Quotient symmetry protected topological phenomena
QQuotient symmetry protected topological phenomena
Ruben Verresen, Julian Bibo, and Frank Pollmann
2, 3 Department of Physics, Harvard University, Cambridge, MA 02138, USA Department of Physics, Technical University of Munich, 85748 Garching, Germany Munich Center for Quantum Science and Technology (MQCST), D-80799 Munich, Germany
Topological phenomena are commonly studied in phases of matter which are separated from atrivial phase by an unavoidable quantum phase transition. This can be overly restrictive, leavingout scenarios of practical relevance—similar to the distinction between liquid water and vapor.Indeed, we show that topological phenomena can be stable over a large part of parameter spaceeven when the bulk is strictly speaking in a trivial phase of matter. In particular, we focus onsymmetry-protected topological phases which can be trivialized by extending the symmetry group.The topological Haldane phase in spin chains serves as a paradigmatic example where the SO (3)symmetry is extended to SU (2) by tuning away from the Mott limit. Although the Haldane phaseis then adiabatically connected to a product state, we show that characteristic phenomena—edgemodes, entanglement degeneracies and bulk phase transitions—remain parametrically stable. Thisstability is due to a separation of energy scales, characterized by quantized invariants which arewell-defined when a subgroup of the symmetry only acts on high-energy degrees of freedom. Thelow-energy symmetry group is a quotient group whose emergent anomalies stabilize edge modes andunnecessary criticality, which can occur in any dimension. INTRODUCTION
In the presence of symmetries, ground states of many-body quantum systems can form symmetry protectedtopological (SPT) phases of matter, which cannot be di-agnosed by conventional local order parameters. Charac-teristic for SPT phases is their anomalous symmetry ac-tion at the edge [1–9] implying gapless edge modes [10],degenerate or gapless entanglement spectra [1, 11, 12],and nonlocal order parameters [13–16].A celebrated one-dimensional (1D) example is the Hal-dane phase of the spin-1 antiferromagnetic Heisenbergchain, protected by, e.g., SO (3) spin rotation symme-try [11, 17, 18]. The edge transforms as a spin-1 / SO (3) symmetry [19]. The samephase arises in a bond-alternating spin-1 / H = (cid:80) j (1 + ( − j δ ) S j · S j +1 . No gapped spin-rotation symmetric Hamiltonian can adiabatically con-nect the two δ < δ > two-site unit cell, the spin rotation symmetry forms an inte-ger SO (3) representation. If lattice sites start at j = 1,then δ > / δ → + ∞ [20].In recent times, it has been realized that the propertythat makes SPT phases non-trivial—the unusual sym-metry action at its boundary—is its Achilles’ heel. AnySPT phase can be trivialized by extending its symmetrygroup [21–25]. For instance, the Haldane phase is trivi-alized by extending SO (3) into SU (2). This correspondsto introducing degrees of freedom on which the − SU (2) acts as an operator, rather than a num-ber (such that it remains SU (2) even after blocking sitesinto unit cells). This can be achieved by driving the spinchain away from its Mott limit, such that 2 π -rotationsequal the fermion parity operator, rather than a clas- − − . . δ U U pert U c U c entanglementtransitionboundary transition FIG. 1.
Emergent Haldane SPT phase in ionic Hub-bard chain with ∆ = 0 . . The Mott limit U → ∞ recoverstwo topologically distinct spin chains which are connected bythe small- U regime. The solid black (white) line is a second(first) order phase transition. The transition line ends in Isingcriticality at U c . The edge mode of the (emergent) Haldanephase is stable until it gaps out at small U (blue dashed line)without a bulk phase transition. The red dashed line indicateswhere degeneracy of lowest entanglement level is lost. sical number. Refs. [26, 27] demonstrated that using afermionic model, one can adiabatically connect the trivialand Haldane phase. In this work, we ask: how immediateis this trivialization? Are topological characteristics—such as edge modes—immediately gapped out by someexponentially-small energy scale? Or do the edge modesremain exact zero-energy edge modes over some finiteregion of parameter space?We show that—remarkably—the latter option is real-ized. When an SPT phase is, strictly speaking, trivial-ized by introducing new degrees of freedom on which the a r X i v : . [ c ond - m a t . s t r- e l ] F e b symmetry group acts in an extended way, we argue thatwhile certain features such as string orders are knownto immediately lose their meaning [26], other salient fea-tures such as edge modes, degeneracies of the low-lyingentanglement spectrum and bulk phase transitions arestable over a finite region in parameter space. We iden-tify quantized invariants which characterize this stabil-ity. Hence, the original symmetry group—which is nowa quotient group of the extended symmetry group—canstill protect certain topological phenomena; we refer tothese as quotient symmetry protected topological (QSPT) phenomena.We illustrate this parametric stability by consideringthe ionic Hubbard chain [28–41] with bond-alternation.In the Mott limit, this reduces to the bond-alternatingHeisenberg chain. As shown in Fig. 1, these two SPTphases can be adiabatically connected by tuning awayfrom the Mott limit. This phase diagram also shows twokey novel features: (i) the edge modes remain stable untilone drives a boundary phase transition, and (ii) the SPTcriticality is stable over a finite region. After explain-ing these phenomena, we put them in the more generalcontext of emergent anomalies, showcasing their applica-bility to general dimensions and symmetry groups. BOND-ALTERNATINGIONIC HUBBARD MODEL
The bond-alternating ionic Hubbard model (BIHM) ofspinful fermions consists of three terms: ˆ H = ˆ H δ + ˆ H ∆ +ˆ H U withˆ H δ = − (cid:88) j,s (cid:104)(cid:0) − j δ (cid:1) ˆ c † j +1 ,s ˆ c j,s + h.c. (cid:105) ˆ H ∆ = ∆2 (cid:88) j,s ( − j ˆ n j,s (1)ˆ H U = U (cid:88) j (cid:18) ˆ n j, ↑ − (cid:19) (cid:18) ˆ n j, ↓ − (cid:19) . Whenever this Hamiltonian has a unique ground state,it is naturally at half-filling [42]; however, this is not es-sential for what we discuss.The symmetry of interest is SU (2) spin-rotation gener-ated by ˆ S γ = x,y,z = (cid:80) s,s (cid:48) ˆ c † n,s σ γs,s (cid:48) ˆ c n,s (cid:48) . A key aspect—which we will often return to—is that 2 π rotations e πi ˆ S γ equal the fermion parity operator ˆ P rather than a clas-sical number ± SU (2)instead of becoming SO (3). The SU (2) symmetry can-not stabilize a non-trivial SPT phase, i.e., it does notadmit any non-trivial projective representations, or interms of group cohomology [43]: H ( SU (2) , U (1)) = 0.In passing we mention that everything we discuss car-ries through if we only consider the discrete quaternionsubgroup Q ⊂ SU (2) generated [44] by the π -rotations ˆ R x and ˆ R y , which also cannot protect the Haldane SPTphase.In the Mott limit U → ∞ , the ground state has exactlyone particle per site. The system is then described byan effective spin-1 / /U is given by the bond-alternating Heisen-berg chain. Fermion parity becomes a classical number: P = − P = 1 for a two-site unit cell. Thesymmetry group per unit cell is the quotient group [45] SO (3) = SU (2) / Z f which is well-known to protect anon-trivial SPT phase, i.e., H ( SO (3) , U (1)) = Z . In-deed, in the introduction we saw how δ > δ <
0) is atopological (trivial) phase. Similarly, Z × Z = Q / Z f also protects the Haldane phase.Group-theoretically, the Haldane phase is thus trivial-ized by including fluctuating charge degrees of freedom.This has been explored before [26, 27] and is a particu-lar instance of how extending a symmetry group (here,extending SO (3) by fermion parity into SU (2)) can triv-ialize any SPT phase [21–25]. The question of interestexplored in this work is to which extent the topologicalfeatures immediately disappear, i.e., are they fine-tunedor not? As a concrete model for exploring this ques-tion, we now show how the ionic Hubbard chain gives agapped symmetric path from the Haldane phase to thetrivial phase. Note that the ionicity ∆ (cid:54) = 0 in Eq. (1) isnecessary for explicitly breaking bond-centered inversion[19], SO (4) and anti-unitary particle-hole symmetry [46]which would otherwise still distinguish two SPT phases.Using the density matrix renormalization group [47–49],we obtain the phase diagram with ionicity ∆ = 0 . δ = 0 [28–41].In the U → + ∞ limit, it reduces to the gapless spin-1 / (cid:54) = 0), the effective spin chainis translation-invariant at all orders in U [28]. Remark-ably, the criticality persists far beyond this perturbativeregime. To highlight this, we denote the point ( U pert )where perturbation theory for the translation invariantHubbard model ( δ = ∆ = 0) in 1 /U diverges [50]. Alongthe line δ = 0 the model shows two quantum phasetransitions: first, a BKT-transition ( U c ) into a sponta-neously dimerized insulator (SDI) phase and afterwards,an Ising transition ( U c ) into a bond insulator (BI) phase.Whereas previous studies of this model have been field-theoretic or numerical, we will explain the persistenceof this critical line as a being a quotient symmetry pro-tected topological phenomenon. We will identify a quan-tized lattice invariant which implies this stability, readilyextending to other symmetry groups.For δ (cid:54) = 0, we observe that this model adiabaticallyconnects the Haldane and trivial SPT phase by passingthrough this fermionic regime. This is similar to the find-ings of Anfuso and Rosch in Ref. [26] based on a Hubbardladder. In this work we study the stability of the edgemodes; we will see that these are stable until encoun-tering a boundary phase transition (blue dashed line in FIG. 2.
QSPT edge phenomena.
We evaluate parity quan-tum numbers for a vertical slice δ = 0 . P ) for a system with open boundaries. We observe aboundary phase transition where the parity gap closes below(above) which the ground state is unique (degenerate). (b)The three lowest entanglement levels and the quantum num-bers of ˆ P = ˆ U x ˆ U y ˆ U † x ˆ U † y where ˆ U γ is the action of ˆ R γ on thespace of dominant eigenstates of the reduced density matrixfor a bipartition of an infinitely-long chain [1]. Fig. 1). A similar behavior is observed for the entangle-ment spectrum. Finally, note that these boundary andentanglement transitions merge at U c , where the criticalline disappears, suggesting a universal reason for thesefeatures. STABILITY OF EDGE MODES
Here we show that the zero-energy edge modes of theHaldane phase are stable until the fermion parity gapcloses for open boundary conditions. Since gaps are para-metrically stable, this implies that the topological edgemodes exist over a finite region of parameter space, i.e.,they do not vanish as soon as U is finite.Per the usual arguments of symmetry fractionaliza-tion, an on-site unitary symmetry ˆ U acts on a symmet-ric gapped chain with edges as ˆ U = ˆ U L ˆ U R [2, 3]. Here,ˆ U L,R are exponentially localized on the boundaries. Asdiscussed above, a fermionic chain with SU (2) symmetrycannot host a non-trivial SPT phase; hence, these frac-tionalized symmetries must obey the same group prop-erties as the bulk symmetry, in particular: ˆ R Lx ˆ R Ly =ˆ P L ˆ R Ly ˆ R Lx . In the Mott limit U → ∞ for δ >
0, we knowthe edge is a spin-1 / P L = − R Lx ˆ R Ly = − ˆ R Ly ˆ R Lx implies a twofold degener-acy. An eigenvalue of ˆ P L cannot immediately jump suchthat its associated twofold degeneracy is parametrically stable. The only way P L can change is if we make U suf-ficiently small such that another (non-degenerate) levelwith P L = +1 crosses it in energy.As an illustration, it is instructive to consider δ = 1,where the edge mode decouples from the bulk. Focusingon the left edge (i.e., site j = 1 in Eq. (1)), we read offthat the energy of a doubly-occupied site (where P L =+1) is U − ∆ whereas the energy of a singly-occupied site(with P L = −
1) is − U − ∆2 . Hence, in this simple case,the edge mode is stable until U = ∆, at which point itundergoes a boundary phase transition (which in 0 + 1 D corresponds to a level crossing).Away from this exactly-solvable limit, we numericallydetermine this boundary transition as shown in Fig. 1.In the many-body system, we cannot directly read off theeigenvalue of ˆ P L , instead we focus on ˆ P = ˆ P L ˆ P R . Theground state always satisfies P = +1 (since both edgeshappen to undergo the boundary transition at the sametime); nevertheless, the above reasoning shows that thegap for this global parity does close with open boundaryconditions (since it must close at each edge) which we ver-ify in Fig. 2(a). The above arguments can be repeatedfor a virtual bipartition of an infinite system, with thedegenerate low-lying entanglement spectrum being sta-bilized by the parity quantum numbers of the dominanteigenstates of the reduced density matrix, confirmed inFig. 2(b). STABILITY OF SPT TRANSITION
Here we explain why the phase transition between atrivial and SPT phase does not immediately gap out af-ter extending the symmetry group. Consequently, thecritical line in Fig. 1 is a generic QSPT phenomenon.Our model has a useful duality symmetry δ → − δ given by the modified (unitary) translation operatorˆ D ˆ c n,s ˆ D † = ( − n ˆ c † n +1 ,s . A direct transition can thusonly occur at δ = 0. In the spin chain limit, ˆ D actsas a single-site translation symmetry and an on-siteunitary: ˆ D ˆ S γj ˆ D † = ˆ R y ˆ S γj +1 ˆ R † y . It is well-known thatin combination with spin-rotation symmetry, this im-plies a Lieb-Schultz-Mattis (LSM) anomaly, disallowinga gapped symmetric ground state [51–57]: in the ab-sence of symmetry breaking, this stabilizes a direct phasetransition. The standard proofs for the LSM anomalyhinge on the fact that on a single site, ˆ R x and ˆ R y anti-commute. This no longer holds when charges fluctuate:ˆ R x ˆ R y ˆ R − x ˆ R − y = ˆ P (and indeed, in Fig. 1 we see thatsmall U admits a gapped phase). Nevertheless, we showthat there is an emergent Lieb-Schultz-Mattis theorem ,enforcing the parametric stability of the phase transition.Above, we understood the parametric stability of edgemodes in terms of parity quantum numbers. Similarly,we now introduce a quantized invariant for a paritystring. As long as fermionic operators remain gapped,the fermion parity string generically has long-range or- . . . . . . . . U . . . . . . O + , O − S D I MIBI O − O + . . U . . O − O + FIG. 3.
QSPT transition and emergent anomaly alongthe self-dual line δ = 0 (with ionicity ∆ = 0 . ). The ob-servable O ± captures whether the fermion parity string oscil-lates or not: it measures the Z invariant θ = 0 , π in Eq. (2).If θ = π (i.e., O − (cid:54) = 0 and O + = 0), there is an emergentLieb-Schultz-Mattis anomaly, preventing a symmetric gappedstate. Inset: the intermediate phase spontaneously breaks theduality symmetry D such that both O ± are nonzero. der. Moreover, at δ = 0, this must have a well-definedmomentum under the duality/translation symmetry D : (cid:104) ˆ P m ˆ P m +1 · · · ˆ P n − ˆ P n (cid:105) ∼ constant × e iθ ( n − m ) . (2)Since parity is a Z symmetry, we obtain a quantizedinvariant θ ∈ { , π } . Eq. (2) can be rigorously derivedfrom symmetry fractionalization (see Appendix A 2). In-tuitively, θ = π formalizes the idea of being close to aMott limit, where fermion parity coincides with the par-ity of the number of sites.We claim that if θ = π , then there is an emergentanomaly that forbids a gapped symmetric ground state.To prove this, suppose there is such a ground state. Sym-metry fractionalization then allows us to factorize a stringof ˆ R x as ˆ R xL ˆ R yR . Now consider the composite symmetryˆ I = ˆ R y ˆ D ˆ R † y ˆ D † . Since D and R y commute, ˆ I = 1. Alter-natively, using standard symmetry fractionalization ar-guments, we derive ˆ I ˆ R Rx ˆ I † = e iθ ˆ R Rx (see Appendix A 3).Taking these two facts together, we see that a gappedsymmetric state implies that e iθ = 1, i.e., θ = 0. If θ = π , the assumption of a gapped symmetric groundstate must be violated.In the Mott limit, we have θ = π , such that we red-erive the known LSM anomaly. However, since θ is arobust quantized invariant, we know that the transitionis parametrically stable to finite U . The only way θ canchange is (1) by closing the fermion parity gap (such thatthe parity string does not have long-range order), or (2)by spontaneously breaking the duality symmetry D , suchthat the charge e iθ is no longer well-defined. This partic-ular model opts for the second option, as we see in Fig. 1(the white line corresponds to spontaneous dimerization).To confirm this interpretation of the phase diagram, we measure θ by plotting O ± := lim | n − m |→∞ |(cid:104) ˆ S ± m ˆ S ± n (cid:105)| with ˆ S ± n := (cid:89) k 1. From Fig. 3, we conclude that the gapless Mottinsulator (MI, U > U c ) has θ = π , the bond insulator(BI, U < U c ) has θ = 0, and the intermediate spon-taneous dimerized insulator (SDI) has no well-defined θ since D is broken. GENERAL EMERGENT ANOMALIES The arguments for the parametric stability of edgemodes and phase transitions readily extend to other sym-metry groups and dimensions. Edge modes of 1D SPTphases are characterized by a non-trivial projective rep-resentation of a symmetry group ˜ G [1–5]. One can alwaysextend the symmetry group by H into a bigger symme-try group G (where ˜ G = G/H ) such that the lifted rep-resentation becomes linear, thereby trivializing the SPTphase [21–25]. However, the quantum numbers of theadditional symmetry group H still label these distinctrepresentations, and the edge mode remains stable untilthe boundary undergoes a phase transition where it closesthe gap to excitations charged under H . In the above ex-ample (where ˜ G = SO (3), H = Z f and G = SU (2)), wehad to close the fermion parity gap at the boundary todestroy the edge mode.SPT transitions are stabilized by the mutual anomalybetween the protecting symmetry group ˜ G and the dual-ity symmetry at the transition (e.g., the Lieb-Schultz-Mattis anomaly of the Heisenberg chain as discussedabove) [58–60]. This anomaly is lifted when we extend˜ G by H , but we propose that there is always an emer-gent anomaly encoded in the symmetry properties of thestring orders associated to the symmetry group H (suchas the fermion parity string in Eq. (2) being odd under D ). It is interesting to observe that since a QSPT tran-sition involves the whole 1D bulk, the emergent anomalyis characterized by a 1D object. In contrast, an edgemode is a zero-dimensional QSPT phenomenon, and cor-respondingly it parametric stability is encoded in a 0Dcharge.Note that the same mechanism of emergent anomaliesin 1D SPT transitions applies to the 1D edge modes of 2DSPT phases. For instance, if one trivializes the Z -SPTin 2D by extending by a second Z symmetry ˆ P into Z ,then we expect that the string order of ˆ P will have a non-trivial charge under Z —encoding the emergent anomaly.In the simplest case, where the (now non-anomalous)edge mode is written as a 1D system, this string ordercan be directly computed and its non-trivial charge is amechanism for the recently-discovered intrinsically gap-less SPT phases [61]. In the full 2D model, it can betricky to directly access this string order (similar to ac-cessing P L directly in Fig. 2), but a clear-cut implicationis that the edge mode has an emergent anomaly that canonly disappear by spontaneously breaking the symme-try or by driving a boundary phase transition where thegap associated to the extending symmetry ˆ P must close.This situation is analogous to what we saw for the SPTtransition in the ionic Hubbard chain. DISCUSSION There are two ways of trivializing an SPT phase with-out driving through a quantum phase transition: eitherone breaks or extends the symmetry group. The for-mer will immediately gap out the edge modes and phasetransitions. In this work, we have shown how the latterapproach—extending the symmetry group—leaves vari-ous topological phenomena intact over a finite region ofthe phase diagram. We characterized this stability interms of discrete observables.This is of particular relevance to physical implemen-tations of SPT phases, where the protecting symmetrygroup is often a low-energy effective quotient group. Forinstance, the spin-rotation symmetry protecting the Hal-dane phase in a spin chain will be derived from a fermionsymmetry of an underlying Hubbard model. The resultsin this work thus show that the experimental realizationof such bosonic SPT phases is meaningful, since manyof their topological phenomena are stable even when thebulk is—strictly speaking—not in a true SPT phase.In the above lattice model, we saw a (parametrically) stable transition and edge mode. It is an open ques-tion how closely these are linked: it is tempting to thinkthat the emergent edge mode cannot disappear beforethe emergent phase transition gaps out. Indeed, in Fig. 1both features terminate at U c . This is intuitive given theinterpretation of edge modes as being spatially-localizedphase transitions [62–65], but it would be interesting tomake this correspondence more exact. 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A brief recap of symmetry fractionalization Let us briefly review the argument for symmetry fractionalization [2], focusing on 1D systems for concreteness.If | ψ (cid:105) is the ground state of a gapped local quantum Hamiltonian, it will have a finite correlation length ξ [68]. Forany block of N sites, we can consider the Schmidt decomposition | ψ (cid:105) = (cid:80) α Λ α | ψ in α (cid:105) ⊗ | ψ out α (cid:105) , where | ψ in α (cid:105) are quantumstates defined purely within the block of N sites. Due to there being a finite correlation length, all the Schmidt states | ψ in α (cid:105) are indistinguishable deep within the block (i.e., they simply look like | ψ (cid:105) ). Moreover, due to locality, the twoedges of the block are effectively independent (up to exponentially small errors). We can thus presume that α can beinterpreted as a super-index which is equivalent to two smaller indices α L , α R such that | ψ in α (cid:105) = | ψ in α L ,α R (cid:105) where thevalue of α L ( α R ) only affects correlation functions on the left (right) end of the block.Let ˆ U = (cid:81) n ˆ U n be a unitary on-site symmetry. If we act with ˆ U only within this block, it leaves the states | ψ out α (cid:105) unaffected. Since the states in a Schmidt decomposition form a complete basis, we can write ˆ U | ψ in α (cid:105) = (cid:80) β U β ; α | ψ in β (cid:105) = (cid:80) β L ,β R U β L ,β R ; α L ,α R | ψ in β L ,β R (cid:105) . Since ˆ U is an on-site operator, it will preserve locality, i.e., α L and β R in U β L ,β R ; α L ,α R will be uncorrelated. More precisely, we can write U β L ,β R ; α L ,α R = U Lβ L ; α L U Rβ R ; α R . In short, wewrite that (cid:81) Ln =1 ˆ U n | ψ (cid:105) = ˆ U L ˆ U R | ψ (cid:105) where ˆ U L,R are exponentially localized near the edges of the block of N sites.The explicit formula in terms of Schmidt states implies that if (cid:81) Li =1 ˆ U i | ψ (cid:105) = ˆ U L ˆ U R | ψ (cid:105) and (cid:81) Li =1 ˆ V i | ψ (cid:105) = ˆ V L ˆ V R | ψ (cid:105) (for a second symmetry ˆ V ), then we also have that (cid:81) Li =1 ˆ V i (cid:81) Li =1 ˆ U i | ψ (cid:105) = ˆ V L ˆ V R ˆ U L ˆ U R | ψ (cid:105) . (This is not as obviousas it might seem on first sight, since the conclusion would not hold if the ˆ U and ˆ V symmetries were acting ondistinct-but-overlapping blocks).The fractionalized symmetries obey the same group relations as the original symmetries up to potential phasefactors. For instance, let us suppose ˆ U and ˆ V commute. Let us also assume that ˆ U L,R are bosonic operators (suchthat ˆ U L commutes with ˆ V R ). Then1 = ˆ U ˆ V ˆ U − ˆ V − = ˆ U L ˆ V L (cid:0) ˆ U L (cid:1) − (cid:0) ˆ V L (cid:1) − × ˆ U R ˆ V R (cid:0) ˆ U R (cid:1) − (cid:0) ˆ V R (cid:1) − . (A1)Since the two factors on the right-hand side act on disjoint regions yet they multiply to the identity, each of the twofactors has to be proportional to a number: ˆ U L ˆ V L (cid:0) ˆ U L (cid:1) − (cid:0) ˆ V L (cid:1) − = e iα . Moreover, using similar manipulations, onecan show that if ˆ U = 1, then e iα = ± 1. More generally, the fractionalized symmetries will form a projective repre-sentation of the original symmetry group. Non-trivial projective representations correspond to non-trivial SPT phasesand imply edge modes in the energy spectrum with open boundary conditions, or degeneracies in the entanglementspectrum for virtual bipartitions (since any projective representation acting on a 1D vector space is trivial). 2. Derivation of the θ invariant It is instructive to derive the invariant θ in Eq. (2) rigorously, which we can do using symmetry fractionalization. Fora parity string which is much longer than the fermionic correlation length, we can write ˆ P m,n := (cid:81) m ≤ k ≤ n ˆ P k = ˆ P Lm ˆ P Rn on the ground state subspace. Since ˆ D ˆ P m,n ˆ D † = ˆ P m +1 ,n +1 , we obtain (cid:16) ˆ D ˆ P Lm ˆ D † (cid:17) (cid:16) ˆ D ˆ P Rn ˆ D † (cid:17) = ˆ P Lm +1 ˆ P Rn +1 ⇒ ˆ D ˆ P Lm ˆ D † = α m,n ˆ P Lm +1 and ˆ D ˆ P Rn ˆ D † = ¯ α m,n ˆ P Rn +1 , (A2)where α m,n is some proportionality factor. Note that the first (second) equation tells us that it cannot depend on n ( m ). I.e., the proportionality factor is a genuine constant. Let us denote it as α n,m = e iθ . Since (cid:104) ˆ P m,n (cid:105) = (cid:104) ˆ P Lm (cid:105)(cid:104) ˆ P Rn (cid:105) (due to locality and the spatial separation between the fractionalized symmetries ˆ P L,R ) and the fact that ˆ D is asymmetry (i.e., (cid:104) ˆ P Lm (cid:105) = (cid:104) ˆ D ˆ P Lm ˆ D † (cid:105) = e iθ (cid:104) ˆ P Lm +1 (cid:105) = e iθk (cid:104) ˆ P Lm + k (cid:105) ), we derive Eq. (2): (cid:104) ˆ P m ˆ P m +1 · · · ˆ P n − ˆ P n (cid:105) = (cid:104) ˆ P m,n (cid:105) = (cid:104) ˆ P Lm (cid:105)(cid:104) ˆ P Rn (cid:105) = e iθ ( n − m ) (cid:104) ˆ P Ln (cid:105)(cid:104) ˆ P Rn (cid:105) , (A3)where n is some reference site that does not depend on n or m . Finally, the quantization of θ follows from ˆ P m,n = 1since then ( ˆ P Rn ) ∝ e iθ = 1, i.e., θ = 0 , π . (As an illustration, note that in the spin chain limit, P Rn = ( − n , which implies θ = π .) 3. The emergent anomaly Here we derive that ˆ I ˆ R Rx ˆ I † = e iθ ˆ R Rx where ˆ I = ˆ R y ˆ D ˆ R † y ˆ D † . This straightforwardly follows from symmetry frac-tionalization. We will need what we derived above: ˆ D ˆ P Ln ˆ D † = e iθ ˆ P Ln +1 . Similary, there is a phase factor e iκ such thatˆ D (cid:2) ˆ R Lx (cid:3) n ˆ D † = e iκ (cid:2) ˆ R Lx (cid:3) n +1 . Lastly, since group relations are always obeyed up to a phase factor, there is phase factor e iµ such that ˆ R † y (cid:2) ˆ R Lx (cid:3) n ˆ R y = (cid:2) ˆ R Ly (cid:3) † n (cid:2) ˆ R Lx (cid:3) n (cid:2) ˆ R Ly (cid:3) n = e iµ ˆ P Ln (cid:2) ˆ R Lx (cid:3) n . (A4)Note that this also implies ˆ R y ˆ P Ln (cid:2) ˆ R Lx (cid:3) n ˆ R † y = e − iµ (cid:2) ˆ R Lx (cid:3) n . Plugging in these identities, we obtainˆ I (cid:2) ˆ R Lx (cid:3) n ˆ I † = ˆ R y ˆ D ˆ R † y (cid:0) ˆ D † (cid:2) ˆ R Lx (cid:3) n ˆ D (cid:1) ˆ R y ˆ D † R † y = e − iκ ˆ R y ˆ D (cid:0) ˆ R † y (cid:2) ˆ R Lx (cid:3) n − ˆ R y (cid:1) ˆ D † R † y (A5)= e i ( µ − κ ) ˆ R y ˆ D ˆ P Ln − (cid:2) ˆ R Lx (cid:3) n − ˆ D † R † y = e i ( µ − κ ) ˆ R y (cid:0) ˆ D ˆ P Ln − ˆ D † (cid:1)(cid:0) ˆ D (cid:2) ˆ R Lx (cid:3) n − ˆ D † (cid:1) R † y (A6)= e i ( µ − κ ) ˆ R y (cid:0) e iθ ˆ P Ln (cid:1)(cid:0) e iκ (cid:2) ˆ R Lx (cid:3) n (cid:1) R † y = e iθ (cid:0) e iµ ˆ R y ˆ P Ln (cid:2) ˆ R Lx (cid:3) n ˆ R † y (cid:1) = e iθ (cid:2) ˆ R Lx (cid:3) n ..