The split and approximate split property in 2D systems: stability and absence of superselection sectors
TThe split and approximate split property in 2Dsystems: stability and absence of superselectionsectors
Pieter Naaijkens and Yoshiko Ogata School of Mathematics, Cardiff University, United Kingdom Graduate School of Mathematical Sciences, The University of Tokyo, Japan
February 16, 2021
Abstract
The split property of a pure state for a certain cut of a quantum spinsystem can be understood as the entanglement between the two subsys-tems being weak. From this point of view, we may say that if it is notpossible to transform a state ω via quasi-local automorphisms into a statesatisfying the split property, then the state ω has a long-range entangle-ment. It is well known that in 1D, gapped ground states have the splitproperty with respect to cutting the system into left and right half-chains.In 2D, however, the split property fails to hold for interesting models suchas Kitaev’s toric code. In fact, we will show that this failure is related tothe existence of anyons.There is a folklore saying that the existence of anyons, like in the toriccode model, implies long-range entanglement of the state. In this paper,we prove this folklore in an infinite dimensional setting. More precisely,we introduce superselection sectors which we regard as anyons. We showthat the existence of a non-trivial superselection sector (=anyon) of a state ω implies that the state ω is long-range entangled.Whilst the split property fails to hold for Kitaev’s toric code, it doessatisfy the weaker approximate split property. Contrary to the fact thatquasi-local automorphism can destroy the strict split property in generalfor 2D, we show this approximate version is stable under it. A pair (
N, M ) of commuting von Neumann algebras is called split if there isa Type I factor F such that N ⊂ F ⊂ M (cid:48) [DL84]. In applications to physicstypically N and M are generated by local observables located in two disjoint(or, in relativistic theories, spacelike separated) regions Λ and Λ . The splitproperty then can be interpreted as a type of statistical independence betweenregions. More precisely, one can locally prepare a normal state ϕ such thatrestricted to measurements in Λ i we have ϕ ( AB ) = ϕ ( A ) ϕ ( B ), for givennormal states ϕ i on the algebra generated by observables localized in Λ i [Wer87].In particular, it means that there is no entanglement between the two parts.1 a r X i v : . [ m a t h - ph ] F e b lternatively, the Type I factor allows us to find a tensor product decompositionof the Hilbert space, with the algebras N and M acting on distinct factors. Sucha decomposition is far from obvious in systems with infinitely many degrees offreedom and may even not exist for a given bipartition of the system. Earlyapplications have been in algebraic quantum field theory [BW86], for examplein the study of entanglement properties of the vacuum [SW88].More recently the split property has found applications in the classificationof phases of 1D gapped quantum spin systems. Under quite general conditionsonce can show that the split property holds in ground state representations. Inparticular, Matsui [Mat13] showed that if ω is a pure ground state of a gappedlocal Hamiltonian (on the chain), it satisfies the split property in the sense that ω is quasi-equivalent to ω L ⊗ ω R . Here ω L (resp. ω R ) is the ground state restrictedto the left (resp. right) half-chain A L ( A R ). In this case this is equivalent tosaying that the inclusion π ω ( A L ) (cid:48)(cid:48) ⊂ π ω ( A R ) (cid:48) is split in the sense above, where π ω is a GNS representation for ω [Mat01] (see also [Oga21a, Remark 1.5]).The split property can then be used to define a H ( G, U (1))-index for aunique gapped ground state on a quantum spin chain with finite group on-site symmetry [Oga20], as well as Z -valued index for reflection symmetry,generalizing a construction by Pollmann et al. [PTBO10] for matrix productstates. The index was used to prove a general Lieb-Schultz-Mattis type theoremsin [OTT20].For fermionic chains, the split property for a unique gapped ground stateis proven in [Mat20]. Bourne and Schulz-Baldes and independently Matsuiintroduced Z -index for fermionic chains without symmetry [BS20, Mat20]. Aclassification of SPT-phases with on-site symmetry in 1D fermionic chain basedon the split property was carried out in [BO20]. There, a Z × H ( G, Z ) × H ( G, U (1) p )-valued index was found using the split property.The split property is essential in all these constructions: it allows one tofactor the Hilbert space into a tensor product with the left half-chain acting onone factor, and the right half-chain on the other. The Type I factor F is suchthat F (cid:39) B ( H L ) ⊗ I with respect to this decomposition. This can then be usedto extend a symmetry β L of the spin chain to an automorphism of F , which byWigner’s theorem can be implemented by a (anti-)unitary. This in turn can beused to define an index.In two dimensional systems, which are the focus of this article, the situationis more complicated. For example, consider Kitaev’s toric code model [Kit03].Then one can consider a cone-like region (extending to infinity) Λ and its com-plement, as an analogue of the two half-chains in 1D. It turns out that thetranslation invariant ground state ω of the toric code is not split with respectto this bipartition [Naa12, FN15], in contrast with the 1D case discussed above.In fact, it is related to the presence of long-range entanglement in the groundstate.It turns out that at least for abelian quantum double models a weakerstatement is true. That is, if one considers a pair of cones Λ ⊂ Λ whoseboundaries are sufficiently far apart, there is a Type I factor F such that π ω ( A Λ ) (cid:48)(cid:48) ⊂ F ⊂ π ω ( A Λ c ) (cid:48) [FN15]. This should be compared with the settingin relativistic quantum field theory mentioned earlier, where the split propertyfails if the intersection of the closures of the two regions has non-empty intersec-tion, but holds when they are spacelike separated. This property is sometimescalled the distal or approximate split property to distinguish it from the sit-2ation in e.g. 1D systems. Despite being weaker than the split property, itstill has important applications. For example, in two dimensional systems theapproximate split property is one of the assumptions used in relating the totalquantum dimension (a property of the superselection sectors) to the index ofa certain subfactor [Naa13]. This result can be used to show one has foundall superselection sectors of a given model. A variant also plays a role in thediscussion of “approximately localized” superselection sectors [CNN20].The interest of this paper is in these split and approximate split propertiesin 2D quantum spin systems. In particular, we regard a state with the splitproperty as having small entanglement with respect to the given cut. Fromthis point of view, a state which cannot be transformed into a split state viaquasi-local automorphisms has long-range entanglement. (See subsection 2.3 forthe definition of quasi-local automorphisms and their importance in the theoryof gapped ground state phases.) There is a folklore saying that the existenceof anyons, like in the toric code model, implies long-range entanglement of thestate. In this paper, we prove this folklore in infinite dimensional setting. Moreprecisely, we introduce superselection sectors which can be regarded as anyons.In fact, the superselection criterion can be relaxed (from unitary equivalence toquasi-equivalence) to accommodate non-abelian quantum double models. Weshow that the existence of a non-trivial superselection sector (=anyon) of a state ω implies that the state ω is long-range entangled. In fact, our superselectionsectors are stable under quasi-local automorphism. A somewhat different ver-sion of superselection sectors was considered in [CNN20], where it is proven thatthe full structure (including e.g. braiding) is stable under quasi-local automor-phisms. For a class of Hamiltonians consisting of local commuting projectors,Haah [Haa16] introduced an ingenious index such that it having a non-trivialvalue implies that one needs a quantum circuit with depth on the order of thesystem size to transform into product states. Our result is in accordance withthese results. In general, the split property itself in 2D is not stable underquasi-local automorphism. We show, however, the approximate split propertyis stable under it.The key ingredient for the proof is a factorization property of quasi-localautomorphisms α s . More precisely, α s can be written as a composition of sev-eral automorphisms, each of which is local with respect to certain partitions ofthe system, followed by conjugation with a unitary in the quasi-local algebra.This factorization property was first used in [Oga20], in proving the stabilityof the index of 1D SPT. Following this idea, in [Moo19] the stability of splitproperty in 1D was shown. Its 2-dimensional version is essential here, but anextra complication is that in 2D or higher, the boundary between the regionswe will consider is infinite. This makes locality estimates much more subtle.Coincidentally, this more complicated geometry is also a key reason why Mat-sui’s result on the split property for 1D spin chains [Mat13] does not generalizeto higher dimensions. A special case of the 2D-version (with respect to conelike regions with common apex) of the factorization property is also used in[Oga21b], to define a H ( G, U (1))-valued index and to show its stability.In Section 2 we fix notation and recall some basic facts about Lieb-Robinsonbounds and quasi-local maps. Then, in Section 3, we prove the factorizationproperty of quasi-local automorphisms in a general setting. In Section 4 weconsider states in 2D which are quasi-equivalent to a product state, and hencesatisfy the strict split property. In particular, we show that the states in this3apped quantum phase have trivial superselection structure. Finally, in Sec-tion 5 we show that our main technical result applies to a natural class ofquasi-local automorphisms, and use this to show that the approximate splitproperty is stable in such models.
Acknowledgments.
PN was supported in part by funding from the EuropeanUnion’s Horizon 2020 research and innovation program under the EuropeanResearch Council (ERC) Consolidator Grant GAPS (No. 648913). YO is sup-ported is supported in part by JSPS KAKENHI Grant Number 16K05171 and19K03534. It was also supported by JST CREST Grant Number JPMJCR19T2.
We first fix the setting and introduce the main definitions. A key part is playedby quasi-local maps and Lieb-Robinson bounds. For a state-of-the-art overviewof the topic see [NSY19]; for our purpose the most relevant facts will be recalledhere. We largely adopt the notation of [NSY19]. We assume basic familiaritywith the operator algebraic formulation of quantum spin systems (see e.g. [BR87,BR97]).Let (Γ , d ) be a countable metric space which is ν -regular i.e.,sup x ∈ Γ | b x ( n ) | ≤ κn ν , ≤ n ∈ N , (2.1)for some constant κ >
0. Here, we used the notation b x ( n ) := { y ∈ Γ | d ( x, y ) ≤ n } . (2.2)In concrete applications we typically consider Γ = Z ν (or its edges) with theusual metric, but for now we keep the discussion as general as possible.Let P (Γ) be the set of all finite subsets of Γ. For Λ ∈ P (Γ) we set A Λ := (cid:79) x ∈ Λ B ( H x ) , (2.3)where H x are finite dimensional Hilbert spaces whose dimensions are uniformlybounded: sup x ∈ Γ dim H x < ∞ . (2.4)If Λ ⊂ Λ there is a natural inclusion of algebras, and hence we can write A locΓ := (cid:91) Λ ∈P (Γ) A Λ (2.5)for the algebra of local observables. To get the C ∗ -algebra A Γ of quasi-localobservables we take the norm closure of A Γ . In general, if Λ ⊂ Γ is any subsetof Γ, A Λ is the norm closure of (cid:83) Λ ⊂ Λ , Λ ∈P (Γ) A Λ . We denote by U ( A Γ ) theset of all unitaries in A Γ .For any subset X of Γ, we denote by Π X the conditional expectation onto A X given by the tracial state on A X c . These maps will be used to approximatequasi-local observables by local ones. For any m ∈ N ∪ { } and X ⊂ Γ, we set X ( m ) := { x ∈ Γ | d ( x, X ) ≤ m } . (2.6)4urthermore, we define∆ X ( m ) := Π X ( m ) − Π X ( m − , m ∈ N , X ⊂ Γ . (2.7)Note that we have (cid:13)(cid:13) ∆ X ( m ) ( A ) (cid:13)(cid:13) ≤ (cid:107) A (cid:107) , A ∈ A Γ , (2.8)since Π is a projection. We will be interested in the split property with respect to different regions ofΓ, leading to the following definition.
Definition 2.1.
Let Γ ⊂ Γ ⊂ Γ and ω a pure state of A Γ . Then we say that ω is split with respect to the inclusion Γ ⊂ Γ if there is a Type I factor F such that π ( A Γ ) (cid:48)(cid:48) ⊂ F ⊂ π ( A Γ ) (cid:48)(cid:48) , (2.9)where π is a GNS representation for ω .Conjugating with a unitary does not affect the split property. Furthermore,one would expect that automorphisms of A Γ and A Γ c have no effect on theexistence of the Type I factor F . We can even allow for a non-trivial automor-phism on a “widening” of the region Γ \ Γ , at the expense of shrinking (resp.growing) the two regions in the definition of the split property. This is the ideabehind the next proposition. Proposition 2.2.
Let Γ ⊂ Γ ⊂ Γ ⊂ Γ be a sequence of subsets in Γ . Let ω be a pure state on A Γ . Let ( H , π, Ω) be a GNS triple of ω . Suppose that thereis a type I factor F such that π ( A Γ ) (cid:48)(cid:48) ⊂ F ⊂ π ( A Γ ) (cid:48)(cid:48) . (2.10) Let α be an automorphism of A Γ . Let α Γ c , α Γ \ Γ , α Γ be automorphisms of A Γ c , A Γ \ Γ , A Γ respectively. Define an automorphism ˜ α of A Γ by ˜ α := α Γ c ⊗ α Γ \ Γ ⊗ α Γ . (2.11) Suppose moreover that there is an automorphism β Γ \ Γ of A Γ \ Γ and a unitary u ∈ A Γ such that α = Ad( u ) ◦ ˜ α ◦ (cid:16) ˜ β Γ \ Γ (cid:17) , (2.12) where ˜ β Γ \ Γ = β Γ \ Γ ⊗ id (Γ \ Γ ) c . Then ω ◦ α is split for the inclusion Γ ⊂ Γ ,and in the GNS representation π ◦ α we can choose ˜ F = Ad( π ( u ))( F ) as theinterpolating Type I factor.Proof. We have π ◦ ˜ α ( A Γ ) (cid:48)(cid:48) = π ◦ α Γ ( A Γ ) (cid:48)(cid:48) ⊂ π ( A Γ ) (cid:48)(cid:48) ⊂ F. (2.13)5e also have ˜ α − ( A Γ ) = A Γ ⊂ A Γ , and hence π ( A Γ ) ⊂ π ◦ ˜ α ( A Γ ). There-fore we have π ◦ ˜ α ( A Γ ) (cid:48) ⊂ π ( A Γ ) (cid:48) ⊂ F (cid:48) , (2.14)and by taking commutants F ⊂ π ◦ ˜ α ( A Γ ) (cid:48)(cid:48) . Hence we obtain π ◦ ˜ α ( A Γ ) (cid:48)(cid:48) ⊂ F ⊂ π ◦ ˜ α ( A Γ ) (cid:48)(cid:48) . (2.15)Note that by assumption and the fact that ˜ β Γ \ Γ acts trivially on A Γ , α ( A Γ ) = Ad( u ) ◦ ˜ α ◦ ˜ β Γ \ Γ ( A Γ ) = Ad( u ) ◦ ˜ α ◦ ( A Γ ) , (2.16)and similar with A Γ replaced by A Γ . Hence we have( π ◦ α ( A Γ )) (cid:48)(cid:48) = Ad ( π ( u )) (cid:0) π ◦ ˜ α ( A Γ ) (cid:48)(cid:48) (cid:1) ⊂ Ad ( π ( u )) ( F ) ⊂ Ad ( π ( u )) (cid:0) ( π ◦ ˜ α ( A Γ )) (cid:48)(cid:48) (cid:1) = ( π ◦ α ( A Γ )) (cid:48)(cid:48) . (2.17)This completes the proof.The main technical contribution of the paper consists in proving that thequasi-local automorphisms α s admit a decomposition as in (2.12) of the propo-sition. The present work is at least partly motivated by superselection sector theory.In two dimensional systems with long-range topological order, there is the pos-sibility of quasi-particles with braided exchange statistics. These charges, oranyons, correspond to superselection sectors of the theory. Typical examplesof such models are Kitaev’s quantum double models [Kit03] and the Levin-Wen string-net models [LW05]. Mathematically, the algebraic properties of theanyons are described by a braided tensor category [Wan10].Typical methods to extract the superselection sector content from a groundstate rely quite heavily on certain properties (e.g. symmetries) of the under-lying model, and are therefore less suitable for a general analysis. In fact, infinite systems it is not always clear how to even define a sector, in particularonce one loses strict locality as a result of perturbations. We therefore take adifferent approach, motivated by sector theory in algebraic quantum field the-ory [Haa92], in which one in principle can recover the full anyon structure froma few general and physically motivated principles. The idea of a superselectionsector stems from the observation that it appears to be impossible to make co-herent superpositions between certain states. Mathematically this phenomenonis related to the existence of non-equivalent representations of the algebra ofobservables. One way to interpret this is to think of charge conservation: withlocal operations it is not possible to change the total charge of the system. Inparticular, say we create a conjugate pair of anyons (thus preserving the totalcharge), and move one far away. Then acting locally the total charge in thatregion cannot be changed.The C ∗ -algebra A Γ has many inequivalent representations, but most of themare not physically relevant. Hence we need a selection criterion to select the rel-evant representations. Based on properties of models such as the toric code,6here single charges can be created by string operators extending to infin-ity [Naa11], and topological charges in algebraic quantum field theory [BF82],we are interested in charges that can be localized in a cone. That is, outside of acone Λ the charged representations should be unitarily equivalent to the groundstate representation. Moreover, we want to be able to move the localizationregion around. This leads to the following criterion: we consider irreduciblerepresentations π that satisfy π |A Λ c ∼ = π |A Λ c , (2.18)for any cone Λ. Here π is the (reference) ground state representation, forexample the unique frustration free ground state in abelian quantum doublemodels. Since the criterion is required to hold for any cone, the localizationregion can be moved around. This is called transportability of the charges, andwe say that the charge is transportable (see e.g. [Haa92, Section IV.2]). A sectoris then a (unitary) equivalence class of representations π satisfying the selectioncriterion. The trivial sector is the equivalence class containing the referencerepresentation π . Later we will slightly relax the criterion (2.18) to requireonly quasi-equivalence.It is perhaps surprising that by just imposing this single selection criterion,we obtain a very rich structure. Using the selection criterion (and a technicalcondition called Haag duality ) one can recover the full algebraic properties of theanyonic charges. In addition, in concrete models such as the toric code thereare natural candidates to construct representations π satisfying the criterion,even without resorting to Haag duality. For example, in the toric code or itsabelian generalizations one can use string operators to create a pair of anyons.To describe a single anyon we can then send one of them off to infinity, and theresulting representation satisfies (2.18) [Naa11, FN15].The split property enters the analysis in various ways. We first note thatthe topological phenomena in our systems of interest, in particular the exis-tence of anyons, are believed to be due to the presence of long-range entangle-ment [CGW10]. Product states exhibit no entanglement, and hence should bein the trivial phase without any anyons. A state with long-range entanglementis then roughly speaking a state that cannot be transformed into a product stateby applying a finite sequence of local unitaries throughout the system. Considerthe case where we have a pure state ω = ω Λ ⊗ ω Λ c that is a product state withrespect to a cone Λ and its complement. It is easy to see (see Section 4) thatin this case π ω ( A Λ ) (cid:48)(cid:48) is a Type I factor and the inclusion π ω ( A Λ ) (cid:48)(cid:48) ⊂ π ω ( A Λ c ) (cid:48) therefore is split. In Section 4 we show that in this case the sector theory istrivial. This corroborates the notion that the sector theory is a good invariantfor topological phases by proving that indeed states without long-range entan-glement have a trivial sector structure. Indeed, we will prove that this still isthe case for pure states ω such that ω ◦ α is quasi-equivalent to a product state.Here, α is a quasi-local automorphism, which can be seen as the generalizationof finite-depth quantum circuits to infinite systems. This result also explainswhy in models such as the toric code, which do have a non-trivial sector theory,we only have a weaker form of the split property, where we have to consider aninclusion Λ ⊂ Λ of cones whose boundaries are sufficiently far apart [Naa12].This weaker form of the split property also plays a role in the analysisin [Naa13], where the index of a certain subfactor is shown to be related to7he total quantum dimension of the sectors. This result can be used to showthat a given list of sectors is complete. It also is necessary in showing that ap-proximately localized sectors, a generalisation of the notion of a sector discussedabove, is stable under applying a path of quasi-local automorphisms [CNN20].In either case, the split property for an inclusion Λ ⊂ Λ allows us to obtaina tensor product decomposition of the ground state Hilbert space such thatobservables in A Λ and those in A Λ c act on the distinct factors. In contrastto finite systems such a decomposition need not exist if the split property failsto hold. This decomposition can then be used to approximately localize endo-morphisms or observables [CNN20]. This plays a crucial role in the proof ofstability of superselection sectors. Although the proof only requires a variant ofthe split property to hold at one point along the path of gapped Hamiltonians, itis nevertheless important to understand the stability of the split property itself. In the classification problem of gapped ground state phases, we say that twostates are in the same phase if they can be realized as ground states of gappedHamiltonians that can be connected via a continuous (or, for technical reasons, C ) path, in such a way that the energy gap does not close along the path.Using the spectral flow [BMNS12], an adaptation of Hastings and Wen’s quasi-adiabatic continuation [HW05] to the thermodynamic limit, one obtains a pathof automorphisms s (cid:55)→ α s relating the ground states along the path of gappedHamiltonians. Its infinite system version, where a uniform gap for the localHamiltonians can be replaced by the spectral gap of the bulk Hamiltonian inthe GNS representation was shown in [MO20]. Quasi-local automorphisms areessential transformation in the theory of gapped ground state phases.A quasi-local map on A Γ is a map that maps strictly localized observables toobservables that can still be approximately localized in a slightly larger region,with error bounds satisfying a Lieb-Robinson type of estimate. Our discussiondraws heavily on [NSY19], which in turn incorporates decades of advancementsin Lieb-Robinson bounds.Typically the quasi-local maps are obtained as the dynamics generated bysome sufficiently local interaction. The notion of “sufficiently local” is madeprecise in the following definitions. Definition 2.3. An F -function F on (Γ , d ) is a non-increasing function F :[0 , ∞ ) → (0 , ∞ ) such that (i) (cid:107) F (cid:107) := sup x ∈ Γ (cid:16)(cid:80) y ∈ Γ F ( d ( x, y )) (cid:17) < ∞ , and (ii) C F := sup x,y ∈ Γ (cid:16)(cid:80) z ∈ Γ F ( d ( x,z )) F ( d ( z,y )) F ( d ( x,y )) (cid:17) < ∞ .These are called uniform integrability and the convolution identity , respectively.For an F -function F on (Γ , d ), define a function G F on t ≥ G F ( t ) := sup x ∈ Γ (cid:88) y ∈ Γ ,d ( x,y ) ≥ t F ( d ( x, y )) , t ≥ . (2.19)8ote that by uniform integrability the supremum is finite for all t .Our goal is to interpolate continuously between two local interactions. Hencewe will mainly be considering paths of local interactions, in the following sense: Definition 2.4.
A norm-continuous interaction on A Γ defined on an interval[0 ,
1] is a map Φ : P (Γ) × [0 , → A locΓ such that (i) for any t ∈ [0 , · ; t ) : P (Γ) → A locΓ is an interaction, and (ii) for any Z ∈ P (Γ), the map Φ( Z ; · ) : [0 , → A Z is norm-continuous.To ensure that the interactions induce quasi-local automorphisms we needto impose sufficient decay properties on the interaction strength. Definition 2.5.
Let F be an F -function on (Γ , d ). We denote by B F ([0 , A Γ defined on an interval [0 ,
1] suchthat the function (cid:107) Φ (cid:107) : [0 , → R defined by (cid:107) Φ (cid:107) ( t ) := sup x,y ∈ Γ F ( d ( x, y )) (cid:88) Z ∈P (Γ) ,Z (cid:51) x,y (cid:107) Φ( Z ; t ) (cid:107) , t ∈ [0 , , (2.20)is uniformly bounded, i.e., sup t ∈ [0 , (cid:107) Φ (cid:107) ( t ) < ∞ . It follows that t (cid:55)→ (cid:107) Φ (cid:107) ( t ) isintegrable, and we set I (Φ) := I , (Φ) := C F (cid:90) dt (cid:107) Φ (cid:107) ( t ) . (2.21)We will need some more notation. For Φ ∈ B F ([0 , ≤ m ∈ R , weintroduce a path of interactions Φ m byΦ m ( X ; t ) := | X | m Φ ( X ; t ) , X ∈ P (Γ) , t ∈ [0 , . (2.22)Next we recall that an interaction gives rise to local (and here, time-dependent)Hamiltonians, via H Λ , Φ ( t ) := (cid:88) Z ⊂ Λ Φ( Z ; t ) , t ∈ [0 , . (2.23)We denote by U Λ , Φ ( t ; s ), the solution of ddt U Λ , Φ ( t ; s ) = − iH Λ , Φ ( t ) U Λ , Φ ( t ; s ) , t ∈ [0 ,
1] (2.24) U Λ , Φ ( s ; s ) = I . (2.25)We define corresponding automorphisms τ (Λ) , Φ t,s , ˆ τ (Λ) , Φ t,s on A Γ by τ (Λ) , Φ t,s ( A ) := U Λ , Φ ( t ; s ) ∗ AU Λ , Φ ( t ; s ) , (2.26)ˆ τ (Λ) , Φ t,s ( A ) := U Λ , Φ ( t ; s ) AU Λ , Φ ( t ; s ) ∗ , (2.27)with A ∈ A Γ . Note that ˆ τ (Λ) , Φ t,s = τ (Λ) , Φ s,t , (2.28)by the uniqueness of the solution of the differential equation. Using standardtechniques one can prove locality estimates for time-evolved local observables inthe form of Lieb-Robinson bounds, which in turn can be used to show that thelocal dynamics τ (Λ) , Φ t,s induce global dynamics. Since we will make use of thesefacts repeatedly we recall the main points here.9 heorem 2.6 ([NSY19]) . Let (Γ , d ) be a countable metric space, and let F bea F -function on (Γ , d ) . Suppose that Φ ∈ B F ([0 , . The following holds: (i) The limit τ Φ t,s ( A ) := lim Λ (cid:37) Γ τ (Λ) , Φ t,s ( A ) , A ∈ A Γ , t, s ∈ [0 ,
1] (2.29) exists and defines a strongly continuous family of automorphisms on A Γ such that τ Φ t,s ◦ τ Φ s,u = τ Φ t,u , τ Φ t,t = id A Γ , t, s, u ∈ [0 , . (2.30) (ii) For any
X, Y ∈ P (Γ) with X ∩ Y = ∅ , and A ∈ A X , B ∈ A Y we have (cid:13)(cid:13)(cid:2) τ Φ t,s ( A ) , B (cid:3)(cid:13)(cid:13) ≤ (cid:107) A (cid:107) (cid:107) B (cid:107) C F (cid:16) e I (Φ) − (cid:17) | X | G F ( d ( X, Y )) . (2.31) If Λ ∈ P (Γ) and X ∪ Y ⊂ Λ , a similar bound holds for τ (Λ) , Φ t,s . (iii) For any X ∈ P (Γ) we have (cid:13)(cid:13) ∆ X ( m ) (cid:0) τ Φ t,s ( A ) (cid:1)(cid:13)(cid:13) ≤ (cid:107) A (cid:107) C F (cid:16) e I (Φ) − (cid:17) | X | G F ( m ) , (2.32) for all Λ ∈ P (Γ) and A ∈ A X . A similar bound holds for τ (Λ) , Φ t,s . (iv) For any X, Λ ∈ P (Γ) with X ⊂ Λ , and A ∈ A X we have (cid:13)(cid:13)(cid:13) τ (Λ) , Φ t,s ( A ) − τ Φ t,s ( A ) (cid:13)(cid:13)(cid:13) ≤ C F (cid:107) A (cid:107) e I (Φ) I (Φ) | X | G F ( d ( X, Γ \ Λ)) . (2.33) Proof.
Item (i) is Theorem 3.5 of [NSY19], while (ii) and (iv) follow fromCorollary 3.6 of the same paper by a straightforward bounding of D ( X, Y )and the summation in eq. (3.80) of [NSY19] respectively. Finally, (iii) can beobtained using (ii) and [NSY19, Cor. 4.4] (see also the proof of Lemma 5.1 inthe same paper).Consider the same notation and assumptions as in Theorem 2.6. To continuewe need to make additional assumptions on the function F . In particular, weassume that there is an α ∈ (0 ,
1) such that ∞ (cid:88) n =0 (1 + n ) ν +1 G F ( n ) α < ∞ , (2.34)where G F is as defined in (2.19). Furthermore, we assume that there is an F -function ˜ F on (Γ , d ) such thatmax F (cid:16) r (cid:17) , ∞ (cid:88) n =[ r ] (1 + n ) ν +1 G F ( n ) α ≤ ˜ F ( r ) . (2.35)With these additional assumptions we can distill the following result. It gives usa way to apply a quasi-local automorphism to a given dynamics. The result willgenerally not be an interaction, since the interaction terms will not localized infinite regions any more. Nevertheless, the theorem shows that we can define aproper interaction that gives the correct local Hamiltonians.10 heorem 2.7. Let (Γ , d ) be a countable ν -regular metric space and F be an F -function on (Γ , d ) such that there are α and ˜ F satisfying (2.34) and (2.35) .Let Φ ∈ B F ([0 , be a path of interactions such that Φ ∈ B F ([0 , , where Φ is defined in (2.22) . Finally, choose an increasing sequence Λ n ∈ P (Γ) suchthat Λ n (cid:37) Γ , and let τ Φ t,s and τ (Λ n ) , Φ t,s be as in Theorem 2.6.Then, with s ∈ [0 , , the right hand side of the following sum Ψ ( s ) ( Z, t ) := (cid:88) m ≥ (cid:88) X ⊂ Z, X ( m )= Z ∆ X ( m ) (cid:0) τ Φ t,s (Φ ( X ; t )) (cid:1) (2.36) defines an interaction Ψ ( s ) ∈ B ˜ F ([0 , . Furthermore, the formula Ψ ( n )( s ) ( Z, t ) := (cid:88) m ≥ (cid:88) X ⊂ Z,X ( m ) ∩ Λ n = Z ∆ X ( m ) (cid:16) τ (Λ n , Φ) t,s (Φ ( X ; t )) (cid:17) (2.37) defines Ψ ( n )( s ) ∈ B ˜ F ([0 , such that Ψ ( n ) ( Z, t ) = 0 unless Z ⊂ Λ n , and satisfies τ (Λ n ) , Φ t,s ( H Λ n , Φ ( t )) = H Λ n , Ψ ( n ) ( t ) . (2.38) For any t, u ∈ [0 , , we have lim n →∞ (cid:13)(cid:13)(cid:13) τ Ψ ( n )( s ) t,u ( A ) − τ Ψ ( s ) t,u ( A ) (cid:13)(cid:13)(cid:13) = 0 , A ∈ A Γ . (2.39) Proof. If Z is a finite set, we see that the right-hand side of (2.36) containsonly finitely many terms and hence is well-defined. Moreover, because of the∆ X ( m ) , it follows that Ψ ( s ) ( Z, t ) ∈ A Z . Since τ t,s is in automorphism we see thatΨ ( s ) ( Z, t ) is self-adjoint, and hence defines an interaction. That this interactionis in B ˜ F ([0 , p = 0 and q = r = 1.Similarly, equation (2.37) defines an interaction, and (2.38) can be verifiedby an explicit calculation, if we note that τ (Λ n ) , Φ t,s (Φ( X ; t )) is in A Λ n . By part (ii)of Theorem 5.17 of [NSY19] it follows that Ψ ( n )( s ) ∈ B ˜ F ([0 , ( n )( s ) converges to Ψ ( s ) in F -norm with respect to ˜ F . Theorem 5.13of [NSY19] implies sup n (cid:90) (cid:13)(cid:13)(cid:13) Ψ ( n )( s ) (cid:13)(cid:13)(cid:13) ˜ F ( t ) dt < ∞ , (2.40)see also [NSY19, eq. (5.101)]. Therefore, from Theorem 3.8 of [NSY19], weobtain (2.39). In this section we give our main technical result. In particular, we study con-ditions under which a quasi-local automorphism τ Φ1 , “factorizes” as in Proposi-tion 2.2, in particular equation (2.12). In the next theorem we give a sufficientcondition in terms of the regions involved and the F -function for Φ.Before we state the full conditions and prove the result, let us briefly outlinethe main steps. The idea behind the proof is to compare the full dynamics11enerated by the interaction Φ with the “decoupled” dynamics Φ (0) . The lattersimply omits all interaction terms of Φ crossing the boundary of Γ \ Γ . Thefirst step is to show that the difference between the dynamics, τ Φ1 , ◦ (cid:16) τ Φ (0) , (cid:17) − is quasi-local, and generated by an interaction as in Theorem 2.7. In the secondstep we show that this interaction can be well approximated by interaction termslocalized in Γ (cid:48) \ Γ (cid:48) , with Γ (cid:48) ⊂ Γ ⊂ Γ ⊂ Γ (cid:48) , in the sense that the contributions outside this region sum up to a bounded operator in A Γ . In Step 3 this is thenused to show that the difference of the full and decoupled dynamics can bewritten as an automorphism of A Γ (cid:48) \ Γ (cid:48) followed by conjugation with a unitary.This ultimately allows us to write the interaction in form that allows us to applyProposition 2.2. Theorem 3.1.
Let (Γ , d ) be a countable ν -regular metric space with constant κ as in (2.1) . Let F be an F -function on (Γ , d ) such that the function G F defined by (2.19) satisfies (2.34) for some α ∈ (0 , . Suppose that there is an F -function ˜ F satisfying (2.35) for this F . Let A Γ be a quantum spin systemgiven by (2.3) and (2.4).Let Φ ∈ B F ([0 , be a path of interactions satisfying Φ ∈ B F ([0 , . (Recallfrom definition (2.22) that this means that X (cid:55)→ | X | Φ( X ; t ) is in B F ([0 , ).Let Γ (cid:48) ⊂ Γ ⊂ Γ ⊂ Γ (cid:48) ⊂ Γ . (3.1) For m ∈ N ∪ { } , x, y ∈ Γ , set f ( m, x, y ) := (cid:88) X (cid:51) x,y,d (( Γ (cid:48) \ Γ (cid:48) ) c ,X ) ≤ m | X | sup t ∈ [0 , (cid:107) Φ( X, t ) (cid:107) . (3.2) We assume that (cid:88) x ∈ Γ (cid:88) y ∈ Γ c + (cid:88) x ∈ Γ \ Γ (cid:88) y ∈ (Γ \ Γ ) c ∞ (cid:88) m =0 G F ( m ) f ( m, x, y ) < ∞ (3.3) Define Φ (0) ∈ B F ([0 , by Φ (0) ( X ; t ) := (cid:40) Φ ( X ; t ) , if X ⊂ Γ or X ⊂ Γ \ Γ or X ⊂ Γ c , otherwise , (3.4) for each X ∈ P (Γ) , t ∈ [0 , . Then there is an automorphism β Γ (cid:48) \ Γ (cid:48) on A Γ (cid:48) \ Γ (cid:48) and a unitary u ∈ A Γ such that τ Φ1 , = Ad( u ) ◦ τ Φ (0) , ◦ (cid:16) ˜ β Γ (cid:48) \ Γ (cid:48) (cid:17) . (3.5) Proof. Step 1.
First we would like to represent τ Φ1 , ◦ (cid:16) τ Φ (0) , (cid:17) − as some quasi-local automorphism, applying Theorem 2.7. Let { Λ n } ∞ n =1 ⊂ P (Γ) be an in-creasing sequence Λ n (cid:37) Γ. We also define Φ (1) ∈ B F ([0 , (1) ( X ; t ) := Φ (0) ( X ; t ) − Φ ( X ; t ) , (3.6)12or each X ∈ P (Γ), t ∈ [0 , t, s ∈ [0 , (1) . Hence we setΨ ( s ) ( Z, t ) := (cid:88) m ≥ (cid:88) X ⊂ Z, X ( m )= Z ∆ X ( m ) (cid:16) τ Φ t,s (cid:16) Φ (1) ( X ; t ) (cid:17)(cid:17) (3.7)andΨ ( n )( s ) ( Z, t ) := (cid:88) m ≥ (cid:88) X ⊂ Z,X ( m ) ∩ Λ n = Z ∆ X ( m ) (cid:16) τ (Λ n )Φ t,s (cid:16) Φ (1) ( X ; t ) (cid:17)(cid:17) . (3.8)Corresponding to (2.38), we obtain τ (Λ n ) , Φ t,s (cid:0) H Λ n , Φ (1) (cid:1) = H Λ n , Ψ ( n )( s ) ( t ) . (3.9)Applying Theorem 2.7, we have Ψ ( n )( s ) , Ψ ( s ) ∈ B ˜ F ([0 , n →∞ (cid:13)(cid:13)(cid:13) τ Ψ ( n )( s ) t,u ( A ) − τ Ψ ( s ) t,u ( A ) (cid:13)(cid:13)(cid:13) = 0 , A ∈ A Γ , t, u ∈ [0 ,
1] (3.10)holds. Note that ddt ˆ τ (Λ n ) , Ψ ( n )( s ) t,s ( A ) = − i (cid:104) H Λ n , Ψ ( n )( s ) ( t ) , ˆ τ (Λ n ) , Ψ ( n )( s ) t,s ( A ) (cid:105) = − i (cid:104) τ (Λ n ) , Φ t,s (cid:0) H Λ n , Φ (1) (cid:1) , ˆ τ (Λ n ) , Ψ ( n )( s ) t,s ( A ) (cid:105) . (3.11)On the other hand, we have ddt τ (Λ n ) , Φ t,s ◦ (cid:16) τ (Λ n ) , Φ (0) t,s (cid:17) − ( A )= τ (Λ n ) , Φ t,s (cid:18) i (cid:20) H Λ n , Φ ( t ) − H Λ n , Φ (0) ( t ) , (cid:16) τ (Λ n ) , Φ (0) t,s (cid:17) − ( A ) (cid:21)(cid:19) = − i (cid:20) τ (Λ n ) , Φ t,s (cid:0) H Λ n , Φ (1) (cid:1) , τ (Λ n ) , Φ t,s ◦ (cid:16) τ (Λ n ) , Φ (0) t,s (cid:17) − ( A ) (cid:21) . (3.12)Hence ˆ τ (Λ n ) , Ψ ( n )( s ) t,s ( A ) and τ (Λ n ) , Φ t,s ◦ (cid:16) τ (Λ n ) , Φ (0) t,s (cid:17) − ( A ) satisfy the same differ-ential equation with the ˆ τ (Λ n ) , Ψ ( n )( s ) s,s ( A ) = τ (Λ n ) , Φ s,s ◦ (cid:16) τ (Λ n ) , Φ (0) s,s (cid:17) − ( A ) = A .Therefore we obtainˆ τ (Λ n ) , Ψ ( n )( s ) t,s ( A ) = τ (Λ n ) , Φ t,s ◦ (cid:16) τ (Λ n ) , Φ (0) t,s (cid:17) − ( A ) , t ∈ [0 , , A ∈ A Γ . (3.13)From the fact that ˆ τ Ψ ( n )( s ) t,u ( A ) = ˆ τ (Λ n ) , Ψ ( n )( s ) t,u = τ (Λ n ) , Ψ ( n )( s ) u,t = τ Ψ ( n )( s ) u,t convergesstrongly to an automorphism τ Ψ ( s ) u,t on A Γ (3.10), we havelim n →∞ (cid:13)(cid:13)(cid:13) ˆ τ Ψ ( n )( s ) t,s ( A ) − τ Ψ ( s ) s,t ( A ) (cid:13)(cid:13)(cid:13) = 0 , A ∈ A Γ . (3.14)On the other hand, by Theorem 2.6, we have for t ∈ [0 ,
1] and A ∈ A Γ lim n →∞ (cid:13)(cid:13)(cid:13)(cid:13) τ (Λ n ) , Φ t,s ◦ (cid:16) τ (Λ n ) , Φ (0) t,s (cid:17) − ( A ) − τ Φ t,s ◦ (cid:16) τ Φ (0) t,s (cid:17) − ( A ) (cid:13)(cid:13)(cid:13)(cid:13) = 0 . (3.15)13herefore, taking n → ∞ limit in (3.13), we obtain τ Ψ ( s ) s,t ( A ) = τ Φ t,s ◦ (cid:16) τ Φ (0) t,s (cid:17) − ( A ) , t, s ∈ [0 , , A ∈ A Γ . (3.16)Hence we have τ Φ s,t = (cid:0) τ Φ t,s (cid:1) − = (cid:16) τ Φ (0) t,s (cid:17) − (cid:16) τ Ψ ( s ) s,t (cid:17) − = τ Φ (0) s,t τ Ψ ( s ) t,s (3.17)In particular, we get τ Φ1 , = τ Φ (0) , τ Ψ (1) , . (3.18) Step 2.
We show that the summation V ( t ) := (cid:88) Z ∈P (Γ) (cid:0) id − Π Γ (cid:48) \ Γ (cid:48) (cid:1) (cid:16) Ψ (1) ( Z, t ) (cid:17) ∈ A Γ (3.19)converges absolutely in the norm topology, and uniformly in t ∈ [0 , V n ( t ) := (cid:88) Z ∈P (Γ) , Z ⊂ Λ n (cid:0) id − Π Γ (cid:48) \ Γ (cid:48) (cid:1) (cid:16) Ψ (1) ( Z, t ) (cid:17) ∈ A Λ n , n ∈ N . (3.20)From the convergence of (3.19) uniform in t , we getlim n →∞ sup t ∈ [0 , (cid:107) V n ( t ) − V ( t ) (cid:107) = 0 . (3.21)To prove the convergence of (3.19), it suffices to provelim n →∞ sup t ∈ [0 , (cid:88) Z ∈P (Γ) , Z ∩ Λ cn (cid:54) = ∅ (cid:13)(cid:13)(cid:13)(cid:0) id − Π Γ (cid:48) \ Γ (cid:48) (cid:1) (cid:16) Ψ (1) ( z, t ) (cid:17)(cid:13)(cid:13)(cid:13) = 0 . (3.22)To prove this, we introduce the following functions. For m ∈ N ∪ { } , n ∈ N ,and x, y ∈ Γ, set f n ( m, x, y ) := (cid:88) X (cid:51) x,y,d ( X, Λ cn ) ≤ m d (( Γ (cid:48) \ Γ (cid:48) ) c ,X ) ≤ m | X | sup t ∈ [0 , (cid:107) Φ( X, t ) (cid:107) . (3.23)Note that f n ( m, x, y ) is bounded by f point-wise (by definition) and converges tozero point-wise, by (3.3). Hence by (3.3) and Lebesgue’s dominated convergencetheorem, we obtainlim n →∞ (cid:88) x ∈ Γ (cid:88) y ∈ Γ c + (cid:88) x ∈ Γ \ Γ (cid:88) y ∈ (Γ \ Γ ) c ∞ (cid:88) m =0 G F ( m ) f n ( m, x, y ) = 0 . (3.24)14e havesup t ∈ [0 , (cid:88) Z ∈P (Γ) , Z ∩ Λ cn (cid:54) = ∅ (cid:13)(cid:13)(cid:13)(cid:0) id − Π Γ (cid:48) \ Γ (cid:48) (cid:1) (cid:16) Ψ (1) ( Z, t ) (cid:17)(cid:13)(cid:13)(cid:13) (3.25) ≤ (cid:88) Z ∈P (Γ) , Z ∩ Λ cn (cid:54) = ∅ (cid:88) m ≥ (cid:88) X ⊂ Z, X ( m )= Z (cid:34) sup t ∈ [0 , (cid:13)(cid:13)(cid:13)(cid:0) id − Π Γ (cid:48) \ Γ (cid:48) (cid:1) ∆ X ( m ) (cid:16) τ Φ t, (cid:16) Φ (1) ( X ; t ) (cid:17)(cid:17)(cid:13)(cid:13)(cid:13)(cid:35) (3.26) ≤ (cid:88) m ≥ (cid:88) X ∈P (Γ) X ( m ) ∩ Λ cn (cid:54) = ∅ sup t ∈ [0 , (cid:13)(cid:13)(cid:13)(cid:0) id − Π Γ (cid:48) \ Γ (cid:48) (cid:1) ∆ X ( m ) (cid:16) τ Φ t, (cid:16) Φ (1) ( X ; t ) (cid:17)(cid:17)(cid:13)(cid:13)(cid:13) (3.27) ≤ (cid:88) m ≥ (cid:88) X ∈P (Γ) X ( m ) ∩ Λ cn (cid:54) = ∅ ,X ( m ) ∩ ( Γ (cid:48) \ Γ (cid:48) ) c (cid:54) = ∅ sup t ∈ [0 , (cid:13)(cid:13)(cid:13) ∆ X ( m ) (cid:16) τ Φ t, (cid:16) Φ (1) ( X ; t ) (cid:17)(cid:17)(cid:13)(cid:13)(cid:13) (3.28) ≤ (cid:88) m ≥ (cid:88) X ∈P (Γ) X ( m ) ∩ Λ cn (cid:54) = ∅ ,X ( m ) ∩ ( Γ (cid:48) \ Γ (cid:48) ) c (cid:54) = ∅ (cid:34) sup t ∈ [0 , (cid:13)(cid:13) Φ (1) ( X ; t ) (cid:13)(cid:13) C F (cid:16) e I (Φ) − (cid:17) | X | G F ( m ) (cid:35) (3.29)= 8 C F (cid:16) e I (Φ) − (cid:17) (cid:88) m ≥ (cid:88) X ∈P (Γ) X ( m ) ∩ Λ cn (cid:54) = ∅ ,X ( m ) ∩ ( Γ (cid:48) \ Γ (cid:48) ) c (cid:54) = ∅ (cid:34) sup t ∈ [0 , (cid:16)(cid:13)(cid:13)(cid:13) Φ (1) ( X ; t ) (cid:13)(cid:13)(cid:13)(cid:17) | X | G F ( m ) (cid:35) (3.30)For the fourth inequality, we used Theorem 2.6 (iii). From the definition ofΦ (1) , we have Φ (1) ( X ; t ) = 0 , unless X has a non-empty intersection with atleast two of Γ , Γ c , Γ \ Γ . In particular, we have Φ (1) ( X ; t ) = 0 , unless X ∩ Γ (cid:54) = ∅ , X ∩ Γ c (cid:54) = ∅ or X ∩ (Γ \ Γ ) (cid:54) = ∅ , X ∩ (Γ \ Γ ) c (cid:54) = ∅ . Therefore,if Φ (1) ( X ; t ) (cid:54) = 0 , there should be x ∈ Γ , y ∈ Γ c with X (cid:51) x, y or x ∈ Γ \ Γ y ∈ (Γ \ Γ ) c with X (cid:51) x, y . We also note that if X ( m ) ∩ Λ cn (cid:54) = ∅ and X ( m ) ∩ (Γ (cid:48) \ Γ (cid:48) ) c , then we have d ( X, Λ cn ) ≤ m and d ( X, (Γ (cid:48) \ Γ (cid:48) ) c ) ≤ m .Therefore we have(3 . ≤ C F (cid:16) e I (Φ) − (cid:17) (cid:88) x ∈ Γ (cid:88) y ∈ Γ c + (cid:88) x ∈ Γ \ Γ (cid:88) y ∈ (Γ \ Γ ) c (3.31) (cid:88) m ≥ (cid:88) X ∈P (Γ) d ( X, Λ cn ) ≤ m, d ( X, ( Γ (cid:48) \ Γ (cid:48) ) c ) ≤ m, X (cid:51) x,y sup t ∈ [0 , (cid:16)(cid:13)(cid:13)(cid:13) Φ (1) ( X ; t ) (cid:13)(cid:13)(cid:13)(cid:17) | X | G F ( m )(3.32)= 8 C F (cid:16) e I (Φ) − (cid:17) (cid:88) x ∈ Γ (cid:88) y ∈ Γ c + (cid:88) x ∈ Γ \ Γ (cid:88) y ∈ (Γ \ Γ ) c (cid:88) m ≥ f n ( m, x, y ) G F ( m ) . (3.33)The last part converges to 0 as n → ∞ because of (3.24). This proves (3.22),and hence that (3.19) converges. 15 tep 3. Next we decompose Ψ (1) into a Γ (cid:48) \ Γ (cid:48) -part˜Ψ( Z, t ) := Π Γ (cid:48) \ Γ (cid:48) (cid:16) Ψ (1) ( Z, t ) (cid:17) (3.34)and the rest. Clearly, we have ˜Ψ ∈ B ˜ F ([0 , H Λ n , ˜Ψ ( t ) + V n ( t ) = H Λ n , Ψ (1) ( t ) . (3.35)As a uniform limit of [0 , (cid:51) t (cid:55)→ V n ( t ) ∈ A Γ , [0 , (cid:51) t (cid:55)→ V ( t ) ∈ A Γ isnorm-continuous. Because of ˜Ψ ∈ B ˜ F ([0 , , (cid:51) t (cid:55)→ τ ˜Ψ t,s ( V ( t )) ∈ A Γ is alsonorm-continuous, for each s ∈ [0 , s ∈ [0 , ,
1] to U ( A Γ ) such that ddt W ( s ) ( t ) = − iτ ˜Ψ t,s ( V ( t )) W ( s ) ( t ) , W ( s ) ( s ) = I . (3.36)The solution is given by W ( s ) ( t ) := ∞ (cid:88) k =0 ( − i ) k (cid:90) ts ds (cid:90) s s ds · · · (cid:90) s k − s ds k τ ˜Ψ s ,s ( V ( s )) · · · τ ˜Ψ s k ,s ( V ( s k )) . (3.37)Analogously, for each s ∈ [0 ,
1] and n ∈ N , we define a unique norm-differentiablemap from [0 ,
1] to U ( A Γ ) such that ddt W ( s ) n ( t ) = − iτ (Λ n ) ˜Ψ t,s ( V n ( t )) W ( s ) n ( t ) , W ( s ) n ( s ) = I . (3.38)This differential equation can be solved similarly as in equation (3.37). By theuniform convergence (3.21), we then havelim n sup t ∈ [0 , (cid:13)(cid:13)(cid:13) W ( s ) n ( t ) − W ( s ) ( t ) (cid:13)(cid:13)(cid:13) = 0 . (3.39)From this and Theorem 2.6 (iv) for Ψ (1) , ˜Ψ ∈ B ˜ F ([0 , n →∞ τ (Λ n ) , ˜Ψ s,t ◦ Ad (cid:16) W ( s ) n ( t ) (cid:17) ( A ) = τ ˜Ψ s,t ◦ Ad (cid:16) W ( s ) ( t ) (cid:17) ( A ) , (3.40)lim n →∞ τ (Λ n ) , Ψ (1) s,t ( A ) = τ Ψ (1) s,t ( A ) , (3.41)for any A ∈ A Γ .Note that for any A ∈ A Γ ddt τ (Λ n ) , ˜Ψ s,t ◦ Ad (cid:16) W ( s ) n ( t ) (cid:17) ( A ) (3.42)= − i (cid:104) H Λ n , ˜Ψ ( t ) , τ (Λ n ) , ˜Ψ s,t ◦ Ad (cid:16) W ( s ) n ( t ) (cid:17) ( A ) (cid:105) − iτ (Λ n ) , ˜Ψ s,t (cid:16)(cid:104) τ (Λ n ) , ˜Ψ t,s ( V n ( t )) , Ad (cid:16) W ( s ) n ( t ) (cid:17) ( A ) (cid:105)(cid:17) (3.43)= − i (cid:104) H Λ n , ˜Ψ ( t ) + V n ( t ) , τ (Λ n ) , ˜Ψ s,t ◦ Ad (cid:16) W ( s ) n ( t ) (cid:17) ( A ) (cid:105) = − i (cid:104) H Λ n , Ψ (1) ( t ) , τ (Λ n ) , ˜Ψ s,t ◦ Ad (cid:16) W ( s ) n ( t ) (cid:17) ( A ) (cid:105) . (3.44)16e used (2.28) for the second equality and (3.35) for the third equality. On theother hand, for any A ∈ A Γ , we have ddt τ (Λ n ) , Ψ (1) s,t ( A ) = − i (cid:104) H Λ n , Ψ (1) ( t ) , τ (Λ n ) , Ψ (1) s,t ( A ) (cid:105) . (3.45)Therefore, τ (Λ n ) , ˜Ψ s,t ◦ Ad (cid:16) W ( s ) n ( t ) (cid:17) ( A ) and τ (Λ n ) , Ψ (1) s,t ( A ) satisfy the same differ-ential equation. Also note that we have τ (Λ n ) , ˜Ψ s,s ◦ Ad (cid:16) W ( s ) n ( s ) (cid:17) ( A ) = τ (Λ n ) , Ψ (1) s,s ( A ) = A. Therefore, we get τ (Λ n ) , ˜Ψ s,t ◦ Ad (cid:16) W ( s ) n ( t ) (cid:17) ( A ) = τ (Λ n ) , Ψ (1) s,t ( A ) . (3.46)By (3.40), we obtain τ ˜Ψ s,t ◦ Ad (cid:16) W ( s ) ( t ) (cid:17) ( A ) = τ Ψ (1) s,t ( A ) , A ∈ A Γ , t, s ∈ [0 , . (3.47)Taking inverse, we getAd (cid:16) W ( s ) ∗ ( t ) (cid:17) ◦ τ ˜Ψ t,s = τ Ψ (1) t,s , t, s ∈ [0 , . (3.48) Step 4.
Combining (3.18) and (3.48) we have τ Φ1 , = τ Φ (0) , τ Ψ (1) , = τ Φ (0) , ◦ Ad (cid:16)(cid:16) W (1) (0) (cid:17) ∗ (cid:17) ◦ τ ˜Ψ0 , . (3.49)Setting β Γ (cid:48) \ Γ (cid:48) := τ ˜Ψ0 , , u := τ Φ (0) , (cid:16)(cid:16) W (1) (0) (cid:17) ∗ (cid:17) (3.50)completes the proof. An interesting problem is to find conditions that lead to a trivial superselectionstructure. Topological order is associated to “long-range entanglement” thatcannot be removed by local operations. This should be contrasted with productstates, which are not entangled at all. Hence one is interested in states thatcannot be transformed into product states by such local operations. The productstates are said to be in the topologically trivial phase [CGW10].The goal of this section is to show that such a topologically trivial stateindeed leads to a trivial superselection structure, at least when we restrict tostrictly localized sectors as in equation (2.18). To make this precise, we recallthat the equivalence relation defined in terms of finite depth quantum circuitsis somewhat too restrictive in the thermodynamic limit, and one has to look atlimits of such automorphisms as well. In addition, we will only require to beable to “decouple” a cone-like region. Because of transportability of the anyonsthat is assumed, the choice of cone is not important. We therefore adopt thefollowing definition. 17 efinition 4.1.
Let A Γ be the quasi-local algebra of a quantum spin systemwith Γ = Z ν . We say that a pure state ω has long-range entanglement (LRE) ifthere is no quasi-local automorphism α ∈ Aut( A Γ ) such that ω ◦ α is a productstate with respect to some cone Λ. Here we say that a state is a product statefor a cone Λ if it is of the form ω = ω Λ ⊗ ω Λ c , with ω Λ is a state on A Λ , andsimilarly for ω Λ c .The condition that Γ = Z ν is not essential. However, in the general caseone should define the appropriate analogue of a cone. This depends on thelocalization properties of the excitations one would want to consider, but forthe definition to be non-trivial a cone should at least have infinitely many sites.Note that for a state to be not long-range entangled, we only require thecondition to hold for a single cone Λ.That is, a state is not long-range entangledif we can disentangle the cone Λ from its complement. Typically the states weare interested in have a large degree of ‘homogeneity’, for example because theywill be translation invariant. Moreover, we will be interested in transportablecharges, in the sense that we can move a charge localized in a specific cone to any with a unitary operator. Thus typically one expects that if it is possible todecouple a single cone in this situation, one can do it for more cones. Since wewill not actually need that, we restrict to this simpler definition.In the following we consider the situation where the pure reference state ω is a product state with respect to a fixed cone Λ, i.e., ω = ω Λ ⊗ ω Λ c for somestates ω Λ and ω Λ c on A Λ and A Λ c respectively. Below we consider general purestates without long-range entanglement.We first recall the following Lemma (compare with e.g. [Mat01, Mat10]). Lemma 4.2.
Let ϕ be a pure state on A Γ and suppose that ϕ is quasi-equivalentto ϕ Λ ⊗ ϕ Λ c , where ϕ Λ := ϕ |A Λ for some cone Λ . Then R Λ := π ϕ ( A Λ ) (cid:48)(cid:48) is afactor of Type I, and so is R c Λ . Moreover, Haag duality holds: R Λ = R (cid:48) Λ c .Proof. Write ( π ϕ , H ϕ , Ω ϕ ) for the GNS representation of ϕ . Because ϕ is pure, π ϕ ( A Γ ) (cid:48)(cid:48) is a Type I factor. Note that R Λ ∨ R Λ c = B ( H ϕ ). Taking the commu-tant of this equation, and noting that by locality we have that R Λ ⊂ R (cid:48) Λ c , oneobtains R (cid:48) Λ ∩ R Λ ⊂ R (cid:48) Λ ∩ R (cid:48) Λ c = C I. Hence R Λ is a factor, and so is R Λ c .Since ϕ is quasi-equivalent to ϕ Λ ⊗ ϕ Λ c it follows that there is a normal iso-morphism τ : π ϕ ( A Γ ) (cid:48)(cid:48) → π ϕ Λ ( A Λ ) (cid:48)(cid:48) ⊗ π ϕ Λ c ( A Λ c ) (cid:48)(cid:48) . Because the tensor productof two von Neumann algebras is Type I if and only if both factors are Type I, itfollows that π ϕ Λ ( A Λ ) (cid:48)(cid:48) must be Type I, and similarly for π ϕ Λ c ( A Λ c ) (cid:48)(cid:48) . Finally,since R Λ is a factor, every subrepresentation of π Λ := π ϕ |A Λ is quasi-equivalentto π Λ itself. This is true in particular for π ϕ Λ , and hence R Λ must be of TypeI as well. The same is true for R Λ c .Finally, since R Λ is of Type I, there are Hilbert spaces H and H and aunitary U : H ϕ → H ⊗ H , with U R Λ U ∗ = B ( H ) ⊗ I, and U R Λ c U ∗ ⊂ I ⊗ B ( H ) . The inclusion follows because R Λ c ⊂ R (cid:48) Λ by locality, and because ( B ( H ) ⊗ I ) (cid:48) = I ⊗ B ( H ). Because R Λ and R Λ c generate B ( H ϕ ), it follows that in fact it mustbe an equality. Therefore R Λ = R (cid:48) Λ c . 18 emark . As is shown in the references cited above, the factors being Type Iimplies that ϕ is quasi-equivalent to a product state. However, Haag dualitydoes not necessarily imply the split property.This allows us to prove that if the reference is a product state with respect toa cone, there are no non-trivial representations that are both strictly localizableand transportable. In other words, the superselection structure is trivial. Wewill in fact slightly relax the superselection criterion, and only assume that therepresentations π of interest are quasi- equivalent to π . More precisely, we willbe interested in representation π such that π |A Λ c ∼ q.e. π |A Λ c , (4.1)for all cones Λ. This is true in particular when π is unitarily equivalent to n · π when restricted to observables outside a cone. Here n · π is the directsum of n copies of π , as usual. The reason to allow this relaxation is thatsuch representations can be constructed naturally when considering non-abelianmodels [SV93, Naa15].The following proof is inspired by Proposition 4.2 of [M¨ug99]. Theorem 4.4.
Let ω be a pure state such that its GNS representation π isquasi-equivalent to π Λ ⊗ π Λ c , with π Λ and π Λ c irreducible representations of A Λ and A Λ c respectively. Consider ω to be the reference state in the superselectioncriterion. Then the corresponding sector theory is trivial, in the sense that eachrepresentation π satisfying the selection criterion (4.1) is quasi-equivalent to π .In particular, if π is irreducible, then π and π are equivalent.Proof. Because π | A Λ c is quasi-equivalent to π | A Λ c , which is quasi-equivalent to π Λ c , and π Λ c is irreducible, there is a Hilbert space K and a unitary W : H →H Λ c ⊗ K such that W π ( B ) W ∗ = π Λ c ( B ) ⊗ I K , B ∈ A Λ c . (4.2)Because π Λ c ( A Λ c ) (cid:48)(cid:48) is a Type I factor, it follows that( π Λ c ( A Λ c ) ⊗ I K ) (cid:48) = I H Λ c ⊗ B ( K ) . By the commutativity of A Λ and A Λ c , it follows that for all A ∈ A Λ , we havethat W π ( A ) W ∗ ∈ ( π Λ c ( A Λ c ) ⊗ I K ) (cid:48) . Thus we see that there is a representation ρ of A Λ on K such that W π ( A ) W ∗ = I H Λ c ⊗ ρ ( A ) , A ∈ A Λ . (4.3)Consider a cone Λ (cid:48) such that Λ ⊂ (Λ (cid:48) ) c . Then, by applying the superselectioncriterion and restricting to the cone Λ, it follows that the representation π | A Λ is quasi-equivalent to π | A Λ , which in turn is quasi-equivalent to the irreduciblerepresentation π Λ . On the other hand, from equation (4.3), ρ is quasi-equivalentto π | A Λ . Hence ρ is quasi-equivalent to the irreducible π Λ . Therefore, there area Hilbert space K and a unitary V : K → H Λ ⊗ K such that V ρ ( A ) V ∗ = π Λ ( A ) ⊗ I K , A ∈ A Λ . (4.4)Hence we get( I Λ c ⊗ V ) W π ( AB ) W ∗ ( I Λ c ⊗ V ) ∗ = π Λ c ( B ) ⊗ π Λ ( A ) ⊗ I K (4.5)for all A ∈ A Λ and B ∈ A Λ c . As the right hand side is quasi-equivalent to π , π is quasi-equivalent to π . 19 emark . Note that the assumption in the theorem is a 2D analogue of thesplit property for 1D spin chains. It should be noted that it does not hold formodels such as the toric code, which have non-trivial excitations (or sectors)localized in cones. The reason is that the ground state has long-range entan-glement and cannot be converted into a product state with local operations.However, as we already mentioned in the introduction, we still have the approx-imate or distal split property: a Type I factor R Λ ⊂ F ⊂ R (cid:48) Λ c exists if theboundary of the cones Λ ⊂ Λ are sufficiently distant [FN15]. In that case,it is no longer possible to decompose the representation as a tensor product ofrepresentations of A Λ and A Λ c .The theorem says that, as expected, the product state does not have anynon-trivial superselection sectors. For a general state without long range entan-glement, we can try use the quasi-local automorphism α from Definition 4.1 torelate the sectors of ω ◦ α with those of ω . In general there is no reason why ω should be quasi-equivalent to ω ◦ α , so it does not follow directly that ω ◦ α has trivial sectors. However if α comes from quasi-local dynamics satisfyingTheorem 3.1, we can relate the sectors of π ω and π ω ◦ α . The key point is thatwe can almost “factorize” the automorphism α into automorphisms acting ona cone Λ and its complement, up to conjugation with a unitary in A Γ and anautomorphism acting non-trivially only near the border of Λ. Definition 4.6.
Let α be an automorphism of A Γ and consider an inclusion ofcones Γ (cid:48) ⊂ Λ ⊂ Γ (cid:48) ⊂ Γ . We say that α is quasi-factorizable with respect to this inclusion if there is aunitary u ∈ A and automorphisms α Λ and α Λ c of A Λ and A Λ c respectively,such that α = Ad( u ) ◦ (cid:101) Ξ ◦ ( α Λ ◦ α Λ c ) , where Ξ is an automorphism on Γ (cid:48) \ Γ (cid:48) .In Section 5 we will show how such automorphisms can be obtained usingTheorem 3.1. Theorem 4.7.
Let ( H , π ) be a representation. Let α be a quasi-local automor-phism such that for every cone Λ , we can find an inclusion of cones Γ ⊂ Λ ⊂ Γ such that α is quasi-factorizable with repsect to this inclusion. Suppose that arepresentation π satisfies the superselection criterion for π in the sense thatfor all cones Λ in Z , we have π | A Λ c ∼ q.e. π | A Λ c . (4.6) Then π ◦ α satisfies the superselection criterion (4.1) for π ◦ α Proof.
Let Λ be a cone. We will show that π ◦ α | A Λ c ∼ q.e. π ◦ α | A Λ c . (4.7)By assumption we can factorize α as α = Ad( u ) ◦ (cid:101) Ξ ◦ ( α Λ ⊗ α Λ c ) , (4.8)20s in Definition 4.6. From this, for any A ∈ A Λ c , we have π ◦ α ( A ) = π ◦ Ad( u ) ◦ (cid:101) Ξ( α Λ c ( A ))= Ad ( π ( u )) ◦ π ◦ (cid:101) Ξ( α Λ c ( A ))= Ad ( π ( u )) ◦ π | A Γ c ◦ (cid:101) Ξ( α Λ c ( A )) . (4.9)This implies π ◦ α | A Λ c ∼ q.e. π | A Γ c ◦ (cid:101) Ξ ◦ α Λ c | A Λ c . (4.10)(In fact this is even a unitary equivalence). Similarly, we have π ◦ α | A Λ c ∼ q.e. π | A Γ c ◦ (cid:101) Ξ ◦ α Λ c | A Λ c . (4.11)Because we have π | A Γ c ∼ q.e. π | A Γ c by virtue of the superselection criterion, weget π | A Γ c ◦ (cid:101) Ξ ◦ α Λ c | A Λ c ∼ q.e. π | A Γ c ◦ (cid:101) Ξ ◦ α Λ c | A Λ c . (4.12)Combining this with (4.10) and (4.11), we get π ◦ α | A Λ c ∼ q.e. π ◦ α | A Λ c . (4.13)This proves the claim.Combining the two theorems in this section then shows that short-rangeentangled states indeed have a trivial sector structure. Corollary 4.8.
Let ( H , π ) be an irreducible representation which factorizesas π = π Λ ⊗ π Λ c for some cone Λ , where ( π Λ , H Λ ) , ( π Λ c , H Λ c ) are irreduciblerepresentations of A Λ , A Λ c respectively. Let α be a quasi-local automorphismwhich is quasi-factorizable for all cones Λ . Suppose that a representation π satisfies the superselection criterion for π ◦ α in the sense that for all cones (cid:101) Λ in Z , we have π | A (cid:101) Λ c ∼ q.e. π ◦ α | A (cid:101) Λ c . (4.14) Then π is quasi-equivalent to π ◦ α . In particular, if π is irreducible, then π and π are equivalent.Proof. If α is a quasi-local automorphism, the same is true for α − , and it isquasi-factorizable as well. Because π satisfies the superselection criterion for π ◦ α and α − is a quasi-local automorphism, by Theorem 4.7, π ◦ α − satisfiesthe superselection criterion for π ◦ α ◦ α − = π . Then by Theorem 4.4, π ◦ α − is quasi-equivalent to π . From this, it follows that π is quasi-equivalent to π ◦ α . Remark . We argued that a state that satisfies the strict split property for agiven cone is trivial in the sense that there are no anyonic excitations (superse-lection sectors). It is however possible to further classify this trivial sector, forexample if there is an on-site symmetry G . In that case, it is natural to demandthat two states are only in the same gapped phase if they can be connected by a21ontinuous path of gapped Hamiltonians respecting the G -symmetry [CGW10].In two dimensions, the set of states that are in the trivial phase (i.e., containingthe product state with respect to each site) can then be classified by a cocyclein H ( G, U (1)) [Oga21b]. However, in our definition, the absence of long-rangeentanglement does not necessarily imply that the state is such a product ofsingle-site states. It seems plausible that if we demand the split property tohold for any cone, this would follow.There is another natural generalization of the superselection criterion (2.18).Given that the spectral flow is quasi-local, it is natural to look at representa-tions that can be localized in cones up to some exponentially decaying error.This leads to the notion of approximately localizable endomorphisms, and onecan develop the full sector theory (including e.g. braiding of charges) usingthem [CNN20]. These properties are stable upon applying the quasi-local spec-tral flow. We should add the caveat that this is a result about approximately localized sectors, i.e. localized up to some exponentially decaying error, and wecannot rule out that despite the absence of strictly localized sectors, there is anon-trivial approximately localized sector. In abelian quantum double models,this can be ruled out by imposing an “energy criterion”, essentially excludingany possible confined charges [CNN20]. We presently do not know if the ab-sence of such sectors can be proven from more fundamental assumptions. Forexample, in the case of strict localization it is not necessary. The results inthis section and in [CNN20] strongly suggest that in a state with short-rangeentanglement, there are no approximately localizable sectors either.
We apply the results of Section 3 to two-dimensional models. In Section 4 wehave already discussed the split property for a cone and its complement. Asalready mentioned, this strong version does not hold for, for example, abelianquantum double models, where only a weaker version is true [FN15, Naa12].This in turn is a key assumption in the stability of superselection sectors analysisin [CNN20]. Although there we only need the approximate split property forthe “unperturbed” model, it is interesting to know if it is in fact a property ofthe whole phase. Hence, in this section, we show that for suitable perturbationsthis is indeed the case, and the perturbed model also satisfies the approximatesplit property.Let us recall that if F is an F -function, F r ( r ) := e − r F ( r ) is also an F -function. This is an example of a weighted F -function in the terminology ofRef. [NSY19]. Such weighted F -functions have favorable decay properties, ascan be seen in the following Lemma. Lemma 5.1.
Let (Γ , d ) be Z with the usual metric. Then there is a C > such that we have the following estimate for all m > √ : G F r ( m ) ≤ CF ( m − √ me − m , (5.1) where G F r is as defined in equation (2.19) .Proof. By translation invariance of the metric and Γ we do not need the supre-22um in equation (2.19). Hence we get G F r ( m ) = (cid:88) | x |≥ m e −| x | F ( | x | ) ≤ π (cid:90) ∞ m re − r + √ F ( r − √ dr ≤ πe √ F ( m − √ (cid:90) ∞ m re − r dr ≤ πe √ F ( m − √ me − m . This can be seen by noting that (cid:90) x +1 x (cid:90) y +1 y e −| ( x,y ) | + √ F ( | ( x, y ) | ) dxdy ≥ e −| ( x,y ) | F ( | ( x, y ) | ) (5.2)for x, y ≥ F is positive and decreasing), and doing a coordinate trans-formation to polar coordinates.It is possible to generalize the lemma to other suitable weightings F g ( r ) := e − g ( r ) F ( r ) (see e.g. [CNN20]). This could be necessary because in applicationsone would need to assume that interactions have finite interaction norm withrespect to the weighted F -function, instead of F itself. Since we will consideronly bounded range interactions, this is not an issue for us and we restrict tothe easier case for simplicity. Theorem 5.2.
Let
Γ = Z with the usual metric d and consider the corre-sponding quantum spin system A Γ , where the local dimension of the spins isuniformly bounded. Let t (cid:55)→ Φ( X ; t ) be a path of dynamics such that (cid:107) Φ( X ; t ) (cid:107) is uniformly bounded both in X and t . Moreover assume that Φ is of boundedrange, and let F be an F -function. Then Φ ∈ B F r ([0 , , and it generatesquasi-local dynamics τ Φ t,s . Assume that Γ ⊂ Γ is an inclusion of cones suchthat their borders are sufficiently far away, in the sense that the lines markingthe boundaries of the cones are not parallel. Then there exist cones Γ (cid:48) ⊂ Γ and Γ (cid:48) ⊃ Γ such that the conditions of Theorem 3.1 are satisfied.Proof. Without loss of generality we may assume that the cones Γ and Γ have their center line in the direction of the positive x -axis. We write α for theopening angle of Γ and β for the opening angle Γ (see Figure 1). The distancebetween their tips will be denoted by d . Let 0 < (cid:15) < β such that α + (cid:15) < π/ (cid:48) and Γ (cid:48) as in the figure. Later in the proof we willprovide convenient values for d and d (cid:48) , but we note that with a little extrawork is is possible to show that any positive value will do.We show that we can apply Theorem 3.1. First note that Γ is 2-regular, sincethe number of points in a disk of radius r scales with the area. Because theinteraction range is uniformly bounded and because of 2-regularity, there areconstants C and d Φ such that Φ( X ; t ) = 0 whenever | X | > C or diam( X ) >d Φ . It follows that Φ ∈ B F r ([0 , G αF r has finite moments for α ∈ (0 ,
1] (in the sense of equation (2.34)) and we canfind a suitable F -function (cid:101) F such that equation (2.35) is satisfied for F r .It remains to be shown that equation (3.3) is satisfied. As a first step westudy the function f ( m, x, y ) of equation (3.2). Note that the summation in the23igure 1: Cones as in Theorem 5.2.definition is over certain subsets of X such that x, y ∈ X . Hence if d ( x, y ) > d Φ we have Φ( X ; t ) = 0 and consequently f ( m, x, y ) = 0. Similarly, the summationis only over X such that d ( X, (Γ (cid:48) \ Γ (cid:48) ) c ) ≤ m . Hence, it follows that f ( m, x, y ) =0 unless d ( x, (Γ (cid:48) \ Γ (cid:48) ) c ) ≤ m + d Φ , or the same is true for y . Or giving a rougherestimate, f ( m, x, y ) = 0 unless d ( x, (Γ (cid:48) \ Γ (cid:48) ) c ) ≤ m + 2 d Φ , regardless of y .Now consider the case where d ( x, y ) ≤ d Φ and m large enough such that d ( x, (Γ (cid:48) \ Γ (cid:48) ) c ) ≤ m + 2 d Φ . In that case, we have f ( m, x, y ) = (cid:88) x,y (cid:51) X | X | sup t (cid:107) Φ( X ; t ) (cid:107) ≤ C M | b ( d Φ ) | , (5.3)where M := sup X sup t ∈ [0 , (cid:107) Φ( X ; t ) (cid:107) , which is finite by assumption. We alsoused translation invariance of the metric (and Γ), and that by the finite rangeassumption any contributing subset X must be contained in b x ( d Φ ). There areat most 2 | b ( d Φ ) | of such subsets, leading to the claimed bound.Next note that Lemma 5.1 gives us the following estimate: ∞ (cid:88) m = k G F r ( m ) ≤ CF ( k − √ ∞ (cid:88) m = k me − m ≤ CF (0) e − k +1 (( e − k + 1)( e − (5.4)whenever k ≥
2. Note in particular the factor of e − k +1 , which will be importantto guarantee convergence in our case.We now return to equation (3.3). Note that d (Γ , Γ c ) = d sin α . If this isgreater than d Φ , by the remarks above the first summation (over x ∈ Γ and y ∈ Γ c ) vanishes. In general, since the cone Γ has a wider opening angle thanΓ , we see that there are only finitely many pairs x ∈ Γ and y ∈ Γ c with d ( x, y ) ≤ d Φ , and hence only finitely many contributions to the summation.Together with equations (5.3) and (5.4) it can be seen that this contribution isfinite.At this point we are left with estimating the following summation: (cid:88) x ∈ Γ \ Γ (cid:88) y ∈ Γ c + (cid:88) y ∈ Γ ∞ (cid:88) m =0 G F r ( m ) f ( m, x, y ) , (5.5)where we have split up the summation over (Γ \ Γ ) c into two parts. Weconsider the summation over Γ c , the other one can be handled in the same24igure 2: Definition of various distances.manner. Note that d (Γ , (Γ (cid:48) ) c ) = d (cid:48) sin( α + (cid:15) ). Similarly, d (Γ ∩ b ( n ) c , Γ (cid:48) ) = d (cid:48) sin( α + (cid:15) ) + n sin( (cid:15) ). Write d Γ ( n ) for the distance between the tip of the coneΓ and the circle of radius n based on the tip of Γ , where we set d Γ ( n ) = 0 ifthey do not intersect (see Fig. 2 for an idea of the various distances we need tointroduce). In case it is non-zero, we see that in fact d Γ ( n ) = (cid:113) n − d (1 − cos β ) − d cos β. Let γ be the distance from the tip of Γ (cid:48) to the intersection of the line per-pendicular to the boundary of Γ (cid:48) and the boundary of Γ . We write d γ forthe distance of the tip of Γ to this intersection. Then for large enough n thedistance of the intersection of the circle of radius n with the boundary of Γ and the boundary of Γ (cid:48) is given by γ ( n ) = γ + ( d Γ ( n ) − d γ ) sin (cid:15). From the geometric situation we see that d Γ ( n + k ) − d Γ ( n ) ≥ k , hence γ ( n )grows at least linearly in n .Let n be the smallest integer such that d := min { d (cid:48) sin( α + (cid:15) ) + n sin( (cid:15) ) , γ ( n ) } > d Φ . (5.6)Write B k := ( b ( d + ( k + 1) / sin( (cid:15) )) \ b ( d + k/ sin( (cid:15) ))). We now rewrite thesummation as (cid:88) x ∈ b ( d ) ∩ (Γ \ Γ ) + ∞ (cid:88) k =0 (cid:88) x ∈ B k ∩ (Γ \ Γ ) (cid:88) y ∈ Γ c ∞ (cid:88) m =0 G F r ( m ) f ( m, x, y ) . For the first summation over all x ∈ Γ \ Γ in the ball around the origin wenote that there are only finitely many such x . We have already seen that for anygiven x , there are only finitely many y (in fact, this number can be boundedfrom above independently of x ) such that f ( m, x, y ) is non-zero. Again byequations (5.3) and (5.4) it follows that the first summation is finite.For the second summation, note that if x ∈ B k ∩ (Γ \ Γ ), then d ( x, (Γ (cid:48) ) c ) ≥ k + 2 d Φ and d ( x, Γ (cid:48) ) ≥ k + 2 d Φ , and hence d ( x, (Γ (cid:48) \ Γ (cid:48) ) c )) ≥ k + 2 d Φ . Bywhat we have seen earlier, this implies that f ( m, x, y ) = 0 if m < k for such x ∈ k ∩ (Γ \ Γ ). Furthermore, because of the finite range assumption, contributingpairs x ∈ B k and y ∈ Γ c must be within a “band” of width d Φ around each sideof the boundary of Γ \ Γ . It follows that we can bound the number of pairs( x, y ) ∈ ( B k ∩ Γ \ Γ ) × Γ c by some constant C p > k . Puttingthis together we can estimate the second summation as follows. ∞ (cid:88) k =0 (cid:88) x ∈ B k ∩ (Γ \ Γ ) (cid:88) y ∈ Γ c ∞ (cid:88) m =0 G F r ( m ) f ( m, x, y ) ≤ ∞ (cid:88) k =0 C p C | b ( d Φ ) | ∞ (cid:88) m = k G F r ( m ) ≤ C (cid:48) ∞ (cid:88) k =0 e − k +1 (( e − k + 1) < ∞ (5.7)for some C (cid:48) >
0. Here we again used the estimates (5.3) and (5.4). Thiscompletes the proof.We expect that with a more careful analysis one could allow for more generalinteractions, as long as they decay sufficiently fast. It does however seem nec-essary that that Γ (cid:48) has a bigger opening angle than Γ , so that towards infinitythe distance between their respective boundaries grows. This is necessary toensure that for x, y far from the origin, f ( m, x, y ) is non-zero only for large m .Together with the decay properties of G F of Lemma 5.1 this ensures that thesum converges.The following now follows immediately from the theorem, by using Proposi-tion 2.2. Corollary 5.3.
Let A Γ and t (cid:55)→ Φ( X ; t ) be as in Theorem 5.2 and τ Φ t,s thecorresponding quasi-local dynamics. Assume that Γ ⊂ Γ is an inclusion ofcones such that their borders are sufficiently far away and in the representation π of A Γ we have the split property with respect to these cones. Then there existcones Γ (cid:48) ⊂ Γ and Γ (cid:48) ⊃ Γ such that π ◦ τ Φ1 , satisfies the split property withrespect to Γ (cid:48) ⊂ Γ (cid:48) . Finally, it allows us to construct examples of quasi-factorizable automor-phisms.
Corollary 5.4.
Let α = τ Φ0 , , with Φ as in Theorem 5.2. Then, for every cone Λ , we can find cones Γ (cid:48) ⊂ Λ ⊂ Γ (cid:48) such that α is quasi-factorizable with respectto this inclusion.Proof. We will apply Theorem 3.1; we shall see later why the conditions aresatisfied. Suppose that the cone Λ has opening angle θ . Fix some cone Λ which has the same apex and central axis as Λ but with a larger angle θ > θ ,satisfying Λ ⊂ Λ . Set Γ := Λ, Γ := Λ . Then, by Theorem 5.2, there arecones Γ (cid:48) ⊂ Γ and Γ (cid:48) ⊃ Γ such that the conditions of Theorem 3.1 are satisfied.Recall that α = τ Φ0 , . Then τ Φ (0) , , in the notation of Theorem 3.1, decomposesas τ Φ (0) , = α Γ ⊗ α Γ \ Γ ⊗ α Γ c (5.8)26here α Γ ∈ Aut( A Γ ), and similar for the others. Moreover, by noting that u ∈ A Γ and taking inverses on both sides of equation (3.5), we obtain fromTheorem 3.1 that there is (cid:101) u ∈ A such that α = τ Φ0 , = Ad(˜ u ) ◦ (cid:16) (cid:101) β − (cid:48) \ Γ (cid:48) ◦ τ Φ (0) , (cid:17) = Ad(˜ u ) ◦ (cid:16) (cid:101) β − (cid:48) \ Γ (cid:48) ◦ α Γ \ Γ (cid:17) ◦ (cid:0) α Γ ⊗ α Γ c (cid:1) = Ad( (cid:101) u ) ◦ (cid:101) Ξ ◦ ( α Λ ⊗ α Λ c ) . Here, Ξ := (cid:101) β − (cid:48) \ Γ (cid:48) ◦ α Γ \ Γ is an automorphism on A Γ (cid:48) = A Λ , α Λ := α Γ is anautomorphism on A Λ = A Γ , and α Λ c := α Γ c ⊗ id Λ \ Γ is an automorphism on A Λ c . References [BF82] Detlev Buchholz and Klaus Fredenhagen. Locality and the structureof particle states.
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