Classification of intrinsically mixed $1+1$D non-invertible Rep$(G) \times G$ SPT phases

Classification of in trinsically mixed 1 + 1 D non-in v ertible Rep( G ) × G SPT phases Y ouxuan W ang 1 1 Dep artment of Physics, The Hong Kong University of Scienc e and T e chnolo gy, Clear Water Bay, Hong Kong, China W e classify 1+1d b osonic SPT phases with non-inv ertible symmetry Rep( G ) × G , equiv alen tly the fusion-category symmetry H = Rep ( G ) ⊠ V ec G . F o cusing on intrinsic al ly mixe d phases (trivial under either factor alone), we use the correspondence betw een H -SPTs, H -mo dules ov er Vec , and fib er functors H → V ec [1, 2] to obtain a complete classification: such phases are parametrized by ϕ ∈ End( G ) / Inn( G ). F or each ϕ we identify the associated condensable (Lagrangian) algebra A ϕ in the bulk Z ( H ) ≃ D 2 G . W e further provide an explicit lattice realization b y modifying Kitaev’s quan tum double mo del with a domain wall B ϕ and smo oth/rough b oundaries [3, 4], and then con tracting to a 1D c hain, yielding a (possibly twisted) group-based cluster state [5] whose ribb on- generated symmetry op erators encode the same ϕ . I. INTR ODUCTION Symmetry-protected top ological (SPT) phases are gapp ed short-range-entangled phases that are adiabatically con- nected to trivial product states once the protecting symmetry is forgotten, but b ecome distinct in the presence of a global symmetry . In one spatial dimension, SPT order is equiv alen tly characterized by the absence of in trinsic bulk top ological order together with robust b oundary degrees of freedom, or by the imp ossibility of disentangling the ground state b y an y symmetry-preserving finite-depth local unitary circuit. F or b osonic on-site gr oup symmetries, 1D SPT phases are classified by pro jective representations—equiv alen tly by the group cohomology class in H 2 ( G, U (1))—as understo od from matrix-pro duct-state approaches and related viewp oin ts [6–13]. More recently , the notion of symmetry itself has b een broadened by the p erspective that symmetries are implemented b y top ological defect operators. In quan tum field theory this includes higher-form symmetries generated by higher- co dimension top ological op erators [14–17]. Beyond inv ertible defects, one encoun ters non-invertible symmetries, whose defects fuse according to a fusion algebra rather than a group law [18–22]. In 1+1D suc h symmetries are naturally enco ded b y a fusion category C , and gapp ed phases with C symmetry admit a categorical/top ological description in terms of mo dule categories, fib er functors, and the bulk Drinfeld center Z ( C ) (the “Symmetry TFT” viewp oin t) [1, 2, 23–25]. This framework is particularly well-suited for constructing microscopic realizations and identifying explicit nonlocal observ ables, as exemplified by recent group-based cluster state constructions with non-in vertible Rep( G ) × G symmetry [5, 26, 27] and by lattice realizations of gapp ed b oundaries/domain walls in quantum double mo dels [3, 4, 28]. Recen t progress has made categorical SPT phases with group-theoretical symmetry data increasingly concrete at the lattice level. In particular, Ref. [5] provides an explicit 1D stabilizer construction realizing a nontrivial G × Rep( G )-protected phase, exhibiting protected edge degrees of freedom, string order, and a quantized charge–flux resp onse. F rom a complementary and more systematic p ersp ectiv e, Ref. [2] formulates 1+1d SPT phases with finite fusion-category symmetry C in the MPO–MPS framework, relating phases to fib er-functor data, deriving an interface algebra whose represen tation theory enforces degenerate interface modes, and identifying S 1 -families via a non-Ab elian Thouless-pump in v arian t [29–32]. These developmen ts motiv ate mo ving b eyond “factorized” settings, where non trivialit y can already b e detected after restricting to a single symmetry factor. F or the pro duct symmetry C = Rep ( G ) ⊠ V ec G , there is a distinguished class of intrinsic al ly mixe d phases: they b ecome trivial upon restriction to either the c harge sector Rep ( G ) or the flux sector V ec G , y et remain non trivial under the full product symmetry . Physically , such phases isolate response/obstruction data that dep ends essentially on the interpla y b et w een charge and flux defects, reminiscen t of the charge–flux structure underlying quantum doubles and their b oundaries/domain walls [33–38]. Accordingly , w e take intrinsically mixed phases as a minimal arena in which Rep ( G ) and Vec G m ust b e treated on equal fo oting, and where gen uinely non- factorizable categorical-SPT phenomena can b e cleanly identified. W e give a complete classification of intrinsic al ly mixe d Rep( G ) × G SPT phases, showing that they are parameterized b y endomorphisms ϕ ∈ End( G ) / Inn( G ) (equiv alen tly , by . . . ). F or eac h ϕ we explicitly construct the corresponding fib er functor / indecomp osable mo dule category , and w e identify the asso ciated Lagrangian (condensable) algebra A ϕ inside Z ( Rep ( G ) ⊠ Vec G ) ≃ D ( G 2 ), together with the pairing rule in Eq. (39). W e interpret A ϕ as the gapp ed domain wall B ϕ in the quan tum double mo del, and deriv e the any on transmission rules across the w all: [ g ] 7→ [ ϕ ( ¯ g )] for fluxes and π 7→ ϕ ∗ ¯ π for charges (with the appropriate charge–flux pairing dictated by Eq. (39)). These intrinsically mixed phases therefore realize domain walls whose action is invisible up on restricting to either the pure charge (Rep( G )) or pure flux ( Vec G ) symmetry alone, but is fully captured by the mixed Rep ( G ) ⊠ V ec G structure. 2 Our strategy is to connect the abstract classification data to an explicit microscopic realization and to directly computable observ ables, while keeping the categorical dictionary C -SPT ⇐ ⇒ C -mo dule ov er Vec ⇐ ⇒ fib er functor C → V ec . Concretely , we start from a mo dified quantum double construction equipp ed with a gapp ed domain w all B ϕ and compatible smo oth/rough b oundaries, and then p erform a controlled reduction to a one-dimensional c hain, which yields a (possibly twisted) group-based cluster state. In this reduction, closed ribb on op erators descend to on- site/non-on-site symmetry operators on the chain: one family implements the G sector (flux-t yp e op erations), while the other realizes the Rep( G ) sector (charge-t ype op erations), and their mutual commutation/attac hmen t relations enco de the same endomorphism data ϕ that lab els our intrinsically mixed phases. On the mathematical side we construct the corresp onding mo dule categories and matc h them to condensable algebras in Z ( C ), leveraging standard results on fusion categories and their mo dule categories [23]. The paper is organized as follows. In Sec. II w e construct the corresp onding 1 + 1d lattice realizations b y con tracting a mo dified Kitaev quantum double mo del with a B ϕ domain wall and smo oth/rough b oundaries, obtaining (p ossibly t wisted) group-based cluster states and their symmetry op erators (closed ribb ons) implementing the G and Rep( G ) actions. In Sec. IV w e characterize each phase by a condensable/Lagrangian algebra A ϕ in the bulk Z ( H ) ≃ D 2 G and derive the pairing rule Eq. (39). Finally , in Sec. V we give the SymTFT interpretation and identify A ϕ with the ph ysical domain wall B ϕ in the quantum double mo del, including the induced any on transmission rules across the w all. W e conclude with a brief discussion and outlook; additional computations are collected in the App endix. Notation. Throughout, G is a finite group and Vec G denotes the fusion category of finite-dimensional G -graded complex vector spaces (with Vec the category of finite-dimensional complex vector spaces). W e write Rep( G ) for finite-dimensional complex representations, and Rep ( G ) for the corresp onding fusion category when viewed as a sym- metry input. The Drinfeld center is denoted by Z ( C ). W e use [ g ] for the simple flux any on lab eled b y a (conjugacy class of ) group elemen t g , and π for a charge any on lab eled by an irreducible representation; a bar indicates inv er- sion/dualization (e.g. ¯ g = g − 1 , ¯ π the dual representation), and ϕ ∗ denotes pullbac k of representations along ϕ . All categories are assumed C -linear, semisimple, and unitary where appropriate. W e will refer to Vec G × G , Rep ( G ) ⊠ Vec G and D 2 G as G , H and D resp ectively in following context. I I. CONSTR UCTION OF THE MODELS In this section, we present a lattice realization of the classified SPT phases based on Kitaev’s quantum double mo del. A. Review of Kitaev’s quan tum double W e consider the (2 + 1)-dimensional Kitaev quan tum double mo del [3] defined on a honeycomb lattice. Giv en a finite group G , a lo cal Hilb ert space H G = span {| g ⟩} g ∈ G is assigned to each oriented edge; reversing the orientation corresp onds to inv erting the group element. W e represen t a physical edge state by an arrow together with a group elemen t g . F or eac h vertex α , the vertex op erator is defined as ˆ V α    α g 1 g 2 g 3    : = δ e,g 3 g 2 g 1    α g 1 g 2 g 3    . (1) F or eac h plaquette β , the plaquette op erator asso ciated with a group elemen t h is defined by ˆ P β ( h )      g 1 g 2 g 5 g 3 g 4 g 6 β      : =       hg 1 hg 2 hg 5 hg 3 hg 4 hg 6 β       , (2) or, equiv alen tly , ˆ P β ( h ) = Q i ∈ β L ( i ) h , where L ( R ) ( i ) h denotes the left (righ t) action of h on edge i , which is equal to the righ t (left) action of ¯ h when the orien tation of edge i is rev ersed. The total plaquette operator is ˆ P β = 1 | G | P h ∈ G ˆ P β ( h ). 3 The Hamiltonian of Kitaev’s quantum double mo del is H = − X vertex α ˆ V α − X plaquette β ˆ P β . (3) Note that ˆ V α and ˆ P β are pro jectors and m utually commute for all α and β . Consequently , the ground-state subspace consists of the common +1 eigenspaces of all ˆ V α and ˆ P β . A quasiparticle excitation is sp ecified b y a conjugacy class [ g ] together with an irreducible representation π of the cen tralizer of a chosen representativ e g of that conjugacy class. In addition, there are non-top ological (lo cal) degrees of freedom: one may lab el them by an element pg ¯ p ∈ [ g ] and a basis vector ˆ e i of the represen tation space, and denote the corresp onding excitation b y ([ g ] , π ) pi (often suppressing the non-top ological index when it is not essential). Quasiparticles are created and transp orted by ribb on op erators. A general ribb on op erator, labeled by a pair of group elemen ts g and h , is defined on a complete path: namely , a path P on the direct lattice (with edge indices i ∈ I ) together with a path P ∗ on the dual lattice (with dual-edge indices i ∗ ∈ I ∗ ) suc h that P ∗ passes through ev ery plaquette immediately to the right of P . The corresp onding ribbon op erator takes the form F h,g ( P ) = δ e, Q i ∗ ∈ I ∗ g i ∗ ·   Y i ∗ ∈ I ∗ , 0 Y i ∈ I i ∗ L ( i ) ( g i ∗ ...g 1 ∗ ) h ( ¯ g 1 ∗ ... ¯ g i ∗ )   , (4) where I i ∗ denotes the set of edges of P b elonging to the plaquette adjacent to the dual edges i ∗ and i ∗ + 1. F or example: F h,g (red path)          y 1 y 2 y 3 y 4 y 5 x 1 x 2 g h i x 1 h i x 2 x 1 h          = δ g ,x 2 x 1          h · y 1 h · y 2 i x 1 h · y 3 i x 2 x 1 h · y 4 i x 2 x 1 h · y 5 x 1 x 2          (5) where i x ( h ) = xh ¯ x . Simple any ons can b e obtained by taking appropriate linear combinations of ribb on op erators. Concretely , one defines W ([ h ] ,π ) pi,q j (path) = X z ∈ Z ( h ) π ( z ) ij F ph ¯ p,qz ¯ p (path) , (6) whic h creates, at the terminating site of the path, a simple an y on lab eled by ([ g ] , π ), and simultaneously creates its dual an y on ([ ¯ g ] , ¯ π ) at the starting site. The labels pi and q j enco de the in ternal non-top ological degrees of freedom at the terminal and initial sites, resp ectively . B. Twisted group-based cluster state from mo dified quantum double In this section we construct a family of one-dimensional qudit chains obtained from a suitably mo dified quan tum double mo del, and clarify their relation to the group-based cluster state [5]. Our construction starts from the defect structures of the quan tum double mo del in troduced in the previous section. the motiv ation for adopting this particular setup will b e explained in Sec. V. W e first fo cus on t wo fundamen tal t ypes of gapp ed boundaries, whic h generalize the smooth and rough boundaries of the Kitaev toric co de [3] and play a central role in our construction [4]. (In the language of the SymTFT framework, these tw o b oundaries corresp ond exactly to the Vec G and Rep ( G ) b oundaries in tro duced earlier.) Concretely , as sho wn in Fig. 1, the “smooth” b oundary is characterized by a mo dified vertex term, while the “rough” boundary is c haracterized by a mo dified plaquette term. In our conv ention, the semi-vertex operators on the smo oth b oundary tak e the form ˆ V α = δ g 5 ,g 6 (W e will work straight forward on the pro jective of this pro jector.) and semi-plaquette op eratores on the rough b oundary take the form ˆ P β ( h ) = Q 4 i =1 L ( i ) h , as sho wn in Fig. 1. 4 β 1 6 5 4 3 2 α FIG. 1. the “smo oth” and “rough” b oundary of quantum double model B ϕ α β smo oth rough 1 9 8 7 5 4 3 2 6 FIG. 2. the domain wall B ϕ b et ween smo oth and rough b oundary Another key defect we will use is a domain wall labeled b y an endomorphism ϕ ∈ End( G ), denoted by B ϕ . Crossing B ϕ , the lo cal constraints of the quantum double mo del are ϕ -twisted: vertex and plaquette terms adjacent to the wall are mo dified so that the group multiplication rules and gauge transformations are acted on by ϕ on one side of the w all. F or instance, in the geometry of Fig. 2, the vertex constraint at α is replaced by ˆ V α = δ e,ϕ ( g 1 g 2 ) g 3 , and a plaquette term crossing the wall acts as ˆ P β ( h ) = L (4) h L (5) h L (6) ϕ ( h ) L (7) ϕ ( h ) L (8) ϕ ( h ) L (9) ϕ ( h ) . In tuitively , edges on one side of the w all transform by h , while edges on the other side transform b y ϕ ( h ). Com bining the tw o gapp ed b oundaries discussed ab ov e with the ϕ -domain wall, we no w take the quasi-1D limit by shrinking the width of the configuration in Fig. 2 to a single strip (one “grid”), yielding an effective (1 + 1)-dimensional mo del shown in Fig. 3. In this limit the Hamiltonian is generated b y tw o families of m utually commuting op erators, whic h we interpret as the effective vertex and plaquette terms along the c hain: ˆ V i : = δ e,ϕ ( ¯ g i ) g i + 1 2 ϕ ( g i +1 ) , ˆ P i : = 1 | G | X h ∈ G R ( i − 1 2 ) ϕ ( ¯ h ) L ( i ) h L ( i + 1 2 ) ϕ ( h ) . Here the half-integer lab els refer to edge degrees of freedom b et ween neighboring plaquette, and L and R denote the left and right actions introduced ab ov e. W e ha v e also identified the tw o adjacent edges meeting at each semi-v ertex, so that eac h integer p osition carries a single effective edge degree of freedom. g i g i +1 g i + 1 2 g i − 1 2 B ϕ ... ... FIG. 3. the contracted 1 + 1 d model The con traction form a 2 + 1 d theory to a 1 + 1 d one seems redundancy . How ev er, this interpretation enco des the ideology of SymTFT whic h is essential for the classification and realization of SPT phases. W e now in tro duce tw o families of op erators—an X -type and a Z -type—which generalize the Pauli op erators ap- p earing in the group-based cluster mo del [5]. They are defined b y ← X h = X g ∈ G | g ¯ h ⟩⟨ g | , → X h = X g ∈ G | hg ⟩⟨ g | , Z Γ = X g ∈ G Γ( g ) | g ⟩⟨ g | , 5 for any h ∈ G and any representation Γ of G . Here → X h and ← X h implemen t the left and right regular actions (i.e., group m ultiplication on the ket from the left or from the righ t), while Z Γ is diagonal in the group-element basis and w eights each basis state | g ⟩ by the representation matrix Γ( g ). In terms of these op erators, the quasi-1D vertex and plaquette constraints derived ab ov e can b e rewritten as ˆ V i = 1 | G | X Γ ∈ irrep( G ) d Γ T r  Z † ( i ) ϕ ∗ Γ · Z ( i + 1 2 ) Γ · Z ( i +1) ϕ ∗ Γ  , ˆ P i = 1 | G | X h ∈ G ← X ( i − 1 2 ) ϕ ( h ) → X ( i ) h → X ( i + 1 2 ) ϕ ( h ) . The first expression is a character/F ourier resolution of the group-v alued constraint app earing in ˆ V i : the sum ov er Γ ∈ irrep( G ), w eigh ted by d Γ , enforces the corresponding δ -type condition through the standard orthogonality relations of irreducible representations. The second expression makes the “twisted” gauge transformation structure manifest: the degree of freedom at the integer site i transforms b y h , while the neighboring half-in teger degrees of freedom transform by ϕ ( h ), with the left/right arrows recording whether the multiplication acts on the ket from the left or from the righ t. Therefore, the Hamiltonian of the effective (1 + 1)-dimensional mo del reads H = − 1 | G | X i   X Γ ∈ irrep( G ) d Γ T r  Z † ( i ) ϕ ∗ Γ · Z ( i + 1 2 ) Γ · Z ( i +1) ϕ ∗ Γ  + X h ∈ G ← X ( i − 1 2 ) ϕ ( h ) → X ( i ) h → X ( i + 1 2 ) ϕ ( h )   . (7) By construction, all terms in this Hamiltonian commute with one another, so the mo del is exactly solv able and its ground states can b e obtained by imp osing the simultaneous +1 eigenv alue conditions of all ˆ V i and ˆ P i . In particular, for an N sites op en chain, one can write an explicit ground state in the group-elemen t basis as | Ω ⟩ = X g 2 ,...,g N − 1 | g 1 ⟩ ⊗ | ϕ ( g 1 ¯ g 2 ) ⟩ ⊗ | g 2 ⟩ ⊗ | ϕ ( g 2 ¯ g 3 ) ⟩ ⊗ ... ⊗ | g N ⟩ . (8) This expression mak es the structure of the quasi-1D constraints transparent: the integer sites carry the v ariables g i , while the half-integer sites store the ϕ -twisted “differences” b etw een neigh b oring group elements. Note that the t wo terminal g 1 and g N are arbitrary , thus the ground space is | G | 2 degeneracy . Tw o limits are particularly instructive. When ϕ = e , the half-integer degrees of freedom are pinned to the identit y elemen t, and the ground state reduces to | Ω ⟩ = X g 2 ,...,g N − 1 | g 1 ⟩ ⊗ | e ⟩ ⊗ | g 2 ⟩ ⊗ | e ⟩ ⊗ ... ⊗ | g N ⟩ , whic h is a simple pro duct state. In con trast, when ϕ = id, the half-integer sites record the un twisted relative group elemen ts g i ¯ g i +1 , and w e obtain | Ω ⟩ = X g 2 ,...,g N − 1 | g 1 ⟩ ⊗ | g 1 ¯ g 2 ⟩ ⊗ | g 2 ⟩ ⊗ | g 2 ¯ g 3 ⟩ ⊗ ... ⊗ | g N ⟩ , whic h is precisely the group-based cluster state introduced in [5]. The global symmetry act on the chain is characterized by tw o families of ribb on op erators corresp onding to the G and Rep( G ) symmetries resp ectiv ely , which are: A g = B ϕ ... ... g = Y i ← X ( i ) g , B Γ αβ = B ϕ ... ... Γ = Y i Z ( i + 1 2 ) Γ ! αβ . (9) Within the ground-state subspace, one can directly chec k that the global symmetry acts effectiv ely as op erators supp orted only at the tw o ends of the chain. In particular, the symmetry generators reduce to the b oundary actions A g = ← X (1) g ← X ( N ) g , B Γ αβ =  Z (1) ϕ ∗ Γ · Z † ( N ) ϕ ∗ Γ  αβ . (10) 6 That is, although the symmetry is defined microscopically as a global transformation, its action on the low-energy (ground) subspace is represented entirely b y local op erators at the left and righ t boundaries. This “symmetry frac- tionalization” on to the ends can be viewed as the appearance of protected edge modes, which is an essen tial diagnostic of SPT order. Notice that the edge modes of G symmetry and Rep( G ) symmetry comm ute within eac h family but do not comm ute with one another, F or example, ← X (1) g Z (1) ϕ ∗ Γ = h Z (1) ϕ ∗ Γ · Γ( ϕg ) i ← X (1) g . (11) Ho wev er, the total edge operators – the pro duct of the left and righ t op erators – still comm ute. This precise serves as an in terpretation of what is called “intrin sically mixed”. Finally , we emphasize that the discussion ab o ve assumes that b oth endp oin ts are in teger sites (i.e., horizontal edges in our quasi-1D geometry). The case in which the endp oin ts lie on half-integer sites (vertical edges) is completely analogous, with the corresp onding b oundary op erators obtained b y the same reduction pro cedure. I II. CLASSIFY OF 1 + 1 D Rep( G ) × G SPT Ha ving constructed, in Sec.II, a family of quasi-1D Hamiltonian from a mo dified quantum double mo del and iden tified its ground state as a ϕ -twisted group-based cluster state, we now turn to a purely categorical classification of the corresponding (1 + 1)D SPT phases. The goal of this section is to explain wh y the lattice constructions obtained from different domain-wall twists ϕ ∈ End( G ) exhaust all Rep( G ) × G (equiv alen tly Rep ( G ) ⊠ V ec G ) SPT phases in (1 + 1) dimensions, and to provide a compact inv ariant that distinguishes them. The guiding principle is that (1 + 1)D b osonic SPT phases with categorical symmetry C are classified by C -mo dule categories o ver Vec , or equiv alen tly b y fib er functors C → Vec [22]. In the present problem the symmetry category is H : = Rep ( G ) ⊠ V ec G , so w e seek all H -module structures on V ec and iden tify the associated monoidal functors. Ph ysically , this classification organizes the possible w a ys in which the Rep( G ) and G symmetry defects can b e consisten tly “glued” on the boundary; mathematically , it c haracterizes the admissible condensations in the stac k ed bulk Z ( H ) ≃ D 2 G that correspond to SPT (i.e., short-range en tangled) phases. This section is structured as follo ws. In Sec. I I I A we classify Rep ( G ) ⊠ V ec G -mo dule structures on Vec by translating the problem into one ab out bimo dules ov er suitable algebra ob jects inside Vec G × G . In Sec. I I I B we compute the corresp onding asso ciator data and extract the fib er functor explicitly , showing that the resulting inv ariant dep ends only on the class of ϕ mo dulo inner automorphisms. These results will then be matched, in Sec. IV, with the domain- w all picture and the condensable algebra A ϕ in D 2 G , thereby connecting the categorical classification to the lattice realization constructed earlier. A. Rep( G ) × G -mo dule structures on Vec Constructing the desired module structure directly from the axiomatic definition of an H -mo dule category is cum- b ersome. Instead, we use a standard identification that realizes H as a bimo dule category inside G . Let G 1 : = { ( g , e ) | g ∈ G } ⊆ G × G, and denote by ˆ G 1 the corresp onding group algebra, viewed as an algebra ob ject in G . Then H is equiv alen t to the category of ˆ G 1 -bimo dules in G . More precisely , there is an equiv alence H ∼ − → Bimo d G ( ˆ G 1 ); Γ × g 7→ M h ( h, g ) ⊕ dim(Γ) , p, q ! ≡ g Γ , where p and q denote the left and right ˆ G 1 -mo dule structures, resp ectively . 7 Concretely , fixing a basis { ˆ e ( h, g ) i } of g Γ (with i = 1 , . . . , d Γ , where d Γ is the dimension of Γ), the left action p is defined b y ˆ G 1 ⊗ g Γ → g Γ; ( h, e ) ⊗ ˆ e ( h 0 , g 0 ) i 7→ X j Γ − 1 ( h ) ij ˆ e ( hh 0 , g 0 ) j , while the righ t action q is given by g Γ ⊗ ˆ G 1 → g Γ; ˆ e ( h 0 , g 0 ) i ⊗ ( h, e ) 7→ ˆ e ( h 0 h, g 0 ) i . In other words, ˆ G 1 acts on the left by left multiplication on the first group comp onent, accompanied by the repre- sen tation matrix Γ − 1 ( h ) on the multiplicit y space, whereas the right action is the regular right m ultiplication on the same comp onen t and is trivial on the multiplicit y index. In diagram: i j g Γ ˆ G 1 ˆ G 1 ( g 2 , 1) ( g, h ) ( g 1 , 1) ( g 2 g g 1 , h ) = i j g Γ ˆ G 1 ˆ G 1 ( g 2 , 1) ( g, h ) ( g 1 , 1) ( g 2 g g 1 , h ) = Γ − 1 ( g 2 ) ij . (12) W e no w inv ok e Theorems 7.12.16 and 7.10.1 of [1]. T aken together, they imply that any mo dule category o ver A C A is equiv alen t to one of the form A C B , where A and B are algebra ob jects in C . Here we use the shorthand A C B for the category Bimo d C ( A, B ) of ( A, B )-bimo dules in C , and we will adopt the same conv en tion throughout. A t this p oin t it is helpful to distinguish t wo closely related notions of “mo dule” that appea r in the literature. The first is a mo dule c ate gory ov er a monoidal category; the second is a c ate gory of mo dule obje cts ov er an algebra ob ject in a monoidal category . In our setting, Theorem 7.10.1 of [1] provides the precise bridge b et w een these viewp oints, allo wing us to mov e freely b etw een them. F or clarity , we will refer to the action of A C A on A C B as the mo dule pr o duct , and to the underlying algebra action on individual ob jects as the mo dule action . Concretely , the module pro duct is the functor ⊙ : A C A × A C B → A C B ; M ⊙ N : = M ⊗ A N , (13) where M ⊗ A N denotes the relativ e tensor pro duct o v er A . In particular, M ⊗ A N can be realized as the image of an idemp oten t endomorphism on M ⊗ N (equiv alen tly , as the co equalizer implementing the identification of the right A -action on M with the left A -action on N ); in the graphical calculus used b elo w, this idemp oten t is represen ted by the follo wing morphism, which is straightforw ard to chec k is a pro jector: P ⊗ A = ˆ G 1 M N (14) Therefore, sp ecifying an H -mo dule structure on Vec is equiv alen t to sp ecifying an algebra ob ject ˆ H in G such that the resulting bimo dule category satisfies ˆ G 1 G ˆ H ≃ Vec . Recall that the simple ob jects in ˆ G 1 G can b e describ ed in terms of G -graded ˆ G -algebras; more concretely , they are represen ted by M g : = M h ∈ G ( g , h ) , 8 with the mo dule structure induced b y group multiplication. In order for ˆ G 1 G ˆ H to collapse to Vec , whic h has a unique simple ob ject, the subgroup H ⊆ G × G underlying ˆ H must satisfy tw o basic constrain ts. First, the pro jection of H onto the second factor must b e surjective. If instead the second component ranges only o ver a prop er subgroup N ⊊ G , then simples in ˆ G 1 G ˆ H w ould b e graded by righ t cosets G/ N , pro ducing more than one simple ob ject. Second, the fib er of this pro jection m ust b e trivial: for each h ∈ G there must b e a unique g ∈ G suc h that ( g , h ) ∈ H . Equiv alen tly , H cannot contain a nontrivial subgroup of the form ( N , 1) with N ⊆ G , since such a stabilizer w ould lead to additional simples lab eled by irreducible representations of N . The only subgroups H ⊆ G × G satisfying b oth conditions are graphs of endomorphisms, H ϕ : = { ( ϕ ( h ) , h ) | h ∈ G } , (15) with ϕ ∈ End( G ). There do exist other subgroup algebras of dimension larger than | G | that can also define bimo dule categories; ho wev er, these necessarily inv olv e t wisted (in particular, anomalous) data. In our setting, such t wists correspond to phases that carry an anomaly under G or Rep( G ) and hence are not in trinsically mixed. W e defer a detailed discussion of this p oin t to Sec. VI. T o summarize, H -mo dule structures on Vec are parametrized (in the unt wisted, anomaly-free sector relev an t here) b y endomorphisms ϕ ∈ End( G ). F or each ϕ , the corresp onding mo dule category is ˆ H ϕ G ˆ G 1 , and it has a single simple ob ject Φ : = M g ,h ( g , h ) , whose left and righ t mo dule actions are given by left and right group multiplication, resp ectively . B. fib er functor In this subsection we determine the explicit mo dule pr o duct of H on the mo dule category constructed in Sec. I II A, and then extract the asso ciated fib er functor. Conceptually , this step connects the “static” classification of mo dule categories (i.e., which H -mo dules ov er Vec exist) to the monoidal data that distinguish SPT phases: the associativity constrain ts of the mo dule pro duct precisely enco de the tensor-compatibility isomorphisms of the fib er functor. Concretely , for a mo dule category M ≃ Vec ov er a monoidal category A , the fib er functor F can b e reconstructed from the action ⊙ by restricting to the distinguished simple ob ject 1 ∈ M . Namely , F ( X ) ≡ X ⊙ 1 ⇐ ⇒ X ⊙ M ≡ F ( X ) ⊗ M , (16) and the coherence of the mo dule asso ciator translates into the monoidal structure of F : X ⊙ ( Y ⊙ 1) ∼ − − − − → m X,Y , 1 ( X ⊗ Y ) ⊙ 1 ⇐ ⇒ F ( X ) ⊗ F ( Y ) ∼ − − − → J X,Y F ( X ⊗ Y ) . (17) Recall from Eq. 13 that the mo dule pro duct is implemented by the relative tensor pro duct. In our realization H ≃ ˆ G 1 G ˆ G 1 acting on the mo dule category ˆ G 1 G ˆ H ϕ , this action tak es the form ˆ G 1 G ˆ G 1 × ˆ G 1 G ˆ H ϕ ⊗ ˆ G 1 − − − → ˆ G 1 G ˆ H ϕ : g Γ ⊗ ˆ G 1 Φ ≃ Φ ⊕ d Γ . (18) Imp ortan tly , the ab o v e isomorphism is not canonical: while the ob ject g Γ ⊗ ˆ G 1 Φ is isomorphic to a direct sum Φ ⊕ d Γ as an ob ject of ˆ G 1 G ˆ H ϕ , the identification dep ends on a c hoice of intert wining data. This non-canonicity is precisely what gives rise to the monoidal structure maps J X,Y in Eq. 17, and hence to the ev entual classification inv ariant v alued in End( G ) / Inn( G ). More concretely , the isomorphism in g Γ ⊗ ˆ G 1 Φ ≃ Φ ⊕ d Γ 9 is not necessarily the identit y map, b ecause the relativ e tensor product inherits a nontrivial left ˆ G 1 -action from g Γ. T o make this explicit, define a basis of g Γ ⊗ ˆ G 1 Φ as follows. Consider the image of ˆ e ( h ′ ¯ h, g ) i × ϕ ( h, ¯ g g ′ ) in the quotient defining ⊗ ˆ G 1 , and denote the resulting basis vectors by ˆ µ ( h ′ , g ′ ) i (here h is irrelev an t mo dulo ˆ G 1 ). With resp ect to this basis, the induced ( ˆ G 1 , ˆ H ϕ )-bimo dule action takes the form ( h 1 , e ) ⊗ ˆ µ ( h, g ) i ⊗ ( ϕ ( h 2 ) , h 2 ) 7→ X j Γ − 1 ( h 1 ) ij ˆ µ ( h 1 hϕ ( h 2 ) , g h 2 ) j , (19) whic h mirrors the defining left action of g Γ. In the next step, we will use a suitable change of basis to isolate d Γ canonical copies of Φ; the resulting comparison maps b et ween different tensor-pro duct orderings will then determine J X,Y . T o trivialize the residual Γ-dep endence in the left ˆ G 1 -action, it is con venien t to p erform a further change of basis. Define ˆ µ ′ ( h, g ) i : = X j Γ( ϕ ( h ) ¯ g ) ij ˆ µ ( h, g ) j , (20) whic h is an inv ertible transformation on the multiplicit y index for each fixed ( h, g ). A direct substitution into the bimo dule action shows that the Γ − 1 ( h 1 ) factor is absorb ed, and the action b ecomes purely regular: ( h 1 , e ) ⊗ ˆ µ ′ ( h, g ) i ⊗ ( ϕ ( h 2 ) , h 2 ) 7→ ˆ µ ′ ( h 1 hϕ ( h 2 ) , g h 2 ) i . (21) In particular, for eac h fixed i , the subspace M g ,h ˆ µ ′ ( h, g ) i is stable under the left ˆ G 1 - and right ˆ H ϕ -actions, and hence forms a copy of Φ as a ( ˆ G 1 , ˆ H ϕ )-bimo dule. Consequen tly , w e obtain a decomp osition g Γ ⊗ ˆ G 1 Φ ≃ Φ ⊕ d Γ , where the d Γ summands are precisely the subspaces lab eled by i . The remaining ambiguit y in identifying these summands across iterated tensor products is what will b e captured b y the coherence data summarized in the follo wing diagram (with L ≡ L g ,h ): Φ L ˆ µ 1 ( h, g 1 g ) ⊕ d Γ 1 Φ ⊕ d Γ 1 L  ˆ µ 2 ( h, g 2 g 1 g ) ⊕ d Γ 2 ⊗ ˆ µ 1 ( h, g 2 g 1 g ) ⊕ d Γ 1  L ˆ µ 2 ( h, g 2 g 1 g ) ⊕ d Γ 2 d Γ 1 Φ ⊕ d Γ 1 d Γ 2 g 1 Γ 1 ⊗ ˆ G 1 g 2 g 1 (Γ 2 ⊗ Γ 1 ) ⊗ ˆ G 1 L Γ 1 ( ϕ ( g 1 g ) ¯ h ) Γ 1 ( ϕ ( g 2 )) g 2 Γ 2 ⊗ ˆ G 1 L  (Γ 2 ⊗ Γ 1 )( ϕ ( g 2 g 1 g ) ¯ h )  L  Γ 2 ( ϕ ( g 2 g 1 g ) ¯ h ) ⊗ I d Γ 1  , Recalling Eq. 17, the comparison b et ween the tw o natural wa ys of identifying g 2 Γ 2 ⊙ ( g 1 Γ 1 ⊙ 1) and ( g 2 Γ 2 ⊗ g 1 Γ 1 ) ⊙ 1 yields the monoidal structure isomorphism J g 2 Γ 2 , g 1 Γ 1 = Γ 1 ( ϕ ( g 2 )) . (22) Equiv alen tly , the nontrivial asso ciativity data of the mo dule pro duct is enco ded by ev aluating the representation Γ 1 on the ϕ -image of the group elemen t g 2 . This is the precise sense in which the endomorphism ϕ controls the “mixed” coupling b et w een the Rep( G ) and G sectors. It follo ws that the fib er functor is characterized, up to equiv alence, by ϕ mo dulo inner automorphisms, namely b y a class in End( G ) / Inn( G ). Indeed, if f ∈ Inn( G ) is an inner automorphism, then for ev ery represen tation Γ the represen tations Γ ◦ f and Γ are equiv alen t, so replacing ϕ by f ◦ ϕ do es not change the resulting J -symbols up to a natural monoidal isomorphism. Consequently , only the outer part of ϕ can lab el distinct H -SPT phases. 10 T o summarize, the classification data of the in trinsically mixed Rep( G ) × G SPT phases constructed here can be pac k aged in to the monoidal structure J X,Y of the associated fib er functor, and this structure is completely determined b y the class of ϕ in End( G ) / Inn( G ). F rom the physical viewp oin t, the same J -symbols provide the top ological resp onse: they dictate how symmetry defects fuse in the effectiv e (1 + 1)D theory , equiv alently the phase factors (or matrix intert winers) acquired when a G -defect is mov ed past a Rep( G )-defect. In the lattice realization, this resp onse manifests as the ϕ -twisted action across the domain wall and, up on dimensional reduction, as the protected edge-mo de transformation la ws enco ded b y the b oundary symmetry op erators. IV. CHARA CTERIZE SPT BY CONDENSABLE ALGEBRAS Ha ving classified the Rep( G ) × G SPT phases in terms of fib er functors (equiv alen tly , mo dule categories ov er H with underlying category Vec ), we now relate this abstract classification to the more physical language of any on condensation and domain walls in the corresp onding (2 + 1)D top ological order. Concretely , we w ould like to identify the condensable (Lagrangian) algebra in the bulk that realizes each SPT phase, and to match it with the boundary data enco ded b y the H -mo dule structure. This step provides a bridge b etw een the categorical inv ariant extracted in the previous subsection (the monoidal data J X,Y , hence ϕ ∈ End( G ) / Inn( G )) and the lattice/domain-wall picture dev elop ed later in Sec. V. Within the SymTFT framew ork, whic h we will review in detail in Sec. V, ev ery 1 + 1 d gapp ed phase with a categorical symmetry C corresp onds one-to-one with a ph ysical gapp ed b oundary in a sandwich construction. In this construction, the bulk top ological order is asso ciated with the top ological skeleton Z ( C ). The categorical description of such a physical b oundary has a “double identit y”: it can b e viewed b oth as a bimo dule category ov er an algebra A in C , and as a righ t mo dule category ov er a Lagrangian algebra A in Z ( C ). Our task in this section is to iden tify these t wo algebras. In ph ysical terms, these corresp ond resp ectiv ely to b eing “gauged by A ” and “condensed by A ”. F or a symmetry-protected top ological phase, no nontrivial order parameters (i.e., an yon lines) can stretch b etw een the t wo b oundaries. Categorically , this is equiv alen t to requiring that the domain wall b et ween C and A C A , namely C A , is trivial (v acuum). This is precisely why categorical 1 + 1 d SPT phases are classified by fib er functors. In our case, where C = H , w e therefore seek algebra ob jects A ∈ H and A ϕ ∈ Z ( H ) ≃ D G ⊠ D G suc h that H A ≃ ˆ G 1 G ˆ H ϕ , (23) ( D G ⊠ D G ) A ϕ ≃ A H A . (24) Let us first fo cus on Eq. 23. T o this end, we briefly recall the notion of the internal hom. F or a left C -mo dule category M and ob jects X , Y ∈ M , the internal hom hom C ( X, Y ) ∈ C is characterized (when it exists) by the natural isomorphism α : hom M ( • ⊙ X, Y ) ≃ hom C ( • , hom C ( X, Y )) . W e denote by ev X,Y : hom C ( X, Y ) ⊙ X → Y the morphism corresp onding, under α , to id hom C ( X,Y ) ; equiv alen tly , ev X,Y is the ev aluation map implementing the universal prop ert y of hom C ( X, Y ). No w sp ecialize to C = H and M = ˆ G 1 G ˆ H ϕ . Applying Theorem 7.10.1 of [1], we obtain the algebra ob ject A = hom H ( G 1 ⊗ H ϕ , G 1 ⊗ H ϕ ) . (25) Moreo ver, A carries a canonical algebra structure induced by comp osition of internal endomorphisms. Concretely , for an y X ∈ M , the multiplication on hom H ( X, X ) is determined b y the comp osite (suppressing iden tity morphisms) hom H ( X, X ) ⊗ hom H ( X, X ) ⊙ X ev X,X ◦ ev X,X − − − − − − − − − → X α X 7− − → hom H ( X, X ) ⊗ hom H ( X, X ) m − → hom H ( X, X ) , (26) whic h is the internal version of the usual multiplication by comp osition of endomorphisms. Using the graphical calculus, we next observe the following diagrammatic isotopy , which implements the univ ersal prop ert y of the relative tensor product and amounts to “sliding” the ˆ G 1 -actions through the pairing. In particular, 11 for X, Y , Z ∈ ˆ G 1 G ˆ H ϕ one has hom ˆ G 1 G ˆ H ϕ ( X ⊗ ˆ G 1 Y , Z ) = ˆ G 1 ˆ G 1 ˆ H ϕ Y X Z ≃ ˆ H ϕ ˆ G 1 ˆ G 1 X Z Y ∗ = hom H ( X, Z ⊗ ˆ H ϕ Y ∗ ) , (27) where Y ∗ denotes the dual of Y as a right ˆ H ϕ -mo dule. In particular, comparing the tw o expressions sho ws that the in ternal hom in H is realized by hom H ( Y , Z ) = Z ⊗ ˆ H ϕ Y ∗ . Applying this to X = Y = Z = G 1 ⊗ H ϕ , w e obtain A = G 1 ⊗ H ϕ ⊗ ˆ H ϕ H ∗ ϕ ⊗ G ∗ 1 = G 1 ⊗ H ∗ ϕ ⊗ G ∗ 1 . Finally , Eq. 26 iden tifies the multiplication on A as the comp osition of internal endomorphisms; in the present realization this reduces to A ⊗ A m H ϕ ◦ ev G 1 − − − − − − − → A (again suppressing iden tity morphisms), where m H ϕ is the group m ultiplication of H ϕ . Next, w e pro ceed to determine the corresponding Lagrangian algebra A ϕ . Before doing so, we note a useful simplification. Since A is an algebra ob ject in H = ˆ G 1 G ˆ G 1 , the bimodule category A H A can b e canonically iden tified with A G A . This iden tification follows from tw o complementary observ ations. On the one hand, given any ob ject in A G A , w e ma y equip it with an ˆ G 1 -bimo dule structure by restricting the A -actions along the unit embedding ˆ G 1 u − → A . The resulting left and righ t ˆ G 1 -actions are automatically compatible with the original A -bimo dule structure, so the ob ject may equally well b e regarded as an ob ject of A H A . On the other hand, any ob ject of A H A admits a canonical image in A G A under the forgetful functor that discards the ˆ G 1 -bimo dule structure while retaining the underlying A - bimo dule in G . These tw o constructions are inv erse to each other up to canonical isomorphism, yielding the claimed iden tification A H A ≃ A G A . As a result, it is technically more conv enien t to work with A G A when computing A ϕ , rather than analyzing A H A directly . It’s kno wn that [1], the Drinfeld centers of C and A C A are equiv alen t in a canonical wa y . More precisely , there is a braided equiv alence Z ( C ) ≃ − → Z ( A C A ) : Z 7→ Z ⊗ A, (28) whic h sends an ob ject Z ∈ Z ( C ) (equipp ed with its half-braiding) to the ob ject Z ⊗ A viewed in Z ( A C A ). Under this equiv alence, the ( A, A )-bimo dule structure on Z ⊗ A is sp ecified as follows. The right A -action is the ob vious one on the last tensor factor, while the left A -action uses the half-braiding of Z . Concretely , the left action is giv en by the comp osite A ⊗ ( Z ⊗ A ) γ A,Z ⊗ id A − − − − − − → Z ⊗ A ⊗ A id Z ⊗ m − − − − − → Z ⊗ A, (29) where γ A,Z denotes the half-braiding isomorphism asso ciated to Z and m is the multiplication of A . In the graphical calculus, these left and right mo dule structures are depicted by: left mo dule: m Z A A ; righ t mo dule: m A A Z . 12 It is also w ell kno wn that the b oundary topological order C can be recov ered from its bulk Z ( C ) by condensing the canonical Lagrangian algebra I C ( 1 ). Equiv alently , C ≃ Z ( C ) I C ( 1 ) . Here I C denotes the right adjoint of the forgetful functor F : Z ( C ) → C , characterized by the adjunction isomorphism hom Z ( C ) ( Z, I C ( c )) ≃ hom C ( F ( Z ) , c ) . (30) W e now return to our setting. Since the unit ob ject 1 A G A of the bimo dule category A G A is simply A itself, Eqs. 28 and 30 together imply that the corresp onding Lagrangian algebra A ϕ ∈ O ( D 2 G ) decomp oses as A ϕ = M Z ∈O ( D 2 G ) Z ⊕ dim ( hom A G A ( Z ⊗ A,A ) ) . (31) Th us, determining A ϕ reduces to computing the multiplicit y space of ( A, A )-bimodule morphisms from Z ⊗ A to A . In other w ords, we must characterize all bimo dule maps F : Z ⊗ A → A ; suc h an F is constrained by the requirement that it in tertwines b oth the left and right A -actions, i.e., it must satisfy: m F A A Z = m F A A Z and m F Z A A = m F Z A A . (32) W e now expand the bimodule constrain ts in Eq. 32 in comp onents. Before doing so, w e fix some notation. Recall that A = G 1 ⊗ H ∗ ϕ ⊗ G ∗ 1 . W e will write A in terms of comp onents A ( g 1 , h 1 , g 2 ) : = ( g 1 , e ) ⊗ ( ϕ ( h 1 ) , h 1 ) ⊗ ( ¯ g 2 , e ) , so that the m ultiplication induced by Eq. 26 takes the form A ( g 1 , h 1 , g 2 ) ⊗ A ( g 3 , h 2 , g 4 ) ≡ δ ¯ g 2 g 3 A ( g 1 , h 1 h 2 , g 4 ) . In addition, we denote an irreducible ob ject of D 2 G b y an any on pair ([ g ] , ρ ) ⊠ ([ h ] , π ), where [ g ] and [ h ] are conjugacy classes and ρ and π are irreducible representations of the cen tralizers of g and h , resp ectively . W e write its basis comp onen ts as ˆ e ( g , h ) i,j , with i and j ranging ov er the dimensions of ρ and π . A first immediate consequence of Eq. 32 is that, whenev er F is nonzero on a simple tensor ˆ e ( g , h ) i,j ⊗ A ( g 1 , h 1 , g 2 ), its image must b e compatible with the delta function in the m ultiplication of A . In particular, F ( ˆ e ( g , h ) i,j ⊗ A ( g 1 , h 1 , g 2 )) m ust b e prop ortional to A ( g 1 , hh 1 , g 2 ). Equating the corresp onding group lab els forces g g 1 ϕ ( h 1 ) g 2 = g 1 ϕ ( hh 1 ) g 2 , otherwise the image under F is necessarily zero. This constraint implies g = g 1 ϕ ( h ) ¯ g 1 , and hence [ g ] = [ ϕ ( h )]. Accordingly , w e may assume without loss of generality that F has the following comp onent form: F ( ˆ e ( g 1 ϕ ( h ) ¯ g 1 , h ) i,j ⊗ A ( g 1 , h 1 , g 2 )) = f i,j ( h, g 1 , h 1 , g 2 ) A ( g 1 , hh 1 , ¯ g 2 ) . (33) F rom the first equation in Eq. 32, we deduce the constraint f i,j ( h, g 1 , h 1 , g 2 ) = f i,j ( h, g 1 , h 1 h 2 , g 4 ) , ∀ h 2 , g 4 , (34) 13 whic h immediately implies that f i,j is independent of h 1 and g 2 . W e may therefore write f i,j ( h, g 1 , h 1 , g 2 ) = f i,j ( h, g 1 ). Substituting this simplification in to the second equation of Eq. 32 yields F ( γ ( A ( g 1 , h 1 , ¯ g 2 ) ⊗ ˆ e ( g 2 ϕ ( h ) ¯ g 2 , h ) i,j ) ⊗ m A ( g 2 , h 2 , ¯ g 3 )) = f i,j ( h, g 2 ) A ( g 1 , h 1 hh 2 , ¯ g 3 ) , (35) where γ denotes the half-braiding and ⊗ m indicates that the tensor pro duct on the right is follow ed by the multipli- cation of A . T o analyze Eq. 35, fix representativ es c and d of the conjugacy classes [ h ] and [ ϕ ( h )], resp ectively . Cho ose sets of elements { p } and { q } such that pc ¯ p (resp ectiv ely q d ¯ q ) runs through all elements of [ c ] (resp ectively [ d ]) exactly once. Let Y = { y } and Z = { z } denote the corresp onding centralizers. Without loss of generality , we adopt the parametrization ϕ ( c ) = d, h = pc ¯ p, h 1 = p 1 y 1 ¯ p, g 2 = q 2 z 2 ϕ ( ¯ p ) , g 1 ϕ ( p 1 y 1 ) = q 1 z 1 , (36) whic h organizes the v ariables app earing in Eq. 35 in to conjugacy-class data and centralizer data. In these v ariables, Eq. 35 reduces to the comp onent relation ρ † ( z 1 ¯ z 2 ) ik π † ( y 1 ) j l f k,l ( p 1 c ¯ p 1 , q 1 z 1 ϕ ( ¯ y 1 ¯ p 1 )) = f i,j ( pc ¯ p, q 2 z 2 ϕ ( ¯ p )) , (37) v alid for arbitrary z 1 , z 2 , y 1 , p, p 1 , q 1 , q 2 . Eliminating v ariables that play no essential role, we find that f i,j ( pc ¯ p, q z ϕ ( ¯ p )) = f i,j ( c, z ) for an y p, q and z . Substituting this back, the condition further simplifies to ρ ( z 2 ¯ z 1 ) · f ( c, z 1 ϕ ( ¯ y )) · ¯ π ( y ) = f ( c, z 2 ) , (38) where ¯ π denotes the complex conjugate representation of π . This equation shows that the existence of a nonzero solution f is equiv alen t to the existence of an intert winer b et ween ρ ◦ ϕ and ¯ π . Equiv alen tly , since ϕ ( y ) ∈ Z for y ∈ Y , ϕ induces the pullback represen tation ϕ ∗ ρ ( y ) : = ρ ( ϕ ( y )), and the intert wining condition may b e stated as ϕ ∗ ρ → ¯ π in Rep( Y ). Consequen tly , the m ultiplicity of the any on pair ([ ϕ ( h )] , ρ ) ⊠ ([ h ] , π ) in A ϕ is giv en by the intert wining num ber b et w een ϕ ∗ ρ and ¯ π , i.e., A ϕ = M O ( D ) ([ g ] , ρ ) ⊠ ([ h ] , π ) ⊕ δ [ g ] , [ ϕ ( h )] · [ ϕ ∗ ρ : ¯ π ] . (39) (Here w e hav e suppressed the equiv alence ˆ H ϕ ≃ ˆ H ∗ ϕ that app ears during the deriv ation, and we also note that ([ h ] , π ) 7→ ([ ¯ h ] , ¯ π ) is tautologically an automorphism of D G .) Example 1. 1 + 1 d Rep( S 3 ) × S 3 SPT. In our notation, S 3 = r, s | r 3 = s 2 = sr sr = 1. There are e igh t any on t yp es in D ( S 3 ). F ollowing the con ven tion in [28], we lab el them as A, B , . . . , H : A , B , and C correspond to the identit y , sign, and tw o-dimensional representation, resp ectiv ely; D and E corresp ond to the conjugacy class [ s ] asso ciated with the + and − representations of Z 2 , resp ectiv ely; F , G , and H corresp ond to the conjugacy class [ r ] asso ciated with the 1 , ω , and ¯ ω representations of Z 3 , resp ectiv ely . a. ϕ ( g ) = 1 In this case, the left any on can only b e A , B , or C , and ρ ◦ c is simply a direct sum of identit y represen tations. Therefore, they m ust pair with A , D , and E , with multiplicit y equal to the dimension of ρ . Th us, A ϕ = ( A 1 ⊕ B 1 ⊕ 2 C 1 ) ⊠ ( A 2 ⊕ D 2 ⊕ F 2 ). As will b e discussed later, this corresp onds to the trivial pro duct phase in ph ysical terms. b. ϕ ( g ) = g Here, each any on can only pair with its antiparticle, i.e., A ϕ = A 1 ⊠ A 2 ⊕ B 1 ⊠ B 2 ⊕ C 1 ⊠ C 2 ⊕ D 1 ⊠ D 2 ⊕ E 1 ⊠ E 2 ⊕ F 1 ⊠ F 2 ⊕ G 1 ⊠ H 2 ⊕ H 1 ⊠ G 2 . As will b e seen in the following, this corresp onds to the cluster phase in ph ysics. c. ϕ (1 , r , r 2 ) = 1 , ϕ ( s, sr, sr 2 ) = s In this case, [ s ] maps to itself while [1] and [ r ] map to [1]. The conjugacy classes [1] and [ s ] b ehav e similarly to the second case, as ϕ preserv es their conjugacy classes and centralizers. How ev er, [ r ] and its centralizer Z 3 are all mapp ed to 1, so on the right side only F remains in the algebra, pairing with A , B , and C with multiplicities equal to their dimensions. Therefore, A ϕ = A 1 ⊠ A 2 ⊕ B 1 ⊠ B 2 ⊕ C 1 ⊠ C 2 ⊕ D 1 ⊠ D 2 ⊕ E 1 ⊠ E 2 ⊕ ( A 1 ⊕ B 1 ⊕ 2 C 1 ) ⊠ F 2 . 14 V. INTERPRET A TION IN SYMMETR Y TFT PICTURE It is natural to ask for the physical meaning of the condensable algebras constructed in Sec. IV. The answ er is simple: they describ e gapp ed domain wal ls in the G -quan tum double top ological order. T o make this picture concrete, let us briefly review SymTFT and the “sandwic h” (or unfolding) construction. Consider a (2 + 1)D top ological order T realized on a spacetime of the form M × I , where I = [0 , 1] is an in terv al. This geometry has tw o spatial b oundaries, M × { 0 } and M × { 1 } , which w e call the left and righ t b oundaries, resp ectively . W e place on the left b oundary (the “reference b oundary”) the categorical symmetry C of interest; physically , C also serv es as the top ological data describing a gapp ed b oundary condition of the bulk T . By the (2 + 1)D b oundary–bulk corresp ondence, the bulk top ological order is then fixed (up to braided equiv alence) to b e the Drinfeld cen ter of C , T ≃ Z ( C ) . If we further imp ose a (p ossibly different) gapp ed b oundary condition on the right b oundary M × { 1 } , the entire configuration can b e viewed as a (1 + 1)D system living along M , equipp ed with symmetry C , obtained by “closing” the bulk b et w een tw o gapp ed b oundaries. W e now apply this general picture to our setting. In our construction, the reference b oundary is H and the bulk is Z ( H ) = D , and both are stacke d (i.e. la yered) top ological orders. This allows us to s eparate the stack into tw o lay ers and then p erform the standard unfolding mo v e across the physic al b oundary , which in our case is the condensed b oundary D A ϕ . After unfolding, the bulk degrees of freedom reorganize into a single interface geometry: the left half supp orts D G , while the right half supports D G (the time-reversed/c hirality-flipped cop y). Their outer b oundaries are the familiar gapp ed b oundaries V ec G and Rep ( G ), resp ectively . In this unfolded picture, the original physical b oundary condition D A ϕ is rein terpreted as a gapp e d domain wal l b et ween D G and D G . W e denote this domain wall by B ϕ (see Fig. 4). B ϕ H D unstac k B ϕ D G D G V ec G Rep ( G ) unfold B ϕ D G D G Rep ( G ) V ec G FIG. 4. the unfolding trick When an an yon ([ g ] , π ) crosses the domain wall B ϕ from righ t to left, its flux lab el (conjugacy class) is mapp ed as [ g ] 7− → [ ϕ ( ¯ g )] . Con versely , when it crosses from left to righ t, its charge lab el transforms b y the pullback π 7− → ϕ ∗ ¯ π , in agreement with the pairing rule in Eq. 39. The app earance of complex conjugation has a clear physical origin: the righ t half-space is D G , i.e. the orientation-rev ersed (time-reversed) copy of D G , so charge lab els are conjugated when transp orted across the in terface. (Equiv alen tly , one may use the canonical braided equiv alence D G ≃ D G giv en by ([ g ] , π ) 7→ ([ g ] , π ∗ ).) Moreo ver, if w e dimensionally reduce the interface geometry—bringing the t wo outer b oundaries close enough that the sandwic hed (2 + 1)D region can b e regarded as “thin”—the resulting effective theory is a (1 + 1)D gapp ed system in the v acuum sector. In this reduced picture, the domain wall data precisely enco des a (1 + 1)D SPT phase protected b y Rep( G ) × G . With this physical interpretation in hand, the lattice realization in tro duced earlier (based on the Kitaev quan- tum double model) pro vides an explicit microscopic implementation of the same domain w all. T o connect this microscopic picture to the categorical computation, consider tw o ribb on op erators meeting at the wall: a creation op erator W ([ g ] ,ρ ) q 2 k,q 1 i (path 1 ) coming from the smo oth side and ending on the domain wall, and another creation op erator 15 W ([ h ] ,π ) pl,p 1 j (path 2 ) coming from the rough side and ending at the same site. The question is when there exists a linear com bination of their intersecting comp onen ts, X q 2 k, pl f q 2 k,pl W ([ g ] ,ρ ) q 2 k,q 1 i (path 1 ) W ([ h ] ,π ) pl,p 1 j (path 2 ) , that comm utes with all mo dified local v ertex/plaquette operators supported on the wall. Imp osing this commutativit y condition yields exactly the same constraint as Eq. 35. In other words, the lattice “gluing” condition for ribb on op erators at the interface repro duces the bimo dule/half-braiding constraint that determines the algebra A ϕ , thereb y matc hing the lattice domain wall B ϕ with the condensable-algebra construction. VI. DISCUSSION Since our construction only pro duces the intrinsic al ly mixe d G × Rep( G ) SPT phases, it is natural to ask ho w far these results can b e generalized. a. R eplacing Rep( G ) by Rep( G ′ ) . A first and rather direct extension is to consider symmetry of the form G × Rep( G ′ ), where G and G ′ are tw o (not necessarily isomorphic) finite groups. In this situation the group homomorphism that con trols the wall/condensable-algebra data should b e replaced by ϕ ∈ hom( G, G ′ ) , and the deriv ations in the previous sections go through with only notational changes: the flux sector is still gov erned b y conjugacy data in G , while the charge sector is gov erned by representations of G ′ , with ϕ mediating ho w the tw o parts are glued across the interface. W e therefore exp ect an analogous pairing/multiplicit y rule, with pullback ϕ ∗ acting on Rep( G ′ ). b. T owar d a ful l classific ation of G × Rep( G ) SPT phases. A more am bitious problem is the complete classification of al l G × Rep( G ) SPT phases, not only the in trinsically mixed sub class. Here w e ha ve to b e candid: the situation b ecomes substantially richer, and a uniform closed-form classification seems harder than one might exp ect. The key p oint is that, in Sec. I I I A, the intrinsically mixed assumption drastically restricts the p ossible c hoices of the condensable algebra ˆ H . Once this constraint is dropp ed, there is a muc h larger family of admissible subalgebras, and corresp ondingly many more mo dule structures on Vec . Ev en b efore mixing is addressed, one must understand the indep enden t Rep( G ) and G SPT building blo cks. On the Rep( G ) side, it is known that (1 + 1)D Rep( G )-SPT phases (equiv alen tly , indecomp osable Rep( G )-mo dule structures on Vec ) are characterized by a pair ( L, ψ ), where L ≤ G is a subgroup and ψ ∈ Z 2 ( L, U (1)) is a 2-co cycle suc h that the twisted group algebra C ψ [ L ] is simple. Equiv alen tly , the corresp onding t wisted representation category Rep ψ ( L ) acts trivially in the sense that Rep ψ ( L ) ≃ Vec . On the G side, ordinary (1 + 1)D G -SPT phases are classified by ω ∈ Z 2 ( G, U (1)), giving the twisted group algebra C ω [ G ]. F or a non-mixe d (decoupled) phase obtained by stacking a Rep( G )-SPT lab eled by ( L, ψ ) with a G -SPT lab eled b y ω , one may take the condensable algebra to factorize as ˆ H ∼ = C ψ [ L ] ⊗ C ω [ G ] , so the discussion essentially reduces to the tw o indep endent classifications. The genuinely mixed phases are more subtle. Two obstructions app ear. First , even the Rep( G )-SPT input ( L, ψ ) is not alwa ys easy to en umerate in a uniform wa y: determining all pairs ( L, ψ ) for which C ψ [ L ] is simple can b e nontrivial and is often handled case-by-case in practice. Se c ond , once mixing is allow ed, the relev an t subgroup data underlying H is no longer a direct pro duct, but rather a “fib er pro duct” (or graph-like) subgroup sp ecified b y N ⊴ G, L ⊴ A ⊆ G, where A is a subgroup of G , and N and L are normal in G and A , resp ectiv ely , such that there is a common quotient group P with G/ N ≃ A/L ≃ P. 16 In terms of cosets labeled by p ∈ P , one may write schematically H ≡ [ p ∈ P pN ⊗ pL. (40) This form reflects the fact that the G and Rep( G ) comp onen ts are correlated through the shared quotien t P , which is precisely what “mixing” means at the level of symmetry defects/domain-wall data. The remaining challenge is then to incorp orate twisting data consistently . In the factorized case one may inde- p enden tly insert co cycles ψ and ω into the tw o tensor factors. F or a fib er-pro duct type H , how ev er, there is in general no c anonic al w ay to embed a given pair of twists ( ψ , ω ) into a single asso ciative algebra structure on H : one exp ects additional compatibilit y conditions and p ossibly extra cohomological data con trolling ho w the tw o twists are coupled along the common quotien t P . When these couplings are forced to b e trivial (for instance, when P , A , and L degenerate in the ab ov e construction), one recov ers the intrinsically mixed family studied in this w ork. A systematic classification of these mixed twists—i.e. a practical set of inv ariants and a uniform construction of the corresponding condensable algebras/domain walls—goes b eyond the scope of the presen t paper. W e leav e a full treatmen t of the general G × Rep( G ) case to future w ork. 17 [1] Shlomo Gelaki, Dmitri Nikshyc h, Victor Ostrik, and Pa vel Etingof, editors. T ensor Categories . 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