduced state in the subalgebra
$\widetilde{\mathcal{A}}_{r}$.

######  Lemma 10.

The reduced state $\widetilde{\rho}_{r}\in\widetilde{\mathcal{A}}_{r}$
satisfying

$\widetilde{\Tr}_{r}(\widetilde{\rho}_{r}\widetilde{\mathcal{O}}_{r})=\Tr(\rho\widetilde{\mathcal{O}}_{r})$
(95)

for all $\widetilde{\mathcal{O}}_{r}\in\widetilde{\mathcal{A}}_{r}$ is of the
form

$\widetilde{\rho}_{r}=\hat{\rho}_{r}\Pi_{\text{GI}},\quad\hat{\rho}_{r}=\sum_{\alpha
ijk}\hat{\rho}_{\alpha ij}\ket{\alpha ik}_{r}\bra{\alpha
jk}\otimes\text{1}_{\overline{r}},$ (96)

with $\hat{\rho}_{\alpha ij}=d_{\alpha}\rho_{\alpha ij}$, where $\rho_{\alpha
ij}$ is defined by

$\rho_{r}=\sum_{\alpha ijk}\rho_{\alpha ij}\ket{\alpha ik}_{r}\bra{\alpha
jk}.$ (97)

###### Proof.

A general gauge-invariant state $\rho\in\widetilde{\mathcal{A}}$ can be
written as

$\rho=\sum_{\alpha\beta ijki^{\prime}j^{\prime}k^{\prime}}\rho_{\alpha\beta
ii^{\prime}jj^{\prime}}\ket{\alpha
ik}_{r}\ket{\overline{\alpha}jk}_{\overline{r}}\bra{\beta
i^{\prime}k^{\prime}}_{r}\bra{\overline{\beta}j^{\prime}k^{\prime}}_{\overline{r}}$
(98)

using the basis states for $\widetilde{\mathcal{H}}$. Tracing over
$\overline{r}$, we find

$\displaystyle\rho_{r}=\Tr_{\overline{r}}\rho$
$\displaystyle=\sum_{\alpha\beta
ijki^{\prime}j^{\prime}k^{\prime}}\rho_{\alpha\beta
ii^{\prime}jj^{\prime}}\expectationvalue{\overline{\beta}j^{\prime}k^{\prime}|\overline{\alpha}jk}_{\overline{r}}\ket{\alpha
i^{\prime}k^{\prime}}_{r}\bra{\beta ik}$ $\displaystyle=\sum_{\alpha
ii^{\prime}jk}\rho_{\alpha\alpha ii^{\prime}jj}\ket{\alpha ik}_{r}\bra{\alpha
i^{\prime}k}.$ (99)

This verifies (97) and determines $\rho_{aij}$.

Now recall that as an element of $\widetilde{\mathcal{A}}_{r}$,
$\widetilde{\rho}_{r}$ must be of the form (96) with some complex coefficients
$\hat{\rho}_{\alpha ij}$. It remains to determine what they are from (95). In
order to impose it, we define the following basis for
$\widetilde{\mathcal{A}}_{r}$:

$\widetilde{\mathcal{O}}^{\alpha ij}_{r}=\hat{\mathcal{O}}^{\alpha
ij}_{r}\Pi_{\text{GI}},\quad\hat{\mathcal{O}}^{\alpha
ij}_{r}=\sum_{k}\ket{\alpha ik}_{r}\bra{\alpha
jk}\otimes\text{1}_{\overline{r}},$ (100)

such that we can rewrite the reduced gauge-invariant density matrix as

$\widetilde{\rho}_{r}=\sum_{\alpha ij}\hat{\rho}_{\alpha
ij}\widetilde{\mathcal{O}}_{\alpha ij}.$ (101)

Note that the basis elements $\widetilde{\mathcal{O}}^{\alpha ij}_{r}$ and
their corresponding basis elements $\hat{\mathcal{O}}^{\alpha
ij}_{r}\in\hat{\mathcal{A}}_{r}$ obey the following relations:

$\widetilde{\mathcal{O}}^{\alpha ij}_{r}\widetilde{\mathcal{O}}^{\beta
i^{\prime}j^{\prime}}_{r}=\delta_{\alpha\beta}\delta_{i^{\prime}j}\widetilde{\mathcal{O}}^{\alpha
ij^{\prime}}_{r},\quad\widetilde{\Tr}_{r}\widetilde{\mathcal{O}}^{\alpha
ij}_{r}=\delta_{ij}$ (102) $\hat{\mathcal{O}}^{\alpha
ij}_{r}\hat{\mathcal{O}}^{\beta
i^{\prime}j^{\prime}}_{r}=\delta_{\alpha\beta}\delta_{i^{\prime}j}\hat{\mathcal{O}}^{\alpha
ij^{\prime}}_{r},\quad\Tr_{r}\hat{\mathcal{O}}^{\alpha
ij}_{r}=d_{\alpha}\delta_{ij}.$ (103)

From these relations we can check both sides of (95) for
$\widetilde{\mathcal{O}}_{r}$ set to one of the basis elements
$\widetilde{\mathcal{O}}^{\alpha ij}_{r}$. The trace in the gauge-invariant
algebra becomes

$\widetilde{\Tr}_{r}\left(\widetilde{\rho}_{r}\widetilde{\mathcal{O}}^{\alpha
ij}_{r}\right)=\widetilde{\Tr}_{r}\left(\sum_{\beta
i^{\prime}j^{\prime}}\hat{\rho}_{\beta
i^{\prime}j^{\prime}}\widetilde{\mathcal{O}}^{\beta
i^{\prime}j^{\prime}}_{r}\widetilde{\mathcal{O}}^{\alpha
ij}\right)=\sum_{\beta i^{\prime}j^{\prime}}\hat{\rho}_{\beta
i^{\prime}j^{\prime}}\delta_{\alpha\beta}\delta_{i^{\prime}j}\delta_{ij^{\prime}}=\hat{\rho}_{\alpha
ji}.$ (104)

We need to equate this with the trace in the pre-gauged algebra, which we
begin to evaluate by simplifying to the trace in $\mathcal{A}_{r}$. We have

$\Tr(\rho\widetilde{\mathcal{O}}^{\alpha
ij}_{r})=\Tr(\rho\hat{\mathcal{O}}^{\alpha
ij}_{r}\Pi_{\text{GI}})=\Tr(\Pi_{\text{GI}}\rho\hat{\mathcal{O}}^{\alpha
ij}_{r})=\Tr(\rho\hat{\mathcal{O}}^{\alpha
ij}_{r})=\Tr_{r}(\rho_{r}\hat{\mathcal{O}}^{\alpha ij}_{r}),$ (105)

where we have used the cyclicity of the trace, the gauge invariance of $\rho$,
and the fact that $\hat{\mathcal{O}}^{\alpha ij}_{r}\in\mathcal{A}_{r}$. We
further simplify this and obtain

$\Tr_{r}(\rho_{r}\hat{\mathcal{O}}^{\alpha ij}_{r})=\Tr_{r}\left(\sum_{\beta
i^{\prime}j^{\prime}}\rho_{\beta
i^{\prime}j^{\prime}}\hat{\mathcal{O}}_{r}^{\beta
i^{\prime}j^{\prime}}\hat{\mathcal{O}}_{r}^{\alpha ij}\right)=\sum_{\beta
i^{\prime}j^{\prime}}d_{\alpha}\rho_{\beta
i^{\prime}j^{\prime}}\delta_{\alpha\beta}\delta_{i^{\prime}j}\delta_{ij^{\prime}}=d_{\alpha}\rho_{\alpha
ji}.$ (106)

Thus we identify the reduced density matrix
$\widetilde{\rho}_{r}\in\widetilde{\mathcal{A}}_{r}$ as a density matrix of
the form (96) with

$\hat{\rho}_{\alpha ij}=d_{\alpha}\rho_{\alpha ij}.$ (107)

∎

##  4 Entropies in the Gauged Random Tensor Network

Having written down the reduced states in $\mathcal{A}_{r}$ and
$\widetilde{\mathcal{A}}_{r}$, we are now ready to compute the von Neumann
entropies with respect to the two algebras. As we will see, the difference
between the two entropies in the gauged random tensor network is precisely
accounted for by an additional contribution to the area operator in the non-
trivial center $\widetilde{\mathcal{Z}}_{r}$.

###  4.1 Entanglement entropy

From (96) and (97), we proceed by defining the reduced states projected onto a
superselection sector $\alpha$:

$\rho^{\alpha}_{r}=\sum_{ijk}\rho_{\alpha ij}|\alpha
ik\rangle_{r}\langle\alpha
jk|,\qquad\hat{\rho}^{\alpha}_{r}=\sum_{ijk}\hat{\rho}_{\alpha ij}|\alpha
ik\rangle_{r}\langle\alpha jk|=d_{\alpha}\rho_{r}^{\alpha}.$ (108)

Note that these density matrices are not properly normalized with respect to
their appropriate traces. The reduced states (96) and (97) can be written as a
direct sum over representations:

$\rho_{r}=\bigoplus_{\alpha}\rho^{\alpha}_{r},\qquad\widetilde{\rho}_{r}=\bigoplus_{\alpha}\hat{\rho}^{\alpha}_{r}\Pi_{\text{GI}}.$
(109)

Furthermore, functions of the reduced states are superselected in the same
way. In particular,

$\rho_{r}\log\rho_{r}=\bigoplus_{\alpha}\rho^{\alpha}_{r}\log\rho^{\alpha}_{r},\qquad\widetilde{\rho}_{r}\log\widetilde{\rho}_{r}=\bigoplus_{\alpha}(\hat{\rho}^{\alpha}_{r}\log\hat{\rho}^{\alpha}_{r})\Pi_{\text{GI}},$
(110)

where we used the fact that $[\hat{\rho}^{\alpha}_{r},\Pi_{\text{GI}}]=0$.

We are now ready to compute the subregion entropies (in the bulk). The von
Neumann entropy of $\rho$ with respect to $\mathcal{A}_{r}$ is simply given by

$S(\rho,\mathcal{A}_{r})=-\Tr_{r}\rho_{r}\log\rho_{r}=-\sum_{\alpha}\Tr_{r}\rho_{r}^{\alpha}\log\rho_{r}^{\alpha}.$
(111)

On the other hand, using the relation between the traces (93), we can write
the von Neumann entropy with respect to $\widetilde{\mathcal{A}}_{r}$ as

$S(\rho,\widetilde{\mathcal{A}}_{r})=-\widetilde{\Tr}_{r}\widetilde{\rho}_{r}\log\widetilde{\rho}_{r}=-\sum_{\alpha}d_{\alpha}^{-1}\Tr_{r}\hat{\rho}_{r}^{\alpha}\log\hat{\rho}_{r}^{\alpha}.$
(112)

Using $\hat{\rho}^{\alpha}_{r}=d_{\alpha}\rho^{\alpha}_{r}$ and
$\Tr_{r}\rho^{\alpha}_{r}=\widetilde{\Tr}_{r}\widetilde{\rho}^{\alpha}_{r}$,
we can rewrite each term in the sum as

$\displaystyle
d_{\alpha}^{-1}\Tr_{r}\hat{\rho}^{\alpha}_{r}\log\hat{\rho}^{\alpha}_{r}$
$\displaystyle=\Tr_{r}\rho^{\alpha}_{r}\log\rho^{\alpha}_{r}+\Tr_{r}\rho^{\alpha}_{r}\log
d_{\alpha}$
$\displaystyle=\Tr_{r}\rho^{\alpha}_{r}\log\rho^{\alpha}_{r}+\widetilde{\Tr}_{r}\widetilde{\rho}^{\alpha}_{r}\log
d_{\alpha}.$ (113)

The von Neumann entropy with respect to $\widetilde{\mathcal{A}}_{r}$ can thus
be written as

$\displaystyle S(\rho,\widetilde{\mathcal{A}}_{r})$
$\displaystyle=-\sum_{\alpha}\left(\Tr_{r}\rho_{r}^{\alpha}\log\rho_{r}^{\alpha}+\widetilde{\Tr}_{r}\widetilde{\rho}_{r}^{\alpha}\log
d_{\alpha}\right)$
$\displaystyle=S(\rho,\mathcal{A}_{r})-\widetilde{\Tr}_{r}\left(\widetilde{\rho}_{r}\Delta\widetilde{A}\right),$
(114)

where we have defined a new “extra area operator” via

$\Delta\widetilde{A}\equiv\bigoplus_{\alpha}\widetilde{P}^{\alpha}\log
d_{\alpha}.$ (115)

The projections $\widetilde{P}^{\alpha}$ are precisely the projections (73)
which generate the center $\widetilde{\mathcal{Z}}_{r}$, so
$\Delta\widetilde{A}$ is manifestly an operator in the center.

We have now arrived at our final relation between the entropies with respect
to $\mathcal{A}_{r}$ and $\widetilde{\mathcal{A}}_{r}$,

$S(\rho,\mathcal{A}_{r})=S(\rho,\widetilde{\mathcal{A}}_{r})+\widetilde{\Tr}_{r}\left(\widetilde{\rho}_{r}\Delta\widetilde{A}\right),$
(116)

which we now use in our two-layer gauged RTN defined in Section 2. In
particular, we would like to derive an FLM formula relating the boundary
entropy with the gauged bulk entropy $S(\rho,\widetilde{\mathcal{A}}_{r})$.
Recall that when we feed any bulk state $\rho$ in the pre-gauged algebra
$\mathcal{A}$ into the RTN, the entropy $S(R)$ of the resulting boundary state
on a boundary subregion $R$ satisfies an FLM formula:

$S(R)=\absolutevalue{\gamma_{R}}\log D+S(\rho,\mathcal{A}_{r}),$ (117)

where the bulk subregion $r$ is chosen to be the entanglement wedge between
$R$ and its minimal surface $\gamma_{R}$. Now specializing to a gauge-
invariant bulk state $\rho\in\widetilde{{\mathcal{A}}}$ and using (116), we
find that the boundary entropy can now be written as a new FLM formula:

$S(R)=\widetilde{\Tr}_{r}\Big{(}\widetilde{\rho}_{r}\widetilde{A}\Big{)}+S\left(\rho,\widetilde{\mathcal{A}}_{r}\right),$
(118)

where the full area operator $\widetilde{A}$ is

$\widetilde{A}\,=\,\absolutevalue{\gamma_{R}}\log
D\,+\,\bigoplus_{\alpha}\widetilde{P}^{\alpha}\log
d_{\alpha}\,=\,\absolutevalue{\gamma_{R}}\log
D\,\,\,+\\!\\!\bigoplus_{\alpha_{1},\cdots,\alpha_{n}}\\!\\!\\!\widetilde{P}^{(\alpha_{1},\cdots,\alpha_{n})}\sum_{i=1}^{n}\log
d_{\alpha_{i}}.$ (119)

Again, we sum over all irreps $\alpha=(\alpha_{1},\cdots,\alpha_{n})$ of
$G^{n}$ acting on the cut, although some $\alpha$ sectors may be emtpy (i.e.,
$n_{\alpha}$ or ${\overline{n}}_{\overline{\alpha}}$ is zero) in which case
$\widetilde{P}^{\alpha}$ vanishes.

This is our main result. We note that this area operator looks like what
arises in a superposition of a stack of standard RTNs with probabilities
determined by the projections $\widetilde{P}^{\alpha}$ and with bond
dimensions augmented by $d_{\alpha_{i}}$.

###  4.2 Rényi entropy and Rényi mutual information

As discussed in Section 2, one can modify the entanglement structure of the
links in the standard RTN to obtain a non-flat Rényi spectrum for boundary
states. However, this is not enough to reproduce the properties of holographic
Rényi entropies on general boundary subregions. In particular, it fails to
account for the lack of backreaction, displayed in the tensor network as a
lack of (Rényi) correlation between disconnected boundary subregions when the
RT surface is in a disconnected phase. This problem becomes clear when one
calculates the Rényi mutual information between two such boundary subregions
$R_{1}$ and $R_{2}$, defined as131313The Rényi index $n$ should not be
confused with the number of vertices on the cut $n=|V_{c}|$.

$I_{n}(R_{1}:R_{2})\equiv S_{n}(R_{1})+S_{n}(R_{2})-S_{n}(R_{1}\cup R_{2}).$
(120)

As the area operator in the original RTN is a $c$-number, using (3) we find
that the area operator contribution cancels out in $I_{n}(R_{1}:R_{2})$ for
all $n$ (as long as the minimal surface $\gamma_{R}$ is in a disconnected
phase), leaving the boundary mutual information equal to the bulk mutual
information:

$I_{n}(R_{1}:R_{2})=I_{n}(r_{1}:r_{2},\mathcal{A}_{r_{1}r_{2}}).$ (121)

This implies that, if one wants a contribution to the Rényi mutual information
of the same order as the area, that is $\mathcal{O}(\log D)$, one must input
by hand a highly entangled bulk state. Doing this is unsatisfying and quite
arbitrary.

Figure 3: A simple gauged RTN in which we compute the Rényi mutual information
between $R_{1}$ and $R_{2}$. The input from the top layer lives on four edges
of a disconnected graph $G$, as we choose to have no matter on any of the
vertices. In the bottom layer, the thick legs have a bond dimension much
larger than that of the thin legs, such that the minimal surfaces for the
three boundary regions $R_{1}$, $R_{2}$, and $R_{1}\cup R_{2}$ only involve
the light internal legs. Consequently, the associated bulk regions will be
$r_{1}$, $r_{2}$, and $r_{1}\cup r_{2}$.

We will now see that our gauged RTN solves this problem in a natural way, due
to our nontrivial area operator. In general, the presence of a nontrivial area
operator will lead to a nontrivial, $n$-dependent boundary Rényi mutual
information, even for states with vanishing bulk Rényi mutual information.

To see how this is realized in the gauged RTN, we will study a simple example
shown in Figure 3, where the top layer is disconnected but the bottom layer is
connected.141414This connection is unnecessary to prove our point, as the
internal leg connecting $r_{1}$ and $r_{2}$ never contributes to the area
term, but it is more intuitively satisfying to discuss a connected spatial
slice for the purposes of demonstrating backreaction. We allow the bond
dimensions in the bottom layer to be different for different links, and in
fact design them so that the minimal surfaces associated with $R_{1}$,
$R_{2}$, and their union $R_{1}\cup R_{2}$ are fixed as we vary the Rényi
index $n$ at $\mathcal{O}(1)$ values. We will feed in a gauge-invariant bulk
state $\rho$ with the following reduced state on $r_{1}\cup r_{2}$:

$\rho_{r_{1}r_{2}}=\sum_{\alpha\beta}(d_{\alpha}d_{\beta})^{-1}P(\alpha,\beta)\sum_{k\ell}\ket{\alpha
ik}_{r_{1}}\ket{\beta j\ell}_{r_{2}}\bra{\alpha ik}_{r_{1}}\bra{\beta
j\ell}_{r_{2}},$ (122)

for some particular choice of $i$, $j$. This state has classical correlations
between $r_{1}$ and $r_{2}$ as described by a probability distribution
$P(\alpha,\beta)$, but has no quantum correlations. For simplicity, we
consider the following distribution $P(\alpha,\beta)$ that has support on only
two superselection sectors $\alpha_{1}$, $\alpha_{2}$ on $r_{1}$ and only two
sectors $\beta_{1}$, $\beta_{2}$ on $r_{2}$:

$P(\alpha_{1},\beta_{1})=p,\quad
P(\alpha_{2},\beta_{1})=P(\alpha_{1},\beta_{2})=p^{\prime},\quad
P(\alpha_{2},\beta_{2})=p^{\prime\prime},$ (123)

subject to the constraint $p+2p^{\prime}+p^{\prime\prime}=1$.

The Rényi entropy of $\rho$ in the pre-gauged algebra
$\mathcal{A}_{r_{1}r_{2}}$ is defined as

$S_{n}(\rho,\mathcal{A}_{r_{1}r_{2}})\equiv\frac{1}{1-n}\log\left(\Tr_{r_{1}r_{2}}\rho_{r_{1}r_{2}}^{n}\right).$
(124)

Using our $\rho_{r_{1}r_{2}}$, we find

$\displaystyle S_{n}(\rho,\mathcal{A}_{r_{1}r_{2}})=\frac{1}{1-n}\log$
$\displaystyle\bigg{(}d_{\alpha_{1}}d_{\beta_{1}}\left(\frac{p}{d_{\alpha_{1}}d_{\beta_{1}}}\right)^{n}+d_{\alpha_{2}}d_{\beta_{1}}\left(\frac{p^{\prime}}{d_{\alpha_{2}}d_{\beta_{1}}}\right)^{n}$
$\displaystyle+d_{\alpha_{1}}d_{\beta_{2}}\left(\frac{p^{\prime}}{d_{\alpha_{1}}d_{\beta_{2}}}\right)^{n}+d_{\alpha_{2}}d_{\beta_{2}}\left(\frac{p^{\prime\prime}}{d_{\alpha_{2}}d_{\beta_{2}}}\right)^{n}\bigg{)}.$
(125)

We can also compute the reduced density matrices on $r_{1}$ and $r_{2}$, as
well as their corresponding Rényi entropies in the pre-gauged algebra. We find
the reduced density matrices to be

$\displaystyle\rho_{r_{1}}$
$\displaystyle=\sum_{k=1}^{d_{\alpha_{1}}}d_{\alpha_{1}}^{-1}(p+p^{\prime})\ket{\alpha_{1}ik}_{r_{1}}\bra{\alpha_{1}ik}+\sum_{k^{\prime}=1}^{d_{\alpha_{2}}}d_{\alpha_{2}}^{-1}(p^{\prime}+p^{\prime\prime})\ket{\alpha_{2}ik^{\prime}}_{r_{1}}\bra{\alpha_{2}ik^{\prime}},$
$\displaystyle\rho_{r_{2}}$
$\displaystyle=\sum_{k=1}^{d_{\beta_{1}}}d_{\beta_{1}}^{-1}(p+p^{\prime})\ket{\beta_{1}jk}_{r_{2}}\bra{\beta_{1}jk}+\sum_{k^{\prime}=1}^{d_{\beta_{2}}}d_{\beta_{2}}^{-1}(p^{\prime}+p^{\prime\prime})\ket{\beta_{2}jk^{\prime}}_{r_{2}}\bra{\beta_{2}jk^{\prime}},$
(126)

and the bulk Rényi entropies are

$\displaystyle S_{n}(\rho,\mathcal{A}_{r_{1}})$
$\displaystyle=\frac{1}{1-n}\log\left(d_{\alpha_{1}}\left(\frac{p+p^{\prime}}{d_{\alpha_{1}}}\right)^{n}+d_{\alpha_{2}}\left(\frac{p^{\prime}+p^{\prime\prime}}{d_{\alpha_{2}}}\right)^{n}\right)$
$\displaystyle S_{n}(\rho,\mathcal{A}_{r_{2}})$
$\displaystyle=\frac{1}{1-n}\log\left(d_{\beta_{1}}\left(\frac{p+p^{\prime}}{d_{\beta_{1}}}\right)^{n}+d_{\beta_{2}}\left(\frac{p^{\prime}+p^{\prime\prime}}{d_{\beta_{2}}}\right)^{n}\right).$
(127)

In the gauge-invariant algebra, the dependence on irrep dimensions drops out
and the Rényi entropies become purely Shannon terms:

$\displaystyle
S_{n}(\rho,\widetilde{\mathcal{A}}_{r_{1}})=S_{n}(\rho,\widetilde{\mathcal{A}}_{r_{2}})$
$\displaystyle=\frac{1}{1-n}\log\left(\left(p+p^{\prime}\right)^{n}+\left(p^{\prime}+p^{\prime\prime}\right)^{n}\right)$
$\displaystyle S_{n}(\rho,\widetilde{\mathcal{A}}_{r_{1}r_{2}})$
$\displaystyle=\frac{1}{1-n}\log\left(p^{n}+2(p^{\prime})^{n}+(p^{\prime\prime})^{n}\right),$
(128)

which we choose to be parametrically suppressed relative to the Rényi
entropies in the pre-gauged algebra.

When the sum inside the logarithm is dominated by one term, we can approximate
it using

$\log\Big({\textstyle\sum_{i}}x_{i}\Big{missing})\approx\log\Big(\max_{i}\left\\{x_{i}\right\\}\Big{missing}).$
(129)

To simplify our calculation, we will enter a parameter regime where all three
(pre-gauged) Rényi entropies satisfy the approximation above and have phase
transitions.

First consider $S_{n}(\rho,\mathcal{A}_{r_{1}})$. We take
$d_{\alpha_{1}}>d_{\alpha_{2}}$. The two terms in the sum are equal at some
critical $n_{*}$, given by

$\left(\frac{p+p^{\prime}}{p^{\prime}+p^{\prime\prime}}\right)^{n_{*}}=\left(\frac{d_{\alpha_{1}}}{d_{\alpha_{2}}}\right)^{n_{*}-1}\quad\Rightarrow\quad\frac{n_{*}}{n_{*}-1}=\frac{\log\left(\frac{d_{\alpha_{1}}}{d_{\alpha_{2}}}\right)}{\log\left(\frac{p+p^{\prime}}{p^{\prime}+p^{\prime\prime}}\right)}.$
(130)

Thus, in order to have a phase transition at $n_{*}>1$ we require

$\log\left(\frac{d_{\alpha_{1}}}{d_{\alpha_{2}}}\right)>\log\left(\frac{p+p^{\prime}}{p^{\prime}+p^{\prime\prime}}\right).$
(131)

The width of this transition is controlled by the corrections to (129). This
depends on the curvature of $S_{n}(\rho,\mathcal{A}_{r_{1}})$ at $n_{*}$;
explicitly we can diagnose this with the following quantity:

$\left.\frac{d^{2}}{dn^{2}}(1-n)S_{n}(\rho,\mathcal{A}_{r_{1}})\right|_{n=n_{*}}=\frac{1}{4}\left(\log\frac{d_{\alpha_{1}}(p^{\prime}+p^{\prime\prime})}{d_{\alpha_{2}}(p+p^{\prime})}\right)^{2}.$
(132)

For fixed $n_{*}$, this quantity increases with increasing
$d_{\alpha_{1}}/d_{\alpha_{2}}$, so we should make this ratio large for a
sharp transition. A simple way to ensure the previous conditions is the
following:

$\frac{d_{\alpha_{1}}}{d_{\alpha_{2}}}\equiv q\gg 1,\quad p\gg
p^{\prime},\quad p^{\prime}\gg p^{\prime\prime}.$ (133)

Furthermore, we impose

$\frac{d_{\alpha_{1}}}{d_{\alpha_{2}}}=\frac{d_{\beta_{1}}}{d_{\beta_{2}}}=q,$
(134)

which forces the phase transitions in $S_{n}(\rho,\mathcal{A}_{r_{1}})$ and
$S_{n}(\rho,\mathcal{A}_{r_{2}})$ to occur at the same critical $n_{*}$.

Now let us examine the phase transition in
$S_{n}(\rho,\mathcal{A}_{r_{1}r_{2}})$. In the limit of sharp transitions we
have

$S_{n}(\rho,\mathcal{A}_{r_{1}r_{2}})\approx\frac{1}{1-n}\log\left(\max\left\\{\frac{p^{n}}{(d_{\alpha_{1}}d_{\beta_{1}})^{n-1}},\frac{(p^{\prime})^{n}}{(d_{\alpha_{2}}d_{\beta_{1}})^{n-1}},\frac{(p^{\prime})^{n}}{(d_{\alpha_{1}}d_{\beta_{2}})^{n-1}},\frac{(p^{\prime\prime})^{n}}{(d_{\alpha_{2}}d_{\beta_{2}})^{n-1}}\right\\}\right).$
(135)

For simplicity, we will choose

$\frac{p}{p^{\prime}}>\frac{p^{\prime}}{p^{\prime\prime}}\gg 1.$ (136)

In this case, we find that $S_{n}(\rho,\mathcal{A}_{r_{1}r_{2}})$ has a phase
transition occurring at a critical $n_{c}$ determined by

$\frac{n_{c}}{n_{c}-1}=\frac{\log(q^{2})}{\log\left(\frac{p}{p^{\prime\prime}}\right)}=\frac{\log(q^{2})}{\log\left(\frac{p}{p^{\prime}}\frac{p^{\prime}}{p^{\prime\prime}}\right)}$
(137)

which satisfies $1<n_{c}<n_{*}$.

We now combine the above results to find the (pre-gauged) Rényi mutual
information

$I_{n}(r_{1}:r_{2},\mathcal{A}_{r_{1}r_{2}})\equiv
S_{n}(\rho,\mathcal{A}_{r_{1}})+S_{n}(\rho,\mathcal{A}_{r_{2}})-S_{n}(\rho,\mathcal{A}_{r_{1}r_{2}}).$
(138)

We find the following phases:

$I_{n}(r_{1}:r_{2},\mathcal{A}_{r_{1}r_{2}})\approx\begin{cases}0&n<n_{c},\\\
\log\left(q^{2}\right)+\frac{n}{1-n}\log\left(\frac{(p+p^{\prime})^{2}}{p^{\prime\prime}}\right)&n_{c}<n<n_{*},\\\
\frac{n}{1-n}\log\left(\frac{(p^{\prime}+p^{\prime\prime})^{2}}{p^{\prime\prime}}\right)&n_{*}<n.\end{cases}$
(139)

Now we rewrite the boundary Rényi mutual information (121) as

$S_{n}(R_{1}:R_{2})=\underbrace{I_{n}(r_{1}:r_{2},\mathcal{A}_{r_{1}r_{2}})-I_{n}(r_{1}:r_{2},\widetilde{\mathcal{A}}_{r_{1}r_{2}})}_{\textrm{area
contribution}}+\underbrace{I_{n}(r_{1}:r_{2},\widetilde{\mathcal{A}}_{r_{1}r_{2}})}_{\textrm{bulk
matter contribution}},$ (140)

where the contribution of the nontrivial area operator to the boundary Rényi
mutual information is identified with the difference of the bulk Rényi mutual
information in the two algebras. As stated previously,
$I_{n}(r_{1}:r_{2},\widetilde{\mathcal{A}}_{r_{1}r_{2}})$ is suppressed
relative to $I_{n}(r_{1}:r_{2},\mathcal{A}_{r_{1}r_{2}})$, so this model
implements phase transitions in the boundary Rényi mutual information without
a large bulk matter contribution (in the gauge-invariant algebra). We plot
these two phase transitions for an example in Figure 4.

Figure 4: Phase transitions in the Rényi mutual information. Here we set
$q=10^{50}$, $p^{\prime}=10^{-16}$, and $p^{\prime\prime}=10^{-24}$. We plot
the dominant contribution to the Rényi mutual information in the three phases
(dashed) as well as the fully analytic interpolating function (solid).

This is a proof of concept showing that adding bulk gauge symmetries to the
RTN in this manner allows the boundary Rényi mutual information to be
nontrivial and $n$-dependent, even for states with small bulk Rényi mutual
information (in the gauge-invariant algebra). In our simple example here, the
minimal surface does not shift—i.e. it is the same for all $n$—but there is no
obstruction to writing a more complicated example in which the location of the
minimal surface changes with $n$ due to the nontrivial area operator.

##  5 Discussion and Outlook

In this work, we have presented a modification of the random tensor network
which allows us to reproduce known features of semiclassical holographic
states. We discuss some open questions and possible future directions below.

We have presented a toy model which, for simple choices of bulk input state,
exhibits sharp phase transitions in the Rényi entropy and Rényi mutual
information. With a sufficiently tuned set of probabilities and irrep
dimensions, one could engineer a smooth varying Rényi entropy that matches
with, for example, the correct one-interval CFT2 Rényi entropy (5). It would
be an even more complicated task to reproduce the correct Rényi entropy for
multiple intervals in the CFT Hartman:2013mia ; Faulkner:2013yia .

The bulk algebras that we encountered in our model are type I von Neumann
algebras. This is in contrast to the type II von Neumann algebras for gravity
constructed using the crossed product Witten:2021unn ; Chandrasekaran:2022eqq
; Jensen:2023yxy . A “type I approximation” to the crossed product was
recently studied in Soni:2023fke . It is thus tempting to incorporate the
crossed product and the resultant birth of a type II algebra into the tensor
network toy models of holography.

Our gauge-invariant subregion algebras generally have nontrivial centers. On
the other hand, a prescription was given in Casini:2013rba  to construct
gauge-invariant subregion algebras with trivial centers in lattice gauge
theory. This prescription involves adding operators to the algebra that we do
not include, so it does not contradict our results in any way.

Here we have implemented a graph version of the lattice gauge theory
construction along the lines of Kogut and Susskind, but crucially without
dynamics, due to the lack of a Hamiltonian. Because of this, our construction
does not have anything more to say about time evolution in tensor networks
than previous models. It would be interesting to understand how to incorporate
a Hamiltonian and the associated time evolution into tensor networks. It would
also be interesting to study the commutators of intersecting area operators in
our gauged RTN, which in standard AdS/CFT do not commute Kaplan:2022orm .

## Acknowledgements

We thank Chris Akers, Horacio Casini, David Grabovsky, Daniel Harlow, Kristan
Jensen, Don Marolf, and Pratik Rath for interesting discussions. This material
is based upon work supported by the Air Force Office of Scientific Research
under Award Number FA9550-19-1-0360. This material is also based upon work
supported by the U.S. Department of Energy, Office of Science, Office of High
Energy Physics, under Award Number DE-SC0011702. SAM would like to thank the
Centro de Ciencias de Benasque Pedro Pascal for their hospitality while a
portion of this work was completed.

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