Symmetry Fractionalized (Irrationalized) Fusion Rules and Two Domain-Wall Verlinde Formulae

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I. INTRODUCTION
Topological orders have been extensively studied and significantly influenced our understanding of quantum phases of matter, due to their exotic and intricate properties, such as fusion and braiding of anyons.Anyon fusion is an ultralocal phenomenon that can hardly be directly detected, while braiding is nonlocal and directly measurable [1][2][3][4][5].The fusion rule and braiding of anyons are related through the Verlinde formula [6].This formula is the very Verlinde formula in conformal field theory due to the correspondence between topological orders and conformal field theory [7][8][9][10][11][12][13].
While single topological orders in 2+1D systems have been rigorously explored, composite topological systems consisting of multiple topological orders separated by gapped domain walls remain largely uncharted.Prior research [14] has brought forward the characteristic properties of such composite systems, notably, the nontrivial braiding between quasiparticles in the gapped domain wall and interdomain excitations labeled by pairs of anyons in different domains (see Fig. 1).It is crucial to understand how this domain-wall S matrix provides key insights into the fundamental structure of composite topological systems, as well as the relationships among various topological orders.
Moreover, topological orders have been effectively used in quantum computation due to their robustness against local perturbations and decoherence [15,16].Recent literature has suggested that topological defects could be a more viable candidate for implementing universal computing [17][18][19][20][21].As a generalization of topological boundaries, gapped domain walls have richer properties and thus would also apply to topological quantum computation.Therefore, exploring the properties of these novel excitations in composite systems mentioned above holds immense practical significance.
In this paper, we discover two domain-wall Verlinde formulae (4) and (5) for the composite systems consisting of two topological orders.These two formulae establish the fusion rules of the interdomain excitations as well as the fusion rules of the domain-wall quasiparticles via the domain-wall S matrix, directly generalizing the defect Verlinde formula that relates the fusion and half-linking of boundary excitations in Ref. [22].
In contrast to the conventional fusion rules, where fusion coefficients are always natural numbers, we discover that interdomain excitations can exhibit fractional fusion rules.These fractionalized fusion rules arise because when anyons carrying internal gauge charges in one domain cross the gapped domain wall into another domain, they may become distinct anyons therein, as these indistinguishable internal gauge charges become global symmetry charges.Consequently, the originally unobservable fractional fusion rules of internal gauge charges of certain anyons become the fractionalized fusion rules of interdomain excitations, which are now physical observables.This phenomenon bears a spatial analogy to symmetry breaking in phase transitions triggered by anyon condensation, so we refer to these unique interdomain fusion rules as symmetry fractionalized fusion rules.
More surprisingly, the interdomain fusion rules can even be irrational.We believe that these irrationalized fusion rules reflect the algebraic symmetries beyond group description [23][24][25][26][27][28][29][30][31].As the full nature of algebraic symmetries remains enigmatic, our insights into irrationalized fusion rules provide a clear lens for examining the characteristics of these emergent symmetries.
The gapped domain wall causes a selection rule of allowed interdomain elementary excitations (a, r), recorded in the branching matrix B BA , whose element B BA ra = 1 if and only if anyon a in domain A can enter and become anyon r in B; otherwise, B BA ra = 0 [36,38,43].We find that B BA can factorize into two matrices B A and B B : ) if and only if A-anyon a (domainwall quasiparticle α) can become domain-wall quasiparticle α (B-anyon r) when entering the gapped domain wall (domain B) and otherwise 0. Hence, an interdomain elementary excitation species (a, r) exists if and only if B BA ra = α∈L DW B B rα B A αa = 0, where L DW denotes the set of domain-wall quasiparticle species.There are as many domain-wall quasiparticle species as interdomain excitation species [14].
An interdomain excitation may braid nontrivially with a domain-wall quasiparticle.The braiding is encoded in an invertible domain-wall S-matrix [14] S DW defined in Fig. 3(a), which is understood (Fig. 3(b)) as the linking of the (spacetime) Wilson loops of interdomain excitation (a, r) and domain-wall quasiparticle α.
Domain-wall S-matrices S DW have been computed in the lattice model of such composite systems for doubled topological orders [14].Here, we can compute S DW from the branching matrices B A and B B and the S-matrices S A and S B of A and B, which may not be doubled, because of the following commutativities we found.
The S DW matrix then reads .

III. DOMAIN-WALL VERLINDE FORMULAE
In a single topological order, e.g., A, the fusion rule N A and the S-matrix S A satisfy the Verlinde formula [6]: A natural question is: Is there a similar formula for our generalized S-matrix S DW ?Indeed, we can imitate formula (3) to define fusion rules N (c,t) (a,r)(b,s) for interdomain excitations and N γ αβ for domain-wall quasiparticles via our domain-wall S-matrix S DW : , (4) Multiplying formula (4) by S DW α(c,t) and then summing over (c, t) ∈ L ID results in  4) and ( 5) -now called the domain-wall Verlinde formulae -generalize the Verlinde formula in a single topological order and the defect Verlinde fromula [22].

IV. INTERDOMAIN FUSION ALGEBRA AND QUANTUM DIMENSIONS
Interdomain excitations extend the notion of anyons in a single topological order, which have quantum dimensions that form a 1-dimensional representation of the fusion algebra of the anyons.Do interdomain excitations also have fusion algebra and quantum dimensions?
In an arbitrary state |ψ of a composite system, we can define a string operator W (a,r) L that creates an interdomain excitation (a, r), with anyons a and r at the two ends of an interdomain path L (Fig. 4).Along a fixed path L, all string operators W (a,r) L form an algebra: Interdomain fusion coefficients N (c,t) (a,r)(b,s) can be expressed in terms of the operator product expansion coefficients f To justify this equation, we define the interdomain loop operators (Fig. 4): Compared to the Fig. 2(a) and 2(b), the interdomain fusion coefficients N (c,t) (a,r)(b,s) are the operator product ex-pansion coefficients of interdomain loop operators: which defines the fusion algebra of interdomain excitations.Equations ( 6), (8), and (9) lead to Eq. ( 7).We define the quantum dimension d (a,r) of interdomain excitation (a, r) by where |0 is the vacuum state, and the equality follows that the trivial domain-wall quasiparticle 1 braids trivially with interdomain excitations.Definition (10) complies with the definition of quantum dimensions in a single topological order.By Eqs. ( 9) and (10), indicating that the quantum dimensions d (a,r) are a 1dimensional representation of the fusion algebra (9).It can be shown that d (a,r) is the largest eigenvalue of ma- . And in a system of infinitely many interdomain excitations (a, r), d (a,r) is the asymptotic dimension of the Hilbert space of each (a, r).
When fusing two interdomain excitations (a, r) and (b, s) in state |ψ , the resulting state W Such a composite system, including the interdomain excitations and their fusion, appears to be describable by a fusion 2-category, which we shall report elsewhere.

V. SYMMETRY FRACTIONALIZED / IRRATIONALIZED FUSION RULES
The fusion coefficients in a single topological order are always natural numbers, and so are the domain-wall fusion coefficients N γ αβ .Nevertheless, interdomain fusion coefficients N (c,t) (a,r)(b,s) can be fractional.For instance, in the composite system of the doubled Ising and Z 2 toric code phases, domain-wall Verlinde Formula (4) leads to The quantum dimensions are ; therefore, fusion rules (13) satisfy Eq. (11).We shall call such fusion rules symmetry fractionalized fusion rules.The symmetry fractionalization is understood by the correspondence between the spatial composite system and temporal anyon condensation as follows.
The Z 2 toric code topological order can originate from the doubled-Ising phase via a phase transition triggered by ψ ψ condensation in the doubled-Ising phase, where an anyon σσ carries internal Z 2 gauge charges [14,45].We denote σσ with Z 2 -charge 0 (1) as σσ 0 (σσ 1 ), which can transform into each other by gauge transformations and are thus unobservable.The ψ ψ condensation breaks this Z 2 gauge invariance to the global Z 2 symmetry of the Z 2 toric code phase, such that as also dictated by branching matrix Eq. ( 2), σσ 0 (σσ 1 ) becomes toric-code anyon m (e), which are now topological observables.We can find that σσ 0 and σσ 1 follow fractional fusion rules: which are likewise unobservable.Branching matrix (2) establishes the following map: (σσ, m) ←→ σσ 0 , (σσ, e) ←→ σσ 1 .
As a result, interdomain excitations (σσ, m) and (σσ, e) fuse in the same manner as σσ 0 and σσ 1 .Nonetheless, m and e are topological observables of interdomain excitations; therefore, the fusion rules (13) are physically measurable and are thus justified to be called symmetry fractionalized fusion rules.
Fractionalized fusion rules occur often in such composite systems, where one domain could arise from the other via anyon condensation, which usually breaks certain gauge invariance.The broken gauge invariance however may not always be describable by a gauge group but rather by certain algebra, such that the resultant global symmetry is also algebraic [23][24][25][26][27][28][29][30][31]; hence, more generally, the interdomain fusion rules can be more complicated than being just fractionalized.For example, in the composite system of su(2) 10 and so(5) 1 phases, the fusion coefficients where a * is the anti-particle of a and |0 is the vacuum state.The path L can be homotopically deformed with its endpoints fixed.Anyon fusion is associative and commutative and can be simply expressed as

Anyons in
Fusion coefficients (N A ) c ab are also the operator product expansion coefficients of loop operators C a L , defining the fusion algebra of topological order A. Here,

The quantum dimension of anyon a
It can be shown that d a is the largest eigenvalue of matrix N A a c b := N A c ab .Thus, in a system with infinitely many anyons a, d a is the asymptotic dimension of the Hilbert space for each anyon a.
Quantum dimensions are a 1-dimensional representation of the fusion algebra: The fusion coefficients and the S-matrix satisfy the following Verlinde Formula [6]:

Appendix B: A Brief Review of Anyon Condensation
We briefly review the phase transition from one topological phase, denoted by parent phase A, to another child phase B triggered by certain anyon condensation in A. An auxiliary intermediate phase T is introduced merely as a method to study the anyon condensation process [46].The anyon condensation in A first leads to intermediate phase T, followed by the transition from the intermediate phase to child phase B (Fig. 6(a)).
The most important data characterizing anyon condensation is the set of condensed anyons in A. These anyons behave like trivial quasiparticles in T, while the other anyons behave like nontrivial quasiparticles.It appears that (i) certain anyons in A, including the condensed ones, may correspond to distinct quasiparticles in T. This phenomenon is called splitting.(ii) Two types of A-anyons may correspond to identical T-quasiparticles if they are related by fusing with a condensed anyon in the parent phase.This phenomenon is called identification.We find that the relations between the anyons in A and the quasiparticles in T can be encoded in the branching matrix B A : An anyon a ∈ L A behaves like quasiparticle α in the intermediate phase if and only if B A αa = 1; otherwise, B A αa = 0.The quasiparticles in intermediate phase T can also fuse.The fusion rule is encoded in the fusion tensor N T , whose component N T γ αβ represents the number of channels that two quasiparticles α and β in T fuse to quasiparticle γ.The fusion tensor commutes with branching matrix B A : where L T denotes the set of quasiparticle species in the gapped domain wall.Not all quasiparticles in T appear in child phase B. If quasiparticle α originates from an A-anyon a with nontrivial braiding with a condensed anyon γ, this quasiparticle is said to be confined in the child phase.Only unconfined quasiparticles are allowed in the child phase as B-anyons.We define the branching matrix B B to record the relations between the quasiparticles in T and the anyons in B, where B B rα = 1 if and only if quasiparticle α in the intermediate phase may become anyon r in child phase B. Composing two branch matrices B A and B B results in the total branching matrix B BA relating the anyon species in A and B [36,38,47]: Branching matrix B BA commutes with the fusion rules and S-matrices: To study anyon condensation, the Hilbert space of child phase B can be embedded in the Hilbert space of the parent phase A [48,49].Specifically, a child excitation state with three anyons r, s, and t is embedded in the Hilbert space of the parent phase, as a linear combination of excitation states with three parent anyons a, b, and c (Fig. 6 an A-anyon a becomes quasiparticle α and then r when moving from domain A across the gapped domain wall into domain B, corresponds to the transformation in anyon condensation, where A-anyon a first becomes quasiparticle α in the intermediate phase and then r in the child phase [14].Hence, the gapped domain wall imposes a selection rule for anyon transformations when entering different domains, which is encoded in the same branching matrices B A , B B , and B BA , as in anyon condensation. We find that in a composite system corresponding to an anyon-condensation-induced phase transition, d (a,r) = d r , and the interdomain fusion rules can be written in terms of vertex lifting coefficients of anyon condensation (Fig. 7): (C1)

Appendix D: Examples
We apply our methods in the article to computing the domain-wall S-matrices, interdomain fusion rules, and domain-wall fusion rules of specific composite systems of two topological orders separated by a gapped domain wall.

FIG. 1 .
FIG.1.A composite system of two topological orders A (red) and B (blue) separated by a gapped domain wall (gray), in which are a domain-wall quasiparticle α and an interdomain excitation with anyon a (r) in phase A (B).

FIG. 4 .
FIG. 4. Interdomain string operator W (a,r) L and interdomain loop operator C (a,r) L acting on state |ψ .
s) L |ψ is a superposition of orthogonal interdomain excitation states W (c,t) L |ψ .The probability of measuring (c, t) is A can fuse.The fusion rule is described by a three-index fusion tensor N A .The component N A c ab ∈ N represents the number of channels where two anyons a and b fuse to anyon c, i,e, the dimension of the Hilbert space spanned by states containing three anyons a * , b * , and c (Fig. 5(b)).

FIG. 5 . 5 . 1 ∈
FIG. 5. (a) Elementary excitation state |a L .At the two ends of path L are two anyons a * and a.(b) Fusing anyons a and b results in a new anyon c, where 1 ≤ µ ≤ N c ab labels different possible fusion channels.

FIG. 6 .FIG. 7 .
FIG. 6.(a) In anyon condensation, anyon a in parent phase A may be tranformed to quasiparticle α in intermediate phase T, and then become anyon r in child phase B. (b) The embedding from the child Hilbert space to the parent Hilbert space.(c) Elementary excitations (a, α) and (α, r) in the system with topological orders A and B separated by a gapped domain wall.

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are called vertex lifting coefficients (VLCs), where µ and ν label bases of the parent and child excitation states.

2 .
The composite system of the su(2)10 and so(5)1 phases There are 11 anyon species in the su(2) 10 topological order, labeled by integers 0 ≤ a ≤ 10, where 0 is the trivial anyon.The fusion tensor N c ab = 1 if and only if (a + b + c) is an even number and |a − b| ≤ c ≤ min(a + b, 20 − a − b); otherwise N c ab = 0.The quantum dimensions of anyons and the S-matrix elements are θ c d c θ a θ b ,