# Entanglement bootstrap approach for gapped domain walls

Bowen Shi  Department of Physics, University of California at San Diego, La
Jolla, CA 92093, USA  Department of Physics, The Ohio State University,
Columbus, OH 43210, USA  Isaac H. Kim  3 Centre for Engineered Quantum
Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia
Stanford University, Stanford, CA 94305, USA

(August 27, 2024)

###### Abstract

We develop a theory of gapped domain wall between topologically ordered
systems in two spatial dimensions. We find a new type of superselection sector
– referred to as the parton sector – that subdivides the known superselection
sectors localized on gapped domain walls. Moreover, we introduce and study the
properties of composite superselection sectors that are made out of the parton
sectors. We explain a systematic method to define these sectors, their fusion
spaces, and their fusion rules, by deriving nontrivial identities relating
their quantum dimensions and fusion multiplicities. We propose a set of axioms
regarding the ground state entanglement entropy of systems that can host
gapped domain walls, generalizing the bulk axioms proposed in [B. Shi, K.
Kato, and I. H. Kim, Ann. Phys. 418, 168164 (2020)]. Similar to our analysis
in the bulk, we derive our main results by examining the self-consistency
relations of an object called information convex set. As an application, we
define an analog of topological entanglement entropy for gapped domain walls
and derive its exact expression.

###### Contents

  1. I Introduction
  2. II Fusion rules from entanglement
    1. II.1 Superselection sectors
    2. II.2 Fusion multiplicities
      1. II.2.1 Merging
      2. II.2.2 Derivation
  3. III Gapped domain walls: a tale of parton sector
    1. III.1 Parton sectors
  4. IV Composite sectors
    1. IV.1 $O$-type sectors
    2. IV.2 Snake sectors
    3. IV.3 $\mathbb{N}$\- and $\mathbb{U}$-type sectors
    4. IV.4 Generalities
    5. IV.5 Summary
  5. V Fusion spaces
    1. V.1 Fusion on gapped domain walls
    2. V.2 Quasi-fusion of parton sectors
  6. VI Fusion rules
    1. VI.1 Fusion rules for $O$-type sectors
      1. VI.1.1 Proofs related to anti-sectors
    2. VI.2 Fusion onto the domain wall
    3. VI.3  Fusions with $\mathbb{N}$-type and $\mathbb{U}$-type sectors
  7. VII Quasi-fusion rules
  8. VIII Domain wall topological entanglement entropy
  9. IX String operators
  10. X Discussion
  11. A Extensions of axioms
  12. B Isomorphism theorem
  13. C Factorization of extreme points
  14. D Fusion space
  15. E Aspects of quasi-fusion
    1. E.1 Quasi-fusion rule
    2. E.2 When quasi-fusion becomes a fusion
  16. F Proof of maximal entropy

##  I Introduction

One of the fundamental discoveries in physics is topologically ordered phases
of matter [1]. These are gapped phases of quantum many-body systems that
possess low-energy excitations with fractional statistics [2, 3, 4, 5, 6]. A
prominent experimental example is the well-known fractional quantum Hall
states [7].

While these systems already exhibit a rich set of phenomena in the bulk of the
material, more new physics can appear on their boundaries. The existence of a
robust gapless boundary mode is well-known [8, 9]. The nontrivial effects of
gapped boundary conditions on the topological ground state degeneracy [10] and
low-energy excitations [11] have also been studied.

More generally, there can be _gapped domain walls_ between two different
topologically ordered mediums [12, 13, 14, 15, 16, 17]. Gapped domain walls
are not just of theoretical interest. When used in conjunction with the low-
energy excitations, the domain walls can complete a universal set of
topologically protected quantum gates [18]. Therefore, studies of gapped
domain walls may lead to new means of building a fault-tolerant quantum
computer [19].

While there have been a number of beautiful prior works that studied gapped
domain walls in various contexts [11, 12, 20, 21, 13, 14, 22, 15, 23, 16, 17,
24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], there are still many unknowns.
For one, less is known about the order parameters that characterize gapped
domain walls. In the bulk of a topologically ordered system, entanglement-
based measures [35, 36, 37] are useful for characterizing the underlying
topological order [38, 39, 40]. However, similar measures for gapped domain
walls are not known to the best of our knowledge.

Moreover, while a theory of gapped domain wall has been proposed already [15],
this theory is based on an assumption about the condensation algebra [22, 12],
which abstracts away the microscopic physics of the underlying many-body
quantum system. The abstractness of this theory is both a blessing and a
curse. It allows us to identify the fundamental data that characterize the
gapped domain wall without ever dealing with the microscopic physics. However,
the downside is that it is not always clear how to extract these data directly
from the original many-body system. Moreover, one may contest that the rules
set out in this theory may not constitute a complete theory of gapped domain
walls. While this is a sentiment that we do not necessarily share, it will
still be desirable to derive these rules from a more microscopic assumption
about the underlying physical system.

To address these issues, we applied a recently discovered approach to studying
topological order [41] to systems separated by a gapped domain wall. In Ref.
[41], we derived the axioms of the fusion theory of anyon and the expression
for the topological entanglement entropy – defined as the subleading
contribution to the ground state entanglement entropy – from a set of simple
assumptions on ground-state entanglement. In this paper, we extend this
analysis to systems that possess a gapped domain wall, by relaxing the set of
assumptions used in Ref. [41] appropriately; see Fig. 1 for the summary of
these assumptions.

Figure 1: The starting assumptions of this paper. Here, $\sigma$ is the ground
state and $S_{A}$ is the entanglement entropy of a subsystem $A$ with respect
to the state $\sigma$. For details, see Section II. (Top) We consider
topologically ordered mediums $P$ (upper half) and $Q$ (lower half) which are
separated by a gapped domain wall. (Bottom) We assume that the ground state
$\sigma$ locally obeys two types of entropic constraints. These constraints
are imposed on balls of bounded radius. While these constraints hold
everywhere in the bulk, the second constraint is relaxed on the domain wall.

From these assumptions, we were able to identify a new set of superselection
sectors localized at the domain wall. These sectors, which we refer to as the
_parton sectors_ , will be the main subject of this paper. These are “parton-
like” in the sense that other superselection sectors are composite objects
made from these sectors. One example of such a composite sector is the
superselection sectors of point excitations on the domain wall, which have
been studied in Refs. [12, 15]. However, there are other types of composite
sectors that are new to the best of our knowledge.

Both the parton and the composite sectors can be “fused” together like the
superselection sectors appearing in the bulk of the topological phase.
However, the ordinary rule of fusion does not always apply. When we say
fusion, we usually mean that there are two sectors, say $a$ and $b$, that
fuses into $c$. The state space in which $a$ and $b$ fusing into $c$ is
isomorphic to the state space of some Hilbert space. However, when we fuse
parton sectors, the state space in which two parton sectors fuse into another
parton sector may not be isomorphic to any such state space. We refer to this
phenomenon as _quasi-fusion_ and later explain how this difference arises.

Another strange thing about the parton sectors is that they should not be
viewed as low-energy excitations. Generally, a single parton by itself cannot
completely specify an excitation. Instead, parton labels should be considered
as quantum numbers that partially determine the excitation.

Despite their bizarre nature, parton sectors are actual physical objects.
There are operators localized on the $N$\- and $U$-shaped regions in Fig. 2
that can measure these sectors. More concretely, for every parton sector,
there is an operator that can unambiguously detect the presence of that
sector. As such, parton sectors should be treated as fundamental objects in
any theory of gapped domain walls.

Figure 2: For every parton sector, there is an operator acting either on the
$N$-shaped(left) or $U$-shaped(right) region that can unambiguously detect the
sector. Figure 3: Subsystems involved in the calculation of the domain wall
topological entanglement entropy.

To examine whether a given microscopic system can host parton sectors,
calculating ground state entanglement can be a fruitful approach. We prove,
starting from a set of assumptions summarized in Fig. 1, that the linear
combination of entanglement entropy in Fig. 3 must be equal to

$\displaystyle S_{\text{topo},N}$ $\displaystyle=2\ln\mathcal{D}_{N},$ (1)
$\displaystyle S_{\text{topo},U}$ $\displaystyle=2\ln\mathcal{D}_{U},$

where $\mathcal{D}_{N}=\sqrt{\sum_{n}d_{n}^{2}}$ and
$\mathcal{D}_{U}=\sqrt{\sum_{u}d_{u}^{2}}$ are the total quantum dimension of
two different types of parton sectors referred to as $U$\- and $N$-sectors. In
analogy with the topological entanglement entropy [35, 36], we refer to these
“order parameters” as _domain wall topological entanglement entropies_. More
discussion on this order parameter will appear in our companion paper [42].

Notwithstanding the rich physics of parton sectors, perhaps the most
remarkable fact of all is that all of these results followed entirely from
Fig. 1. No assumption on the parent Hamiltonian was necessary. The notion of
superselection sectors was derived, instead of being imposed. The existence of
fusion spaces was, again, derived. These facts compel us to name our approach
as _entanglement bootstrap,_ in analogy with the conformal bootstrap program
[43, 44].

While there are many conclusions one can make from this work, the following
two stand out. First, in the presence of gapped domain walls, there is a new
type of superselection sector called parton sector. Parton sectors are more
fundamental than the other sectors in the sense that they subdivide the other
sectors. These findings suggest that there is more to be understood about
gapped domain walls than previously thought.

The second lesson is somewhat philosophical. We often do physics by beginning
with a specific Lagrangian/Hamiltonian in mind and then computing various
properties of the theory from those objects. Alternatively, one may write a
set of consistency equations coming from the underlying symmetry of the theory
[44]. Our work shows that there is a third possibility, namely a possibility
to study the theory from the properties of ground state entanglement. Let us
again emphasize that, in our study, we did not invoke any assumption about the
action or the symmetry. All that was required was the set of consistency
equations coming from the property of the ground state entanglement. The fact
that a new physics can be uncovered this way is, in our opinion, surprising
and certainly warrants further exploration.

The rest of this paper is structured as follows. In Section II, we review Ref.
[41], focusing on the key ideas that are used in this work. In Section III, we
explain how the assumptions used in Ref. [41] are modified in the presence of
gapped domain walls. In particular, we deduce the existence of the parton
sectors, which are the central objects of this paper. In Section IV, we study
the composite superselection sectors that are made out of the parton sectors.
We begin with a few examples and conclude with the general lessons. In Section
V, we introduce a method to construct the fusion spaces of these sectors. In
Section VI, we study the fusion rules. In particular, we derive a number of
nontrivial identities relating the fusion multiplicities to the quantum
dimensions. In Section VII, we discuss the _quasi-fusion rules_ of the parton
sectors, which generalize the ordinary fusion rules. In general, more than one
fusion space is needed to describe a quasi-fusion process, even if both the
parton sectors before and after the quasi-fusion are completely specified. In
Section VIII, we derive various expressions for the topological entanglement
entropies of domain walls. In Section IX, we discuss the properties of the
string-like operators that can create the superselection sectors we have
studied in this paper. We conclude in Section X, listing some open problems
and directions to pursue.

##  II Fusion rules from entanglement

Our theory of gapped domain walls rests on our recent work on anyons [41].
Before this study, the theory of anyons was based on a mathematical framework
called unitary modular tensor category theory [9]. However, in Ref. [41], many
basic rules of that framework emerged from a generic property of entanglement
in gapped ground states. In this section, we provide a brief overview of this
work, focusing on the parts relevant to this paper.

To start with, we explain an important concept called _information convex set_
[41, 45, 46]. The information convex set is essential in understanding Ref.
[41] because the key physical objects of interest emerge from this definition.
To explain this concept, let us consider a subsystem of a two-dimensional
lattice, denoted as $\Omega$. Let $\Omega^{\prime}\supset\Omega$ be a
subsystem obtained by enlarging $\Omega$ along its boundary by an amount large
compared to the correlation length. The information convex set is defined as
follows:

$\Sigma(\Omega):=\left\\{{\rm
Tr}_{\Omega^{\prime}\setminus\Omega}\left(\rho_{\Omega^{\prime}}\right)|\rho_{b}=\sigma_{b}\quad\forall
b\in\mathcal{B}(r),b\subset\Omega^{\prime}\right\\},$ (2)

where $\sigma$ is some fixed _global reference state_. It is helpful to think
of this state as a ground state of some gapped Hamiltonian, although we do not
make use of that fact. Here, $\mathcal{B}(r)$ is the set of balls of bounded
radius $r=\mathcal{O}(1)$, where $r$ is chosen to be large compared to the
correlation length.

As it stands, aside from the fact that it is convex, the information convex
set does not have any particularly noteworthy structure. However, much more
can be said about this set once we incorporate physically motivated axioms on
the reference state $\sigma$. To that end, Ref. [41] advocated two physical
axioms. Specifically, the axioms state that

$\displaystyle(S_{C}+S_{BC}-S_{B})_{\sigma}$ $\displaystyle=0$ (3)
$\displaystyle(S_{BC}+S_{CD}-S_{B}-S_{D})_{\sigma}$ $\displaystyle=0$

over a set of subsystems depicted in Fig. 4, where $S(\rho)=-{\rm
Tr}(\rho\ln\rho)$ is the von Neumann entropy of $\rho$. Here, we specified
$\sigma$ in the subscript of the parenthesis because the underlying global
state is the same for all the entanglement entropies in the linear
combination. The subscript of $S$ represents the relevant subsystem. For
instance, $S_{B}$ appearing in an expression like
$(\ldots+S_{B}+\ldots)_{\sigma}$ represents $S(\sigma_{B})$.

Figure 4: The axioms A0 and A1 of Ref. [41] and their consequences. These
axioms, which are defined on a region of size $\mathcal{O}(1)$, imply that the
same entropic conditions hold at larger length scales; see the first figure in
the “consequences.” The isomorphism and the factorization property holds if
the subsystems’ thicknesses are larger than $2r$. Here, $r$ is the radius of
the disks on which the axioms are imposed. While we only depicted annuli in
this figure, the same consequences apply to _any_ sufficiently smooth
subsystems.

Equation. (3) is a reasonable assumption because it follows from the well-
known expression for the ground state entanglement entropy of gapped systems
[35, 36]:

$S(\sigma_{A})=\alpha|\partial A|-\gamma+\ldots,$ (4)

where $A$ is a simply connected subsystem, $\alpha$ is a non-universal
constant, $\gamma$ is the topological entanglement entropy, and the ellipsis
is the subleading term that vanishes in the $|\partial A|\to\infty$
limit.111While Eq. (3) must be assumed to hold exactly in Ref. [41], we expect
the arguments of the paper to go through even if we the conditions only hold
approximately. In the absence of subsystem symmetries [47], Eq. (4) is
expected to hold. Therefore, the fact that Eq. (3) follows from Eq. (4)
justifies the physical relevance of our axioms.

Figure 5: The technical consequences in Fig. 4 lead to the structural
statements about the information convex sets. The proof of the first statement
is reproduced in Section II.1. For the proof of the second statement, see Ref.
[41]. Here, $X$ is an annulus and $Y$ is a two-hole disk.
$\mathcal{S}(\mathbb{V}_{ab}^{c})$ is the set of density matrices acting on a
finite dimensional Hilbert space $\mathbb{V}_{ab}^{c}$.

These axioms lead to three important consequences, summarized in Fig. 4. We
will focus on discussing their meanings, referring Ref. [41] for the proof.

The first consequence is that Eq. (3) holds at larger length scales. Recall
that the axioms only apply to balls of bounded radius. The same set of
constraints hold on arbitrarily large subsystems.

The second consequence is the isomorphism theorem.

######  Theorem II.1 (Isomorphism theorem [41]).

If $\Omega^{0}$ and $\Omega^{1}$ are connected by a path
$\\{\Omega^{t}\\}_{t\in[0,1]}$, there is an isomorphism $\Phi$ between
$\Sigma(\Omega^{0})$ and $\Sigma(\Omega^{1})$ uniquely determined by the path.
Moreover, this isomorphism preserves the distance and entropy difference
between two elements of the information convex sets: for any
$\rho,\lambda\in\Sigma(\Omega^{0})$,

$\boxed{\begin{aligned} D(\rho,\lambda)&=D(\Phi(\rho),\Phi(\lambda))\\\
S(\rho)-S(\lambda)&=S(\Phi(\rho))-S(\Phi(\lambda)),\end{aligned}}$ (5)

where $D(\cdot,\cdot)$ is any distance measure that is non-increasing under
completely-positive trace preserving maps.

Here we say that a path exists between two subsystems if they can be smoothly
deformed into each other without changing the topology of the subsystem.222In
order to not run into any pathological counterexamples, it is convenient to
only consider subsystems whose thicknesses are at least a few times larger
compared to $r$. This theorem implies that there are “conserved quantities”
which remain invariant under deformations of the subsystems. These quantities
include the distance between two states in the information convex set and
their entropy difference.

The third consequence concerns the factorization property of the extreme
points. Let $\Omega$ be an arbitrary subsystem. Consider a subsystem
$\Omega^{\prime}\supset\Omega$ that can be smoothly deformed into $\Omega$,
where $\Omega^{\prime}\setminus\Omega$ is a “shell” that covers the boundary
of $\Omega$. We shall refer to $\Omega^{\prime}\setminus\Omega$ as the
_thickened boundary_ of $\Omega^{\prime}$. This will be an important concept
that will be used throughout this paper. Let $\rho^{\langle
e\rangle}_{\Omega^{\prime}}\in\Sigma(\Omega^{\prime})$ be an extreme point.
Then we have

$\boxed{(S_{\Omega}+S_{\Omega^{\prime}}-S_{\Omega^{\prime}\setminus\Omega})_{\rho^{\langle
e\rangle}}=0.}$ (6)

To see why we refer to Eq. (6) as the factorization property, consider a
purification of $\rho^{\langle e\rangle}_{\Omega^{\prime}}$, which we denote
as $|\rho^{\langle e\rangle}\rangle_{\Omega^{\prime}\Omega^{\prime c}}$ where
$\Omega^{\prime c}$ is the purifying system of $\Omega^{\prime}$. By using the
fact that the von Neumann entropy of a state is equal to that of its purifying
space, we can conclude

$\displaystyle I(\Omega:\Omega^{\prime c})_{|\rho^{\langle
e\rangle}\rangle_{\Omega^{\prime}\Omega^{\prime c}}}$
$\displaystyle=(S_{\Omega}+S_{\Omega^{\prime}}-S_{\Omega^{\prime}\setminus\Omega})_{\rho^{\langle
e\rangle}}$ (7) $\displaystyle=0,$

where $I(A:B)_{\rho}:=(S_{A}+S_{B}-S_{AB})_{\rho}$ is the mutual information
between $A$ and $B$ over a state $\rho$. In other words, $|\rho^{\langle
e\rangle}\rangle_{\Omega^{\prime}\Omega^{\prime c}}$, upon tracing out
$\Omega^{\prime}\setminus\Omega$, becomes a product state over $\Omega$ and
$\Omega^{\prime c}$.

These three consequences are the main workhorses of our theory. Below, we will
see the power of these consequences in action, by deriving several nontrivial
facts about anyon theory. We urge the readers to carefully digest the ensuing
material before moving to the rest of the paper, as the key ideas remain the
same while the setup becomes more intricate as we move forward.

###  II.1 Superselection sectors

In the theory of anyon, a superselection sector is a topological charge
associated with a point-like excitation. In our theory, we define the
superselection sectors as the extreme points of an information convex set over
an annulus. In this section, we justify this definition by showing that
different extreme points are orthogonal to each other. In particular,
different extreme points can be perfectly distinguished from each other by
some physical experiment.

To prove this fact, we set up the notation as follows. Consider an annulus $X$
and two additional annuli $X^{\prime}$ and $X^{\prime\prime}$ such that
$X^{\prime}\supset X$ and $X^{\prime}X^{\prime\prime}$ is again an annulus;
see Fig. 6. Without loss of generality, consider two extreme points of
$\Sigma(X)$, denoted as $\rho_{X}$ and $\rho^{\prime}_{X}$.

The key idea is to map these extreme points to the extreme points in
$\Sigma(XX^{\prime\prime})$ by using Theorem II.1. Then, we apply the
factorization property of extreme points to argue that these extreme points
must factorize over $X$ and $X^{\prime\prime}$. Our claim will be an immediate
consequence of this last fact.

As a first step, note that any distance measure between $\rho_{X}$ and
$\rho^{\prime}_{X}$ is invariant under the isomorphism associated with a
smooth deformation of $X$; see Theorem II.1. In particular, the fidelity
between the two satisfies the following identity

$F(\rho_{X},\rho_{X}^{\prime})=F(\rho_{X^{\prime}X^{\prime\prime}},\rho_{X^{\prime}X^{\prime\prime}}^{\prime}),$
(8)

where $\rho_{X^{\prime}X^{\prime\prime}}$ and
$\rho_{X^{\prime}X^{\prime\prime}}^{\prime}$ are the extreme points of
$\Sigma(X^{\prime}X^{\prime\prime})$ obtained from the isomorphism. Moreover,
because fidelity is non-decreasing under partial trace, we conclude 333While
fidelity is not a distance measure, one can relate it to a distance measured
called Bures distance, defined as $\sqrt{1-F(\cdot,\cdot)}$.

$\displaystyle F(\rho_{X},\rho_{X}^{\prime})$
$\displaystyle=F(\rho_{X^{\prime}X^{\prime\prime}},\rho_{X^{\prime}X^{\prime\prime}}^{\prime})$
(9) $\displaystyle\leq
F(\rho_{XX^{\prime\prime}},\rho_{XX^{\prime\prime}}^{\prime}).$

Figure 6: Subsystems involved in the proof of the orthogonality of extreme
points in $\Sigma(X)$. The annulus enclosed in the dotted region of the second
figure represents the blue annulus $X$ in the top figure. By applying the
isomorphism theorem (Theorem II.1), the annulus $X$ is deformed into a larger
annulus $X^{\prime}\supset X$ and the union of $X^{\prime}$ (dark blue) with
$X^{\prime\prime}$ (light blue).

Secondly, by the factorization of the extreme points, we have

$(S_{X}+S_{X^{\prime}}-S_{X^{\prime}\setminus X})_{\rho}=0.$ (10)

By using the strong subadditivity of entropy (SSA) [48], we get

$\displaystyle I(X:X^{\prime\prime})_{\rho}$
$\displaystyle\leq(S_{X}+S_{X^{\prime}}-S_{X^{\prime}\setminus X})_{\rho}$
(11) $\displaystyle=0.$

Therefore, $\rho_{X^{\prime}X^{\prime\prime}}$, upon restricting to
$XX^{\prime\prime}$, becomes a product state over $X$ and $X^{\prime\prime}$.
The same conclusion applies to $\rho^{\prime}_{X^{\prime}X^{\prime\prime}}$
because $\rho^{\prime}$ is an extreme point too.

Combining these two observations, we conclude

$\displaystyle F(\rho_{X},\rho_{X}^{\prime})\leq
F(\rho_{X},\rho_{X}^{\prime})F(\rho_{X^{\prime\prime}},\rho^{\prime}_{X^{\prime\prime}}).$
(12)

Again using the isomorphism theorem (Theorem II.1), we get

$F(\rho_{X},\rho_{X}^{\prime})\leq F(\rho_{X},\rho_{X}^{\prime})^{2}.$ (13)

Since $F(\rho_{X},\rho_{X}^{\prime})\in[0,1]$ by the definition of fidelity,
the only allowed values are $F(\rho_{X},\rho_{X}^{\prime})\in\\{0,1\\}$.
Therefore, given any two extreme points, they must be either orthogonal to
each other or exactly equal to each other, thus proving our claim.

Therefore, without loss of generality, we can characterize $\Sigma(X)$ as a
simplex with orthogonal extreme points.444Here, the orthogonality means that
the Hilbert-Schmidt inner product of two-density matrices is $0$. Namely,
${\rm Tr}(\rho^{\dagger}\sigma)=0$. Specifically, we have

$\Sigma(X)=\left\\{\bigoplus_{a}p_{a}\rho_{X}^{a}:\sum_{a}p_{a}=1,p_{a}\geq
0\right\\},$ (14)

where different extreme points $\\{\rho_{X}^{a}\\}$ are supported on
orthogonal subspaces. Provided that the underlying Hilbert space is finite-
dimensional, $a$ belongs to a finite set

${\mathcal{C}}=\\{1,a,b,\cdots\\},$ (15)

where “1” is the vacuum sector, the extreme point of which is obtained by
restricting $\sigma$ to $X$.

To each of the extreme points, we can define a notion of _quantum dimension_.
Let $\rho_{X}^{a}$ be an extreme point of $\Sigma(X)$. We _define_ the quantum
dimension of the superselection sector $a$ as

$\boxed{d_{a}:=\exp\left(\frac{S(\rho_{X}^{a})-S(\rho_{X}^{1})}{2}\right),}$
(16)

Note that, even though we have not specified the annulus here, this definition
is still well-defined because of the isomorphism theorem (Theorem II.1). It
follows from this definition that $d_{1}=1$ and $d_{a}>0$ for all
$a\in\mathcal{C}$.

Our definition of the quantum dimension is not standard. However, this
definition is equivalent to the more widely-held definition [41]. We prove
this in Section II.2 by showing that our quantum dimensions are completely
determined by the fusion multiplicities, as is the case in the more
conventional theory of anyon [9].

###  II.2 Fusion multiplicities

In this section, we derive the following fundamental equation

$\boxed{d_{a}d_{b}=\sum_{c\in\mathcal{C}}N_{ab}^{c}d_{c},}$ (17)

where $N_{ab}^{c}$ is the dimension of the fusion space and $d_{a}$ is defined
in the previous section [46, 41]. This equation implies that the quantum
dimensions _are_ the quantum dimensions of anyons.

Figure 7: By restricting the elements of $\Sigma_{ab}^{c}(Y)$ to three
different annuli, we obtain extreme points of $\Sigma(X)$ for some annulus
$X$. These extreme points correspond to the superselection sectors $a,b,$ and
$c$.

We begin by briefly explaining what we mean by a fusion space, deferring the
proof of Eq. (17) for the moment. The fusion space is defined in terms of the
information convex set of a two-hole disk, say $Y$. Reference [41] completely
characterized this set, under the same set of assumptions we have used so far.
Specifically,

$\Sigma(Y)=\left\\{\bigoplus_{a,b,c}p_{ab}^{c}\rho_{Y}^{abc}:\rho_{Y}^{abc}\in\Sigma_{ab}^{c}(Y)\right\\},$
(18)

where $\\{p_{ab}^{c}\\}$ is a probability distribution and
$\Sigma_{ab}^{c}(Y)$ is a set of states in $\Sigma(Y)$ whose reduced density
matrices on the three annuli are the extreme points associated with
superselection sectors $a,b,$ and $c$; see Fig. 7. Importantly,
$\Sigma_{ab}^{c}(Y)$ is isomorphic to the state space of some finite-
dimensional Hilbert space. This is our definition of the fusion space. The
dimension of this fusion space is $N_{ab}^{c}$.

Below, we focus on the derivation of Eq. (17), by first explaining the
_merging technique_ , and then applying this technique to our setup.

####  II.2.1 Merging

Equation (17) follows from an extremely useful technique called _merging_. The
merging technique addresses the following problem. Let $\rho\in\Sigma(\Omega)$
and $\lambda\in\Sigma(\Omega^{\prime})$ be the elements belonging to two
different information convex sets. Can we construct a state in
$\Sigma(\Omega\cup\Omega^{\prime})$ that is consistent with both $\rho$ and
$\lambda$? Obviously, this is not always possible because such a state may not
even exist.555As a simple example, let $\rho$ be a maximally entangled state
between two subsystems, say $A$ and $B$, and $\lambda$ be a maximally
entangled state between $B$ and $C$. By the monogamy of entanglement, there
cannot be any tripartite state over $ABC$ that is consistent with both $\rho$
and $\lambda$. Moreover, even if there exists a state consistent with both
$\rho$ and $\lambda$, that state may not belong to
$\Sigma(\Omega\cup\Omega^{\prime})$. With the merging technique, we can ensure
both.

The following two statements are the key. First, for general quantum states,
we have the following _merging lemma._

######  Lemma II.2 (Merging Lemma [49]).

If there is a pair of quantum states $\rho_{ABC}$ and $\lambda_{BCD}$
satisfying $\rho_{BC}=\lambda_{BC}$ and
$I(A:C|B)_{\rho}=I(B:D|C)_{\lambda}=0$, there exists a unique quantum state
$\tau_{ABCD}$ such that

$\displaystyle{\rm Tr}_{D}\tau_{ABCD}$ $\displaystyle=$
$\displaystyle\rho_{ABC}$ $\displaystyle{\rm Tr}_{A}\tau_{ABCD}$
$\displaystyle=$ $\displaystyle\lambda_{BCD}$ $\displaystyle I(A:CD|B)_{\tau}$
$\displaystyle=$ $\displaystyle I(AB:D|C)_{\tau}=0.$

Here $I(A:C|B)_{\rho}:=(S_{AB}+S_{BC}-S_{B}-S_{ABC})_{\rho}$ is the
_conditional mutual information_.

Second, with an additional assumption, elements of the information convex sets
are “closed” under the merging operation. Specifically, the density matrices
belonging to information convex sets can merge into an element of another
information convex set. We refer to this fact as the _merging theorem_.

######  Theorem II.3 (Merging Theorem [41]).

Consider two density matrices $\rho_{ABC}\in{\Sigma}(ABC)$ and
$\lambda_{BCD}\in{\Sigma}(BCD)$. Consider the following three conditions:

  1. 1.

$\rho_{BC}=\lambda_{BC}$ and $I(A:C|B)_{\rho}=I(B:D|C)_{\lambda}=0$.

  2. 2.

There exists a partition $B^{\prime}C^{\prime}=BC$, such that no disk of
radius $r$ overlaps with both $AB^{\prime}$ and $CD$.

  3. 3.

$I(A:C^{\prime}|B^{\prime})_{\rho}=I(B^{\prime}:D|C^{\prime})_{\lambda}=0$.

If these three conditions hold, the resulting density matrix generated by
$\rho_{ABC}$ and $\lambda_{BCD}$ using the merging lemma (Lemma II.2) belongs
to ${\Sigma(ABCD)}$.

In this paper, to ensure the merged state is in an information convex set, we
shall exclusively use the merging theorem. If the conditions in the merging
theorem are satisfied, we shall denote the _merged state_ of $\rho$ and
$\sigma$ as:

$\rho\Join\lambda.$ (19)

Merged states are useful because they satisfy the following nontrivial
identities:

$\boxed{\begin{aligned}
I(A:CD|B)_{\rho\Join\lambda}&=I(AB:D|C)_{\rho\Join\lambda}=0,\\\
I(A:C^{\prime}D|B^{\prime})_{\rho\Join\lambda}&=I(AB^{\prime}:D|C^{\prime})_{\rho\Join\lambda}=0,\\\
I(A:D|BC)_{\rho\Join\lambda}&=0,\end{aligned}}$ (20)

which implies that $\rho\Join\lambda$ is the _maximum-entropy state_
consistent with both $\rho$ and $\lambda$. This fact follows from SSA [48].
Moreover,

$\boxed{S(\rho\Join\lambda)-S(\rho^{\prime}\Join\lambda)=S(\rho)-S(\rho^{\prime}),}$
(21)

where we implicitly assumed that both $\rho$ and $\rho^{\prime}$ can be merged
with $\lambda$.

To explain the utility of the merging theorem, we discuss a simple example. We
will discuss merging two density matrices in a toy setup, focusing on the
logic behind why they can be merged.

Consider an annulus $ABC$ and a disk-like region $BCD$ that overlap on a disk-
like region; see Fig. 8. We will consider density matrices
$\rho\in\Sigma(ABC)$ and $\lambda\in\Sigma(BCD)$.666In fact,
$\lambda_{BCD}=\sigma_{BCD}$ on disk $BCD$. This is because any element of the
information convex set is indistinguishable with the reference state on any
disk [41]. These two density matrices have identical reduced density matrix on
disk $BC$. Provided that the overlapping region is sufficiently thick so that
the distance between $A$ and $D$ is large, the requisite conditions in the
merging theorem (Theorem II.3) can be satisfied with an appropriate choice of
$B$ and $C$.

The first condition can be satisfied by partitioning the overlapping region as
in Fig. 8. We can use SSA and the extensions of axioms to derive the
conditional independence condition. For example, in order to prove
$I(A:C|B)_{\rho}=0$, consider an auxiliary subsystem $E$ introduced in Fig. 9.
By using the isomorphism theorem, one can extend $\rho$ to a density matrix
$\rho^{\prime}$ in $\Sigma(ABCE)$. Such density matrix on a disk-like region
$BCE$ is consistent with the reference state $\sigma$. By the extension of
axiom, one can thus see that

$\displaystyle I(A:C|B)_{\rho}$
$\displaystyle\leq(S_{BC}+S_{CE}-S_{B}-S_{E})_{\rho^{\prime}}$ (22)
$\displaystyle=0.$

Conditional independence of other sets of subsystems can be obtained in a
similar way.

Figure 8: Two ways of partitioning the overlapping subsystem ($BC$). Because
the overlapping region was chosen so that $A$ and $D$ are sufficiently far
apart from each other, we can choose $B^{\prime}$ and $C$ to be separated by
more than $2r$, where $r$ is the radius of the disk on which our axioms are
postulated (red). Figure 9: A partition of an annulus (blue) into $A,B,$ and
$C$. Here, $E$ is an auxiliary subsystem used in the proof of the vanishing
conditional mutual information.

Within this example, the conditions in Theorem II.3 can be satisfied because
the overlapping region separates the non-overlapping parts sufficiently far
apart from each other. This observation holds quite generally, as we shall
repeatedly see throughout this paper.

####  II.2.2 Derivation

Armed with the merging technique, we are now in a position to derive Eq. (17).
To do so, we will merge two density matrices with supports overlapping on a
disk-like region. Partition these annuli into $ABC$ and $BCD$, similar to the
partition we had before; see Fig. 10.

Figure 10: The set of subsystems used in merging the extreme points of
$\Sigma(ABC)$ and $\Sigma(BCD)$. Here, $a$ and $b$ are the superselection
sectors that the respective extreme points represent. While we also require
$BC$ to be partitioned into $B^{\prime}C^{\prime}$ such that the requisite
conditions in Theorem II.3 are satisfied, once these conditions are verified,
we will not make use of this partition. This is why we did not describe these
subsystems in this figure.

We can merge the density matrices in $\Sigma(ABC)$ with the density matrices
in $\Sigma(BCD)$ provided that the distance between $A$ and $D$ is
sufficiently large. To see why, first note that these density matrices have
identical density matrices on the overlapping region. Secondly, one can prove
the requisite conditions on the conditional mutual information, again by
utilizing the auxiliary subsystem introduced in Fig. 9.

While one can merge any pair of density matrices from $\Sigma(ABC)$ and
$\Sigma(BCD)$, we will merge the extreme points. Without loss of generality,
let $\rho_{ABC}^{a}\in\Sigma(ABC)$ and $\lambda_{BCD}^{b}\in\Sigma(BCD)$ be a
pair of extreme points associated with the superselection sectors $a$ and $b$.
The merged state,

$\tau_{ABCD}^{a\Join b}:=\rho^{a}_{ABC}\Join\lambda^{b}_{BCD},$ (23)

according to Eq. (20), obeys the following identity:

$\displaystyle(S_{ABCD})_{\tau^{a\Join b}}$
$\displaystyle=(S_{ABC}+S_{BCD}-S_{BC})_{\tau^{a\Join b}}$ (24)
$\displaystyle=2\ln(d_{a}d_{b})+(S_{ABC}+S_{BCD}-S_{BC})_{\sigma}$

In the first line, we used the property of the merged state. In the second
line, we used the definition of the quantum dimensions. The second term of the
second line can be interpreted as the entropy of the merged state
$\tau_{ABCD}^{1\Join 1}$, which is equal to the reference state restricted to
$ABCD$.777This is a fact proved in Ref. [41]. Therefore,

$S(\tau_{ABCD}^{a\Join b})=S(\sigma_{ABCD})+2\ln(d_{a}d_{b}).$ (25)

Note that $\tau_{ABCD}^{a\Join b}$ is the maximum-entropy state of
$\Sigma(ABCD)$ that is consistent with the density matrices of the two annuli.
We can solve this maximization problem directly, which, by definition, must
agree with Eq. (25).

For this derivation, we use the structure of the information convex set of a
two-hole disk summarized at the bottom of Fig. 5.888The proof of this
statement also follows from the axioms in Fig. 4; see Ref. [41] for more
detail. We shall refer to this two-hole disk as $Y:=ABCD$. Because
$\tau_{ABCD}^{a\Join b}\in\Sigma(Y)$, without loss of generality,

$\tau_{Y}^{a\Join
b}=\bigoplus_{c}p_{c}\,\rho^{abc}_{Y},\quad\rho^{abc}_{Y}\in\Sigma_{ab}^{c}(Y),$
(26)

for some probability distribution $\\{p_{c}\\}$. Density matrices in different
$\Sigma_{ab}^{c}(Y)$ are mutually orthogonal to each other. Because
$\tau_{Y}^{a\Join b}$ maximizes the entropy, its entropy is

$\displaystyle S(\tau_{Y}^{a\Join b})$
$\displaystyle=\max_{\begin{subarray}{c}\\{p_{c}\\},\\\
\\{\rho^{abc}_{Y}\\}\end{subarray}}\left(H\left(\\{p_{c}\\}\right)+\sum_{c}p_{c}S(\rho^{abc}_{Y})\right)$
(27)
$\displaystyle=\max_{\\{p_{c}\\}}\left(H\left(\\{p_{c}\\}\right)+\sum_{c}p_{c}\max_{\rho^{abc}_{Y}}S(\rho^{abc}_{Y})\right)$

where $H(\\{p_{c}\\})=-\sum p_{c}\ln p_{c}$ is the Shannon entropy of the
probability distribution $\\{p_{c}\\}$.

This is the key identity:

$\max_{\rho^{abc}_{Y}\in\Sigma_{ab}^{c}(Y)}S(\rho^{abc}_{Y})=S(\sigma_{Y})+\ln
N_{ab}^{c}+\ln(d_{a}d_{b}d_{c}),$ (28)

which we derive in two steps. First, we show that the extreme points within
$\Sigma_{ab}^{c}(Y)$ have identical entropies. Because this space is
isomorphic to the state space of a $N_{ab}^{c}$-dimensional Hilbert space, the
maximum is attained by taking $\rho_{Y}^{abc}$ to be a uniform mixture of
$N_{ab}^{c}$ orthogonal extreme points within $\Sigma_{ab}^{c}(Y)$. Second, we
show that the entropy of the extreme points are equal to
$S(\sigma_{Y})+\ln(d_{a}d_{b}d_{c})$.

For the first step, we use the factorization property of the extreme points.
Recall that any extreme point $\rho_{Y^{\prime}}^{\langle
e\rangle}\in\Sigma_{ab}^{c}(Y^{\prime})$ satisfies

$(S_{Y}+S_{Y^{\prime}}-S_{Y^{\prime}\setminus Y})_{\rho^{\langle
e\rangle}}=0,$ (29)

where $Y^{\prime}\supset Y$ is a two-hole disk that is expanded along the
boundary of $Y$ by an amount large compared to the correlation length. By
using the fact that the reduced density matrices of the elements in
$\Sigma_{ab}^{c}(Y^{\prime})$ on $Y^{\prime}\setminus Y$ are identical and the
fact that the entropy difference over $Y$ and $Y^{\prime}$ are equal, we can
conclude that the entropy of extreme points are identical.

In the second step, we seek to prove

$S(\rho^{\langle e\rangle abc}_{Y})=S(\sigma_{Y})+\ln(d_{a}d_{b}d_{c})$ (30)

for any extreme point $\rho^{\langle e\rangle abc}_{Y}\in\Sigma_{ab}^{c}(Y)$.
We can derive this fact by comparing the entropy of $\rho_{Y}^{\langle
e\rangle abc}$ to $\sigma_{Y}$. More specifically, again consider
$Y^{\prime}\supset Y$ which is obtained by enlarging $Y$ along its boundary by
a large enough amount.999The thickness of $Y^{\prime}\setminus Y$ should be
greater than $2r$ so that the simplex theorem applies to the three annuli
subsystems of $Y^{\prime}\setminus Y$. The extreme points of
$\Sigma_{ab}^{c}(Y)$ are extended into the extreme points of
$\Sigma_{ab}^{c}(Y^{\prime})$ by using the isomorphism theorem. By the
factorization of the extreme points (Eq. (6)), we obtain:

$\displaystyle S(\rho^{\langle e\rangle abc}_{Y})+S(\rho^{\langle e\rangle
abc}_{Y^{\prime}})-S(\rho^{\langle e\rangle abc}_{Y^{\prime}\setminus Y})$
$\displaystyle=0,$ (31) $\displaystyle
S(\sigma_{Y})+S(\sigma_{Y^{\prime}})-S(\sigma_{Y^{\prime}\setminus Y})$
$\displaystyle=0,$

where in the second line we used the fact that $\sigma_{Y}$ is an extreme
point of $\Sigma_{11}^{1}(Y)$.101010This is a fact discussed in Ref. [41]. Our
axioms imply that $\Sigma_{11}^{1}(Y)$ contains a unique element. Since
$\sigma_{Y}$ trivially belongs to $\Sigma_{11}^{1}(Y)$, it must be an extreme
point. By subtracting the second equation from the first, we obtain

$(S_{Y}+S_{Y^{\prime}})_{\rho^{\langle e\rangle
abc}}-(S_{Y}+S_{Y^{\prime}})_{\sigma}=2\ln(d_{a}d_{b}d_{c}).$ (32)

The nontrivial part lies on obtaining the right hand side of Eq. (32). This
expression can be derived by noting that $Y^{\prime}\setminus Y$ is a union of
three annuli and the fact that the reduced density matrix of any element of
$\Sigma_{ab}^{c}(Y^{\prime})$ over $Y^{\prime}\setminus Y$ is a tensor product
of the extreme points associated with definite superselection sectors $a$,
$b$, and $c$.111111Technically speaking, one also needs to prove this fact. We
gloss over this subtlety here, referring the readers to Ref. [41] for the
rigorous proof. The first two terms in the left-hand side of Eq. (32) are
actually equal to each other due to the isomorphism theorem. Thus, we have
proved Eq. (30).

Plugging Eq. (28) into Eq. (27) we obtain

$\displaystyle S(\tau_{Y}^{a\Join b})-S(\sigma_{Y})=$
$\displaystyle\max_{\\{p_{c}\\}}\left(H\left(\\{p_{c}\\}\right)+\sum_{c}p_{c}\ln(N_{ab}^{c}d_{c})\right)$
(33) $\displaystyle+\ln(d_{a}d_{b}).$

This maximization problem can be solved by minimizing the free energy of a
fictitious Hamiltonian $H(c):=-\ln(N_{ab}^{c}d_{c})$ that depends on the
superselection sector $c$ with respect to a “temperature” of $T=1$. The
partition function of this fictitious Hamiltonian is
$Z=\sum_{c}N_{ab}^{c}d_{c}$. Therefore, the free energy, defined as
$F=\sum_{c}p_{c}H(c)-TH(\\{p_{c}\\})$, is minimized as
$\min_{\\{p_{c}\\}}F=-T\ln Z=-\ln(\sum_{c}N_{ab}^{c}d_{c})$. The minimum is
obtained when $p_{c}=\frac{N_{ab}^{c}d_{c}}{d_{a}d_{b}}$. 121212A
mathematically equivalent fact is that relative (Shannon) entropy is
nonnegative. For two probability distributions $\\{p_{i}\\}$ and
$\\{q_{i}\\}$,
$\min_{\\{p_{i}\\}}\left(\sum_{i}p_{i}\ln(p_{i}/q_{i})\right)=0$. The minimum
is obtained if and only if the two probability distributions are identical.
Therefore, we obtain

$S(\tau_{Y}^{a\Join
b})=S(\sigma_{ABCD})+\ln(d_{a}d_{b})+\ln(\sum_{c}N_{ab}^{c}d_{c}).$ (34)

By comparing Eq. (25) to Eq. (34), we conclude that

$\ln(d_{a}d_{b})+\ln(\sum_{c}N_{ab}^{c}d_{c})=2\ln(d_{a}d_{b}),$ (35)

which, after rearranging the terms, becomes Eq. (17).

To conclude, we have sketched the proof of Eq. (17). The key idea was to merge
density matrices associated with two superselection sectors. The entropy of
the merged state can be calculated in two different ways, one that is obtained
from the entropy of the reduced density matrices over smaller regions and
another obtained by directly maximizing the entropy. The gist of the second
calculation follows from Eq. (6).

The ideas sketched above are, in fact, powerful enough to derive a whole slew
of consistency relations, such as [41]

$\displaystyle N_{ab}^{c}$ $\displaystyle=N_{ba}^{c}$ (36) $\displaystyle
N_{1a}^{c}$ $\displaystyle=N_{a1}^{c}=\delta_{a,c}$ $\displaystyle\forall
a,\exists!\,\bar{a}\,\,\text{ s.t. }\,\,N_{ab}^{1}$
$\displaystyle=\delta_{b,\bar{a}}$ $\displaystyle N_{ab}^{c}$
$\displaystyle=N_{\bar{b}\bar{a}}^{\bar{c}}$
$\displaystyle\sum_{i}N_{ab}^{i}N_{ic}^{d}$
$\displaystyle=\sum_{j}N_{aj}^{d}N_{bc}^{j}.$

A generalization of these identities will be discussed in Section VI.

##  III Gapped domain walls: a tale of parton sector

The results we sketched in Section II follow from the axioms described in Fig.
4. However, these assumptions become inadequate in the vicinity of gapped
domain walls. On a domain wall, we need to relax these assumptions
appropriately.

Figure 11: If we use the ansatz for the entanglement entropy in Eq. (37), the
linear combination of entanglement entropy
$(S_{BC}+S_{CD}-S_{B}-S_{D})_{\sigma}$ becomes $0$ for the first two choices.
However, the same conclusion does not generally hold for the third choice.

Here is a heuristic discussion on this issue. Consider two topologically
ordered mediums that are separated by a gapped domain wall. We shall refer to
the bulk phases lying on different sides of the domain wall as $P$ and $Q$.
Suppose that the entanglement entropy of a region $A$ has the following form:

$S(\sigma_{A})=\alpha|\partial A|-\gamma(A)+\ldots,$ (37)

where the first term is the leading area law term that can be canceled from an
appropriate linear combination, the second term $\gamma(A)$ is a constant that
depends on $A$, and the ellipses represent the subleading correction that
vanishes in the $|\partial A|\to\infty$ limit. Based on the study of
entanglement entropy in the bulk [41], we can make a somewhat speculative but
reasonable assumption about $\gamma(A)$: that it is invariant under smooth
deformations of $A$.131313Unlike the subleading contribution in the bulk [35,
36], it is unclear if $\gamma(A)$ can be always obtained from a linear
combination of entanglement entropies. Therefore, it is unclear whether
individual $\gamma(A)$ has a well-defined physical meaning. However, as we
shall see in Section VIII, certain linear combinations of entanglement
entropies do have clear physical meanings. By smooth deformation, we mean any
deformation that retains the topology of $A$ and its restrictions to $P$ and
$Q$.

Once we accept this hypothesis, we can immediately verify that
$(S_{BC}+S_{CD}-S_{B}-S_{D})_{\sigma}=0$ for the first two choices of
subsystems described in Fig. 11. This is because
$\gamma(BC)=\gamma(CD)=\gamma(B)=\gamma(D)$ for these subsystem choices.
However, this hypothesis does not imply that the same linear combination of
entanglement entropy vanishes for the third choice. For the third choice,
$\gamma(BC)$ and $\gamma(B)$ are allowed to take different values since $B$
cannot be smoothly deformed into $BC$.

This observation motivates a relaxed set of axioms to study gapped domain
walls, summarized in Fig. 12. We emphasize that the boundary between $B$ and
$D$ can deform arbitrarily so long as they do not cross the domain wall. We
will not make any assumption about the value of
$(S_{BC}+S_{CD}-S_{B}-S_{D})_{\sigma}$ for the rightmost subsystems in Fig.
11. Remarkably, its value is highly constrained, as we explain in Section
VIII.

Figure 12: On top of the bulk axioms in $P$ and $Q$, we assume that
$(S_{C}+S_{BC}-S_{B})_{\sigma}=0$ (red) and
$(S_{BC}+S_{CD}-S_{B}-S_{D})_{\sigma}=0$ (green) on the domain wall. The
subsystems are allowed to be deformed as long as the boundaries between $B$
and $D$ do not cross the domain wall.

The new axioms in Fig. 12 directly lead to a definition of _parton sectors._
This is a new type of superselection sector that is localized on either side
of the domain wall. We refer to these sectors as parton sectors because they
subdivide the known superselection sectors of point-like excitations on the
domain wall [12, 15]. As in the discussion in Section II.1, the properties of
the parton sectors can be derived from three important consequences: extension
of axioms, isomorphism theorem, and factorization of extreme points.

Let us formally state these consequences below, deferring the proofs to
Appendixes A, B, and C. First, the axioms can be extended to arbitrarily large
regions. Secondly, a generalization of the isomorphism theorem holds.

######  Theorem III.1 (Isomorphism Theorem).

Consider a reference state for which the axioms in Fig. 12 apply. If
$\Omega^{0}$ and $\Omega^{1}$ are connected by a path
$\\{\Omega^{t}\\}_{t\in[0,1]}$, there is an isomorphism $\Phi$ between
$\Sigma(\Omega^{0})$ and $\Sigma(\Omega^{1})$ uniquely determined by the path.
Moreover, this isomorphism preserves the distance and entropy difference
between two elements of the information convex sets: for
$\rho,\lambda\in\Sigma(\Omega^{0})$,

$\boxed{\begin{aligned} D(\rho,\lambda)&=D(\Phi(\rho),\Phi(\lambda)),\\\
S(\rho)-S(\lambda)&=S(\Phi(\rho))-S(\Phi(\lambda)),\end{aligned}}$ (38)

where $D(\cdot,\cdot)$ is any distance measure that is non-increasing under
completely-positive trace preserving maps.

In order to understand this theorem, it is important to understand what a
modified definition of the “path” means in the presence of a domain wall. This
is most convenient to understand in the continuum limit. We say that two
subsystems are connected by a path if they can be continuously deformed into
each other. Specifically, let $\mathcal{M}$ be the manifold, which is divided
into $\mathcal{M}=\mathcal{M}_{P}\cup\mathcal{M}_{Q}$, where $\mathcal{M}_{P}$
is the part that hosts the topological phase $P$ and $\mathcal{M}_{Q}$ is the
part that hosts the phase $Q$. We say that there is a _path_ between
$\Omega^{0}\subset\mathcal{M}$ and $\Omega^{1}\subset\mathcal{M}$ if there is
a one-parameter family of homeomorphism
$\phi_{t}:\Omega^{0}\hookrightarrow\mathcal{M}$ such that
$\phi_{t}(\Omega^{0})$ restricted to $P$ and $Q$ are both homeomorphisms for
$t\in[0,1]$, $\phi_{0}(\Omega^{0})=\Omega^{0}$, and
$\phi_{1}(\Omega^{0})=\Omega^{1}$; see Fig. 13. Note that this definition of
path is a refinement of that in the bulk.

Figure 13: The subsystem on the left side is homeomorphic to a disk. However,
it is not connected to a disk in $P$ (or $Q$) by any path because its part
that lies on $Q$ is a union of two disks. There is no homeomorphism that maps
those disks to either a single disk or an empty set.

Third, we can extend Eq. (6) to the gapped domain wall. Specifically, the
exact same equation holds, irrespective of the presence of the domain wall. We
restate this fact here for readers’ convenience. Given an extreme point of an
information convex set over $\Omega$, let $\Omega^{\prime}\supset\Omega$ be a
region obtained from $\Omega$ by enlarging141414An enlargement is associated
with a path connecting $\Omega$ and $\Omega^{\prime}$. By the isomorphism
theorem, there is an isomorphism between $\Sigma(\Omega)$ and
$\Sigma(\Omega^{\prime})$. $\Omega$ along the boundary, such that
$\Omega^{\prime}\setminus\Omega$ is the thickened boundary of
$\Omega^{\prime}$. Then, for any extreme point $\rho^{\langle
e\rangle}_{\Omega^{\prime}}\in\Sigma(\Omega^{\prime})$, we have:

$\boxed{(S_{\Omega}+S_{\Omega^{\prime}}-S_{\Omega^{\prime}\setminus\Omega})_{\rho^{\langle
e\rangle}}=0.}$ (39)

###  III.1 Parton sectors

Now we are ready to define the parton sectors, the fundamental objects of our
theory. To define these sectors, we choose the subsystems described in Fig.
14. We will refer to the left diagram as an $N$-shaped region and the right
diagram as a $U$-shaped region. Their information convex sets form simplices
with orthogonal extreme points, each of which labels a parton sector.

There are two types of parton sectors, one associated with the $N$-shaped
region and the one associated with the $U$-shaped region. We shall refer to
the former as an $N$-type superselection sector and the latter as a $U$-type
superselection sector to evoke the shape of the underlying regions.

Let us first explain why these sectors are well-defined. We focus only on the
$N$-type superselection sector. Because the same argument applies to the
$U$-type superselection sector, we omit that discussion.

Figure 14: Subsystems associated with the $N$-type (left) and $U$-type (right)
superselection sectors. We use $n$ and $u$ denote the respective sectors.

Our proof is based on the following choice of subsystems, which we depict in
Fig. 15. Without loss of generality, consider an $N$-shaped region $N$. Let
$N^{\prime}\supset N$ be a subsystem obtained by enlarging $N$ along its
boundary. Let $N^{\prime\prime}$ be another $N$-shaped region disjoint from
$N^{\prime}$ such that $N^{\prime}N^{\prime\prime}$ is again an $N$-shaped
region.

Figure 15: Subsystems involved in the proof of the orthogonality of the
extreme points in $\Sigma(N)$. Here, $N^{\prime}\supset N$ (dark blue) is
obtained from $N$ by enlarging $N$ along the boundary. $N^{\prime\prime}$
(light blue) is chosen in such a way that $N^{\prime}N^{\prime\prime}$ is
again a _N_ -shaped subsystem. On the right, the dotted region represents $N$.

Consider a pair of extreme points $\rho_{N},\rho^{\prime}_{N}\in\Sigma(N)$. As
discussed in Section II.1, the key idea is to map these extreme points to the
extreme points in $\Sigma(NN^{\prime\prime})$ by using the isomorphism theorem
(Theorem III.1). These extreme points must be factorized over $N$ and
$N^{\prime\prime}$, which immediately implies our claim.

Let $\rho_{N^{\prime}N^{\prime\prime}}$ and
$\rho_{N^{\prime}N^{\prime\prime}}^{\prime}$ be the extreme points of
$\Sigma(N^{\prime}N^{\prime\prime})$ obtained by applying the isomorphism
theorem (Theorem III.1) between $\Sigma(N)$ and
$\Sigma(N^{\prime}N^{\prime\prime})$ to $\rho_{N}$ and $\rho_{N}^{\prime}$
respectively. Their fidelity is equal to the fidelity between $\rho_{N}$ and
$\rho_{N}^{\prime}$.

$F(\rho_{N},\rho_{N}^{\prime})=F(\rho_{N^{\prime}N^{\prime\prime}},\rho_{N^{\prime}N^{\prime\prime}}^{\prime}).$
(40)

Because fidelity is non-decreasing under partial trace, we have

$\displaystyle F(\rho_{N},\rho_{N}^{\prime})$
$\displaystyle=F(\rho_{N^{\prime}N^{\prime\prime}},\rho_{N^{\prime}N^{\prime\prime}}^{\prime})$
(41) $\displaystyle\leq
F(\rho_{NN^{\prime\prime}},\rho_{NN^{\prime\prime}}^{\prime}).$

Because both $\rho_{NN^{\prime\prime}}$ and
$\rho_{NN^{\prime\prime}}^{\prime}$ are extreme points, they factorize over
$N$ and $N^{\prime\prime}$.

$F(\rho_{NN^{\prime\prime}},\rho_{NN^{\prime\prime}}^{\prime})=F(\rho_{N},\rho_{N}^{\prime})F(\rho_{N^{\prime\prime}},\rho_{N^{\prime\prime}}^{\prime}).$
(42)

In particular, by the isomorphism theorem, we get

$F(\rho_{N},\rho_{N}^{\prime})\leq F(\rho_{N},\rho_{N}^{\prime})^{2}.$ (43)

Therefore $F(\rho_{N},\rho_{N}^{\prime})$ is either $0$ or $1$.

Therefore we can characterize $\Sigma(N)$ as

$\Sigma(N)=\left\\{\bigoplus_{n}p_{n}\rho^{n}_{N}:\sum_{n}p_{n}=1,p_{n}\geq
0\right\\},$ (44)

where different extreme points $\\{\rho^{n}_{N}\\}$ are supported on
orthogonal subspaces. The same argument applies to $\Sigma(U)$:

$\Sigma(N)=\left\\{\bigoplus_{u}p_{u}\rho^{u}_{U}:\sum_{u}p_{u}=1,p_{u}\geq
0\right\\}.$ (45)

We shall formally denote these superselection sectors as

$\displaystyle{\mathcal{C}}_{N}$ $\displaystyle=\\{1,n,\cdots\\},$ (46)
$\displaystyle{\mathcal{C}}_{U}$ $\displaystyle=\\{1,u,\cdots\\}.$

Like in the bulk, we will define the quantum dimensions of the parton sectors
as

$\boxed{\begin{aligned}
d_{n}&:=\exp\left(\frac{S(\rho_{N}^{n})-S(\rho_{N}^{1})}{2}\right),\\\
d_{u}&:=\exp\left(\frac{S(\rho_{U}^{u})-S(\rho_{U}^{1})}{2}\right),\end{aligned}}$
(47)

where $N$ and $U$ are $N$\- and $U$-shaped regions respectively, and the “1”
in the superscript means that the density matrix is obtained by tracing out
all but the region in the subscript over the global reference state $\sigma$.

Readers may wonder whether the parton sectors we identify can be understood
using the “folding technique” [11, 12, 15]. The folding technique turns a
system $P$ and $Q$ separated by a domain wall into a system $P\otimes\bar{Q}$
(upper half-plane) and the vacuum (lower half-plane) separated by a gapped
boundary. (Here $\bar{Q}$ is $Q$ reflected along the domain wall.)
Historically, the folding technique was useful in understanding the
superselection sectors of point excitations on the gapped domain wall (i.e.,
the $O$-type sectors we shall discuss below in Sec. IV.1) by identifying them
to with the superselection sectors of point excitations on the gapped boundary
of the folded non-chiral system. However, we are not aware of any way to
understand the parton sectors by directly looking at the folded system. The
intuitive reason behind this is that the $N$-shaped subsystem cannot be
obtained by unfolding any subsystem.

Let us comment on the physical interpretation of the parton sector. We
emphasize that a parton sector generally does not specify a localized
excitation. Specifically, if the reference state is a ground state of some
local Hamiltonian, its low-energy excitation is not always uniquely determined
by a single parton sector. Often extra information is required to specify such
an excitation, as we explain in Section IV.

Instead, it is better to view them as “quantum numbers” that partially specify
excitations. Because the extreme points of $\Sigma(N)$ (as well as
$\Sigma(U)$) are orthogonal to each other, there is a set of projectors that
project out a unique sector. In principle, one should be able to measure these
projectors, thereby obtaining these “quantum numbers.”

##  IV Composite sectors

In this section, we will study the _composite superselection sectors_. These
are superselection sectors localized on the domain wall that can carry
multiple parton labels:

${\mathcal{C}}_{\textrm{composite}}=\bigcup_{\begin{subarray}{c}n_{1},n_{2},\cdots\in\mathcal{C}_{N}\\\
u_{1},u_{2},\cdots\in\mathcal{C}_{U}\end{subarray}}{\mathcal{C}}_{\textrm{composite}}^{[n_{1},u_{1},n_{2},u_{2},\cdots]}.$
(48)

As before, a superselection sector is associated with some region. This region
may contain $N$\- and $U$-shaped regions as its subsystems, which contain
partial information about the composite sectors. Specifically, recall that
there are projectors localized on $N$\- and $U$-shaped subsystems that can
measure the $N$\- and $U$-type superselection sectors. One can measure those
projectors to determine the parton labels.

There can be multiple composite sectors that carry the same parton labels. In
other words, the collection of parton sectors do not uniquely specify a
composite sector. This is actually not a strange phenomenon. If the domain
wall is trivial, the parton sectors are also trivial. Because the underlying
subsystem is topologically a disk, its information convex set has a unique
element [41]. However, we can instead consider an annulus, which clearly has
_N_ \- and _U_ -shaped regions as its subsystems. The information convex sets
of these subsystems are trivial, but the information convex set of an annulus
is not; see Section II.1 for the discussion. Therefore, even after specifying
the parton sectors, there is a leftover degree of freedom that remains
unspecified.

This “composition rule” of superselection sectors is somewhat mundane in the
bulk. However, in the presence of a gapped domain wall, we can have a much
richer structure. In Section IV.1, we shall study a composite sector that can
be identified with the point-like excitations studied in Refs. [12, 15].
However, we shall see in Section IV.2 and IV.3 that there are other types of
composite sectors as well. They are new to the best of our knowledge. While we
do not believe that we have an exhaustive list of composite sectors, we expect
to be able to characterize any reasonable composite superselection sectors by
using the general observations summarized in Section IV.4.

Before we delve into these details, let us make a remark on our convention. We
will frequently use the following short-hand notation for the merged state:

$\tau^{a\Join b}:=\rho^{a}\Join\lambda^{b},$ (49)

where $\rho^{a}$ and $\lambda^{b}$ are associated with superselection sectors
$a$ and $b$. Both $\rho^{a}$ and $\lambda^{b}$ are elements of some
information convex sets. The choice of these sets will depend on the context.

###  IV.1 $O$-type sectors

The first of these composite sectors is the $O$-type superselection sector.
These sectors correspond to the extreme points of an annulus on the gapped
domain wall; see Fig. 16. These extreme points are orthogonal to each other
because the exposition in Section II.1 applies here as well. Physically, these
sectors are the superselection sectors of the point-like excitations on the
gapped domain wall, studied in Refs. [12, 15].

Figure 16: (Left) A subsystem associated with the $O$-type superselection
sector. (Right) Upon tracing out the annulus, one can obtain a _N_ -shaped and
_U_ -shaped subsystems, which must lie on the information convex set of
$\Sigma(N)$ and $\Sigma(U)$, respectively.

We shall denote the set of $O$-type superselection sectors as

$\mathcal{C}_{O}=\\{1,\alpha,\beta,\cdots\\},$ (50)

where we use Greek letters starting from $\alpha$ to denote these sectors.

Let us explain in what sense the $O$-type sectors are composite. Consider an
extreme point on the annulus that represents the sector
$\alpha\in{\mathcal{C}}_{O}$. Upon tracing out a disk-like region on $Q$, we
get a density matrix over an $N$-shaped subsystem. Similarly, by tracing out a
disk-like region on $P$, we obtain a density matrix over a _U_ -shaped
subsystem. Moreover, these density matrices are elements of $\Sigma(N)$ and
$\Sigma(U)$, respectively.

The elements we obtain this way are not just any element; they are extreme
points. To see why, consider an extreme point on the annulus that represents a
$O$-type sector. As we discussed in Section II.1, if we extend an annulus to a
thicker annulus and trace out the middle of the thicker annulus to obtain two
annuli, these two annuli are decoupled. Importantly, subsystems of the two
annuli must be also decoupled.

Figure 17: (Left) We begin with an extreme point corresponding to
$\alpha\in\mathcal{C}_{O}$. (Right) By using the isomorphism theorem, the
density matrix is extended to a larger annulus. Upon tracing out the middle of
this larger annulus, we obtain two annuli. By the factorization of the extreme
points, the state over these two annuli is a product state. Consequently, the
_N_ -shaped subsystems in these two annuli must also be in a product state.

In particular, the state over the two $N$-shaped regions in these annuli is
factorized; see Fig. 17. This is the key reason why the state is an extreme
point. Let the density matrix in one of these two $N$-shaped subsystems (say
$N$) to be

$\rho_{N}=\bigoplus_{n\in\mathcal{C}_{N}}p_{n}\rho^{n}_{N}.$ (51)

By the isomorphism theorem, the density matrix over $NN^{\prime}$ is

$\rho_{NN^{\prime}}=\bigoplus_{n\in\mathcal{C}_{N}}p_{n}\rho^{n}_{N}\otimes\rho^{n}_{N^{\prime}},$
(52)

where $N^{\prime}$ is the other $N$-shaped subsystem in Fig. 17 separated from
$N$. Therefore, the mutual information between the two regions is

$I(N:N^{\prime})=H(\\{p_{n}\\}).$ (53)

This has to be zero because the underlying state is a product state. The only
possibility is that $p_{n}$ must be equal to $1$ for some $n$ and $0$ for
other elements in $\mathcal{C}_{N}$. Therefore, the reduced density matrix
over $N$ is an extreme point of $\Sigma(N)$. Similarly, the reduced density
matrix over $U$ is an extreme point of $\Sigma(U)$.

Therefore, $\mathcal{C}_{O}$ must be a disjoint union of the following form:

$\mathcal{C}_{O}=\bigcup_{\begin{subarray}{c}n\in\mathcal{C}_{N}\\\
u\in\mathcal{C}_{U}\end{subarray}}\mathcal{C}_{O}^{[n,u]},$ (54)

where $\mathcal{C}_{O}^{[n,u]}\subset\mathcal{C}_{O}$ is a subset in which the
$N$\- and $U$-type superselection sectors are fixed to $n$ and $u$.

The quantum dimension of this sector, which we define as

$\boxed{d_{\alpha}:=\exp\left(\frac{S(\rho^{\alpha}_{O})-S(\rho^{1}_{O})}{2}\right),}$
(55)

has a nontrivial relation with the quantum dimension of the parton sectors.
Specifically,

$\boxed{d_{n}^{2}d_{u}^{2}=\frac{\sum_{\alpha\in\mathcal{C}_{O}^{[n,u]}}d_{\alpha}^{2}}{\sum_{\alpha\in\mathcal{C}_{O}^{[1,1]}}d_{\alpha}^{2}}.}$
(56)

Figure 18: Merging a pair of parton sectors to obtain an $O$-type composite
sector. The subsystem $N=ABC$ carries $n\in{\mathcal{C}}_{N}$ and the
subsystem $U=BCD$ carries $u\in{\mathcal{C}}_{U}$. In the main text, $O$ is
defined to be $ABCD$ in this figure.

To derive this relation, we use the merging technique used in Section II.2.
Specifically, we merge extreme points of $\Sigma(N)$ and $\Sigma(U)$ to obtain
an element in $\Sigma(O)$, where $O$ is an annulus on the domain wall; see
Fig. 18. Without loss of generality, let us refer to these extreme points as
$\rho^{n}_{N}$ and $\rho^{u}_{U}$. For the merged state $\tau^{n\Join
u}_{O}:=\rho^{n}_{N}\Join\rho_{U}^{u}$, its entropy is equal to

$S(\tau^{n\Join u}_{O})=\ln(d^{2}_{n}d^{2}_{u})+(S_{N}+S_{U}-S_{N\cap
U})_{\sigma}.$ (57)

On the other hand, we can directly obtain the maximum entropy consistent with
the given extreme points in $\Sigma(N)$ and $\Sigma(U)$:

$\displaystyle S(\tau^{n\Join u}_{O})$
$\displaystyle=\max_{\\{p_{\alpha}\\}}\left(H(\\{p_{\alpha}\\})+\sum_{\alpha\in\mathcal{C}_{O}^{[n,u]}}p_{\alpha}\ln
d^{2}_{\alpha}\right)+S_{O}(\sigma)$ (58)
$\displaystyle=\ln\left(\sum_{\alpha\in\mathcal{C}_{O}^{[n,u]}}d_{\alpha}^{2}\right)+S(\sigma_{O}).$

Let $\tau^{1\Join 1}_{O}$ be the state merged from extreme points
$1\in\mathcal{C}_{N}$ and $1\in\mathcal{C}_{U}$. We get

$\displaystyle S(\tau^{n\Join u}_{O})-S(\tau^{1\Join 1}_{O})$
$\displaystyle=\ln\left(d_{n}^{2}d_{u}^{2}\right)$ (59)
$\displaystyle=\ln\left(\frac{\sum_{\alpha\in\mathcal{C}_{O}^{[n,u]}}d_{\alpha}^{2}}{\sum_{\alpha\in\mathcal{C}_{O}^{[1,1]}}d_{\alpha}^{2}}\right),$

which leads to Eq. (56).

###  IV.2 Snake sectors

While the $O$-type sector has appeared in the literature already, there are
other composite sectors that are new to the best of our knowledge. One such
example is the _snake sector_ , or alternatively, a _S_ -type sector. This is
a superselection sector associated with the “snake”-shaped regions, e.g., $S$
and $S^{\prime}$ in Fig. 19. The information convex sets of these subsystems
are isomorphic to a simplex with a finite number of orthogonal extreme points.
These snake sectors are again composite sectors of $N$-type and $U$-type
sectors, and therefore many of the discussions about ${\mathcal{C}}_{O}$ in
Section IV.1 apply here as well.

Figure 19: Snake-shaped subsystems.

Let $S$ be the simplest snake-shaped region in Fig. 19. The set of snake
sectors is a disjoint union of the following form.

${\mathcal{C}}_{S}=\bigcup_{\begin{subarray}{c}n\in\mathcal{C}_{N}\\\
u\in\mathcal{C}_{U}\end{subarray}}{\mathcal{C}}_{S}^{[n,u]},$ (60)

where the extreme points associated with ${\mathcal{C}}_{S}^{[n,u]}$ carry
$n\in\mathcal{C}_{N}$ and $u\in\mathcal{C}_{U}$.

Also, we can define the quantum dimensions as follows

$\boxed{d_{s}:=\exp\left(\frac{S(\rho_{S}^{s})-S(\rho^{1}_{S})}{2}\right),}$
(61)

where $\rho_{S}^{s}$ is an extreme point of $\Sigma(S)$.

There is a nontrivial identity between $\\{d_{s}\\}$ and the quantum dimension
of the parton sectors:

$\boxed{d_{n}^{2}d_{u}^{2}=\sum_{s\in{\mathcal{C}}_{S}^{[n,u]}}d_{s}^{2}.}$
(62)

The proof is essentially the same as the proof of Eq. (56), the only
difference being that $\sum_{s\in{\mathcal{C}}_{S}^{[1,1]}}d_{s}^{2}=1$.

This last fact follows from the fact that
$|{\mathcal{C}}_{S}^{[n,1]}|=|{\mathcal{C}}_{S}^{[1,u]}|=1$. That
${\mathcal{C}}_{S}^{[n,1]}$ has a unique element follows from the observation
that an element of $\Sigma(S)$ that carries parton sector $u=1$ is the reduced
density matrix of a certain element in $\Sigma(SD)$, where $SD$ is an
$N$-shaped subsystem; see Fig. 20. Here, $D$ is a disk on the domain wall,
which fills the “slot” in Fig. 20 and turns the $U$-shaped arc into a disk on
the domain wall. (In more detail, we need to divide $S$ into $ABC$ in an
obvious way, in which $BCD$ is another disk on the domain wall. Then, we use
the merging theorem. Note that the merging is possible because $u=1$.) The
proof of $|{\mathcal{C}}_{S}^{[1,u]}|=1$ is analogous.

Figure 20: For the proof of $|{\mathcal{C}}_{S}^{[n,1]}|=1$. A snake-shaped
subsystem $S$ and an $N$-shaped subsystem $SD$. Here, $D$ is a disk on the
domain wall and it fills a slot.

###  IV.3 $\mathbb{N}$\- and $\mathbb{U}$-type sectors

There are composite sectors that play a crucial role in studying the fusion
space of the aforementioned superselection sectors. These are the
$\mathbb{N}$\- and $\mathbb{U}$-type sectors; see Fig. 21. The underlying
subsystems are annuli on the domain wall which are not path-connected to any
$O$-shaped subsystem. It should be obvious – from the discussion about the
bulk superselection sectors and the parton sectors – that the information
convex set associated with this subsystem is also isomorphic to a simplex
formed by a finite number of mutually orthogonal extreme points. Moreover,
these are composite sectors in a sense that, upon tracing out the appropriate
subsystems, one can obtain two $N$\- and $U$-shaped subsystems. Moreover, the
argument that leads to Eq. (54) also applies here, which implies that
$\mathcal{C}_{\mathbb{N}}$ is a disjoint union of sets labeled by
$n,n^{\prime},u,$ and $u^{\prime}$, where $n,n^{\prime}\in\mathcal{C}_{N}$ and
$u,u^{\prime}\in\mathcal{C}_{U}$. However, we will not use this fact in this
paper.

Figure 21: Subsystem choices for ${\mathcal{C}}_{\mathbb{N}}$ and
${\mathcal{C}}_{\mathbb{U}}$.

We define the quantum dimensions of these sectors as follows:

$\boxed{\begin{aligned}
d_{\mathcal{N}}&=\exp\left(\frac{S(\rho_{\mathbb{N}}^{\mathcal{N}})-S(\rho^{1}_{\mathbb{N}})}{2}\right),\\\
d_{\mathcal{U}}&=\exp\left(\frac{S(\rho^{\mathcal{U}}_{\mathbb{U}})-S(\rho^{1}_{\mathbb{U}})}{2}\right),\end{aligned}}$
(63)

where $\mathcal{N}\in\mathcal{C}_{\mathbb{N}}$ and
$\mathcal{U}\in\mathcal{C}_{\mathbb{U}}$ are the $\mathbb{N}$\- and
$\mathbb{U}$-type superselection sectors. We again use the superscript “1” to
denote the extreme point obtained from the reference state $\sigma$.

There is a natural notion of embedding:

$\displaystyle\eta_{N}$
$\displaystyle:\mathcal{C}_{N}\hookrightarrow\mathcal{C}_{\mathbb{N}}$ (64)
$\displaystyle\eta_{U}$
$\displaystyle:\mathcal{C}_{U}\hookrightarrow\mathcal{C}_{\mathbb{U}},$

which is defined by tracing out the interior of the _N_ -shaped (or _U_
-shaped) subsystem; see Fig. 22.

Figure 22: Given an extreme point of $\Sigma(N)$, we can obtain an extreme
point of $\Sigma(\mathbb{N})$ by tracing out the middle part.

In Eq. (64), we are implicitly asserting that an extreme point of $\Sigma(N)$,
upon traced out the middle part, becomes an extreme point of an information
convex set of a $\mathbb{N}$-shaped region. Below, we briefly sketch the
underlying reason.

Consider an $N$-shaped subsystem $N$, which is partitioned into $N^{\prime}$
and
$\mathbb{N}=\mathbb{N}_{\text{in}}\mathbb{N}_{\text{middle}}\mathbb{N}_{\text{out}}$,
where $\mathbb{N}_{\text{in}},\mathbb{N}_{\text{middle}},$ and
$\mathbb{N}_{\text{out}}$ are non-overlapping $\mathbb{N}$-shaped regions.
Specifically, we have the following sequence of $N$-shaped regions:

$\displaystyle N^{\prime}$ $\displaystyle\subset
N^{\prime}\mathbb{N}_{\text{in}}$ (65) $\displaystyle\subset
N^{\prime}\mathbb{N}_{\text{in}}\mathbb{N}_{\text{middle}}$
$\displaystyle\subset
N^{\prime}\mathbb{N}_{\text{in}}\mathbb{N}_{\text{middle}}\mathbb{N}_{\text{out}},$

and the following sequence of $\mathbb{N}$-shaped regions:

$\displaystyle\mathbb{N}_{\text{in}}$
$\displaystyle\subset\mathbb{N}_{\text{in}}\mathbb{N}_{\text{middle}}$ (66)
$\displaystyle\subset\mathbb{N}_{\text{in}}\mathbb{N}_{\text{middle}}\mathbb{N}_{\text{out}}$
$\displaystyle=\mathbb{N}.$

Let $\rho^{\langle e\rangle}_{N}$ be an extreme point of $\Sigma(N)$. By the
factorization property of the extreme point, we have

$I(N^{\prime}\mathbb{N}_{\text{in}}:\mathbb{N}_{\text{out}})_{\rho^{\langle
e\rangle}}=0.$ (67)

By the monotonicity of the mutual information, we get

$I(\mathbb{N}_{\text{in}}:\mathbb{N}_{\text{out}})_{\rho^{\langle
e\rangle}}=0.$ (68)

This is possible only if the reduced density matrix of $\rho^{\langle
e\rangle}_{N}$ over $\mathbb{N}$ is an extreme point.

Moreover, using the factorization property, we can derive:

$\boxed{\begin{aligned} d_{\eta_{N}(n)}&=d_{n}^{2},\\\
d_{\eta_{U}(u)}&=d_{u}^{2}.\end{aligned}}$ (69)

To see why, without loss of generality, consider the subsystems described in
Fig. 23. Here, both $N$ and $N^{\prime}$ are $N$-shaped subsystems.
Importantly, $N^{\prime}\setminus N$ is a $\mathbb{N}$-shaped subsystem. Using
the factorization property of the extreme points, we get:

$\displaystyle(S_{N}+S_{N^{\prime}}-S_{N^{\prime}\setminus
N})_{\rho_{N^{\prime}}^{n}}$ $\displaystyle=0,$ (70)
$\displaystyle(S_{N}+S_{N^{\prime}}-S_{N^{\prime}\setminus
N})_{\rho_{N^{\prime}}^{1}}$ $\displaystyle=0,$

where $\rho_{N^{\prime}}^{n}\in\Sigma(N^{\prime})$ is an extreme point
associated with the sector $n\in\mathcal{C}_{N}$. From these equations and the
definition of the quantum dimension, Eq. (69) follows.

Figure 23: Subsystems involved in the proof of Eq. (69)

Later in Section VI.3, we shall see that there is a one-to-one map between the
set of $\mathbb{N}$-type sectors and the set of $\mathbb{U}$-type sectors. We
denote this fact as follows:

$\varphi:{\mathcal{C}}_{\mathbb{N}}\to{\mathcal{C}}_{\mathbb{U}},$ (71)

where $\varphi$ is a bijection. Later, we will show that this map preserves
the quantum dimensions, namely

$d_{\mathcal{N}}=d_{\varphi({\mathcal{N}})}.$ (72)

This would be certainly true if the domain wall is trivial since both
subsystems can be smoothly deformed to an annulus. However, because
$\mathbb{N}$ and $\mathbb{U}$ cannot be smoothly deformed into each other, Eq.
(72) is a nontrivial fact in general.

To summarize, different sets of superselection sectors are related to each
other in the following way:

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While the cardinality of $\mathcal{C}_{N}$ is generally different from that of
$\mathcal{C}_{U}$, those two sets may be indirectly related to each other via
$\mathcal{C}_{\mathbb{N}}$ and $\mathcal{C}_{\mathbb{U}}$.

###  IV.4 Generalities

In this section, we introduce general facts about superselection sectors.
First, we explain an all-encompassing recipe to show that an information
convex set of a subsystem is isomorphic to a simplex with orthogonal extreme
points. The following discussion will assume the continuum limit, in which the
familiar notion of topology is well-defined.

The following definition will be important.

######  Definition IV.1 (Sectorizable Region).

A subsystem $R$ is _sectorizable_ if there is a region $\widehat{R}$ such
that:

  1. 1.

$\widehat{R}$ contains disjoint regions $R$ and $R^{\prime\prime}$ and

  2. 2.

both $R$ and $R^{\prime\prime}$ can be connected to $\widehat{R}$ by a path,
where the path is a sequence of extensions.

This definition is important because the information convex set of any
sectorizable region is a simplex with orthogonal extreme points.

######  Lemma IV.1.

Let $R$ be a sectorizable region. Then

$\Sigma(R)=\left\\{\bigoplus_{I}p_{I}\rho^{I}_{R}:\sum_{I}p_{I}=1,p_{I}\geq
0\right\\},$ (74)

where $\\{\rho^{I}_{R}\\}$ is a set of density matrices that are mutually
orthogonal to each other.

The proof of this lemma is straightforward, because it is a simple
generalization of what we have been discussing so far. For completeness, we
sketch the proof below. First, extend $R$ to $\widehat{R}$ using the
isomorphism theorem, and then trace out $\widehat{R}\setminus(R\cup
R^{\prime\prime})$. We can obtain the following inequality:

$F(\rho_{R},\rho^{\prime}_{R})\leq
F(\rho_{RR^{\prime\prime}},\rho^{\prime}_{RR^{\prime\prime}}),$ (75)

where $\rho_{R}$ and $\rho^{\prime}_{R}$ are two extreme points of
$\Sigma(R)$, and $\rho_{RR^{\prime\prime}}$ and
$\rho^{\prime}_{RR^{\prime\prime}}$ are obtained from the former density
matrices by an extension to $\widehat{R}$ and a partial trace over
$\widehat{R}\setminus(R\cup R^{\prime\prime})$. Note that Eq. (9) and Eq. (41)
are special cases of Eq. (75). By the factorization property, we get
$F(\rho_{R},\rho^{\prime}_{R})\leq F(\rho_{R},\rho^{\prime}_{R})^{2}$.
Therefore, $F(\rho_{R},\rho^{\prime}_{R})$ must be either $0$ or $1$. This
proves Lemma IV.1.

Second, for two sectorizable subsystems, the set of extreme points obeys the
“product rule.”

######  Lemma IV.2 (Product rule).

Let $\Omega_{1}$ and $\Omega_{2}$ be sectorizable subsystems which are
disjoint from each other. Then the subsystem $\Omega_{1}\Omega_{2}$ is another
sectorizable subsystem and

$\boxed{\mathcal{C}_{\Omega_{1}\Omega_{2}}\cong\mathcal{C}_{\Omega_{1}}\times\mathcal{C}_{\Omega_{2}},}$
(76)

where $\mathcal{C}_{\Omega_{1}}$, $\mathcal{C}_{\Omega_{2}}$ and
$\mathcal{C}_{\Omega_{1}\Omega_{2}}$ are the set of superselection sectors
associated with sectorizable subsystems $\Omega_{1}$, $\Omega_{1}$ and
$\Omega_{1}\Omega_{2}$ respectively. Moreover, every extreme point of
$\Sigma(\Omega_{1}\Omega_{2})$ is a tensor product of extreme points of
$\Sigma(\Omega_{1})$ and $\Sigma(\Omega_{2})$.

###### Proof.

First, the fact that $\Omega_{1}\Omega_{2}$ is again a sectorizable subsystem
is easy to verify. The two conditions in Definition IV.1 are verified by
letting
$\widehat{\Omega_{1}\Omega_{2}}=\widehat{\Omega}_{1}\widehat{\Omega}_{2}$ and
$(\Omega_{1}\Omega_{2})^{\prime\prime}=\Omega^{\prime\prime}_{1}\Omega^{\prime\prime}_{2}$.

Second, note that any extreme point of $\Sigma(\Omega_{1}\Omega_{2})$, once
restricted to either $\Omega_{1}$ or $\Omega_{2}$, becomes an extreme point of
$\Sigma(\Omega_{1})$ and $\Sigma(\Omega_{2})$ respectively. This is because,
once we extend an extreme point in $\Sigma(\Omega_{1}\Omega_{2})$ to an
element of $\Sigma(\widehat{\Omega_{1}\Omega_{2}})$ by using the isomorphism
theorem and tracing out the appropriate subsystems, the mutual information of
this state between $\Omega_{1}\Omega_{2}$ and
$\Omega^{\prime\prime}_{1}\Omega^{\prime\prime}_{2}$ is zero. This fact
follows from the factorization property of the extreme points. Therefore, the
state must be factorized over $\Omega_{1}$ and $\Omega^{\prime\prime}_{1}$.
The same factorization holds over $\Omega_{2}$ and
$\Omega^{\prime\prime}_{2}$. Such factorization is possible only if the
reduced density matrix of any extreme point of $\Sigma(\Omega_{1}\Omega_{2})$
over $\Omega_{1}$ and $\Omega_{2}$ are extreme points.

Now, we can use the factorization property of the extreme point of
$\Sigma(\Omega_{1})$ as follows. Note that the extreme points of
$\Sigma(\Omega_{1}\Omega_{2})$, restricted to
$\Omega_{1}\setminus\partial\Omega_{1}$, where $\partial\Omega_{1}$ is the
thickened boundary of $\Omega_{1}$, must be factorized with anything that is
outside of $\Omega_{1}$. Therefore, these extreme points must be factorized
between $\Omega_{1}\setminus\partial\Omega_{1}$ and $\Omega_{2}$. Using the
isomorphism theorem, we conclude that the extreme points over
$\Sigma(\Omega_{1}\Omega_{2})$ must be factorized over $\Omega_{1}$ and
$\Omega_{2}$. ∎

###  IV.5 Summary

We have so far studied the parton sectors and its (derivative) composite
sectors. Below, we summarize our key results for the readers’ convenience.
First, we have summarized these superselection sectors in Fig. 24. Note that
the set of composite sectors can be decomposed further into a disjoint union
of sets, each of which is labeled by the parton sectors. For instance, $O$\-
and $S$-type sectors are labeled by an $N$\- and a $U$-type sector. On the
other hand, $\mathbb{N}$\- and $\mathbb{U}$-type sectors are labeled by two
$N$\- and two $U$-type sectors.

Figure 24: A list of subsystem topologies and the corresponding superselection
sector labels.

The quantum dimensions of these sectors are all defined in the same way, in
terms of the entanglement entropy of the extreme point associated with the
superselection sector.

We derived the following identities:

$\displaystyle d_{n}^{2}d_{u}^{2}$
$\displaystyle=\frac{\sum_{\alpha\in\mathcal{C}_{O}^{[n,u]}}d_{\alpha}^{2}}{\sum_{\alpha\in\mathcal{C}_{O}^{[1,1]}}d_{\alpha}^{2}}$
(77) $\displaystyle=\sum_{s\in\mathcal{C}_{S}^{[n,u]}}d_{s}^{2}.$

Moreover, we studied the maps $\eta_{N}$ (as well as $\eta_{U}$) and $\varphi$
which has the following properties. The maps $\eta_{N}$ and $\eta_{U}$ are
embeddings from $\mathcal{C}_{N}$ to $\mathcal{C}_{\mathbb{N}}$ and from
$\mathcal{C}_{U}$ to $\mathcal{C}_{\mathbb{U}}$ respectively, such that

$\displaystyle d_{\eta_{N}(n)}$ $\displaystyle=d_{n}^{2}$ (78) $\displaystyle
d_{\eta_{U}(u)}$ $\displaystyle=d_{u}^{2}.$

$\varphi$ is a bijection between $\mathcal{C}_{\mathbb{N}}$ and
$\mathcal{C}_{\mathbb{U}}$ such that

$\displaystyle d_{\varphi(\mathcal{N})}$ $\displaystyle=d_{\mathcal{N}}$ (79)
$\displaystyle d_{\varphi^{-1}(\mathcal{U})}$ $\displaystyle=d_{\mathcal{U}}.$

Finally, we mention that anti-sectors are well defined for
${\mathcal{C}}_{N}$, ${\mathcal{C}}_{U}$ and ${\mathcal{C}}_{O}$. The quantum
dimension of every sector is equal to that of its anti-sector.

$\displaystyle d_{\bar{n}}$
$\displaystyle=d_{n},\quad\bar{\bar{n}}=n,\quad\forall n\in{\mathcal{C}}_{N},$
(80) $\displaystyle d_{\bar{u}}$
$\displaystyle=d_{u},\quad\bar{\bar{u}}=u,\quad\forall u\in{\mathcal{C}}_{U},$
$\displaystyle d_{\bar{\alpha}}$
$\displaystyle=d_{\alpha},\quad\bar{\bar{\alpha}}=\alpha,\quad\forall\alpha\in{\mathcal{C}}_{O},$

where we have used a “bar” over a sector label to denote its anti-sector. We
will prove these identities later.

##  V Fusion spaces

In this section, we define and study the fusion spaces of the superselection
sectors introduced in Section III and IV. To understand our definition of
fusion space, it will be instructive to recall the definition of fusion space
in the theory of anyon. In the anyon theory, a fusion space is a _Hilbert
space_. Specifically, when a sector $a$ and $b$ fuse into another sector $c$,
there is a leftover degree of freedom, described by a state space of some
Hilbert space. This underlying Hilbert space _is_ the fusion space. In this
paper, we will adhere strictly to this rule and ascribe a fusion space to any
space isomorphic to a state space of some Hilbert space.

Without loss of generality, consider an information convex set
$\Sigma(\Omega)$ associated with a subsystem $\Omega$. To characterize
$\Sigma(\Omega)$, it will be helpful to study the information convex set of
its thickened boundary $\partial\Omega$; see Fig. 25. Because $\partial\Omega$
is a sectorizable region, $\Sigma(\partial\Omega)$ is a simplex with
orthogonal extreme points; see Definition IV.1 and Lemma IV.1. Moreover, the
factorization property of the extreme points implies that every extreme point
of $\Sigma(\Omega)$ reduces to an extreme point of $\Sigma(\partial\Omega)$.
Therefore, elements of $\Sigma(\Omega)$ can be divided further in terms of the
extreme points of $\Sigma(\partial\Omega)$:

$\boxed{\Sigma(\Omega)=\left\\{\bigoplus_{I\in\mathcal{C}_{\partial\Omega}}p_{I}\rho_{\Omega}^{I}:\sum_{I}p_{I}=1,p_{I}\geq
0\right\\},}$ (81)

where $\mathcal{C}_{\partial\Omega}$ is a set of superselection sectors
associated with the extreme points of $\Sigma(\partial\Omega)$ and
$\rho_{\Omega}^{I}$ is an element of $\Sigma(\Omega)$ that, upon restricting
to $\partial\Omega$, becomes an extreme point
$\rho^{I}_{\partial\Omega}\in\Sigma(\partial\Omega)$. The set of
$\rho_{\Omega}^{I}$ with a fixed $I$ forms a convex subset of
$\Sigma(\Omega)$, which we shall denote as $\Sigma_{I}(\Omega)$.

Figure 25: An example of thickened boundaries. Here, $\Omega$ is a
sufficiently thick and smooth two-hole disk. The thickened boundary
$\partial\Omega$ is a sectorizable subsystem. Furthermore, $\partial\Omega$ is
a union of three disjoint annuli, each of which is a sectorizable subsystem.

It remains to characterize $\mathcal{C}_{\partial\Omega}$ and
$\Sigma_{I}(\Omega)$. For $\mathcal{C}_{\partial\Omega}$, we can use the
general strategy explained in Section IV. For instance, if $\partial\Omega$
has multiple connected components, the set of superselection sectors
$\mathcal{C}_{\partial\Omega}$ obeys the product rule (Lemma IV.2). For
example, in Fig. 25, $\partial\Omega$ is the union of three disjoint annuli.
In this case, $I\in\mathcal{C}_{\partial\Omega}$ is a triple of superselection
sectors of the three annuli, i.e., $\\{I\\}\cong\\{(a,b,c)\\}$ where $a,b,$
and $c$ belong to the set of superselection sectors associated with an
annulus.

For $\Sigma_{I}(\Omega)$, we can prove the following fact:

$\boxed{\Sigma_{I}(\Omega)\cong\mathcal{S}(\mathbb{V}_{I}),}$ (82)

where $\mathcal{S}(\mathbb{V}_{I})$ is the state space of a finite-dimensional
Hilbert space $\mathbb{V}_{I}$, which generally depends on the choice of $I$.
Combined with Eq. (81), this implies that one can assign a fusion space to any
sufficiently smooth and thick subsystem. The dimension of the Hilbert space,
$N_{I}=\dim\mathbb{V}_{I}$, is a non-negative integer known as the _fusion
multiplicity_.

As a sanity check, we can see that the fusion space defined in Eq. (82)
produces sensible results in known setups. In Fig. 25, $N_{I}$ is simply
$N_{ab}^{c}$, the multiplicity for the fusion of anyons $a$ and $b$ into an
anyon $c$. In that context, Eq. (82) was derived in Theorem 4.5 of Ref. [41].

The proof of Eq. (82) for general subsystems can be done similarly as the
proof of Theorem 4.5 of Ref. [41]. Moreover, we provide an alternative proof
which is simpler; see the _Hilbert space theorem_ (Theorem D.5) in Appendix D.

###  V.1 Fusion on gapped domain walls

In this section, we list a few examples of fusion spaces on gapped domain
walls. A (partial) list of relevant subsystems is described in Fig. 26. For
example, we can consider a two-hole disk on the domain wall, both of the holes
sitting on the domain wall; see the first figure in Fig. 26. The thickened
boundary of that region is a union of three disjoint annuli on the domain
wall, with the extreme points labeled by $\mathcal{C}_{O}$. Hence, the fusion
space of the two-hole disks on the domain wall can be labeled by a triple
$(\alpha,\beta,\gamma)$, where $\alpha,\beta,\gamma\in\mathcal{C}_{O}$. We may
formally denote the fusion space as $\mathbb{V}_{\alpha\beta}^{\gamma}$ and
the fusion multiplicity as $N_{\alpha\beta}^{\gamma}$.

The other examples listed in Fig. 26 can be understood in a similar way. While
we have discussed our notation of superselection sectors in Section IV, we
restate it below for the readers’ convenience:

$\displaystyle\alpha,\beta,\gamma$ $\displaystyle\in{\mathcal{C}}_{O},$ (83)
$\displaystyle a$ $\displaystyle\in{\mathcal{C}}_{P},$ $\displaystyle x$
$\displaystyle\in{\mathcal{C}}_{Q},$ $\displaystyle\mathcal{U}$
$\displaystyle\in{\mathcal{C}}_{\mathbb{U}},$ $\displaystyle\mathcal{N}$
$\displaystyle\in{\mathcal{C}}_{\mathbb{N}},$

where ${\mathcal{C}}_{P}$ and ${\mathcal{C}}_{Q}$ denote the set of anyon
labels in phases $P$ and $Q$ respectively.

Figure 26: Subsystems that are relevant to the study of fusion spaces on
gapped domain walls. Also shown are the labels for the fusion multiplicities.

In Section VI, we will in fact derive the fusion _rules_ that these fusion
spaces must obey and derive intricate constraints on the fusion
multiplicities. Let us briefly mention these results, deferring the details to
Section VI. We can formally express the following fusion processes:

$\boxed{\begin{aligned}
\alpha\times\beta&=\sum_{\gamma}N_{\alpha\beta}^{\gamma}\gamma\\\
a&=\sum_{\alpha}N_{a}^{\alpha}\alpha\\\
x&=\sum_{\alpha}N_{x}^{\alpha}\alpha\\\ a\times
x&=\sum_{\alpha}N_{ax}^{\alpha}\alpha.\end{aligned}}$ (84)

Here, for any choice of sectors on the left-hand side there exists at least
one fusion result on the right-hand side.

However, the same cannot be said about the fusion processes involving
multiplicities $N^{\alpha}_{\mathcal{U}}$ and
$N^{\alpha}_{\mathcal{N}\mathcal{U}}$. For example, for a particular choice of
$\mathcal{N}$ and $\mathcal{U}$, there may not be an $\alpha$ such that
$N_{\mathcal{N}\mathcal{U}}^{\alpha}\neq 0$. We will revisit this issue in
Section VI.

###  V.2 Quasi-fusion of parton sectors

We have seen examples of fusion spaces in Fig. 26. They involve composite
sectors on the domain wall as well as the superselection sectors of anyons in
the 2D bulk $P$ and $Q$. One may wonder whether there is a similar
generalization of fusion spaces to parton sectors. What happens if we “fuse” a
pair of parton sectors (say $n,n^{\prime}\in{\mathcal{C}}_{N}$) together? Can
they fuse into another parton sector $n^{\prime\prime}\in{\mathcal{C}}_{N}$?
Can we define a fusion space ($\mathbb{V}_{nn^{\prime}}^{n^{\prime\prime}}$)
associated with a triple of parton sectors?

Surprisingly, the answer to the last question is “no.” To see why, let us
first formalize the problem. Consider a $M$-shaped subsystem ($M$) shown in
Fig. 27. There are three $N$-shaped subsystems, associated with superselection
sectors $n,n^{\prime},$ and $n^{\prime\prime}$ without loss of generality. The
question is whether the state space with a fixed choice of $n,n^{\prime},$ and
$n^{\prime\prime}$ is isomorphic to a state space of some Hilbert space.

Figure 27: An $M$-shaped region. Three parton sectors
$n,n^{\prime},n^{\prime\prime}\in{\mathcal{C}}_{N}$ can be detected from the
three $N$-shaped subsystems (darker color) of $M$.

It turns out that this is not the case. What _is_ correct is the fact that
$n,n^{\prime},$ and $n^{\prime\prime}$ partially characterize the extreme
points of $\Sigma(\partial M)$, where $\partial M$ is the thickened boundary
of $M$. However, they do not characterize the extreme points of
$\Sigma(\partial M)$ completely. This is because $\partial M$ is not a union
of three $N-$shaped regions; the $N$-shaped regions associated with
$n,n^{\prime},$ and $n^{\prime\prime}$ are part of $\partial M$ but not all of
it. Therefore, even after specifying $n,n^{\prime},$ and $n^{\prime\prime},$
one may have more than one fusion space, each labeled by an extreme point of
$\Sigma(\partial M)$. We shall refer to this phenomenon as _quasi-fusion_ of
parton sectors.

Figure 28: If $Q$ has trivial anyon content, we can apply topology-changing
operation to the subsystems in $Q$ side without affecting the structure of the
information convex sets. In this case, ${\mathcal{C}}_{U}=\\{1\\}$ and
${\mathcal{C}}_{N}\cong{\mathcal{C}}_{O}$. Moreover, the quasi-fusion rule of
$N$-type parton sectors is identical to the conventional fusion rule of point
excitations on the domain wall.

However, when one side of the bulk phase, say $Q$, has a trivial anyon
content, there is a unique fusion space (which can be labeled as
$\mathbb{V}_{nn^{\prime}}^{n^{\prime\prime}}$) for each choice of
$n,n^{\prime},n^{\prime\prime}\in{\mathcal{C}}_{N}$. In this specific
instance, the conventional fusion rule applies to the parton sectors as well.

Importantly, our statement applies even if the bulk phase with a trivial anyon
content has a nonzero chiral central charge. A nontrivial example is the so-
called $E_{8}$ state [9]. A proof of our claim is presented in Appendix E.2.
The key idea is that trivial anyon content implies a new type of entropic
constraint. This new constraint allows us to prove a strengthening of the
isomorphism theorem in which the underlying subsystems can undergo a topology
change. We sketched this idea in Fig. 28, deferring the details to Appendix
E.2.

##  VI Fusion rules

So far, we have defined a number of different superselection sectors and their
fusion spaces. In this section, we will study their fusion rules.

To put our work into a context, let us recall the fusion rules in the bulk.
Formally, we can write

$a\times b=\sum_{c}N_{ab}^{c}c,$ (85)

where $a,b,$ and $c$ are superselection sectors in the bulk; as we discussed
in Section II, these are associated with the extreme points of the information
convex sets of an annulus. $N_{ab}^{c}$ is the fusion multiplicity of $a$ and
$b$ fusing into $c$.

In Ref. [41], we were able to derive the following facts.

$\displaystyle N_{ab}^{c}$ $\displaystyle=N_{ba}^{c}$ (86) $\displaystyle
N_{1a}^{c}$ $\displaystyle=N_{a1}^{c}=\delta_{a,c}$ $\displaystyle\forall
a,\exists!\,\bar{a}\,\,\text{ s.t. }\,\,N_{ab}^{1}$
$\displaystyle=\delta_{b,\bar{a}}$ $\displaystyle N_{ab}^{c}$
$\displaystyle=N_{\bar{b}\bar{a}}^{\bar{c}}$
$\displaystyle\sum_{i}N_{ab}^{i}N_{ic}^{d}$
$\displaystyle=\sum_{j}N_{aj}^{d}N_{bc}^{j}.$

The first line says the fusion rule is commutative. The second line says the
fusion with the vacuum is trivial. The third line implies that anti-sector is
unique. The fourth line is a symmetry of the fusion multiplicity involving the
replacement of sectors with their anti-sectors. The last line says the
composition of fusion multiplicities is associative.

Furthermore, the quantum dimensions – defined in terms of the entropy
difference Eq. (16) – are constrained by the fusion multiplicities by the
following equation:

$d_{a}d_{b}=\sum_{c}N_{ab}^{c}d_{c}.$ (87)

In fact, this equation completely determines the set of (positive) quantum
dimensions because the fusion multiplicities satisfy Eq. (86). It follows from
this constraint that $d_{1}=1$ and $d_{a}=d_{\bar{a}}\geq 1$ for any $a$.
Furthermore, $d_{a}$ is quantized in the sense that it cannot take an
arbitrary value; for example, it cannot take any value in the interval
$(1,\sqrt{2})$.

The primary purpose of this section is to derive identities on the fusion
multiplicities analogous to these equations. We further derive the
quantization of the quantum dimensions of parton sectors by relating them to
these fusion multiplicities. We shall go through the fusion spaces described
in Fig. 26 and derive their respective fusion rules.

###  VI.1 Fusion rules for $O$-type sectors

As a starter, let us first consider the fusion space formed by two sectors in
$\mathcal{C}_{O}$ fusing into another sector in $\mathcal{C}_{O}$. We shall
refer to these sectors as $\alpha,\beta,$ and $\gamma$. This fusion space can
be defined over the information convex set over the blue subsystem described
in Fig. 29, with the appropriately chosen superselection sectors. Formally, we
can write this as

$\alpha\times\beta=\sum_{\gamma}N_{\alpha\beta}^{\gamma}\,\gamma.$ (88)

Figure 29: The subsystem choice and sector labels relevant to the fusion space
$\mathbb{V}_{\alpha\beta}^{\gamma}$.

The fusion rules of the point-like superselection sectors on the domain wall
are very similar to those of the bulk superselection sectors. We first
summarize the results and provide some basic explanations. A discussion on the
proof will then follow.

The following facts about the fusion multiplicities
$\\{N_{\alpha\beta}^{\gamma}\\}$ are derived from our assumptions.

$\displaystyle N_{1\alpha}^{\gamma}$ $\displaystyle=N_{\alpha
1}^{\gamma}=\delta_{\alpha,\gamma}$ (89)
$\displaystyle\forall\alpha,\exists!\,\bar{\alpha}\,\,\text{ s.t.
}\,\,N_{\alpha\beta}^{1}$
$\displaystyle=\delta_{\beta,\bar{\alpha}}=\delta_{\alpha,\bar{\beta}}$
$\displaystyle N_{\alpha\beta}^{\gamma}$
$\displaystyle=N_{\bar{\beta}\bar{\alpha}}^{\bar{\gamma}}$
$\displaystyle\sum_{i\in{\mathcal{C}}_{O}}N_{\alpha\beta}^{i}N_{i\gamma}^{\delta}$
$\displaystyle=\sum_{j\in{\mathcal{C}}_{O}}N_{\alpha
j}^{\delta}N_{\beta\gamma}^{j}.$

First, let us compare these identities with the bulk identities in Eq. (86).
Every identity in Eq. (89) has an analogous identity in the bulk. However, one
bulk identity is generally violated in this context. Specifically,
$N_{\alpha\beta}^{\gamma}\neq N_{\beta\alpha}^{\gamma}$ in general, in
contrast to the identity $N_{ab}^{c}=N_{ba}^{c}$. Intuitively, this is because
there is no room to permute two domain wall sectors.151515This does not imply
domain wall sectors are confined onto the domain wall. They are not. See
Section IX for an explanation of this point.

There is an identity which relates the quantum dimensions $\\{d_{\alpha}\\}$
to the fusion multiplicities $\\{N_{\alpha\beta}^{\gamma}\\}$:

$d_{\alpha}d_{\beta}=\sum_{\gamma}N_{\alpha\beta}^{\gamma}d_{\gamma}.$ (90)

This identity is analogous to Eq. (87). It completely determines the set of
quantum dimensions $\\{d_{\alpha}\\}$ because the fusion multiplicities
satisfy Eq. (89). Then it follows that $d_{1}=1$ and
$d_{\alpha}=d_{\bar{\alpha}}\geq 1$ for $\forall\alpha\in{\mathcal{C}}_{O}$.
Furthermore, the quantum dimension $d_{\alpha}$ is quantized, just like its
bulk counterpart. This completes the summary of the fusion properties of
$O$-type superselection sectors.

In terms of proofs, Eq. (90) follows from the same line of argument explained
in Section II. Also, the proofs of the triviality of the vacuum and the
associativity relation, [the first and fourth lines of Eq. (89)], are
identical to their bulk counterparts. We refer the readers to Ref. [41] for
these proofs.

However, the proofs on the two properties involving the anti-sectors [the
second and the third lines of Eq. (89)] need to be modified a bit.

####  VI.1.1 Proofs related to anti-sectors

Below, we derive the fact that, for each $\alpha\in{\mathcal{C}}_{O}$, there
is a unique anti-sector $\bar{\alpha}\in{\mathcal{C}}_{O}$, such that

$\displaystyle N_{\alpha\beta}^{1}$
$\displaystyle=\delta_{\beta,\bar{\alpha}}=\delta_{\alpha,\bar{\beta}}$ (91)
$\displaystyle N_{\alpha\beta}^{\gamma}$
$\displaystyle=N_{\bar{\beta}\bar{\alpha}}^{\bar{\gamma}}.$

To prove these facts, it will be convenient to instead prove the following
weaker statements.

  1. (i)

$\forall\alpha\in\mathcal{C}_{O}$,
$\exists!\overrightarrow{\alpha},\overleftarrow{\alpha}\in\mathcal{C}_{O}$
s.t.
$N_{\alpha\beta}^{1}=\delta_{\beta,\overrightarrow{\alpha}}=\delta_{\alpha,\overleftarrow{\beta}}$.

  2. (ii)

$N_{\alpha\beta}^{\gamma}=N_{\overrightarrow{\beta}\overrightarrow{\alpha}}^{\overrightarrow{\gamma}}$.

Statement (i) means any $\alpha\in\mathcal{C}_{O}$ has a “left anti-sector”
$\overleftarrow{\alpha}$ and a “right anti-sector” $\overrightarrow{\alpha}$.

These two statements as a whole is weaker than Eq. (91). Nevertheless, with
the established triviality of the vacuum, $N_{1\alpha}^{\gamma}=N_{\alpha
1}^{\gamma}=\delta_{\alpha,\gamma}$, we can derive Eq. (91) from these two
weaker statements. To see why, first note that
$\overleftarrow{\overrightarrow{\alpha}}=\overrightarrow{\overleftarrow{\alpha}}=\alpha$,
which follows from statement (i) alone; this is because statement (i) implies
$N^{1}_{\alpha\overrightarrow{\alpha}}=N^{1}_{\overleftarrow{\overrightarrow{\alpha}}\overrightarrow{\alpha}}=1$
and
$N^{1}_{\overleftarrow{\alpha}\alpha}=N^{1}_{\overleftarrow{\alpha}\overrightarrow{\overleftarrow{\alpha}}}=1$.
Moreover, statement (i) and the triviality of the vacuum imply that
$1=\overrightarrow{1}=\overleftarrow{1}$; this is because statement (i)
implies $N^{1}_{1\overrightarrow{1}}=N^{1}_{\overleftarrow{1}1}=1$. Next, we
choose $\gamma=1$ and $\alpha=\overleftarrow{\beta}$ for statement (ii). We
see that
$N^{1}_{\overleftarrow{\beta}\beta}=N^{1}_{\overrightarrow{\beta}\beta}=1,\forall\beta$.
Thus, $\overleftarrow{\alpha}=\overrightarrow{\alpha}$,
$\forall\alpha\in{\mathcal{C}}_{O}$. In other words, the left anti-sector and
the right anti-sector are identical. Therefore, there is a unique anti-sector
for every superselection sector in $\mathcal{C}_{O}$. We denote the unique
anti-sector of $\alpha$ as $\bar{\alpha}$. Plugging this result into the two
statements, we arrive at Eq. (91).

We have seen that we only need to prove the two statements above in order to
derive Eq. (91). Below, we provide these proofs.

Figure 30:  (a) Merging extreme points associated with $1\in{\mathcal{C}}_{O}$
and $\alpha\in{\mathcal{C}}_{O}$. (b) Merging extreme points associated with
$1\in{\mathcal{C}}_{O}$ and $\beta\in{\mathcal{C}}_{O}$.

Let us first focus on statement (i), the uniqueness of the anti-sector maps
$\alpha\to\overrightarrow{\alpha}$ and $\alpha\to\overleftarrow{\alpha}$. The
idea is similar to the proof of Proposition 4.9 of Ref. [41]. Specifically, we
can merge an extreme point carrying the sector $\alpha\in{\mathcal{C}}_{O}$
with another extreme point carrying the sector $1\in{\mathcal{C}}_{O}$. See
Fig. 30(a). Here, the annulus that carries the sector $\alpha$ is inside the
annulus that carries $1$. The existence of the merged state implies that
$\forall\alpha,\,\exists\beta$ s.t. $N^{1}_{\alpha\beta}\geq 1$. The entropy
of the merged state can be obtained in two different ways, leading to the
following equation:

$2\ln d_{\alpha}=\ln
d_{\alpha}+\ln(\sum_{\beta}N_{\alpha\beta}^{1}d_{\beta}),$ (92)

where we have used the fact that $N_{1\beta}^{1}=\delta_{\beta,1}$. Equation.
(92) further simplifies into

$d_{\alpha}=\sum_{\beta}N_{\alpha\beta}^{1}d_{\beta}.$ (93)

With the exact same approach, we can derive the following identity:

$d_{\beta}=\sum_{\alpha}N_{\alpha\beta}^{1}d_{\alpha}$ (94)

by considering the merging process in Fig. 30(b).161616Unlike the bulk version
of the proof, we cannot “rotate” the 2-hole disk on the domain wall to switch
the two holes. This is why we need the merging process Fig. 30(b). For a
chosen $\alpha$, we pick a $\beta$ such that $N^{1}_{\alpha\beta}\geq 1$. (As
we have discussed, such a choice always exists.) For such $\beta$,
$d_{\alpha}\geq d_{\beta}$. Similarly, for a chosen $\beta$ there must be at
least one $\alpha$ such that $N_{\alpha\beta}^{1}\geq 1$ and $d_{\beta}\geq
d_{\alpha}$. Therefore, $d_{\alpha}=d_{\beta}$ if $N_{\alpha\beta}^{1}\geq 1$.
Obviously, these two quantum dimensions cannot be equal to each other if for
the chosen $\alpha$, $N_{\alpha\beta}^{1}\geq 1$ for more than one choice of
$\beta$, nor can this happen if $N^{1}_{\alpha\beta}>1$. Therefore, we
conclude that statement (i) is true.

As a byproduct of this analysis, we have also found that

$d_{\alpha}=d_{\overleftarrow{\alpha}}=d_{\overrightarrow{\alpha}}.$ (95)

Figure 31: The overall picture of the proof of statement (ii).

Now, let us prove statement (ii), namely
$N_{\alpha\beta}^{\gamma}=N_{\overrightarrow{\beta}\overrightarrow{\alpha}}^{\overrightarrow{\gamma}}$.
This proof is similar to the proof of the bulk version (Proposition 4.10 of
Ref. [41]), but with some modifications.

The overall picture of the derivation is depicted in Fig. 31. The density
matrix in Fig. 31(a) is the unique element of
$\Sigma^{1}_{\alpha\overrightarrow{\alpha}}(G)$, where $G$ is the depicted
subsystem. After taking a partial trace, we merge this density matrix with the
unique element of $\Sigma^{1}_{\beta\overrightarrow{\beta}}(G^{\prime})$,
where $G^{\prime}$ is the subsystem on the right side of Fig. 31(b). The
resulting 4-hole disk $W$ is depicted in Fig. 31(c). The key object in the
proof is the density matrix obtained from this merging process, which we shall
refer to as
$\rho_{W}^{(\alpha\overrightarrow{\alpha};\beta\overrightarrow{\beta})}$.

To see why the merged state
$\rho_{W}^{(\alpha\overrightarrow{\alpha};\beta\overrightarrow{\beta})}$ helps
in the proof of statement (ii), we consider its reduced density matrices on
subsystems $G_{L}$, $G_{M}$ and $G_{R}$ depicted in Fig. 31(d). In general,
while we have fixed the sectors
$\alpha,\overrightarrow{\alpha},\beta,\overrightarrow{\beta}$, the sectors on
the outer boundary of $G_{L}$ and $G_{R}$ are, in general, a mixture. By
inspecting the subsystem $G_{M}$, we see that the superselection sectors on
the outer boundary of $G_{L}$ and $G_{R}$ must fuse to identity. Thus, we can
denote the sectors as $\gamma$ and $\overrightarrow{\gamma}$ respectively.
While there can be multiple possible choices of $\gamma$, we measure the
sector $\overrightarrow{\gamma}$ on the outer boundary of $G_{R}$ whenever we
measure the sector $\gamma$ on the outer boundary of $G_{L}$. Therefore, the
probability of finding the sector $\gamma$ on the outer boundary of $G_{L}$
equals to the probability of finding the sector $\overrightarrow{\gamma}$ on
the outer boundary of $G_{R}$. Formally, we can write this fact as

$P(\gamma|\rho_{G_{L}}^{(\alpha\overrightarrow{\alpha};\beta\overrightarrow{\beta})})=P(\overrightarrow{\gamma}|\rho^{(\alpha\overrightarrow{\alpha};\beta\overrightarrow{\beta})}_{G_{R}}).$
(96)

The next step is to calculate both sides of Eq. (96). The key observation is
the fact that the reduced density matrices of
$\rho_{W}^{(\alpha\overrightarrow{\alpha};\beta\overrightarrow{\beta})}$ on
$G_{L}$ and $G_{R}$ are the maximmum-entropy states with the respective
superselection sector choices. In other words,

$\displaystyle{\rm Tr}_{W\setminus
G_{L}}\,\rho_{W}^{(\alpha\overrightarrow{\alpha};\beta\overrightarrow{\beta})}$
$\displaystyle=\rho_{G_{L}}^{\alpha\Join\beta}$ (97) $\displaystyle{\rm
Tr}_{W\setminus
G_{R}}\,\rho_{W}^{(\alpha\overrightarrow{\alpha};\beta\overrightarrow{\beta})}$
$\displaystyle=\rho_{G_{R}}^{\overrightarrow{\beta}\Join\overrightarrow{\alpha}},$

where $\rho_{G_{L}}^{\alpha\Join\beta}$ and
$\rho_{G_{R}}^{\overrightarrow{\beta}\Join\overrightarrow{\alpha}}$ can be
obtained by merging two annuli associated with the specified superselection
sectors.

Let us study the consequence of Eq. (97), deferring the proof of Eq. (97) to
Appendix F. By using the fact that $\rho_{G_{L}}^{\alpha\Join\beta}$ is the
maximum-entropy state consistent with the chosen sectors $\alpha$ and $\beta$,
we obtain

$P(\gamma|\rho_{G_{L}}^{(\alpha\overrightarrow{\alpha};\beta\overrightarrow{\beta})})=\frac{N_{\alpha\beta}^{\gamma}d_{\gamma}}{d_{\alpha}d_{\beta}}.$
(98)

Similarly,

$P(\overrightarrow{\gamma}|\rho^{(\alpha\overrightarrow{\alpha};\beta\overrightarrow{\beta})}_{G_{R}})=\frac{N_{\overrightarrow{\beta}\overrightarrow{\alpha}}^{\overrightarrow{\gamma}}d_{\overrightarrow{\gamma}}}{d_{\overrightarrow{\beta}}d_{\overrightarrow{\alpha}}}.$
(99)

Because $d_{\alpha}=d_{\overrightarrow{\alpha}}$ (see Eq. (95)), we conclude
$N_{\alpha\beta}^{\gamma}=N_{\overrightarrow{\beta}\overrightarrow{\alpha}}^{\overrightarrow{\gamma}}$,
as we claimed. This completes the proof of statement (ii).

In conclusion, we have justified Eq. (89).

###  VI.2 Fusion onto the domain wall

In this section, we discuss the fusion rules of anyons, i.e., the bulk
superselection sectors, onto the domain wall. We will use
$a,b,c,\ldots\in\mathcal{C}_{P}$ to denote the anyons on the $P$ side and
$x,y,z\ldots\in\mathcal{C}_{Q}$ to denote the anyons on the $Q$ side.

The fusion of anyons onto domain walls gives rise to fusion spaces that
involve the superselection sectors of the bulk and the domain wall. One may
consider moving an anyon $a\in\mathcal{C}_{P}$ onto the domain wall; moving an
anyon $x\in\mathcal{C}_{Q}$ onto the domain wall; or alternatively, bringing a
pair of anyons $a\in\mathcal{C}_{P}$ and $x\in\mathcal{C}_{Q}$ onto the domain
wall. These processes can be formally written as

$\displaystyle a$ $\displaystyle=$
$\displaystyle\sum_{\alpha}N_{a}^{\alpha}\,\alpha,$ (100) $\displaystyle x$
$\displaystyle=$ $\displaystyle\sum_{\alpha}N_{x}^{\alpha}\,\alpha,$ (101)
$\displaystyle a\times x$ $\displaystyle=$
$\displaystyle\sum_{\alpha}N_{ax}^{\alpha}\,\alpha,$ (102)

where the fusion multiplicities $\\{N_{a}^{\alpha}\\}$, $\\{N_{x}^{\alpha}\\}$
and $\\{N_{ax}^{\alpha}\\}$ are again non-negative integers. Here, $\alpha$ is
in $\mathcal{C}_{O}$.

The fusion multiplicities have the following physical interpretation.
$N_{a}^{1}$ is relevant to the process of condensing anyon $a$ onto the domain
wall. What we mean by condensing is that if $N_{a}^{1}\geq 1$, it is possible
to move an anyon $a$ onto the domain wall and annihilate it by a local
process. This also means that if $N_{a}^{1}\geq 1$, we can create a single
anyon $a$ in the bulk with a string operator attached to the domain wall. The
physical interpretation of $N_{x}^{1}$ is similar. A pair $(a,x)$ with
$N_{ax}^{1}\geq 1$ can be simultaneously annihilated (or created) in the
vicinity of the gapped domain wall. Similarly, $N_{a}^{\alpha}$ determines
whether it is possible to fuse an anyon $a$ onto the domain wall and turn it
into a domain wall sector $\alpha\in\mathcal{C}_{O}$.

Figure 32: Three basic subsystem types that are relevant to the fusion of
anyons onto the domain wall: (a) is relevant to $\\{N_{a}^{\alpha}\\}$, (b) is
relevant to $\\{N_{x}^{\alpha}\\}$ and (c) is relevant to
$\\{N_{ax}^{\alpha}\\}$.

The concrete rules that govern these processes can be deduced from three types
of subsystems described in Fig. 32. Repeating the analysis in Section VI.1, we
obtain the following results.

  * •

Fusion of the vacuum:

$\displaystyle\textrm{among }\\{N_{a}^{\alpha}\\}:$
$\displaystyle\,\,\,\,N_{1}^{\alpha}=\delta_{\alpha,1},$ (103)
$\displaystyle\textrm{among }\\{N_{x}^{\alpha}\\}:$
$\displaystyle\,\,\,\,N_{1}^{\alpha}=\delta_{\alpha,1},$ (104)
$\displaystyle\textrm{among }\\{N_{ax}^{\alpha}\\}:$
$\displaystyle\,\,\,\,N_{a1}^{\alpha}=N_{a}^{\alpha},\,\,\,\,N_{1x}^{\alpha}=N_{x}^{\alpha}.$
(105)

  * •

Relations between quantum dimensions and fusion multiplicities

$\displaystyle d_{a}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}N_{a}^{\alpha}d_{\alpha}$ (106) $\displaystyle
d_{x}$ $\displaystyle=$ $\displaystyle\sum_{\alpha}N_{x}^{\alpha}d_{\alpha}$
(107) $\displaystyle d_{a}d_{x}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}N_{ax}^{\alpha}d_{\alpha}.$ (108)

  * •

Relation between a fusion space and an fusion space formed by the anti-
sectors:

$\displaystyle N_{a}^{\alpha}$ $\displaystyle=$ $\displaystyle
N_{\bar{a}}^{\bar{\alpha}}$ (109) $\displaystyle N_{x}^{\alpha}$
$\displaystyle=$ $\displaystyle N_{\bar{x}}^{\bar{\alpha}}$ (110)
$\displaystyle N_{ax}^{\alpha}$ $\displaystyle=$ $\displaystyle
N_{\bar{a}\bar{x}}^{\bar{\alpha}}.$ (111)

  * •

Associativity conditions:

$\displaystyle N_{ax}^{\alpha}$ $\displaystyle=$
$\displaystyle\sum_{\beta,\gamma}N_{a}^{\beta}N_{x}^{\gamma}N_{\beta\gamma}^{\alpha}$
(112) $\displaystyle N_{ax}^{\alpha}$ $\displaystyle=$
$\displaystyle\sum_{\beta,\gamma}N_{x}^{\beta}N_{a}^{\gamma}N_{\beta\gamma}^{\alpha}$
(113) $\displaystyle\sum_{c}N_{ab}^{c}N_{c}^{\gamma}$ $\displaystyle=$
$\displaystyle\sum_{\alpha,\beta}N_{a}^{\alpha}N_{b}^{\beta}N_{\alpha\beta}^{\gamma}$
(114) $\displaystyle\sum_{z}N_{xy}^{z}N_{z}^{\gamma}$ $\displaystyle=$
$\displaystyle\sum_{\alpha,\beta}N_{x}^{\alpha}N_{y}^{\beta}N_{\alpha\beta}^{\gamma}$
(115)
$\displaystyle\sum_{\beta,\gamma}N_{\beta\gamma}^{\alpha}N_{ax}^{\beta}N_{by}^{\gamma}$
$\displaystyle=$ $\displaystyle\sum_{c,z}N_{cz}^{\alpha}N_{ab}^{c}N_{xy}^{z}$
(116)

These identities are consistent with the proposal in the existing literature.
For example, Eq. (116) implies the so called _stable condition_ of Ref. [16].

###  VI.3  Fusions with $\mathbb{N}$-type and $\mathbb{U}$-type sectors

In this section, we discuss a few things related to the fusion of
$\mathbb{N}$\- and $\mathbb{U}$-type sectors. Specifically, we consider the
fusion spaces $\\{N_{\mathcal{N}\mathcal{U}}^{\alpha}\\}$ and
$\\{N_{\mathcal{U}}^{\alpha}\\}$, defined in Figs. 33(a) and (b),
respectively.

Figure 33:  (a) A subsystem relevant to the fusion involving the sectors
$\mathcal{N}\in\mathcal{C}_{\mathbb{N}}$,
$\mathcal{U}\in\mathcal{C}_{\mathbb{U}}$ and $\alpha\in\mathcal{C}_{O}$. (b) A
subsystem relevant to the fusion involving the sector
$\mathcal{U}\in\mathcal{C}_{\mathbb{U}}$ and $\alpha\in\mathcal{C}_{O}$.

These fusions are a bit abstract, so one may wonder why we consider them in
the first place. A simple reason is that the constraints on these fusion
spaces enable us to prove some fundamental properties of the simpler $N$-type
and $U$-type parton sectors. These sectors do not obey the ordinary fusion
rule, as we have briefly discussed in Section V.2. However, we can still
derive nontrivial facts about their fusion by embedding those sectors into the
$\mathbb{N}$\- and $\mathbb{U}$-type sectors (see Section IV.3).

The most notable implication is Proposition VI.5, which implies that the
quantum dimensions of the partons, _i.e.,_ $\\{d_{n}\\}$, are uniquely
determined by two sets of fusion multiplicities
$\\{N_{\alpha\beta}^{\gamma}\\}$ and $\\{N_{\mu(n)}^{\alpha}\\}$. In that
sense, these quantum dimensions are “quantized.” Moreover, unlike the $N$\-
and $U$-type sectors, the $\mathbb{N}$\- and $\mathbb{U}$-sectors do allow a
conventional definition of fusion space.

To study these fusion rules, let us begin by showing some simple properties
when one of the sectors involved is the vacuum sector.

######  Proposition VI.1.

Among the fusion multiplicities $\\{N_{\mathcal{N}\mathcal{U}}^{\alpha}\\}$,
we have

$N_{1\mathcal{U}}^{1}=\delta_{\mathcal{U},1}$ (117)

and

$N_{\mathcal{N}1}^{1}=\delta_{\mathcal{N},1}.$ (118)

Furthermore, among the fusion multiplicities $\\{N_{\mathcal{U}}^{\alpha}\\}$,
we have

$N_{1}^{\alpha}=\delta_{\alpha,1}.$ (119)

Figure 34: (a) A two-step merging process for $\mathcal{N}=1$. It is useful in
the proof of $N^{1}_{1\mathcal{U}}=\delta_{\mathcal{U},1}$. (b) A merging
process for $\mathcal{U}=1$. It is useful in the proof of
$N^{\alpha}_{\mathcal{U}=1}=\delta_{\alpha,1}$.

The key idea of the proof is to use the properties of the vacuum sector to
design merging processes that can “fill” a hole. See Fig. 34 for an
illustration of the relevant merging processes.

###### Proof.

Let us first prove Eqs. (117) Eq. (118). These two proofs are analogous to
each other, so we only discuss the proof of Eq. (117). Recall that the fusion
multiplicities $\\{N_{\mathcal{N}\mathcal{U}}^{\alpha}\\}$ are associated with
the subsystem in Fig. 33(a), where the superselection sectors involved are
$\alpha\in{\mathcal{C}}_{O}$, $\mathcal{N}\in{\mathcal{C}}_{\mathbb{N}}$ and
$\mathcal{U}\in{\mathcal{C}}_{\mathbb{U}}$. To show
$N_{1\mathcal{U}}^{1}=\delta_{\mathcal{U},1}$, it suffices to prove the
following statement. If $\alpha=1$ and $\mathcal{N}=1$, the density matrix of
the blue region in Fig. 33(a) is equal to the reduced density matrix obtained
from $\sigma$.

To see why this is the case, we consider the two-step merging process shown in
Fig. 34(a). This merging process is possible when $\mathcal{N}=1$ is the
vacuum sector. The first step “fills” the $N$-shaped hole with the vacuum, by
merging density matrix over the subsystem on the top of Fig. 34 to the density
matrix obtained from the reference state. This is possible because
$\mathcal{N}=1$; the density matrix on the surrounding $\mathbb{N}$-shaped
region is identical to that of the reference state, satisfying the requisite
condition for the merging theorem (Theorem II.3). The second step fills part
of the $U$-shaped hole and turns it into a point-like area intersecting with
the domain wall. This step is possible because $\mathcal{N}=1$ implies that
one of the parton labels of the sector $\mathcal{U}$ must be the vacuum. (More
precisely, $\mathcal{U}$ carries a pair of $N$-type parton sectors and a pair
of $U$-type parton sectors. The $N$-type parton sector on the left is the
vacuum sector due to $\mathcal{N}=1$.)

After the two-step merging process, we obtain an element of the information
convex set on an $O$-shaped region. Therefore the same $O$-type sector
$\alpha\in{\mathcal{C}}_{O}$ must appear on both boundaries of the $O$-shaped
region, i.e., the annulus on the bottom of Fig. 34(a). If $\alpha=1$, the hole
can be filled. The end result is the reduced density matrix of the reference
state on a disk-like region on the domain wall. Therefore we must have
$\mathcal{U}=1$. Furthermore, the density matrix labeled by $\alpha=1$,
$\mathcal{N}=1$ and $\mathcal{U}=1$ in Fig. 33(a) is unique. This completes
the proof of Eq. (117).

The proof of Eq. (119) follows from a similar line of reasoning. The merging
process in Fig. 34(b) is possible when $\mathcal{U}=1$. The end result is a
disk on the domain wall. This implies that the $\alpha\in{\mathcal{C}}_{O}$ in
the original density matrix must be the vacuum sector. Moreover, this density
matrix is unique. Therefore Eq. (119) holds. This completes the proof. ∎

Figure 35: (a) A merging process involving a pair of regions, which carry
$1\in\mathcal{C}_{O}$ and $\mathcal{N}\in{\mathcal{C}}_{\mathbb{N}}$
respectively. (b) A merging process involving a pair of regions, which carry
$1\in\mathcal{C}_{O}$ and $\mathcal{U}\in{\mathcal{C}}_{\mathbb{U}}$
respectively.

Next, we show that there is a nontrivial isomorphism between
$\mathcal{C}_{\mathbb{N}}$ and $\mathcal{C}_{\mathbb{U}}$. If the gapped
domain wall is trivial, this result would be trivially true because the
$\mathbb{N}$-shaped and $\mathbb{U}$-shaped subsystems can be smoothly
deformed into each other. However, this fact is less obvious when the domain
wall is nontrivial.

######  Proposition VI.2.

There is an isomorphism171717It is possible to define another isomorphism
between ${\mathcal{C}}_{\mathbb{N}}$ and ${\mathcal{C}}_{\mathbb{U}}$ by
considering the mirror image of Fig. 35. This isomorphism can be different
from $\varphi$.

$\varphi:{\mathcal{C}}_{\mathbb{N}}\to{\mathcal{C}}_{\mathbb{U}}$ (120)

such that

$N^{1}_{\mathcal{NU}}=\delta_{\mathcal{U},\varphi(\mathcal{N})},$ (121)

and

$d_{\mathcal{N}}=d_{\varphi(\mathcal{N})}.$ (122)

###### Proof.

We will use the merging processes described in Fig. 35, using the logic used
in the proof of Proposition 4.9 of Ref. [41].

We consider the merging process in Fig. 35(a), which involves two sectors
$1\in{\mathcal{C}}_{O}$ and $\mathcal{N}\in{\mathcal{C}}_{\mathbb{N}}$. Note
that any $\mathcal{N}\in{\mathcal{C}}_{\mathbb{N}}$ is allowed in the merging
process. This implies that
$\sum_{\mathcal{U}}N^{1}_{\mathcal{N}\mathcal{U}}\geq
1,\,\forall\mathcal{N}\in{\mathcal{C}}_{\mathbb{N}}$. By calculating the
entropy difference [between two sector choices (1, $\mathcal{N}$) and (1,
1)]in two different ways, we find

$2\ln d_{\mathcal{N}}=\ln
d_{\mathcal{N}}+\ln\sum_{\mathcal{U}}N_{\mathcal{N}\mathcal{U}}^{1}d_{\mathcal{U}}.$
(123)

The left-hand side is the entropy difference based on the entropy on the
$\mathbb{N}$-shaped subsystem. The right-hand side is obtained by solving a
maximization problem on the merged region. In the calculation of the right-
hand side, we have applied Eq. (117), which implies
$\sum_{\mathcal{U}}N_{1\mathcal{U}}^{1}d_{\mathcal{U}}=1$. There is an
analogous equation for the merging process in Fig. 35(b). By simplifying these
two equations, we find

$\displaystyle d_{\mathcal{N}}$
$\displaystyle=\sum_{\mathcal{U}}N^{1}_{\mathcal{N}\mathcal{U}}d_{\mathcal{U}},$
(124) $\displaystyle d_{\mathcal{U}}$
$\displaystyle=\sum_{\mathcal{N}}N^{1}_{\mathcal{N}\mathcal{U}}d_{\mathcal{N}}.$

Equation (124) is a strong constraint. For a chosen $\mathcal{N}$, pick a
sector $\mathcal{U}$ that satisfies $N^{1}_{\mathcal{N}\mathcal{U}}\geq 1$. It
follows from Eq. (124) that $d_{\mathcal{N}}\geq d_{\mathcal{U}}$ and
$d_{\mathcal{U}}\geq d_{\mathcal{N}}$. We have used the fact that the
multiplicities are nonnegative integers and that the quantum dimensions are
positive. Therefore, for every choice of $\mathcal{N}$, there is a unique
$\mathcal{U}$ for which the fusion multiplicity obeys
$N^{1}_{\mathcal{N}\mathcal{U}}=1$. Moreover, we have
$d_{\mathcal{U}}=d_{\mathcal{N}}$ for this choice. For the same $\mathcal{N}$,
a different choice of $\mathcal{U}$ gives $N^{1}_{\mathcal{N}\mathcal{U}}=0$.

Let $\varphi$ be the map from ${\mathcal{C}}_{\mathbb{N}}$ to
${\mathcal{C}}_{\mathbb{U}}$, mapping a sector
$\mathcal{N}\in{\mathcal{C}}_{\mathbb{N}}$ to the unique sector
$\mathcal{U}\in{\mathcal{C}}_{\mathbb{U}}$ satisfying
$N^{1}_{\mathcal{N}\mathcal{U}}=1$. $\varphi$ is bijective because there is an
inverse map obtained by the same argument, choosing $\mathcal{U}$ instead of
$\mathcal{N}$ first. This completes the proof. ∎

For later purpose, it will be convenient to consider an embedding
$\mu:{\mathcal{C}}_{N}\hookrightarrow{\mathcal{C}}_{\mathbb{U}}$, defined as
$\mu=\varphi\circ\eta_{N}$. Here, $\eta_{N}$ is the embedding defined in Eq.
(64). From Eqs. (69) and (122), it follows that

$d_{\mu(n)}=d_{n}^{2},\quad\forall\,n\in{\mathcal{C}}_{N}.$ (125)

There is an important subtlety about the fusion space
$N_{\mathcal{U}}^{\alpha}$. (See Fig. 33 for the relevant subsystems.) For
some $\mathcal{U}\in\mathcal{C}_{\mathbb{U}}$,
$\sum_{\alpha}N_{\mathcal{U}}^{\alpha}$ may vanish. This can happen when
${\mathcal{C}}_{U}$ contains two or more elements. Recall that each
$\mathcal{U}\in{\mathcal{C}}_{\mathbb{U}}$ can be labeled by four parton
sectors, two of which are $U$-type parton sectors. If a sector $\mathcal{U}$
labels an extreme point of the subsystem in Fig. 33(b), the two $U$-type