Stability of the bulk gap for frustration-free topologically ordered quantum lattice systems
SSTABILITY OF THE BULK GAP FOR FRUSTRATION-FREETOPOLOGICALLY ORDERED QUANTUM LATTICE SYSTEMS
BRUNO NACHTERGAELE, ROBERT SIMS, AND AMANDA YOUNG
Abstract.
We prove that uniformly small short-range perturbations do not close the bulk gapabove the ground state of frustration-free quantum spin systems that satisfy a standard local topo-logical quantum order condition. In contrast with earlier results, we do not require a positive lowerbound for finite-system Hamiltonians uniform in the system size. To obtain this result, we adaptthe Bravyi-Hastings-Michalakis strategy to the GNS representation of the infinite-system groundstate. Introduction
One of the characteristic properties of gapped topologically ordered ground state phases of quan-tum many-body systems is the stability of the spectral gap above the ground state with respectto small perturbations of the Hamiltonian. Over the past decade, several results proving thestability of the spectral gap for more or less general classes of models have appeared in the lit-erature [7, 8, 10, 11, 14, 21]. One obstacle has been the common occurrence of gapless edge statesin topological insulators. Spectral analysis for interacting many-body systems is usually carriedout for finite systems and edge states typically imply that there is no spectral gap uniform in thesystem size. Nevertheless, there may be a bulk gap, meaning excitations away from the boundaryof the system have energy bounded below uniformly in the system size. The goal of this work isto prove stability for the bulk gap in a way that does not require the assumption of a uniformpositive lower bound in the spectrum of finite systems. Previously, it was shown how certain casescan be handled by considering sequences of finite systems with suitable boundary conditions. Forexample, such an approach may work if the edge states are absent in the model considered withperiodic boundary conditions [21, 25]. In general, however, we may not know a suitable boundarycondition that ‘gaps out’ the boundary modes. In our approach here we only assume that the in-finite system described in the GNS representation of the ground state has a gap above the groundstate. Under natural assumptions similar to those in previous works, we prove that sufficientlysmall but extensive perturbations do not close the gap.We follow the strategy of Bravyi, Hastings, and Michalakis [7, 8, 21] and use the techniques wedeveloped in [25, 26] to handle the infinite system setting. From a certain perspective, and apartfrom the technical aspects to deal with unbounded Hamiltonians, the infinite system setting allowsfor a simplification in the statement of conditions and the main result. For concreteness, we workin the quantum spin system setting, but our approach is applicable to lattice fermion systems too.We do not provide new applications of gap stability in this work and just mention that theassumption that the bulk Hamiltonian has a gap in the spectrum above the ground state appearsin several important recent works. For example, the construction of an index for the classificationof symmetry-protected topological phases in the works of Ogata and co-authors makes use of thisassumption [27–30]. Other examples are in [22] and the recent work on adiabatic theorems forinfinite many-body systems [16].
Date : February 16, 2021. a r X i v : . [ m a t h - ph ] F e b B. NACHTERGAELE, R. SIMS, AND A. YOUNG Setup and statement of the main results
Setup and notation.
As the reference system (before perturbation), we consider a frustration-free, finite-range model defined on a discrete metric space (Γ , d ). For x ∈ Γ and n ≥
0, define b x ( n ) = { y ∈ Γ | d ( x, y ) ≤ n } . We assume that (Γ , d ) is ν -regular for some ν >
0, meaning there isa constant κ such that | b x ( n ) | ≤ κn ν , for all x ∈ Γ and n ≥
1. For Λ ∈ P (Γ), the finite subsets ofΓ, and n ≥
0, we define the sets Λ( n ) by(2.1) Λ( n ) = (cid:91) x ∈ Λ b x ( n ) . The algebra of local observables of the system is the usual A loc = (cid:83) Λ ∈P (Γ) A Λ . Here, A Λ is thematrix algebra (cid:78) x ∈ Λ M d x with d x the dimension of the spin at x . The C ∗ -algebra of quasi-localobservables A is the completion of A loc with respect to the operator norm. For A ∈ A loc , the support of A , denoted by supp A , is the smallest X ⊂ Γ such that A ∈ A X .We are specifically interested in systems defined on infinite Γ and will frequently consider A ∈ A that are limits of A n ∈ A Λ n , where Λ n ∈ P (Γ) is an increasing sequence of finite volumes suchthat (cid:83) n Λ n = Γ. We call such a sequence (Λ n ) an increasing and absorbing sequence (IAS). Anoften used example is Λ n = b x ( n ), the sequence of balls centered at x ∈ Γ of radius n = 0 , , , . . . .For any A ∈ A and any IAS (Λ n ), there are A n ∈ A Λ n such that lim n A n = A . It will oftenbe important to have an estimate for the speed of this convergence in terms of a non-increasingfunction g : [0 , ∞ ) → (0 , ∞ ) that vanishes at infinity. We will call any such function g a decayfunction . Given a decay function g , we will call A ∈ A g -local if for all x ∈ Γ, there exists
C > A n ∈ A b x ( n ) , for all n ≥
1, with the property that(2.2) (cid:107) A − A n (cid:107) ≤ Cg ( n ) . Define A g to be the set of all g -local elements of A . Since we assumed g ( r ) > r ≥
0, it iseasy to see that A loc ⊂ A g ⊂ A . In fact, A g is a dense ∗ -subalgebra of A .We will often also assume that a decay function g is uniformly summable over Γ, i.e.,(2.3) sup x ∈ Γ (cid:88) y ∈ Γ g ( d ( x, y )) < ∞ , and additionally, that there is a constant C > (cid:88) z ∈ Γ g ( d ( x, z )) g ( d ( z, y )) ≤ Cg ( d ( x, y )) , for all x, y ∈ Γ . Any decay function g satisfying (2.3) and (2.4) will be called an F -function. For example, withΓ = Z ν , the following is a family of examples of F -functions appearing in this work:(2.5) F ( r ) = 1(1 + r ) ζ e − ar θ , ζ > ν, a ≥ , θ ∈ (0 , . For a discussion of these examples and some basic inequalities, see [26, Appendix].We assume the initial model is defined by a finite-range, uniformly bounded, frustration-freeinteraction. We denote this interaction by η and describe the corresponding interaction terms by afamily { h x } x ∈ Γ satisfying:i. There is a number R ≥
0, called the interaction radius, for which h ∗ x = h x ∈ A b x ( R ) for all x ∈ Γ.ii. These terms are uniformly bounded in the sense that(2.6) (cid:107) η (cid:107) ∞ = sup x ∈ Γ (cid:107) h x (cid:107) < ∞ . TABILITY OF THE BULK GAP 3 iii. The interaction η is frustration-free in the sense that h x ≥ x ∈ Γ and for anyΛ ∈ P (Γ),(2.7) min spec( H Λ ) = 0 where H Λ = (cid:88) x ∈ Λ: supp( h x ) ⊂ Λ h x . It follows from this frustration-free condition that the ground state is ker( H Λ ) ⊆ (cid:78) x ∈ Λ C d x for anyfinite volume Λ ∈ P (Γ). Moreover, ψ ∈ ker( H Λ ) if and only if ψ ∈ ker( h x ) for each x ∈ Γ withsupp( h x ) ⊂ Λ. Thus, if we denote by P Λ the orthogonal projection onto ker( H Λ ), then for anyfinite subsets Λ ⊂ Λ ⊂ Γ we have that(2.8) P Λ P Λ = P Λ P Λ = P Λ . In the above, we have identified H Λ with H Λ ⊗ Λ \ Λ ∈ A Λ whenever Λ ⊂ Λ.Given such an interaction η , the infinite system dynamics on Γ is obtained through the derivation δ defined for A ∈ A loc by(2.9) δ ( A ) = (cid:88) x ∈ Λ( R ) [ h x , A ] for any A ∈ A Λ and Λ ∈ P (Γ) . It is a standard result that there is a closed derivation extending δ , which we also denote by δ ,with domain dom( δ ) for which A loc is a core [6, Theorem 6.2.4] (note that the factor i is absorbedin the definition of the derivation in this reference). The system dynamics is then the stronglycontinuous one-parameter group of C ∗ -automorphisms { τ (0) t | t ∈ R } satisfying(2.10) ddt τ (0) t ( A ) = iτ (0) t ( δ ( A )) for all A ∈ A loc . In fact, this differential equation holds for all A ∈ dom( δ ). Two other general properties are:i. τ (0) t (dom( δ )) ⊂ dom( δ ) for all t ∈ R ;ii. τ (0) t ( δ ( A )) = δ ( τ (0) t ( A )) for all A ∈ dom( δ ) and t ∈ R .More generally, quantum spin models can be defined by an interaction on Γ which, by definition,is a map Φ : P (Γ) → A loc , with the property that Φ( X ) ∗ = Φ( X ) ∈ A X for all X ∈ P (Γ). Forany decay function g , an interaction norm is defined by(2.11) (cid:107) Φ (cid:107) g = sup x,y ∈ Γ g ( d ( x, y )) − (cid:88) X ∈P (Γ): x,y ∈ X (cid:107) Φ( X ) (cid:107) . When the above quantity is finite for some interaction Φ, the function g is said to measure thedecay of Φ. If g is an F -function, the norm (cid:107) · (cid:107) g is called an F-norm . If g is summable, in thesense of (2.3), and (cid:107) Φ (cid:107) g < ∞ , then a closable derivation on A loc can be defined by setting δ ( A ) = (cid:88) Y,Y ∩ X (cid:54) = ∅ [Φ( Y ) , A ] for A ∈ A loc with supp( A ) ⊂ X ∈ P (Γ) . One can prove conditions that guarantee that the derivation δ defined on A loc is a generator of astrongly continuous dynamics given by automorphisms of A [5,6]. In practice, however, one usuallydirectly proves the existence of the thermodynamic limit of the Heisenberg dynamics. Standardresults along these lines prove the existence of the dynamics for Φ in a suitable Banach spaceof interactions [6, 32, 33] starting from a convergent series for small | t | . An alternative approach,based on Lieb-Robinson bounds [20], was introduced by Robinson [31]. Lieb-Robinson boundscan be derived for any interaction Φ with a finite F -norm [23], and this allows one to extend theresults for existence of the dynamics beyond the Banach spaces of interactions B λ introduced byRuelle [32]. These ideas are important for the construction of the spectral flow automorphisms [3].This and some other generalizations relevant for the present work are discussed in detail in [26]. B. NACHTERGAELE, R. SIMS, AND A. YOUNG
In this work, we consider an infinite-volume, zero-energy, ground state of the frustration freemodel defined by δ , denoted by ω . Let ( H , π , Ω) be the GNS triple of ω . This means that π : A → B ( H ) is a representation of the C ∗ -algebra A on a Hilbert space H for which { π ( A )Ω | A ∈A loc } is dense in H . Moreover, the normalized vector Ω ∈ H is such that ω ( A ) = (cid:104) Ω , π ( A )Ω (cid:105) forall A ∈ A . Recall that infinite-volume ground states associated to δ are defined as those states ω on A which satisfy(2.12) ω ( A ∗ δ ( A )) ≥ A ∈ A loc . For the GNS representation of a ground state, as in (2.12), there exists a unique, non-negativeself-adjoint operator H on H , with dense domain dom H , satisfying H Ω = 0 and(2.13) π ( τ (0) t ( A )) = e itH π ( A ) e − itH for all A ∈ A and t ∈ R . The full domain of H is seldom described explicitly. However, for all systems we consider in thispaper, π ( A loc )Ω is a core for H .Frustration-free models are characterized by the property that they have a zero-energy groundstate, meaning there is a state ω on A such that ω ( h x ) = 0 for all x ∈ Γ. It is easy to see that azero-energy ground state is a ground state in the sense of (2.12). Frustration-free models may haveother ground states but thermodynamic limits of ground states of(2.14) H Λ n = (cid:88) x ∈ Λ n : supp( h x ) ⊂ Λ n h x automatically have the zero-energy property. Conversely, any zero-energy infinite volume groundstate is a thermodynamic limit of finite-volume zero-energy ground states. We will refer to theGNS representation of a zero-energy ground state of a frustration-free model as a frustration-freerepresentation .2.2. Main results.
We now state the assumptions for the main results in this paper. We say thatthe ground state ω is gapped if there exists γ > , γ ) ∩ spec( H ) = ∅ . The bulk gap of the model in the state ω is defined as(2.15) γ = sup { γ > | (0 , γ ) ∩ spec( H ) = ∅} . If the set on the RHS is empty, one defines γ = 0. The equivalence of the following two conditionsis easy to verify:i. For some γ > ω satisfies(2.16) ω ( A ∗ δ ( A )) ≥ γω ( A ∗ A ) for all A ∈ A loc with ω ( A ) = 0;ii. The ground state of the GNS Hamiltonian H is unique and the bulk gap satisfies γ ≥ γ .In this work, we assume positivity of the bulk gap for the reference model. Assumption . Let ω be an infinite-volume, zero-energy ground state of a finite-range, uniformly bounded, frustration free model as above. The bulk gap of the associated GNSHamiltonian H , see (2.15), is strictly positive.We also need to impose a condition that the local gaps do not close too fast. There generally issome freedom in choosing the family of finite volumes on which to impose this condition. We willassume that we have a family S = { Λ( x, n ) | n ≥ , x ∈ Γ } ⊂ P (Γ), with b x ( n ) ⊂ Λ( x, n ) for all x and n , and moreover, there is an associated family of partitions of Γ which separates S and hasat most polynomial growth . Concretely, this means there is a family of sets T = {T n | n ≥ } andpositive numbers c and ζ , such that for each n ≥ T n = {T in : i ∈ I n } is a partition of Γ satisfying |I n | ≤ cn ζ and(2.17) Λ( x, n ) ∩ Λ( y, n ) = ∅ for all x, y ∈ T in with x (cid:54) = y. TABILITY OF THE BULK GAP 5
In such cases, we say that T is of ( c, ζ )-polynomial growth.As an example, in the case of Γ = Z ν , we may take for Λ( x, n ) the (cid:96) ∞ -ball of radius n centeredat x , define I n = Λ(0 , n ) and, for each i ∈ I n , set(2.18) T in = { x ∈ Z ν | x j = i j mod 2 n + 1 , i = 1 , . . . , ν } . Assumption . Let η be a finite-range, uniformly bounded frustration-free interactionon Γ. Associated to η , there exists a family S = { Λ( x, n ) | n ≥ , x ∈ Γ } ⊂ P (Γ) with b x ( n ) ⊂ Λ( x, n ) for all x and n , a family T = {T n : n ≥ } of partitions of Γ which separates S and isof ζ -polynomial growth, and an exponent α ≥ γ >
0, such that the finite-volumeHamiltonians associated to η satisfy:(2.19) spec( H Λ( x,n ) ) ⊂ { } ∪ [ n − α γ , ∞ ) for all x ∈ Γ and n ≥ R, where R is the interaction radius of η .It is important here that the local gaps are allowed to vanish in the limit of infinite system size.For example, certain types of topologically ordered two-dimensional systems are expected to havechiral edge modes with an energy of order 1 /L on a finite volume of diameter L . Whether or notsuch edge modes occur in frustration-free systems, however, is not clear. For the class of systemsstudied in [19], the authors find that finite-volume gaps of a system with gapless edge modes in thethermodynamic limit would have to decay at least as fast as L − / . Other results of this type arein [1, 12, 17]. This is consistent with the gapless boundary modes found in a class of toy modelscalled Product Vacua with Boundary States which are of order L − [2, 4]. In any case, regardlessof the possible values of the exponent α , we will prove stability of the bulk gap.The next assumption was introduced in the form we use here in [21] where it is called LocalTopological Quantum Order (LTQO).
Assumption . Let ω be an infinite-volume, zero-energy ground state of a finite-range,uniformly bounded, frustration free model. There is a non-increasing function G : [0 , ∞ ) → [0 , ∞ ),for which(2.20) (cid:88) n ≥ n q G ( n ) < ∞ , for some q > ν + 1 + 2( ζ + α ) , and such that for all m ≥ k ≥ , x ∈ Γ, and A ∈ A b x ( k ) , we have(2.21) (cid:107) P b x ( m ) AP b x ( m ) − ω ( A ) P b x ( m ) (cid:107) ≤ (cid:107) A (cid:107) (1 + k ) ν G ( m − k ) . As explained in detail in [25, Section 8], if both the initial Hamiltonian and the perturbation (seebelow) have a local gauge symmetry, only observables A that commute with this symmetry need tosatisfy (2.21). Other discrete symmetries can be treated similarly (see [25, Section 8]). Therefore,the stability results proved here (Theorems 2.7 and 2.8) will also hold for symmetry-protectedtopological phases.It is an interesting observation that the GNS Hamiltonians associated to frustration free modelswhich satisfy Assumption 2.3 automatically have a unique ground state. This is the content of thefollowing propostion. Proposition 2.4.
Let ω be an infinite-volume, zero-energy ground state of a frustration-free modelsatisfying Assumption 2.3. The kernel of the GNS Hamiltonian H is one-dimensional.Proof. By way of contradiction, let us assume there is a unit vector ψ ∈ ker( H ) with (cid:104) ψ, Ω (cid:105) = 0.To simplify notation, for any A ∈ A , let us denote by ˜ A := π ( A ) the representative of A in theGNS space. The LTQO condition (2.21), in B ( H ), can be stated as: for all m ≥ k ≥ x ∈ Γ, and A ∈ A b x ( k ) ,(2.22) (cid:107) ˜ P b x ( m ) ˜ A ˜ P b x ( m ) − (cid:104) Ω , ˜ A Ω (cid:105) ˜ P b x ( m ) (cid:107) ≤ (cid:107) A (cid:107) (1 + k ) ν G ( m − k ) . B. NACHTERGAELE, R. SIMS, AND A. YOUNG
First, since π ( A loc )Ω is dense in H , there is x ∈ Γ and operators A n ∈ A b x ( n ) for which ψ n = ˜ A n Ω → ψ . In this case,(2.23) lim n →∞ (cid:104) Ω , ˜ A n Ω (cid:105) = lim n →∞ (cid:104) Ω , ψ n (cid:105) = (cid:104) Ω , ψ (cid:105) = 0 . Thus, there is n ∈ N so that |(cid:104) Ω , ˜ A n Ω (cid:105)| ≤ / n ≥ n .Next, since ker( H ) ⊂ ker( ˜ H b x ( m ) ) for all m ≥ x ∈ Γ, we have(2.24) (cid:104) ψ, ˜ P b x ( m ) ˜ A n ˜ P b x ( m ) Ω (cid:105) = (cid:104) ψ, ˜ A n Ω (cid:105) = (cid:104) ψ, ψ n (cid:105) for all m ≥
0. Since 1 = lim n →∞ (cid:104) ψ, ψ n (cid:105) , there is some n ∈ N for which |(cid:104) ψ, ψ n (cid:105)| ≥ / n ≥ n . Combining this with (2.24), we find that for any m ≥ n ≥ n ,(2.25) 2 / ≤ (cid:107) ˜ P b x ( m ) ˜ A n ˜ P b x ( m ) (cid:107) . It follows that for n ≥ max { n , n } ,(2.26) 1 / ≤ (cid:107) ˜ P b x ( m ) ˜ A n ˜ P b x ( m ) − (cid:104) Ω , ˜ A n Ω (cid:105) ˜ P b x ( m ) (cid:107) , a contradiction to (2.22), as the decay function G satisfies G ( m ) → m → ∞ . (cid:3) Next, we turn to the perturbations of the Hamiltonian H . We consider perturbations definedby an anchored interaction Φ on Γ with terms given by { Φ( x, n ) | x ∈ Γ , n ≥ } . This means thatΦ( x, n ) ∗ = Φ( x, n ) ∈ A b x ( n ) for all x ∈ Γ and n ≥
0. By regrouping, we need only consider thoseterms with n ≥ R the interaction radius of η . Assumption . There is a constant (cid:107) Φ (cid:107) ≥ a >
0, and θ ∈ (0 , x ∈ Γ(2.27) (cid:107) Φ( x, n ) (cid:107) ≤ (cid:107) Φ (cid:107) e − an θ for all n ≥ R. Remark 2.6.
Assume Φ satisfies Assumption 2.5. As indicated in (2.5), for any 0 < a (cid:48) < a and ζ > ν , the function F : [0 , ∞ ) → (0 , ∞ ) given by(2.28) F ( r ) = e − a (cid:48) r θ (1 + r ) ζ for all r ≥ , is an F -function on Γ. Let δ = a − a (cid:48) > x, y ∈ Γ with d ( x, y ) ≥ R , we have (cid:88) z ∈ Γ (cid:88) n ≥ R : x,y ∈ b z ( n ) (cid:107) Φ( z, n ) (cid:107) = (cid:88) n ≥ d ( x,y ) (cid:88) z ∈ b y ( n ) ∩ b x ( n ) (cid:107) Φ( z, n ) (cid:107)≤ κ (cid:107) Φ (cid:107) (cid:88) n ≥ d ( x,y ) n ν e − an θ ≤ (cid:107) Φ (cid:107) F F ( d ( x, y ))(2.29)with (cid:107) Φ (cid:107) F = κ (cid:107) Φ (cid:107) (cid:80) n ≥ (1 + n ) ν + ζ e − δn θ < ∞ . Thus any such Φ has a finite F -norm with F as in(2.28).The focus of this work is to analyze the stability of the bulk gap under the presence of perturba-tions whose terms decay as in (2.27). Given an anchored interaction Φ satisfying Assumption 2.5,the main focus of this work will be perturbed Hamiltonians of the form(2.30) H (Λ , s ) = H + sπ ( V Λ ) , s ∈ R where, for any finite volume Λ ∈ P (Γ),(2.31) V Λ = (cid:88) x ∈ Λ (cid:88) n ≥ R : b x ( n ) ⊂ Λ Φ( x, n ) . TABILITY OF THE BULK GAP 7
Clearly, V Λ ∈ A Λ is bounded and self-adjoint, and so H (Λ , s ) defines for all s ∈ R a self-adjointHamiltonian on H with the same dense domain as H .The core result we will prove in the next several sections is the following theorem which establishesthat the spectral gap of H (Λ , s ) remains open for small | s | uniform in the finite volume Λ. Theorem 2.7 (Stability of the gap uniformly in the perturbation region) . Let ω be an infinite-volume, zero-energy ground state of a finite-range, uniformly bounded, frustration free model definedby an interaction η . Suppose Assumptions 2.1, 2.2, and 2.3 hold, and let Φ be an anchored inter-action for which Assumption 2.5 holds. Then, for all γ ∈ (0 , γ ) , there exists s ( γ ) > , such thatfor all real s , | s | < s ( γ ) , and Λ ∈ P (Γ) , we have (2.32) spec H (Λ , s ) ⊂ { E (Λ , s ) } ∪ [ E (Λ , s ) + γ, ∞ ) with H (Λ , s ) as in (2.30) and (2.33) E (Λ , s ) = inf spec H (Λ , s ) . We remark that the quantity s ( γ ) only depends on the values of κ and ν of the lattice, (cid:107) η (cid:107) ∞ ,the gap γ , the parameters in Assumption 2.2, the decay function in Assumption 2.3, and a suitable F -norm of the perturbation Φ. From the arguments in this paper, one can derive an explicit lowerbound for s ( γ ) in terms of these quantities, see Section 6We also investigate the situation where the perturbation region Λ tends to all of Γ. Considerany IAS (Λ n ), and let Φ be an anchored interaction satisfying Assumption 2.5. We will denoteby τ (Λ n ,s ) t the dynamics on A generated by the interaction η + s Φ (cid:22) Λ n , which is generated by thederivation(2.34) δ Λ n s ( A ) = δ ( A ) + [ sV Λ n , A ] for A ∈ A loc . As discussed in [26, Definition 3.7], the sequence of interactions η + s Φ (cid:22) Λ n converges locallyin F -norm to the interaction η + s Φ; here one may use any F -function with the form (2.28).Using [26, Theorem 3.8], we conclude local convergence in the sense that(2.35) lim n →∞ τ (Λ n ,s ) t ( A ) = τ ( s ) t ( A ) for all A ∈ A loc as well as(2.36) lim n →∞ δ Λ n s ( A ) = δ s ( A ) for all A ∈ A loc with τ ( s ) t (respectively, δ s ) being the a priori well-defined strongly continuous dynamics on A (respectively, the closure of the derivation restricted to A loc ) generated by the interaction η + s Φ.Neither of these limits depend on the choice of IAS sequence Λ n .Our second result is then concerned with the ground state and its gap for a family of extensiveperturbations. In particular, the uniformity of the stability result in Theorem 2.7 allows one toprove, almost as a corollary, that for all | s | ≤ s ( γ ) there is a gapped ground state ω s of δ s in thesense of (2.16). To make these precise, we introduce the limiting spectral flow. For any γ > n ), take(2.37) α s ( A ) = lim n α Λ n s ( A ) for all A ∈ A , where the spectral flows α Λ n s will be introduced in more detail in the next section, see (3.25).For now, it suffices to observe that this limit exists and is independent of the choice of IAS. Infact, the interactions defining the spectral flows α Λ n s converge locally in F -norm by argumentsas in [26, Section VI.E.2]. This limiting spectral flow α s defines a strongly continuous co-cycle ofautomorphisms of A , and moreover, under the assumptions we have made, for A ∈ A loc , s (cid:55)→ α s ( A )is differentiable to all orders. B. NACHTERGAELE, R. SIMS, AND A. YOUNG
Theorem 2.8 (Stability of the bulk gap) . Under the assumptions of Theorem 2.7, let γ ∈ (0 , γ ) and take s with | s | < s ( γ ) ; the quantity whose existence is guaranteed by Theorem 2.7. For thelimiting spectral flow α s , as in (2.37) above, the state ω s = ω ◦ α s is a gapped ground state of theperturbed infinite dynamics δ s , i.e. (2.38) ω s ( A ∗ δ s ( A )) ≥ γω s ( A ∗ A ) for all A ∈ A loc with ω s ( A ) = 0 . In particular, the GNS Hamiltonian H s of ω s has a one-dimensional kernel and spec H s has a gapabove its ground state bounded below by γ . Quasi-locality, Domains and Local Decompositions
In this section, we set the stage for proving stability in the infinite-system setting. Quasi-localityon A is the topic of Section 3.1. In Section 3.1.1, we recall general methods for making strictlylocal approximations of both quasi-local observables and maps. Useful examples, as well as explicitestimates, are contained in Sections 3.1.2 - 3.1.5. We prepare results about frustration-free GNSrepresentations we will need in Section 3.2. Lastly, to deal with unbounded operators, we musthave some control over certain vectors in their domain, which is dealt with in Section 3.3.3.1. Quasi-Locality.
We will frequently need controlled strictly local approximations of globalquantities, which we refer to in general as quasi-locality estimates. We first recall some generalfeatures of quasi-locality estimates. Important examples which will be relevant in this work aredescribed in detail in Sections 3.1.2 - 3.1.5.3.1.1.
Quasi-locality estimates.
Let A be a C ∗ -algebra of quasi-local observables over Γ as discussedin Section 2. For any X ∈ P (Γ), a conditional expectation Π X : A → A is defined with respect tothe tracial state ρ on A by setting(3.1) Π X = ρ (cid:22) Γ \ X ⊗ id A X . Here we identify elements of the sub-algebra A X with their representatives in A loc ⊂ A . Thus, inparticular, if A ∈ A Λ and X ⊂ Λ, then Π X ( A ) is just given by the normalized partial trace.It is often the case that for some A ∈ A , there are X ∈ P (Γ) and (cid:15) > (cid:107) [ A, B ] (cid:107) ≤ (cid:15) (cid:107) B (cid:107) whenever B ∈ A locΓ \ X . We refer to such an observable A as quasi-local . In fact, a general result from [9, 24] shows that anysuch quasi-local observable A also satisfies(3.3) (cid:107) A − Π X ( A ) (cid:107) ≤ (cid:15) , i.e. A has a quantifiable, strictly local approximation; namely, Π X ( A ).Similar notions hold for maps. A linear map K : A → A is said to be quasi-local if there are C ≥ p ≥
0, and a decay function G for which(3.4) (cid:107) [ K ( A ) , B ] (cid:107) ≤ C | X | p (cid:107) A (cid:107)(cid:107) B (cid:107) G ( d ( X, Y )) for all A ∈ A X and B ∈ A Y . For a quasi-local map K , the image K ( A ) of a strictly local observable A can be approximatedby a strictly local observable using (3.3). In the present context, it suffices to localize observableson balls. For any quasi-local map K on A satisfying (3.4), (3.3) shows that for any n ≥ A ∈ A b x ( k ) ,(3.5) (cid:107)K ( A ) − Π b x ( k + n ) ( K ( A )) (cid:107) ≤ C | b x ( k ) | p (cid:107) A (cid:107) G ( n ) . When the corresponding decay function G is summable, such an estimate guarantees the (absolute)convergence of telescopic sums, i.e. for any n ≥ K ( A ) = Π b x ( k + n ) ( K ( A )) + ∞ (cid:88) n = n +1 (cid:0) Π b x ( k + n ) − Π b x ( k + n − (cid:1) ( K ( A )) TABILITY OF THE BULK GAP 9 since the terms are easily seen to satisfy(3.7) (cid:107) (cid:0) Π b x ( k + n ) − Π b x ( k + n − (cid:1) ( K ( A )) (cid:107) ≤ C | b x ( k ) | p (cid:107) A (cid:107) G ( n −
1) for n ≥ . A common choice is n = 0 and we adopt the notation(3.8) K ( A ) = (cid:88) n ≥ k ∆ nb x ( k ) ( K ( A )) where ∆ nb x ( k ) = (cid:40) Π b x ( k ) , n = k Π b x ( n ) − Π b x ( n − , n ≥ k + 1 . We now briefly review four quasi-local maps as well as a few of their important, well-knownproperties that are necessary for our later arguments.3.1.2.
The Unperturbed Dynamics.
Let τ (0) t be the dynamics generated by the finite-range, uni-formly bounded, frustration-free interaction η as discussed in Section 2. As is well-known, τ (0) t satisfies an exponential Lieb-Robinson bound, i.e. for every µ > C µ > v µ > X, Y ∈ P (Γ), we have that(3.9) (cid:107) [ τ (0) t ( A ) , B ] (cid:107) ≤ C µ min( | X | , | Y | ) (cid:107) A (cid:107)(cid:107) B (cid:107) e − µ ( d ( X,Y ) − v µ | t | ) for all A ∈ A X , B ∈ A Y , and t ∈ R .3.1.3. The Perturbed Dynamics.
Let η be a finite-range, uniformly bounded, frustration free inter-action and Φ an anchored interaction satisfying Assumption 2.5. For each Λ ∈ P (Γ) and s ∈ R , wehave denoted by τ (Λ ,s ) t the strongly continuous dynamics generated by η + s Φ (cid:22) Λ . Since η is finite-range and uniformly bounded and Φ has a finite F -norm for any F -function of the form (2.28), itis clear that η + s Φ (cid:22) Λ has a finite F -norm as well. In fact, the F -norm of this interaction has anupper bound which is uniform with respect to Λ ∈ P (Γ) and s in compact sets. We conclude thatthere are C F > v F > X, Y, Λ ∈ P (Γ),(3.10) (cid:107) [ τ (Λ ,s ) t ( A ) , B ] (cid:107) ≤ C F (cid:107) A (cid:107)(cid:107) B (cid:107) e v F | t | (cid:88) x ∈ X (cid:88) y ∈ Y F ( d ( x, y ))for all A ∈ A X , B ∈ A Y , and t ∈ R .It will also be useful for us to observe that this perturbed dynamics τ (Λ ,s ) t may equivalently beviewed as the dynamics generated by the closure of the derivation δ Λ s defined by A loc by(3.11) δ Λ s ( A ) = δ ( A ) + [ sV Λ , A ] for all A ∈ A loc . Since each sV Λ is bounded and self-adjoint, [6, Proposition 5.4.1] implies that(3.12) τ (Λ ,s ) t ( A ) = (Γ (Λ ,s ) t ) ∗ τ (0) t ( A )Γ (Λ ,s ) t for all A ∈ A and t ∈ R where { Γ (Λ ,s ) t | t ∈ R } is a one-parameter family of unitaries on A which are uniquely defined as the A -valued solution of(3.13) ddt Γ (Λ ,s ) t = − iτ (0) t ( sV Λ )Γ (Λ ,s ) t with Γ (Λ ,s )0 = 1l . Additionally, these unitaries are quasi-local as, for any A ∈ A loc and t > (cid:107) [Γ (Λ ,s ) t , A ] (cid:107) = (cid:107) (Γ (Λ ,s ) t ) ∗ A Γ (Λ ,s ) t − A (cid:107) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) t ddu (cid:16) (Γ (Λ ,s ) u ) ∗ A Γ (Λ ,s ) u (cid:17) du (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:90) t (cid:107) [ τ (0) u ( sV Λ ) , A ] (cid:107) du . (3.14)An application of (3.9) then shows that for any µ > A ∈ A X with X ∈ P (Γ \ Λ),(3.15) (cid:107) [Γ (Λ ,s ) t , A ] (cid:107) ≤ C µ | s || Λ |(cid:107) V Λ (cid:107)(cid:107) A (cid:107)| t | e µv µ | t | e − µd (Λ ,X ) for any s, t ∈ R . We conclude, using e.g. (3.3), that Γ (Λ ,s ) t ∈ A g for some g of exponential decay. Weighted Integral Operators.
Under the assumptions of Theorem 2.7, fix γ >
0. For eachΛ ∈ P (Γ) and s ∈ R , we define two weighted integral operators F Λ s : A → A and G Λ s : A → A bysetting(3.16) F Λ s ( A ) = (cid:90) ∞−∞ τ (Λ ,s ) t ( A ) w γ ( t ) dt and G Λ s ( A ) = (cid:90) ∞−∞ τ (Λ ,s ) t ( A ) W γ ( t ) dt , for all A ∈ A , where the real-valued functions w γ and W γ , both of which are in L ( R ), are asdefined in [26, Section VI.B]. By strong-continuity of the perturbed dynamics, both of the integralsappearing in (3.16) are well-defined in the sense of Bochner, see e.g. [34]. Both these maps dependon the choice of γ through their weight functions, w γ and W γ respectively, but we suppress this inthe notation. Arguing as in [26, Section VI.E.1], see also [25, Section 4.3.2], we find that for all A ∈ A (3.17) (cid:107)F Λ s ( A ) (cid:107) ≤ (cid:107) A (cid:107) and (cid:107)G Λ s ( A ) (cid:107) ≤ (cid:107) W γ (cid:107) (cid:107) A (cid:107) , i.e. these maps are bounded uniformly with respect to s ∈ R and Λ ∈ P (Γ). Moreover, they areuniformly quasi-local in the sense that for each K ∈ {F , G} there is a decay function G K such that:for any choice of X, Y ∈ P (Γ), we have(3.18) sup s ∈ [ − , (cid:107) [ K Λ s ( A ) , B ] (cid:107) ≤ (cid:107) A (cid:107)(cid:107) B (cid:107)| X | G K ( d ( X, Y ))for all A ∈ A X and B ∈ A Y . As in Lemma 6.10 and Lemma 6.11 of [26], one can be quite explicit inestimating the decay of these functions G K . For our purposes here, we need only stress that thesedecay functions are independent of Λ ∈ P (Γ) and s ∈ [ − , µ ≥ ∞ (cid:88) n =1 ( n + 1) µ G K ( n ) < ∞ . The Spectral Flow.
Under the assumptions of Theorem 2.7, fix γ >
0. For each Λ ∈ P (Γ)and s ∈ R , denote by(3.20) D (Λ , s ) = G Λ s ( V Λ ) = (cid:90) ∞−∞ τ (Λ ,s ) t ( V Λ ) W γ ( t ) dt with G Λ s the weighted integral operator defined in (3.16) above. From (3.17), it is clear that s (cid:55)→ D (Λ , s ) is uniformly bounded. One further checks that for each s ∈ R , D (Λ , s ) is self-adjoint.We now show that s (cid:55)→ D (Λ , s ) is norm continuous. For t ∈ R fixed, the strong differentiability of s (cid:55)→ τ (Λ ,s ) t is proven in [26, Proposition 2.7], in particular, it is shown that(3.21) dds τ (Λ ,s ) t ( A ) = i (cid:90) t τ (Λ ,s ) r ([ V Λ , τ (Λ ,s ) t − r ( A )]) dr . Using (3.21), we find that (cid:107) D (Λ , s ) − D (Λ , s ) (cid:107) ≤ (cid:90) ∞−∞ (cid:107) τ (Λ ,s ) t ( V Λ ) − τ (Λ ,s ) t ( V Λ ) (cid:107)| W γ ( t ) | dt ≤ (cid:107) V Λ (cid:107) | s − s | (cid:90) ∞−∞ | t || W γ ( t ) | dt. (3.22)The norm-continuity follows from this estimate. Again we refer to [26, Section VI.B] for details.Given these properties of D (Λ , s ), there is a unique solution of(3.23) dds u (Λ , s ) = − iD (Λ , s ) u (Λ , s ) with u (Λ ,
0) = 1l , which is given by unitaries in A . TABILITY OF THE BULK GAP 11
Two important consequences of (3.23) follow. First, as we saw with the unitaries defined by(3.13), this solution is quasi-local. In fact, similar arguments show that for s > (cid:107) [ u (Λ , s ) , A ] (cid:107) ≤ (cid:90) s (cid:107) [ G Λ r ( V Λ ) , A ] (cid:107) dr ≤ s (cid:107) A (cid:107)| Λ |(cid:107) V Λ (cid:107) G G ( d ( X, Λ))for any A ∈ A X with X ∈ P (Γ \ Λ). For the above we have used (3.20), i.e. that D (Λ , s ) = G Λ s ( V Λ ),and (3.18). The function G G decays faster than any stretched exponential. Again, we concludethat u (Λ , s ) ∈ A g for some g with finite moments of all orders.Using (3.23) we define the spectral flow as the inner automorphisms on A given by(3.25) α Λ s ( A ) = u (Λ , s ) ∗ Au (Λ , s ) for all A ∈ A . Quasi-locality of this map follows by re-writing the generator as(3.26) D (Λ , s ) = G Λ s ( V Λ ) = (cid:88) x ∈ Λ (cid:88) n ≥ R : b x ( n ) ⊂ Λ G Λ s (Φ( x, n ))using (2.31). Applying the conditional expectations and telescopic sum from (3.8), we further write(3.27) D (Λ , s ) = (cid:88) x ∈ Λ (cid:88) m ≥ R Ψ Λ ( x, m, s ) where Ψ Λ ( x, m, s ) = (cid:88) R ≤ n ≤ m : b x ( n ) ⊂ Λ ∆ mb x ( n ) ( G Λ s (Φ( x, n ))) . Arguing as in [25, Appendix A], there is a decay function G Ψ and a positive number (cid:107) Ψ (cid:107) G Ψ suchthat for all Λ ∈ P (Γ), s ∈ [ − , x ∈ Λ and k ≥ R ,(3.28) (cid:88) m ≥ k (cid:107) Ψ Λ ( x, m, s ) (cid:107) ≤ (cid:107) Ψ (cid:107) G Ψ G Ψ ( k ) . Again, as in [25, Corollary A.3], one can be explicit about estimates for G Ψ , but for our purposeshere, let us only remark that is has finite moments of all orders, as in (3.19). Given (3.27) and(3.28), well-known Lieb-Robinson bounds (for time-dependent interaction) apply to the spectralflow. As a result, there is a decay function G α so that for any choice of X, Y ∈ P (Γ),(3.29) (cid:107) [ α Λ s ( A ) , B ] (cid:107) ≤ s (cid:107) A (cid:107)(cid:107) B (cid:107)| X | G α ( d ( X, Y ))for all A ∈ A X , B ∈ A Y , and s ∈ R . G α is independent of Λ ∈ P (Γ) and has finite moments of allorders.3.2. In the GNS space.
The proofs of our main results will be primarily carried out in the GNSrepresentation of the reference state ω .As discussed in Section 2, let η be a finite-range, uniformly bounded, frustration free interactionon Γ. Denote by δ the corresponding closed derivation with dense domain dom( δ ) ⊂ A having A loc as a core, and let { τ (0) t | t ∈ R } , denote the strongly continuous, system dynamics. For any infinite-volume, zero-energy ground state ω , let ( H , π , Ω) denote the corresponding GNS triple. Thequasi-local maps discussed in the previous subsection can be represented in the GNS representationand we now present the necessary properties we will need in this setting.
The Unperturbed Dynamics:
Since ω is a ground state of δ , there is a unique, non-negative,self-adjoint operator H with dense domain dom( H ) ⊂ H satisfying H Ω = 0 and(3.30) π ( τ (0) t ( A )) = e itH π ( A ) e − itH for all A ∈ A and t ∈ R . Moreover, π ( A loc )Ω is a core for H . The Perturbed Dynamics:
Let Φ be an anchored interaction satisfying Assumption 2.5. For anyΛ ∈ P (Γ), take V Λ as in (2.31) and recall that for any s ∈ R ,(3.31) H (Λ , s ) = H + π ( sV Λ ) defines a self-adjoint operator on H with the same dense domain as H . As before, let us alsointroduce the perturbed dynamics τ (Λ ,s ) t on A , see (3.11). We claim that for all Λ ∈ P (Γ) and s ∈ R ,the representation of this perturbed dynamics in the GNS space coincides with the Heisenbergdynamics, generated by the Hamiltonian H (Λ , s ), on π ( A ) in the sense that(3.32) π ( τ (Λ ,s ) t ( A )) = e itH (Λ ,s ) π ( A ) e − itH (Λ ,s ) for all A ∈ A and t ∈ R . In fact, using the interaction picture representation (3.12), as well as (3.30), we find that in B ( H ),(3.33) π ( τ (Λ ,s ) t ( A )) = ( π (Γ (Λ ,s ) t )) ∗ e itH π ( A ) e − itH π (Γ (Λ ,s ) t ) . Then, (3.32) follows by observing that(3.34) e − itH (Λ ,s ) = e − itH π (Γ (Λ ,s ) t ) , as by (3.13) one readily checks that ˜Γ (Λ ,s ) t := π (Γ (Λ ,s ) t ) is the unique, unitary solution of(3.35) ddt ˜Γ (Λ ,s ) t = − ie itH π ( sV Λ ) e − itH ˜Γ (Λ ,s ) t with ˜Γ (Λ ,s )0 = 1l . Weighted Integral Operators:
For any γ >
0, Λ ∈ P (Γ), and s ∈ R we can map the weighted integraloperators, see (3.16), to the GNS space by setting ˜ F Λ s : B ( H ) → B ( H ) and ˜ G Λ s : B ( H ) → B ( H ) as(3.36)˜ F Λ s ( A ) = (cid:90) ∞−∞ e itH (Λ ,s ) Ae − itH (Λ ,s ) w γ ( t ) dt and ˜ G Λ s ( A ) = (cid:90) ∞−∞ e itH (Λ ,s ) Ae − itH (Λ ,s ) W γ ( t ) dt for all A ∈ B ( H ). Using (3.32), it is clear that(3.37) π ( F Λ s ( A )) = ˜ F Λ s ( π ( A )) and π ( G Λ s ( A )) = ˜ G Λ s ( π ( A )) for all A ∈ A . The Spectral Flow:
As with the weighted integral operators discussed above, we can also map thespectral flow, as defined in (3.25), to the GNS space. In fact, for any γ >
0, Λ ∈ P (Γ), and s ∈ R take U (Λ , s ) = π ( u (Λ , s )) ∈ B ( H ) where u (Λ , s ) is the unique, unitary solution of (3.23) in A .Since π is a representation, each U (Λ , s ) is unitary on H and hence, the automorphisms(3.38) ˜ α Λ s ( A ) = U (Λ , s ) ∗ AU (Λ , s )define the spectral flow for all A ∈ B ( H ). In this case, of course, we again have that(3.39) π ( α Λ s ( A )) = ˜ α Λ s ( π ( A )) for all A ∈ A . Let us now explain the rationale for calling these automorphisms the spectral flow. For anyΛ ∈ P (Γ) and s ∈ R , we are considering the perturbed Hamiltonian H (Λ , s ) as in (3.31). As in(2.33), denote the ground state energy of H (Λ , s ) by(3.40) E (Λ , s ) = inf spec( H (Λ , s )) . Now regard Λ ∈ P (Γ) as fixed. It is clear that H (Λ ,
0) = H is the unperturbed Hamiltonianwith E (Λ ,
0) = 0. Under our assumptions, Proposition 2.4 holds and so the kernel of H is one-dimensional. Moreover, by Assumption 2.1, the bulk gap associated to H is strictly positive.By standard results, see e.g [18], for | s | sufficiently small the kernel of H (Λ , s ) − E (Λ , s )1l is one-dimensional and the ground state gap does not immediately close. More precisely, denote theground state gap of H (Λ , s ) by(3.41) gap( H (Λ , s )) = sup { δ > E (Λ , s ) , E (Λ , s ) + δ ) ∩ spec( H (Λ , s )) = ∅} . By assumption, the bulk gap γ = gap( H (Λ , H ) >
0. Fix 0 < γ < γ . In this case, forany Λ ∈ P (Γ), there is s Λ0 ( γ ) > H (Λ , s )) ≥ γ for all | s | ≤ s Λ0 ( γ ) . TABILITY OF THE BULK GAP 13
As a result, an application of [26, Theorem 6.3] shows that there is a family of automorphisms of B ( H ), which we denote by ˆ α Λ s , that satisfy(3.43) ˆ α Λ s ( P Λ ( s )) = P Λ (0) whenever | s | ≤ s Λ0 ( γ ) , where by P Λ ( s ) we denote the orthogonal projection onto the ground state space of H (Λ , s ). Thisis Hastings’ quasi-adiabatic evolution [13,15]. In fact, this family of automorphisms is implementedby the unique, unitary B ( H )-valued solution of(3.44) dds ˜ U (Λ , s ) = − i ˜ D (Λ , s ) ˜ U (Λ , s ) with ˜ U (Λ ,
0) = 1l , with(3.45) ˜ D (Λ , s ) = ˜ G Λ s ( π ( V Λ )) = (cid:90) ∞−∞ e itH (Λ ,s ) π ( V Λ ) e − itH (Λ ,s ) W γ ( t ) dt . Comparing this unique solution with the representative of the unique solution of (3.23), we concludethat ˆ α Λ s = ˜ α Λ s as defined in (3.38).For any Λ ∈ P (Γ) and s ∈ R , the state ω Λ s : A → C given by(3.46) ω Λ s ( A ) = ω ( α Λ s ( A )) for all A ∈ A is a vector state in the GNS space:(3.47) ω Λ s ( A ) = (cid:104) Ω(Λ , s ) , π ( A )Ω(Λ , s ) (cid:105) for all A ∈ A , where Ω(Λ , s ) = U (Λ , s )Ω ∈ H . By our assumptions, the ground state of H is unique, and hence, P Λ (0) = | Ω (cid:105)(cid:104) Ω | . An application of (3.43) then shows that(3.48) P Λ ( s ) = U (Λ , s ) | Ω (cid:105)(cid:104) Ω | U (Λ , s ) ∗ = | Ω(Λ , s ) (cid:105)(cid:104) Ω(Λ , s ) | for all | s | ≤ s Λ0 ( γ ) , and thus Ω(Λ , s ) is the ground state of H (Λ , s ); in particular, H (Λ , s )Ω(Λ , s ) = E (Λ , s )Ω(Λ , s ).Given this, (3.43) can be written somewhat differently. Let 0 < γ < γ , Λ ∈ P (Γ), and consider | s | ≤ s Λ0 ( γ ), then we have that(3.49) ˜ α Λ s ( | Ω(Λ , s ) (cid:105)(cid:104) Ω(Λ , s ) | ) = | Ω (cid:105)(cid:104) Ω | . Finally, recall that with the parameters γ and s as above that the weighted integral operator ˜ F Λ s from (3.36) satisfies the relation(3.50) (cid:104) ˜ F Λ s ( A ) , | Ω(Λ , s ) (cid:105)(cid:104) Ω(Λ , s ) | (cid:105) = 0 for all A ∈ B ( H ) . See, e.g. [26, Lemma 6.8], for a proof of this property.3.3.
On Domains.
In this section, we describe the action of certain unbounded Hamiltonians onspecific dense subspaces of vectors in their domain.We begin with a basic observation about general derivations. Let (Γ , d ) be a ν -regular metricspace, g an F -function on (Γ , d ), and Φ an interaction on Γ with (cid:107) Φ (cid:107) g < ∞ . As in Section 2, corre-sponding to any such Φ there is well-defined, closed derivation δ Φ with dense domain dom( δ Φ ) ⊂ A which can be obtained as the closure of the linear map δ Φ : A loc → A defined by(3.51) δ Φ ( A ) = (cid:88) Y ∈P Y ∩ X (cid:54) = ∅ [Φ( Y ) , A ] for any A ∈ A X . Although the sum on the right-hand-side above may be infinite, it is absolutely convergent whenΦ has a finite F -norm. In fact, δ Φ is locally bounded:(3.52) (cid:107) δ Φ ( A ) (cid:107) ≤ D | X |(cid:107) A (cid:107) for all X ∈ P (Γ) , A ∈ A X , with D = 2 (cid:107) F (cid:107)(cid:107) Φ (cid:107) F , see Example 4.7 of [26, Section IV.B.1]. We also have the following lemma. Lemma 3.1.
Let (Γ , d ) be ν -regular and g be an F -function on (Γ , d ) with a finite ν -moment, i.e., (3.53) ∞ (cid:88) n =1 ( n + 1) ν g ( n ) < ∞ . For any interaction Φ on Γ with (cid:107) Φ (cid:107) g < ∞ , we have that A g ⊂ dom( δ Φ ) .Proof. For n ≥
1, and A ∈ A g , there is some x ∈ Γ, C ≥
0, and observables A n ∈ A b x ( n ) satisfying (cid:107) A − A n (cid:107) ≤ Cg ( n ). In this case, the bound (cid:107) A n +1 − A n (cid:107) ≤ Cg ( n ) is clear. Using (3.52) and ν -regularity of Γ, we conclude(3.54) (cid:107) δ Φ ( A n +1 ) − δ Φ ( A n ) (cid:107) ≤ κCD ( n + 1) ν g ( n ) . Thus, for all m < n ,(3.55) (cid:107) δ Φ ( A n ) − δ Φ ( A m ) (cid:107) ≤ κCD n − (cid:88) k = m ( k + 1) ν g ( k ) . Since we assumed that g has a finite ν -moment, this implies that δ Φ ( A n ) is a Cauchy sequence.Since A n → A and A loc is a core for δ Φ , it follows that A ∈ dom( δ Φ ). (cid:3) Given our assumptions, Lemma 3.1 clearly applies to the derivation δ defined by the unperturbedinteraction η . Using that H Ω = 0, one readily checks the relation(3.56) e itH π ( A )Ω = π ( τ (0) t ( A ))Ω , from which the inclusion π (dom( δ ))Ω ⊂ dom( H ) is clear. As a result, if g is an F -function witha finite ν -moment, then for any A ∈ A g , we have that π ( A )Ω ∈ dom( H ) = dom( H (Λ , s )) for anyΛ ∈ P (Γ) and s ∈ R .Now, under Assumption 2.1, the bulk gap γ >
0. For any 0 < γ < γ and Λ ∈ P (Γ), thequantity s Λ0 ( γ ) is defined so that the gap constraint, see (3.42), is satisfied. In this case, uniquenessof the corresponding ground state implies that whenever | s | ≤ s Λ0 ( γ ),(3.57) (cid:104) ψ, ( H (Λ , s ) − E (Λ , s )1l) ψ (cid:105) ≥ γ (cid:104) ψ, ψ (cid:105) for all ψ ∈ dom( H ) with (cid:104) Ω(Λ , s ) , ψ (cid:105) = 0. Recalling our notation for the perturbed state ω Λ s , see(3.46) and (3.47), we conclude that for any A ∈ A loc ,(3.58) 0 = ω Λ s ( A ) = (cid:104) Ω(Λ , s ) , ψ (cid:105) where ψ = π ( A )Ω(Λ , s ) . Since Ω(Λ , s ) = U (Λ , s )Ω, U (Λ , s ) = π ( u (Λ , s )), and u (Λ , s ) satisfies the quasi-locality estimate in(3.24), we conclude that each such ψ = π ( Au (Λ , s ))Ω ∈ dom( H (Λ , s )) by Lemma 3.1. As a result,whenever A ∈ A loc and | s | ≤ s Λ0 ( γ ), if ω Λ s ( A ) = 0, then the estimate(3.59) ω Λ s ( A ∗ δ Λ s ( A )) ≥ γω Λ s ( A ∗ A )follows from (3.57).It will be important that on an appropriate dense domain, the action of the unbounded Hamil-tonians can be expressed as a limit of finite-volume quantities. This is the content of the nextlemma. Lemma 3.2.
Let (Γ , d ) be ν -regular, g be an F -function on (Γ , d ) with a finite ν -moment, and ω be an infinite-volume, zero energy, ground state of the frustration free model defined by δ . Denoteby ( H , π , Ω) the frustration free representation and by H the corresponding GNS Hamiltonian.Then, for any IAS (Λ n ) , (3.60) lim n →∞ π ( H Λ n ) ψ = H ψ for all ψ ∈ π ( A g )Ω , where H Λ n ∈ A Λ n is as in (2.14). TABILITY OF THE BULK GAP 15
Proof.
Let (Λ n ) be an IAS and take ψ ∈ π ( A g )Ω. Since ψ = π ( A )Ω for some A ∈ A g , it is clearthat ψ ∈ dom( H ) by Lemma 3.1 and the above discussion. Using that A ∈ A g , there is some x ∈ Γand observables A m ∈ A b x ( m ) with (cid:107) A − A m (cid:107) ≤ Cg ( m ) for all m ≥
1. Considering the vectors ψ m = π ( A m )Ω ∈ H , we have that(3.61) (cid:107) ψ − ψ m (cid:107) ≤ Cg ( m ) and (cid:107) ψ m − ψ k (cid:107) ≤ Cg ( k ) whenever 1 ≤ k < m . Since the frustration-free property implies that(3.62) H π ( B )Ω = π ( δ ( B ))Ω for all B ∈ dom δ . Arguing as in (3.55) above, we conclude that, again for k < m ,(3.63) (cid:107) H ψ m − H ψ k (cid:107) ≤ (cid:107) δ ( A m ) − δ ( A k ) (cid:107) ≤ κCD m − (cid:88) (cid:96) = k ( (cid:96) + 1) ν g ( (cid:96) )Note that here we may take D = 2 κR ν (cid:107) η (cid:107) ∞ . The bound(3.64) (cid:107) H ψ − H ψ k (cid:107) ≤ κCD ∞ (cid:88) (cid:96) = k ( (cid:96) + 1) ν g ( (cid:96) )follows. By the zero-energy property of the ground state, we similarly see that for any n ≥ (cid:107) π ( H Λ n ) ψ m − π ( H Λ n ) ψ k (cid:107) ≤ (cid:107) [ H Λ n , A m − A k ] (cid:107) ≤ κCD m − (cid:88) (cid:96) = k ( (cid:96) + 1) ν g ( (cid:96) )for all 1 ≤ k < m and therefore, we obtain(3.66) (cid:107) π ( H Λ n ) ψ − π ( H Λ n ) ψ k (cid:107) ≤ κCD ∞ (cid:88) (cid:96) = k ( (cid:96) + 1) ν g ( (cid:96) )Now for any n ≥ k ≥
1, we find that (cid:107) π ( H Λ n ) ψ − H ψ (cid:107) ≤ (cid:107) π ( H Λ n ) ψ − π ( H Λ n ) ψ k (cid:107) + (cid:107) π ( H Λ n ) ψ k − H ψ k (cid:107) + (cid:107) H ψ k − H ψ (cid:107) (3.67)For k ≥ k , the middle term vanishes for n sufficiently large; here weused the zero energy property of the ground state. This completes the proof. (cid:3) Lemma 3.2 also trivially applies to the perturbed system in the GNS space. In fact, for Λ ∈ P (Γ)and s ∈ R , under assumptions as above, a direct application of Lemma 3.2 shows that we also have(3.68) lim n π ( H Λ n + sV Λ ) ψ = H ψ + sπ ( V Λ ) ψ = H (Λ , s ) ψ for all ψ ∈ π ( A g )Ω . Remark 3.3.
An analogue of Lemma 3.2 holds more generally. In fact, if (Γ , d ) is ν -regular and g is an F -function on (Γ , d ) with a finite ν -moment, then for any frustration free interaction Φwith (cid:107) Φ (cid:107) g < ∞ , the GNS Hamiltonian in any frustration free representation satisfies (3.60). Theargument is identical to the above except that one uses the more general estimate in Lemma 3.1and bounds the middle term in (3.67) by(3.69) (cid:88) X ∈P X ∩ b x ( k ) (cid:54) = ∅ ,X ∩ Λ cn (cid:54) = ∅ (cid:107) π (Φ( X )) ψ k (cid:107) ≤ (cid:107) Φ (cid:107) g sup m ≥ (cid:107) A m (cid:107) (cid:88) y ∈ b x ( k ) ,z ∈ Λ cn g ( d ( y, z ))For fixed k , the above is the sum of finitely many ‘tails’ of the uniformly summable function g . We now investigate how the weighted integral operator ˜ F Λ s , see (3.36), can be applied to theunbounded Hamiltonian H (Λ , s ). Recall that for any γ > s ∈ R , and Λ ∈ P (Γ), we have defineda weighted integral operator ˜ F Λ s : B ( H ) → B ( H ) by setting(3.70) ˜ F Λ s ( A ) = (cid:90) ∞−∞ e itH (Λ ,s ) Ae − itH (Λ ,s ) w γ ( t ) dt for all A ∈ B ( H ) . It is clear that(3.71) ˜ F Λ s ( e iuH (Λ ,s ) ) = e iuH (Λ ,s ) for all u, s ∈ R , since the dynamics leaves this bounded operator invariant and w γ integrates to 1. We will need adifferential version of this fact, which we turn to next.First, we prove an analogue of the desired statement for the unperturbed dynamics; this isLemma 3.4 below. To this end, assume w : R → R satisfies(3.72) (cid:90) R w ( u ) du = 1 and (cid:90) R | u | ν | w ( u ) | du < ∞ , and define a weighted integral operator ˜ F : B ( H ) → B ( H ) by setting(3.73) ˜ F ( A ) = (cid:90) ∞−∞ e iuH Ae − iuH w ( u ) du for all A ∈ B ( H ) . To simplify notation, let us also write(3.74) ˜ τ (0) u ( A ) = e iuH Ae − iuH . Our first result is as follows.
Lemma 3.4.
Let (Γ , d ) be ν -regular, g be an F -function on (Γ , d ) with a finite ν -moment, and w : R → R satisfy (3.72). For each choice of IAS (Λ n ) , the weighted integral operator ˜ F , as in(3.73), satisfies (3.75) lim n →∞ ˜ F ( π ( H Λ n )) ψ = H ψ for all ψ ∈ π ( A g )Ω , where H Λ n ∈ A Λ n is as in (2.14).Proof. Fix an IAS (Λ n ) and take ψ = π ( A )Ω for some A ∈ A g . We can rewrite the convergenceclaimed in (3.75) as the convergence of integrals of a sequence of functions f n : R → H given by(3.76) f n ( u ) = w ( u )˜ τ (0) u ( π ( H Λ n )) ψ for all u ∈ R . Since H Ω = 0, the above can be re-written as(3.77) f n ( u ) = w ( u ) e iuH π ( H Λ n ) π ( τ (0) − u ( A ))Ωusing (3.30). We claim that there is an F -function g τ with a finite ν -moment such that τ (0) − u ( A ) ∈ A g τ for all u ∈ R . Given this, Lemma 3.2 applies and we find that(3.78) lim n →∞ f n ( u ) = w ( u ) e iuH H π ( τ (0) − u ( A ))Ω = w ( u ) H ψ for all u ∈ R . By the normalization of w , see (3.72), the integral of this limit coincides with the right-hand-sideof (3.75). Therefore, to complete the proof we only need to justify an application of dominatedconvergence.Let us first prove the existence of an F -function g τ as claimed. Fix A ∈ A g . In this case, thereis x ∈ Γ, C ≥
0, and observables A m ∈ A b x ( m ) for which (cid:107) A − A m (cid:107) ≤ Cg ( m ) for all m ∈ N . Let u ∈ R and for any n ∈ N , set(3.79) A n ( u ) = Π n ( τ (0) u ( A (cid:100) n/ (cid:101) )) ∈ A b x ( n )TABILITY OF THE BULK GAP 17 where, to ease notation, we have written Π n = Π b x ( n ) , for the conditional expectation from Sec-tion 3.1.1. A straightforward estimate shows that for any µ > (cid:107) τ (0) u ( A ) − A n ( u ) (cid:107) ≤ (cid:107) τ (0) u ( A ) − τ (0) u ( A (cid:100) n/ (cid:101) ) (cid:107) + (cid:107) τ (0) u ( A (cid:100) n/ (cid:101) ) − A n ( u )) (cid:107)≤ Cg ( n/
2) + κC µ e µ ( v µ | u | +1) (sup m ≥ (cid:107) A m (cid:107) )( n/ ν e − µn/ (3.80)where, for the final bound above, we used (3.9) and (3.5). The claimed F -function g τ can now beobtained by estimating with the minimum of g ( n/
2) and ( n/ ν e − µn/ .We now turn to finding a dominating function for f n . Recall that each A ∈ A g corresponds to asequence of observables A m as described above. In this case, for any m ∈ N , A can be written asan absolutely convergent, telescopic sum:(3.81) A = A m + ∞ (cid:88) k = m +1 B k where B k = A k − A k − and (cid:107) B k (cid:107) ≤ Cg ( k − . Inserting this decomposition of A into (3.77), we find that for any n ∈ N and each u ∈ R :(3.82) (cid:107) f n ( u ) (cid:107) ≤ | w ( u ) | (cid:107) π ( H Λ n ) π ( τ (0) − u ( A m ))Ω (cid:107) + ∞ (cid:88) k = m +1 (cid:107) π ( H Λ n ) π ( τ (0) − u ( B k ))Ω (cid:107) . Now, by the zero-energy property of the ground state, it is clear that(3.83) π ( H Λ n ) π ( A )Ω = π ([ H Λ n , A ])Ω for all A ∈ A loc , and thus, in particular, we have the bound(3.84) (cid:107) π ( H Λ n ) π ( A )Ω (cid:107) ≤ κD ( k + R ) ν (cid:107) A (cid:107) for all A ∈ A b x ( k ) , which we stress is uniform in n . This suggests a mechanism for bounding the first term in (3.82)above. In fact, let (cid:96) ≥ m and use conditional expectations, as in (3.8), to write(3.85) τ (0) − u ( A m ) = (cid:88) (cid:96) ≥ (cid:96) ∆ (cid:96)(cid:96) ( τ (0) − u ( A m )) , where again we have used a short-hand ∆ (cid:96)m for ∆ (cid:96)b x ( m ) as in (3.8). For (cid:96) = (cid:96) , the bound (cid:107) π ( H Λ n ) π (∆ (cid:96) (cid:96) ( τ (0) − u ( A m )))Ω (cid:107) = (cid:107) π ( H Λ n ) π (Π (cid:96) ( τ (0) − u ( A m )))Ω (cid:107)≤ κD ( (cid:96) + R ) ν (cid:107) A m (cid:107) (3.86)follows from (3.84). For (cid:96) ≥ (cid:96) , the estimate (cid:107) π ( H Λ n ) π (∆ (cid:96)(cid:96) ( τ (0) − u ( A m )))Ω (cid:107) ≤ κD ( (cid:96) + R ) ν (cid:107) ∆ (cid:96)(cid:96) ( τ (0) − u ( A m )) (cid:107)≤ κ D C µ m ν ( (cid:96) + R ) ν (cid:107) A m (cid:107) e − µ ( (cid:96) − m − − v µ | u | ) (3.87)follows from another application of (3.84), the general estimate (3.7) for these differences, and thequasi-locality estimate for the unperturbed dynamics. We conclude that (cid:107) π ( H Λ n ) π ( τ (0) − u ( A m ))Ω (cid:107) ≤ (cid:88) (cid:96) ≥ (cid:96) (cid:107) π ( H Λ n ) π (∆ (cid:96)(cid:96) ( τ (0) − u ( A m )))Ω (cid:107)≤ κD (cid:107) A m (cid:107) ( (cid:96) + R ) ν + 2 κC µ m ν (cid:88) (cid:96) ≥ (cid:96) +1 ( (cid:96) + R ) ν e − µ ( (cid:96) − m − − v µ | u | ) . (3.88)If we now take (cid:96) = (cid:100) v µ | u | + m (cid:101) , then we have found that there is K ≥ (cid:107) π ( H Λ n ) π ( τ (0) − u ( A m ))Ω (cid:107) ≤ K (cid:107) A m (cid:107) (cid:0) m ν + | u | ν + ( m | u | ) ν + 1 (cid:1) and here K = K ( κ, µ, ν, R ). Now, we claim that the terms B k in (3.82) can be estimated similarly. In fact, regarding k as m ,argue as in (3.85) - (3.88) with some (cid:96) ≥ k . A bound analogous to (3.89) can be found; of course,here one replaces (cid:107) A m (cid:107) with (cid:107) B k (cid:107) . Since (cid:107) B k (cid:107) ≤ Cg ( k −
1) and g has a finite 2 ν -moment, wehave obtained a bound on the right-hand-side of (3.82) with the form:(3.90) (cid:107) f n ( u ) (cid:107) ≤ ˜ K (1 + | u | ν ) | w ( u ) | for all u ∈ R . By the assumption on w , i.e. (3.72), the above is a dominating function for the sequence f n . Thisjustifies dominated convergence and completes the proof. (cid:3) We will also need a version of Lemma 3.4 for the perturbed system.
Lemma 3.5.
Let (Γ , d ) be ν -regular, g be an F -function on (Γ , d ) with a finite ν -moment, and w : R → R satisfy (3.72). Let Λ ∈ P (Γ) and take s ∈ R . For each choice of IAS (Λ n ) , the weightedintegral operator ˜ F Λ s , as in (3.70), satisfies (3.91) lim n →∞ ˜ F Λ s ( π ( H Λ n + sV Λ )) ψ = H (Λ , s ) ψ for all ψ ∈ π ( A g )Ω , with H Λ n ∈ A Λ n as in (2.14) and V Λ as in (2.31).Proof. Fix an IAS (Λ n ) where we assume for convenience that Λ ⊂ Λ . As in the proof of Lemma3.4, take ψ = π ( A )Ω with A ∈ A g , and for each n ∈ N , consider f n : R → H given by(3.92) f n ( u ) = w ( u )˜ τ (Λ ,s ) u ( π ( H Λ n + sV Λ )) π ( A )Ω for all u ∈ R , where, in analogy to (3.74), we have set(3.93) ˜ τ (Λ ,s ) t ( A ) = e itH (Λ ,s ) Ae − itH (Λ ,s ) for all A ∈ B ( H ) and t ∈ R . Using (3.32), (3.12), and (3.30), we may write(3.94) ˜ τ (Λ ,s ) u ( π ( H Λ n + sV Λ )) = π (Γ (Λ ,s ) u ) ∗ e iuH π ( H Λ n + sV Λ ) e − iuH π (Γ (Λ ,s ) u )for all u ∈ R . In this case, we find that(3.95) f n ( u ) = w ( u ) π (Γ (Λ ,s ) u ) ∗ e iuH π ( H Λ n + sV Λ ) π ( τ (0) − u (Γ (Λ ,s ) u A ))Ω . Arguing as before, here we will also need (3.15), one shows that there is an F -function g (cid:48) with afinite ν -moment such that τ (0) − u (Γ (Λ ,s ) u A ) ∈ A g (cid:48) . As a result, the point-wise limit(3.96) lim n f n ( u ) = w ( u ) π (Γ (Λ ,s ) u ) ∗ e iuH ( H + sπ ( V Λ )) π ( τ (0) − u (Γ (Λ ,s ) u A ))Ω = w ( u ) H (Λ , s ) ψ is clear using properties of the interaction picture dynamics, see discussion following (3.12).The argument demonstrating that we can apply the dominated convergence theorem also pro-ceeds as in the proof of Lemma 3.4. Since the differences stemming from the presence of the u -dependence in the operators A m and B k are minor, we leave the details to the reader. (cid:3) Construction of a unitarily equivalent perturbed system
Construction of a relatively bounded perturbation.
The crux of the proof of stabilityintroduced in [7], is the construction of a unitarily equivalent perturbed system using the spectralflow (aka quasi-adiabatic evolution) for which one can prove a relative form bound. In the infinite-system setting, this means transforming the unbounded Hamiltonian H (Λ , s ), see (2.30), by thespectral flow to achieve an equivalent Hamiltonian of the form(4.1) U (Λ , s ) ∗ H (Λ , s ) U (Λ , s ) ψ = H ψ + W (Λ , s ) ψ + E (Λ , s ) ψ for all ψ ∈ π ( A loc )Ω , with W (Λ , s ) a bounded operator with an explicit, Λ-independent form-bound with respect to H and E (Λ , s ) the ground state energy of H (Λ , s ) as in (2.33). In previous work, we established quasi-locality estimates for this spectral flow when the perturbation is of sufficiently rapid decay. GivenAssumption 2.5, the resulting quasi-locality can be used to establish that each ψ ∈ π ( A loc )Ω is in TABILITY OF THE BULK GAP 19 the domain of the unitarily evolved Hamiltonian in (4.1) whenever Λ ∈ P (Γ) as U (Λ , s ) ψ ∈ A g foran appropriate choice of g .The proof of Theorem 2.7 will be obtained as a direct consequence of two results. The firstresult establishes that W (Λ , s ) can be expressed in a suitable form for deriving a relative formbound (Theorem 4.1). The second result is the relative form bound itself (Theorem 5.1 in Section5). Theorem 4.1.
If Assumptions 2.1-2.5 are satisfied, for any γ ∈ (0 , γ ) and | s | ≤ s Λ0 ( γ ) , there isa familiy of self-adjoint observables Φ (2) ( x, m, s ) ∈ A b x ( m ) , for each x ∈ Γ and m ≥ R , with thefollowing properties: (i) Φ (2) ( x, m, s ) P b x ( m ) = P b x ( m ) Φ (2) ( x, m, s ) = 0 ; (ii) (cid:107) Φ (2) ( x, m, s ) (cid:107) ≤ sG (2)Λ ( x, m ) with (4.2) G (2)Λ ( x, m ) = G Λ ( x, m/
2) + 2 G (1)Λ ( x, m + 1) + 2 G (1)Λ ( x, R ) (cid:112) (1 + m ) ν G ( m/ where G Λ ( x, m ) is as in Theorem 4.2 below and G (1)Λ ( x, m ) = (cid:80) n ≥ m G Λ ( x, n ) . Furthermore, W (Λ , s ) is given by the absolutely convergent sum (4.3) W (Λ , s ) = (cid:88) x ∈ Γ (cid:88) m ≥ R π (Φ (2) ( x, m, s )) . The operator W (Λ , s ) is a priori defined in the GNS representation and the identity (4.1) onlymakes sense in that representation. A posteriori , we find via (4.3) that W (Λ , s ), in fact, correspondsto a quasi-local observable in A .We prove the decomposition from Theorem 4.1 in two steps. First, we use quasi-locality andconditional expectations to prove that for all | s | ≤ s Λ0 ( γ ), the action of the spectral flow on theGNS Hamiltonian H (Λ , s ) can be again realized as a perturbation of H . Namely, we show thatfor all ψ ∈ π ( A loc )Ω(4.4) U (Λ , s ) ∗ H (Λ , s ) U (Λ , s ) ψ = H ψ + (cid:88) x ∈ Γ (cid:88) m ≥ R ˜Φ (1) ( x, m, s ) ψ where the perturbation terms ˜Φ (1) ( x, m, s ) ∈ π ( A b x ( m ) ) are self-adjoint, absolutely summable, andsatisfy a norm bound that is linear in s , see Theorem 4.2 below. In the second step, the finalform of (4.3) from Theorem 4.1 follows from using the frustration-free and LTQO ground stateproperties to produce a refined decomposition of the perturbation terms from (4.4). We now turnto establishing the first step in this decomposition, and prove Theorem 4.1 in Section 4.2. Theorem 4.2.
Fix any finite volume Λ ⊂ Γ . Under the conditions of Theorem 4.1, there exists afunction G Λ : Γ × [0 , ∞ ) → [0 , ∞ ) for which (4.5) (cid:88) x ∈ Γ (cid:88) m ≥ R G Λ ( x, m ) < ∞ and a self-adjoint operator ˜Φ (1) ( x, m, s ) ∗ = ˜Φ (1) ( x, m, s ) ∈ π ( A b x ( m ) ) for each x ∈ Γ and m ≥ R ,such that (cid:107) ˜Φ (1) ( x, m, s ) (cid:107) ≤ sG Λ ( x, m ) and (4.6) W (Λ , s ) + E (Λ , s )1l = (cid:88) x ∈ Γ (cid:88) m ≥ R ˜Φ (1) ( x, m, s ) . Moreover, for each x ∈ Γ , the global operator ˜Φ (1) ( x, s ) = (cid:80) m ≥ R ˜Φ (1) ( x, m, s ) is quasi-local, i.e.belongs to π ( A ) , and commutes with the ground state projection | Ω (cid:105)(cid:104) Ω | . To prove this result, we will use two quasi-local maps K i, Λ s , i = 1 ,
2, which are defined in termsof the examples introduced in Section 3.1. To see how these maps emerge, we show how to relate(4.1) to a sequence of finite-volume Hamiltonians H Λ n + sV Λ . This will also be the initial step fordefining the global operators ˜Φ (1) ( x, s ) from Theorem 4.2.By Lemma 3.2, it is clear that(4.7) ( W (Λ , s ) + E (Λ , s )1l) ψ := U (Λ , s ) ∗ H (Λ , s ) U (Λ , s ) ψ − H ψ is well-defined for all ψ ∈ π ( A loc )Ω. Here, we use that U (Λ , s ) ψ ∈ π ( A g )Ω for a function g of theform (2.5) with ζ > ν + 1. Fixing any IAS (Λ n ), Lemmas 3.4 and 3.5 then imply(4.8) ( W (Λ , s ) + E (Λ , s )1l) ψ = lim n →∞ (cid:16) U (Λ , s ) ∗ ˜ F Λ s ( π ( H Λ n + sV Λ )) U (Λ , s ) ψ − ˜ F ( π ( H Λ n )) ψ (cid:17) where we choose ˜ F = ˜ F Λ0 .Each term in the limit above is the sum of bounded operators. Using the relationship betweenthe quasi-local maps in the GNS representation to those on the C ∗ -algebra, i.e. (3.37)-(3.39), wecan rewrite each of these terms as follows:˜ α Λ s ◦ ˜ F Λ s ( ˜ H Λ n + s ˜ V Λ ) − ˜ F ( ˜ H Λ n ) = (cid:88) x ∈ Λ n π (cid:0) α Λ s ◦ F Λ s ( h x ) − F ( h x ) (cid:1) + (cid:88) x ∈ Λ (cid:88) k ≥ R : b x ( k ) ⊆ Λ π (cid:0) sα Λ s ◦ F Λ s (Φ( x, k )) (cid:1) . (4.9)Given (4.9), for i = 1 , K i, Λ s : A → A by(4.10) K , Λ s ( A ) = α Λ s ( F Λ s ( A )) − F ( A ) and K , Λ s ( A ) = sα Λ s ( F Λ s ( A )) . It was proved, e.g. in [25, Lemma 4.4], that both of these maps satisfy a local bound and quasi-localestimate that is independent of the finite volume Λ. Specifically, for each i = 1 , p i , q i and C i , and a decay function G K i (all independent of Λ) for which (cid:107)K i, Λ s ( A ) (cid:107) ≤ sC i | X | p i (cid:107) A (cid:107) (4.11) (cid:107) [ K i, Λ s ( A ) , B ] (cid:107) ≤ s | X | q i (cid:107) A (cid:107)(cid:107) B (cid:107) G K i ( d ( X, Y ))(4.12)hold for any
X, Y ∈ P (Γ), A ∈ A X , B ∈ A Y , and s ∈ R . In fact, p = q = 2, p = 0 and q = 1.One can make explicit estimates on the decay of G K i , see e.g. [25, Remark 4.7]. However, it sufficesto note that they each have finite moments of all orders in the sense of (3.19).Given the limit n → ∞ in (4.8), to prove that the decomposition in (4.6) is absolutely summable,we will need local bound and quasi-local estimates for K i, Λ s as in (4.11)-(4.12) that also decay inthe distance d ( X, Λ). This is the content of Lemmas 4.3 and 4.4 below. We proceed with the proofof Theorem 4.2 with the knowledge that such bounds do hold.
Proof of Theorem 4.2.
Fix γ ∈ (0 , γ ), Λ ∈ P (Γ), and take any IAS (Λ n ) for which Λ ⊆ Λ n for all n . Define the spectral flow α Λ s and the weighted integral operators F Λ s , F = F Λ0 with respect tothe specific choices of γ and Λ, see (3.25) and (3.16).Given K i, Λ s , i = 1 ,
2, as in (4.10), for each x ∈ Γ and | s | ≤ s Λ0 ( γ ) we then define the self-adjointoperator ˜Φ (1) ( x, s ) = π (Φ (1) ( x, s )) ∈ B ( H ) by(4.13) Φ (1) ( x, s ) = K , Λ s ( h x ) + χ Λ ( x ) (cid:88) k ≥ R : b x ( k ) ⊂ Λ K , Λ s (Φ( x, k )) ∈ A where χ Λ is the characteristic function of Λ ⊂ Γ. We first show that each term ˜Φ (1) ( x, s ) commuteswith the ground state projector | Ω (cid:105)(cid:104) Ω | , and then turn to the decomposition from (4.6). TABILITY OF THE BULK GAP 21
Fix | s | ≤ s Λ0 ( γ ) and recall that Ω(Λ , s ) = U (Λ , s )Ω. Combining this with the relations (3.37)-(3.39) shows that for all A ∈ A (cid:2) π ( α Λ s ( F Λ s ( A ))) , | Ω (cid:105)(cid:104) Ω | (cid:3) = (cid:104) U (Λ , s ) ∗ ˜ F Λ s ( π ( A )) U (Λ , s ) , | Ω (cid:105)(cid:104) Ω | (cid:105) = U (Λ , s ) ∗ (cid:104) ˜ F Λ s ( π ( A )) , | Ω(Λ , s ) (cid:105)(cid:104) Ω(Λ , s ) | (cid:105) U (Λ , s ) = 0 , (4.14)where for the final equality, we used (3.50). Considering (4.10), this relation implies that(4.15) (cid:2) π ( K , Λ s ( A )) , | Ω (cid:105)(cid:104) Ω | (cid:3) = (cid:2) π ( K , Λ s ( A )) , | Ω (cid:105)(cid:104) Ω | (cid:3) = 0 . from which it is immediate that [ ˜Φ (1) ( x, s ) , | Ω (cid:105)(cid:104) Ω | ] = 0 for all x ∈ Γ and | s | ≤ s Λ0 ( γ ) as claimed.We now establish (4.6). Using conditional expectations, see Section 3.1, each term Φ (1) ( x, s )may be further decomposed as(4.16) Φ (1) ( x, s ) = (cid:88) m ≥ R Φ (1) ( x, m, s )where, for each m ≥ R , we have introduced the self-adjoint term Φ (1) ( x, m, s ) ∈ A b x ( m ) given by(4.17) Φ (1) ( x, m, s ) = ∆ mb x ( R ) ( K , Λ s ( h x )) + χ Λ ( x ) (cid:88) R ≤ k ≤ m : b x ( k ) ⊆ Λ ∆ mb x ( k ) ( K , Λ s (Φ( x, k ))) . With respect to this notation, (4.8), (4.9) and (4.13) show that( W (Λ , s ) + E (Λ , s )1l) ψ = lim n →∞ (cid:88) x ∈ Λ n ˜Φ (1) ( x, s ) ψ = lim n →∞ (cid:88) x ∈ Λ n (cid:88) m ≥ R ˜Φ (1) ( x, m, s ) ψ. (4.18)for all ψ ∈ π ( A loc )Ω. Since this set is dense in H , the equality in (4.6) will follow once we establishabsolute summability of these terms. We do so by demonstrating the existence of a function(4.19) G Λ ( x, m ) = χ Λ( R ) ( x ) G ( m ) + χ Γ \ Λ( R ) ( x ) G ( d ( b x ( R ) , Λ) , m )which bounds the norms of these terms and satisfies (4.5). Here, we note that Λ( R ) is as in (2.1),and the functions G : [0 , ∞ ) → [0 , ∞ ) and G : [0 , ∞ ) × [0 , ∞ ) → [0 , ∞ ) will be independent of Λ.First, suppose x ∈ Γ \ Λ( R ) where R ≥ (1) ( x, m, s ) = ∆ mb x ( R ) ( K , Λ s ( h x )) . Then, applying Lemmas 4.3 and 4.4 with the local approximation bound (3.7) one finds(4.21) (cid:107) ˜Φ (1) ( x, m, s ) (cid:107) = (cid:107) Φ (1) ( x, m, s ) (cid:107) ≤ sG ( d ( b x ( R ) , Λ) , m )where for any fixed δ ∈ (0 , G is(4.22) G ( l, m ) = C · (cid:26) F K ( l ) , if m = R, F δ ( l ) · F − δ ( m − R − , if m ≥ R + 1 . Here, C = κ R ν (cid:107) η (cid:107) ∞ , F K is the function from Lemma 4.3, and(4.23) F δ ( l ) = max (cid:110) ( F K ( l )) δ , ( G K ( l )) δ (cid:111) . More specifically, the bound in (4.21) for m = R is a direct application of Lemma 4.3 while thebound for m ≥ R + 1 follows from the quasi-local estimate in Lemma 4.4 and the subsequent bound(4.46) coupled with (3.4)-(3.7). Summability of G Λ , see (4.19), over the sites x ∈ Γ \ Λ( R ) follows from observing that (cid:88) x ∈ Λ( R ) c (cid:88) m ≥ R G ( d ( b x ( R ) , Λ) , m ) = C (cid:88) x ∈ Λ( R ) c F K ( d ( b x ( R ) , Λ))+ 4 C (cid:88) x ∈ Λ( R ) c F δ ( d ( b x ( R ) , Λ)) (cid:88) m ≥ R +1 F − δ ( m − R − < ∞ (4.24)as both F K and G K (and, thus, F δ for all 0 < δ <
1) have finite moments of all orders. Inparticular, for any decay function F : [0 , ∞ ) → [0 , ∞ ) with a finite ν -moment, (cid:88) x ∈ Λ( R ) c F ( d ( b x ( R ) , Λ)) ≤ (cid:88) n ≥ R (cid:88) x ∈ Λ( n +1) \ Λ( n ) F ( n − R ) ≤ κ | Λ | (cid:88) n ≥ R ( n + 1) ν F ( n − R ) < ∞ . We now turn to the sites x ∈ Λ( R ), for which we demonstrate that(4.25) (cid:107) Φ (1) ( x, m, s ) (cid:107) ≤ sG ( m )where G is a summable function. Consider (4.17) when m = R . Clearly, (cid:107) Φ (1) ( x, R, s ) (cid:107) ≤ (cid:107)K , Λ ( h x ) (cid:107) + (cid:107)K , Λ (Φ( x, R )) (cid:107)≤ sC κ R ν (cid:107) η (cid:107) ∞ + sC (cid:107) Φ (cid:107) e − aR θ (4.26)where we have used the local bounds (4.11), the uniform bound (2.6), and the interaction boundin Assumption 2.5.For m ≥ R + 1, (4.17) can be estimated as: (cid:107) Φ (1) ( x, m, s ) (cid:107) ≤ (cid:107) ∆ mb x ( R ) ( K , Λ s ( h x )) (cid:107) + χ Λ ( x ) (cid:88) R ≤ k ≤ m : b x ( k ) ⊆ Λ (cid:107) ∆ mb x ( k ) ( K , Λ s (Φ( x, k ))) (cid:107)≤ sκ R ν (cid:107) η (cid:107) ∞ G K ( m − R −
1) + 2 sκ (cid:88) R ≤ k ≤ m : b x ( k ) ⊆ Λ k ν (cid:107) Φ( x, k ) (cid:107) G K ( m − k − (cid:88) R ≤ k ≤ m : b x ( k ) ⊆ Λ k ν (cid:107) Φ( x, k ) (cid:107) G K ( m − k − ≤ (cid:107) Φ (cid:107) m (cid:88) k = R k ν e − ak θ G K ( m − k − ≤ (cid:107) Φ (cid:107) G K ( m/ m/ − (cid:88) k = R k ν e − ak θ + G K (0) (cid:88) k ≥ m/ k ν e − ak θ . (4.28)To summarize, for x ∈ Λ( R ) we have proven (4.25) with the decay function G defined as(4.29) G ( R ) = C κ R ν (cid:107) η (cid:107) ∞ + C (cid:107) Φ (cid:107) e − aR θ and for m ≥ R + 1,(4.30) G ( m ) = 2 CG K ( m − R −
1) + 2 κ (cid:107) Φ (cid:107) ( M Φ ( R ) G K ( m/
2) + G K (0) M Φ ( m/ . where we have used the notation(4.31) M Φ ( r ) := (cid:88) k ≥ r k ν e − ak θ TABILITY OF THE BULK GAP 23 to denote the moments associated to the decay of Φ. Since each of the decay functions in (4.30)has finite moments of all orders, it is clear that (cid:80) m ≥ R G ( m ) < ∞ . As a consequence, G Λ as in(4.19) satisfies (cid:88) x ∈ Λ( R ) (cid:88) m ≥ R G Λ ( x, m ) ≤ κR ν | Λ | (cid:88) m ≥ R G ( m ) < ∞ . This demonstrates absolute summability of the terms in (4.6), and hence, completes the proof ofTheorem 4.2. (cid:3)
Given that we are working in the infinite volume setting, in the case of K , Λ s one needs a morerefined version of the estimates in (4.11) - (4.12) to obtain the absolute summability of the terms in(4.6). In the situation under consideration, these refined bounds are a consequence of perturbation V Λ being locally supported as this implies that the spectral flow α Λ s is approximately the identity farfrom Λ. We now prove the necessary distance-dependent locality bound and quasi-local estimates(namely, Lemma 4.3 and Lemma 4.4 below) which were used in the proof of Theorem 4.2. Lemma 4.3 (Distance Locality Bound for K s ) . There exists a decay function F K , with finitemoments of all orders, for which: given any Λ ∈ P (Γ) , X ∈ P (Γ) with d ( X, Λ) > , A ∈ A X ,and any s ∈ R , we have the following local bound (4.32) (cid:107)K , Λ s ( A ) (cid:107) ≤ s | X | (cid:107) A (cid:107) F K ( d ( X, Λ)) . Before we prove this lemma, let F be any decay function with finite ν -moment and take (cid:15) ∈ (0 , M (cid:15)F is given by(4.33) M (cid:15)F ( r ) = (cid:88) n ≥ r ( n + 1) ν F ( (cid:15)n ) for any r ≥ . The proof below shows that one may take(4.34) F K ( r ) = 2 κ (cid:107) Ψ (cid:107) G Ψ (cid:16) M − (cid:15)G Ψ ( r ) + 2 G Ψ ( R ) M (cid:15)G F ( r ) (cid:17) where G F and G Ψ are the decay functions previously discussed in (3.18) and (3.28). Since G F and G Ψ both have finite moments of all orders, the same is true for G .Let us also consider any non-negative, non-increasing function F with a finite ν -moment. Wewill use the following bound in our proof of Lemma 4.3. Let (cid:15) ∈ (0 , ∈ P (Γ), and let X ∈ P (Γ) with d ( X, Λ) >
0. The following estimate holds:(4.35) (cid:88) z ∈ Λ F ( (cid:15)d ( z, X )) ≤ κ | X | M (cid:15)F ( d ( X, Λ)) . A simple argument for this is: (cid:88) z ∈ Λ F ( (cid:15)d ( z, X )) ≤ (cid:88) z ∈ Γ: d ( z,X ) ≥ d ( X, Λ) F ( (cid:15)d ( z, X )) ≤ (cid:88) n ≥ d ( X, Λ) (cid:88) z ∈ Γ: n ≤ d ( z,X ) 1, the set X ( n ) satisfies | X ( n ) | ≤ κn ν | X | by ν -regularity, see (2.1). Proof of Lemma 4.3: Recall that K , Λ s is as defined in (4.10). Since α = id and F = F Λ0 , one findsthat(4.37) K , Λ s ( A ) = (cid:90) s ddr α Λ r ( F Λ r ( A )) dr = i (cid:90) s α Λ r ([ D (Λ , r ) , F Λ r ( A )]) dr for all A ∈ A X with X ∩ Λ = ∅ . Here D (Λ , s ) is the generator of the spectral flow, see (3.20), andmoreover, we have used (3.21) and [26, (6.37)] to obtain dds F Λ s ( A ) = i (cid:90) ∞−∞ (cid:90) t τ (Λ ,s ) r ([ V Λ , τ (Λ ,s ) t − r ( A )]) dr w γ ( t ) dt = i (cid:90) ∞−∞ (cid:90) t τ (Λ ,s ) r ([ V Λ , τ (Λ ,s ) t − r ( A )]) dr (cid:18) − ddt W γ ( t ) + δ ( t ) (cid:19) dt = i (cid:90) ∞−∞ τ (Λ ,s ) t ([ V Λ , A ]) W γ ( t ) dt = 0 , (4.38)where we integrated by parts and used that [ V Λ , A ] = 0 since the support of these observables isdisjoint.Returning to (4.37), we expand the generator as in (3.27) to write(4.39) [ D (Λ , r ) , F Λ r ( A )] = (cid:88) z ∈ Λ (cid:88) n ≥ R [Ψ Λ ( z, n, r ) , F Λ r ( A )]Fix (cid:15) ∈ (0 , z ∈ Λ, set k z ( (cid:15) ) = (cid:15)d ( z, X ). For each term in (4.39), we approximate F Λ r ( A ) with a strictly local approximation:(4.40) [Ψ Λ ( z, n, r ) , F Λ r ( A )] = [Ψ Λ ( z, n, r ) , Π X ( k z ( (cid:15) )) ( F Λ r ( A )) + (cid:0) F Λ r ( A ) − Π X ( k z ( (cid:15) )) ( F Λ r ( A )) (cid:1) ]where we use the conditional expectations as described in Section 3.1.1 with respect to the inflatedset X ( k z ( (cid:15) )). Using the quasi-locality bounds for F Λ r , see (3.18), coupled with (3.5), we find that(4.41) (cid:107) [Ψ Λ ( z, n, r ) , F Λ r ( A ) − Π X ( k z ( (cid:15) )) ( F Λ r ( A ))] (cid:107) ≤ (cid:107) A (cid:107)| X |(cid:107) Ψ Λ ( z, n, r ) (cid:107) G F ( k z ( (cid:15) )) . As a result, we find that(4.42) (cid:88) z ∈ Λ (cid:88) n ≥ R (cid:107) [Ψ Λ ( z, n, r ) , F Λ r ( A ) − Π X ( k z ( (cid:15) )) ( F Λ r ( A ))] (cid:107) ≤ κ (cid:107) A (cid:107)| X | (cid:107) Ψ (cid:107) G Ψ G Ψ ( R ) M (cid:15)G F ( d ( X, Λ))by using (3.28) and (4.35).To estimate the remaining terms, note that for each z ∈ Λ, b z ( n ) ∩ X ( k z ( (cid:15) )) (cid:54) = ∅ only when n ≥ k z (1 − (cid:15) ). As a result, (cid:88) z ∈ Λ (cid:88) n ≥ R (cid:107) [Ψ Λ ( z, n, r ) , Π X ( k z ( (cid:15) )) ( F Λ r ( A ))] (cid:107) = (cid:88) z ∈ Λ (cid:88) n ≥ k z (1 − (cid:15) ) (cid:107) [Ψ Λ ( z, n, r ) , Π X ( k z ( (cid:15) )) ( F Λ r ( A ))] (cid:107)≤ (cid:107) A (cid:107)(cid:107) Ψ (cid:107) G Ψ (cid:88) z ∈ Λ G Ψ ( k z (1 − (cid:15) )) ≤ κ (cid:107) A (cid:107)| X |(cid:107) Ψ (cid:107) G Ψ M − (cid:15)G Ψ ( d ( X, Λ))(4.43)with arguments similar to the prior estimate.The bound claimed in (4.32), with the specific decay function in (4.34), now follows by inserting(4.39) into (4.37) and using the estimates found in (4.42) and (4.43) above. (cid:3) An improvement on the quasi-locality bound for (cid:107) [ K , Λ s ( A ) , B ] (cid:107) which decays in both the distancebetween the supports of the observables A and B , as well as the distance between Λ and the supportof A follows by combining the estimates in Lemma 4.3 and the original quasi-locality bound in(4.12). We now state and prove this. Lemma 4.4 (Distance Quasi-Locality for K ) . There exists a decay function G : [0 , ∞ ) × [0 , ∞ ) → [0 , ∞ ) for which: given any Λ ∈ P (Γ) , X ∈ P (Γ) with d ( X, Λ) > , and Y ∈ P (Γ) , the bound (4.44) (cid:107) [ K , Λ s ( A ) , B ] (cid:107) ≤ s (cid:107) A (cid:107)(cid:107) B (cid:107)| X | G ( d ( X, Λ) , d ( X, Y )) TABILITY OF THE BULK GAP 25 holds for all A ∈ A X , B ∈ A Y , and s ∈ R . More precisely, for any δ ∈ (0 , , one may choose (4.45) G ( m, n ) = max (cid:110) F δ K ( m ) F − δ K ( n ) , G δ K ( m ) G − δ K ( n ) (cid:111) where F K and G K are the decay functions from Lemma 4.3 and (4.12) , respectively. Here, we usethe notation G δ ( m ) = ( G ( m )) δ . In applications, it is often convenient to have a factorized version of the bound in (4.44). Thisis easily achieved. In fact, with δ ∈ (0 , 1) as in (4.45),(4.46) G ( m, n ) ≤ F δ ( m ) · F − δ ( n ) with F δ ( m ) = max { F δ K ( m ) , G δ K ( m ) } . Proof. Fix 0 < δ < 1. We consider two cases. First, suppose d ( X, Λ) ≤ d ( X, Y ). The quasi-localityestimate (4.12) shows that(4.47) (cid:107) [ K , Λ s ( A ) , B ] (cid:107) ≤ s | X | (cid:107) A (cid:107)(cid:107) B (cid:107) G δ K ( d ( X, Λ)) G − δ K ( d ( X, Y ))where we have used that G K is non-increasing.Now, suppose d ( X, Λ) > d ( X, Y ). The local bound from Lemma 4.3, i.e. (4.32), implies(4.48) (cid:107) [ K , Λ s ( A ) , B ] (cid:107) ≤ (cid:107)K , Λ s ( A ) (cid:107)(cid:107) B (cid:107) ≤ s | X | (cid:107) A (cid:107)(cid:107) B (cid:107) F K ( d ( X, Λ)) . Since F K is also non-increasing, the bound F K ( d ( X, Λ)) ≤ F δ K ( d ( X, Λ)) F − δ K ( d ( X, Y )) followsin this case. The bound claimed in (4.44) with the decay function from (4.45) is then a consequenceof (4.47) and (4.48). (cid:3) Final step in the proof of Theorem 4.1. In the previous section, we proved Theorem 4.2.A consequence of the equality in (4.7) is that (cid:104) Ω , W (Λ , s )Ω (cid:105) = 0. Then, denoting by ˜Φ (1) ( x, m, s ) = π (Φ (1) ( x, m, s )) is clear from Theorem 4.2 that(4.49) U (Λ , s ) ∗ H (Λ , s ) U (Λ , s ) ψ − E (Λ , s ) ψ = H ψ + (cid:88) x ∈ Γ (cid:88) m ≥ R ˜Φ (1) ω ( x, m, s ) ψ, for all ψ ∈ π ( A loc )Ω where we have set(4.50) ˜Φ (1) ω ( x, m, s ) = ˜Φ (1) ( x, m, s ) − (cid:104) Ω , ˜Φ (1) ( x, m, s )Ω (cid:105) . The analogous relation in A holds, namely(4.51) Φ (1) ω ( x, m, s ) = Φ (1) ( x, m, s ) − ω (Φ (1) ( x, m, s ))1l . It will also be convenient for us to introduce the notation(4.52) ˜Φ (1) ω ( x, s ) = (cid:88) m ≥ R ˜Φ (1) ω ( x, m, s )which should be compared with (4.16). We begin with a lemma which makes use of Assumption 2.3. Lemma 4.5. Denote by ˜ P b x ( n ) = π ( P b x ( n ) ) ∈ B ( H ) the representative of the ground state projection P b x ( n ) in the GNS space. If (2.21) holds, then (4.53) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m (cid:88) k = R ˜Φ ω ( x, k, s ) ˜ P b x ( n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ s (cid:16) G (1)Λ ( x, m + 1) + G (1)Λ ( x, R ) (cid:112) (1 + m ) ν G ( n − m ) (cid:17) where G (1)Λ ( x, m ) = (cid:80) k ≥ m G Λ ( x, k ) for G Λ as in Theorem 4.2, and G is the decay function fromAssumption 2.3. Proof. We first begin by using (2.21) to establish the following inequality: for any A ∈ A b x ( m ) (4.54) (cid:12)(cid:12)(cid:12) (cid:107) ˜ A ˜ P b x ( n ) (cid:107) − (cid:107) ˜ AP Ω (cid:107) (cid:12)(cid:12)(cid:12) ≤ (cid:107) A (cid:107) (cid:112) (1 + m ) ν G ( n − m ) . where we have set P Ω = | Ω (cid:105)(cid:104) Ω | and ˜ A = π ( A ). From the simple inequality | a − b | ≤ | a − b | for a, b ≥ , it follows that (cid:12)(cid:12)(cid:12) (cid:107) ˜ A ˜ P b x ( n ) (cid:107) − (cid:107) ˜ AP Ω (cid:107) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) (cid:107) ˜ A ˜ P b x ( n ) (cid:107) − (cid:107) ˜ AP Ω (cid:107) (cid:12)(cid:12)(cid:12) . The second term on the right-hand-side above is (cid:107) ˜ AP Ω (cid:107) = (cid:104) Ω , π ( A ∗ A )Ω (cid:105) (cid:107) P Ω (cid:107) = ω ( A ∗ A ) (cid:107) P b x ( n ) (cid:107) . As π is norm-preserving, we also have that (cid:107) ˜ A ˜ P b x ( n ) (cid:107) = (cid:107) π ( AP b x ( n ) ) (cid:107) = (cid:107) P b x ( n ) A ∗ AP b x ( n ) (cid:107) . Given these observations, we conclude that for A ∈ A b x ( m ) (cid:12)(cid:12)(cid:12) (cid:107) ˜ A ˜ P b x ( n ) (cid:107) − (cid:107) ˜ AP Ω (cid:107) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12) (cid:107) P b x ( n ) A ∗ AP b x ( n ) (cid:107) − ω ( A ∗ A ) (cid:107) P b x ( n ) (cid:107) (cid:12)(cid:12) ≤ (cid:107) P b x ( n ) A ∗ AP b x ( n ) − ω ( A ∗ A ) P b x ( n ) (cid:107)≤ (cid:107) A (cid:107) (1 + m ) ν G ( n − m ) , (4.55)as a consequence of (2.21); this is (4.54).Now using (4.54) with ˜ A = (cid:80) mk = R ˜Φ (1) ω ( x, k, s ) ∈ π ( A b x ( m ) ), we find that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m (cid:88) k = R ˜Φ (1) ω ( x, k, s ) ˜ P b x ( n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m (cid:88) k = R ˜Φ (1) ω ( x, k, s ) P Ω (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m (cid:88) k = R ˜Φ (1) ω ( x, k, s ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) (1 + m ) ν G ( n − m ) . (4.56)Since (cid:107) ˜Φ (1) ω ( x, k, s ) (cid:107) ≤ (cid:107) ˜Φ (1) ( x, k, s ) (cid:107) ≤ sG ( x, k ), the final bound following from Theorem 4.2,the second term in (4.56) can be estimated as in (4.53).For the first norm in (4.56), we bound with(4.57) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m (cid:88) k = R ˜Φ (1) ω ( x, k, s ) P Ω (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) ˜Φ (1) ω ( x, s ) P Ω (cid:13)(cid:13)(cid:13) + ∞ (cid:88) k = m +1 (cid:13)(cid:13)(cid:13) ˜Φ (1) ω ( x, k, s ) P Ω (cid:13)(cid:13)(cid:13) . where we have used (4.52). Clearly,(4.58) ∞ (cid:88) k = m +1 (cid:13)(cid:13)(cid:13) ˜Φ (1) ω ( x, k, s ) P Ω (cid:13)(cid:13)(cid:13) ≤ ∞ (cid:88) k = m +1 (cid:13)(cid:13)(cid:13) ˜Φ (1) ( x, k, s ) (cid:13)(cid:13)(cid:13) ≤ sG (1)Λ ( x, m + 1) . We now claim that the remaining term is zero. As a result of Theorem 4.2, we have that[ ˜Φ (1) ω ( x, s ) , P Ω ] = [ ˜Φ (1) ( x, s ) , P Ω ] = 0 . As a result, considering the definition of ˜Φ (1) ω ( x, s ), see (4.52) and also (4.50),˜Φ (1) ω ( x, s ) P Ω = P Ω ˜Φ (1) ω ( x, s ) P Ω = (cid:68) Ω , ˜Φ (1) ω ( x, s )Ω (cid:69) P Ω = 0 , (4.59)where we use (4.52). This completes the proof of (4.53). (cid:3) We conclude this section with the proof of Theorem 4.1. TABILITY OF THE BULK GAP 27 Proof of Theorem 4.1. Fix x ∈ Γ. Since [ ˜Φ (1) ω ( x, s ) , P Ω ] = 0, we can write˜Φ (1) ω ( x, s ) = P Ω ˜Φ (1) ω ( x, s ) P Ω + (1l − P Ω ) ˜Φ (1) ω ( x, s )(1l − P Ω ) = (1l − P Ω ) ˜Φ (1) ω ( x, s )(1l − P Ω )where we have applied (4.59). The terms ˜Φ (2) ( x, m, s ) will be defined by decomposing 1l − P Ω interms of the finite volume ground state projections ˜ P n := ˜ P b x ( n ) ∈ π ( A b x ( n ) ).The finite volume ground state projections ˜ P n converge strongly to P Ω for all ψ ∈ H . As aconsequence, the collection of operators˜ E n = (cid:40) − ˜ P R , n = R ˜ P n − − ˜ P n , n ≥ R + 1(4.60)forms a family of orthogonal projections that are mutually orthogonal and sum to 1l − P Ω . That is,(4.61) ˜ E ∗ n = ˜ E n , ˜ E n ˜ E m = δ m,n ˜ E n , and (1l − P Ω ) ψ = (cid:88) n ≥ R ˜ E n ψ ∀ ψ ∈ H . In fact, the frustration-free property yields ˜ P n ˜ P m = ˜ P m for m ≥ n , and moreover,(4.62) 1l − ˜ P m = m (cid:88) n = R ˜ E n is clear. Using (4.52), the above properties imply that for all ψ, φ ∈ H , (cid:68) φ, (1l − P Ω ) ˜Φ (1) ω ( x, s )(1l − P Ω ) ψ (cid:69) = (cid:88) k,m,n ≥ R (cid:68) φ, ˜ E n ˜Φ (1) ω ( x, k, s ) ˜ E m ψ (cid:69) (4.63)We note that the triple sum of operators converges absolutely in norm, and so the operator equalityactually holds in the norm sense.Each term Φ (2) ( x, m, s ) will be defined as a sum of two self-adjoint terms, i.e.Φ (2) ( x, m, s ) = Θ ( x, m, s ) + Θ ( x, m, s ) , each of which is annihilated by the ground state projection P m . Fix k ≥ R . Using the propertiesin (4.61)-(4.62) we write(4.64) (cid:88) n,m ≥ R ˜ E n ˜Φ (1) ω ( x, k, s ) ˜ E m = (1l − ˜ P k ) ˜Φ (1) ω ( x, k, s )(1l − ˜ P k ) + (cid:88) m> k ˜Φ k,m , where ˜Φ k,m = ˜Φ ∗ k,m is defined by(4.65) ˜Φ k,m = ˜ E m ˜Φ (1) ω ( x, k, s )(1l − ˜ P m − ) + (1l − ˜ P m ) ˜Φ (1) ω ( x, k, s ) ˜ E m ∈ π ( A b x ( m ) ) . Self-adjointness follows from noting 1l − ˜ P m = 1l − ˜ P m − + ˜ E m .For each k ≥ R , we define Θ ( x, k, s )(4.66) Θ ( x, k, s ) = (1l − ˜ P k ) ˜Φ (1) ω ( x, k, s )(1l − ˜ P k ) ∈ π ( A b x (2 k ) ) , and define Θ ( x, m, s ) = 0 for all other values of m . Clearly, these operators satisfy Θ ( x, m, s ) ˜ P m =˜ P m Θ ( x, m, s ) = 0, and moreover, by Theorem 4.2 that (cid:107) Θ ( x, m, s ) (cid:107) ≤ sG Λ ( x, m/ 2) as(4.67) (cid:107) Θ ( x, k, s ) (cid:107) ≤ (cid:107) Φ (1) ω ( x, k, s ) (cid:107) ≤ sG Λ ( x, k ) . Summing the remaining terms ˜Φ k,m over k , we define the Θ terms by exchanging summationsas follows: (cid:88) k ≥ R (cid:88) m> k ˜Φ k,m = (cid:88) k ≥ R (cid:88) m> R ˜Φ k,m χ { m> k } = (cid:88) m> R (cid:88) R ≤ k 2) + 2 G (1)Λ ( x, m + 1) + 2 G (1)Λ ( x, R ) (cid:112) (1 + m ) ν G ( m/ . The absolute summability of the series in (4.70) is a direct consequence of G being summableas well as that both G Λ and G (1)Λ satisfy (4.5). For G (1)Λ this can easily be seen from the fact that G Λ is a combination of functions with finite moments of all orders, see specifically (4.19), (4.22)and (4.30). (cid:3) TABILITY OF THE BULK GAP 29 Proof of the form bound for the GNS Hamiltonian In this section, we generalize the statement and proof of Theorem 3.8 in [25] (which, in turn, isbased on a theorem in [21]), so that it is applicable to our setting for infinite systems.We work in the setting described in Section 2. In particular, under Assumption 2.2, thereis a family of sets S = { Λ( x, n ) | x ∈ Γ , n ≥ } , accompanied with a family T = {T n : n ≥ } of separating partitions of ( c, ζ )-polynomial growth, for which the associated finite volumeHamiltonians satisfy(5.1) H Λ( x,n ) ≥ γ ( n ) P Λ( x,n ) , for all n ≥ R . Recall also that the uniform bound on the local gaps, i.e. γ ( n ) above, has been further assumed tosatisfy γ ( n ) ≥ γ /n α for some α ≥ Theorem 5.1. Let V be an anchored interaction on (Γ , d ) with the form: (5.2) V = (cid:88) x ∈ Γ (cid:88) n ≥ R Φ( x, n ) with Φ( x, n ) ∗ = Φ( x, n ) ∈ A b x ( n ) for all x ∈ Γ and n ≥ R . Suppose that V is absolutely summable, i.e. (5.3) (cid:88) x ∈ Γ (cid:88) n ≥ R (cid:107) Φ( x, n ) (cid:107) < ∞ , that the terms of V annihilate the local ground states, i.e. (5.4) Φ( x, n ) P b x ( n ) = P b x ( n ) Φ( x, n ) = 0 for all x ∈ Γ and n ≥ R , and let G : [0 , ∞ ) → [0 , ∞ ) denote any uniform bound on these terms, i.e. (5.5) sup x ∈ Γ (cid:107) Φ( x, n ) (cid:107) ≤ G ( n ) . Then, for all ψ ∈ dom H , (5.6) |(cid:104) ψ, π ω ( V ) ψ (cid:105)| ≤ β (cid:104) ψ, H ψ (cid:105) where (5.7) β = c (cid:88) n ≥ R n ζ G ( n ) γ ( n ) . In keeping with the notation from the previous sections, and under Assumption 2.2, for eachΛ( x, n ) ∈ S , let us denote by ˜ H Λ( x,n ) , ˜ P Λ( x,n ) , and ˜ Q Λ( x,n ) the representations of H Λ( x,n ) , P Λ( x,n ) ,and Q Λ( x,n ) = 1l − P Λ( x,n ) as elements of B ( H ω ); similarly we write ˜Φ( x, n ) = π ω (Φ( x, n )). Theproof of Theorem 5.1 will follow closely the argument which proves Theorem 3.8 in [25], with theproviso that we must take care to ensure that the infinite sums replacing the finite ones in [25] arewell-defined.To this end, let us define some convenient families of orthogonal projections which act on H ω and are associated to the unperturbed Hamiltonian. Let n ≥ R , and consider the n -th separatingpartition T n = {T in : i ∈ I n } . For each i ∈ I n and any choice of x, y ∈ T in , it is clear that(5.8) [ H Λ( x,n ) , H Λ( y,n ) ] = 0 and [ P Λ( x,n ) , P Λ( y,n ) ] = 0as these are operators with disjoint support in the algebra. Moreover, the GNS representationpreserves this commutativity:(5.9) [ ˜ H Λ( x,n ) , ˜ H Λ( y,n ) ] = 0 and [ ˜ P Λ( x,n ) , ˜ P Λ( y,n ) ] = 0 . With n ≥ R and i ∈ I n fixed, let us denote by C in the collection of all configurations associated to T in , more precisely,(5.10) C in = { , } T in = (cid:8) σ : σ = { σ x } where σ x ∈ { , } for all x ∈ T in (cid:9) . For each σ ∈ C in (5.11) | σ | = (cid:88) x ∈T in σ x and note that(5.12) | σ | < ∞ if and only if |{ x ∈ T in : σ x = 1 }| < ∞ . Recall that D ω = { π ω ( A )Ω ω : A ∈ A loc } is a dense subspace of H ω . Let ψ ∈ D ω . Then ψ = ˜ A Ω ω for some ˜ A = π ω ( A ) with A ∈ A X . In this case, we have that(5.13) ˜ P Λ( x,n ) ψ = ψ and ˜ Q Λ( x,n ) ψ = 0whenever Λ( x, n ) satisfies Λ( x, n ) ∩ X = ∅ . It is also clear that for any X ∈ P (Γ), { x ∈ T in :Λ( x, n ) ∩ X (cid:54) = ∅} is finite. Thus, for any ψ ∈ D ω , we define(5.14) S ( σ ) ψ = (cid:40) (cid:81) x ∈T in (cid:104) σ x ˜ Q Λ( x,n ) + (1 − σ x ) ˜ P Λ( x,n ) (cid:105) ψ if | σ | < ∞ , . Note that, if | σ | < ∞ , then at most finitely many of these factors act non-trivially, and moreover,by (5.9), all factors above commute. Since D ω is dense, there is a unique extension of S ( σ ) to anelement of B ( H ω ) for each σ ∈ C in . One checks that these operators satisfy:(5.15) S ( σ ) ∗ = S ( σ ) , S ( σ ) S ( σ (cid:48) ) = δ σ,σ (cid:48) S ( σ ) , and (cid:88) σ ∈ C in S ( σ ) = 1l . Now since P Λ( x,n ) is the orthogonal projection onto the kernel of H Λ( x,n ) , one clearly has that(5.16) [ ˜ Q Λ( x,n ) , ˜ P Λ( y,n ) ] = 0 for all x, y ∈ T in . As a result, we find that(5.17) [ ˜ Q Λ( x,n ) , S ( σ )] = 0 for each x ∈ T in and all σ ∈ C in . In fact, one need only consider σ ∈ C in with | σ | < ∞ , as otherwise the result is trivial. For suchconfigurations, one readily checks that (5.16) implies 0 = [ ˜ Q Λ( x,n ) , S ( σ )] ψ for all ψ ∈ D ω , and thisrelation uniquely defines this bounded commutator. Combining (5.15) and (5.17), we find that(5.18) S ( σ ) ˜ Q Λ( x,n ) S ( σ (cid:48) ) = δ σ,σ (cid:48) ˜ Q Λ( x,n ) S ( σ ) = δ σ,σ (cid:48) (1l − ˜ P Λ( x,n ) ) S ( σ ) = δ σ,σ (cid:48) σ x S ( σ )for all σ, σ (cid:48) ∈ C in and each x ∈ T in .Given that these families of orthogonal projections exist, we can now present our proof. Proof of Theorem 5.1: Since V is absolutely summable, see (5.3), its terms can be rearranged. Moreprecisely, we can write(5.19) ˜ V = (cid:88) x ∈ Γ (cid:88) n ≥ R ˜Φ( x, n ) = (cid:88) n ≥ R (cid:88) i ∈I n ˜ V in as a bounded operator acting on the GNS space. Here we have used the family of separatingpartitions T = {T n : n ≥ R } , as in Assumption 2.2, and denoted by(5.20) ˜ V in = (cid:88) x ∈T in ˜Φ( x, n ) . As a result, for any ψ ∈ H ω , the following naive bound(5.21) |(cid:104) ψ, ˜ V ψ (cid:105)| ≤ (cid:88) n ≥ R (cid:88) i ∈I n |(cid:104) ψ, ˜ V in ψ (cid:105)| with |(cid:104) ψ, ˜ V in ψ (cid:105)| ≤ (cid:88) x ∈T in |(cid:104) ψ, ˜Φ( x, n ) ψ (cid:105)| certainly holds. TABILITY OF THE BULK GAP 31 Using now that we have assumed b x ( n ) ⊂ Λ( x, n ) for all Λ( x, n ) ∈ S , the frustration-free propertyof the ground-state projections implies(5.22) P Λ( x,n ) = P b x ( n ) P Λ( x,n ) = P Λ( x,n ) P b x ( n ) . By our assumptions on the terms Φ( x, n ), see (5.2) and (5.4), we find that(5.23) [ ˜Φ( x, n ) , ˜ P Λ( y,n ) ] = 0 for all x, y ∈ T in . Arguing as above, we find then that(5.24) [ ˜Φ( x, n ) , S ( σ )] = 0 for each x ∈ T in and all σ ∈ C in , and moreover, for all σ, σ (cid:48) ∈ C in and each x ∈ T in , we have that(5.25) S ( σ ) ˜Φ( x, n ) S ( σ (cid:48) ) = δ σ,σ (cid:48) S ( σ ) ˜Φ( x, n ) = δ σ,σ (cid:48) σ x S ( σ ) ˜Φ( x, n ) . In fact, using (5.18), the above may also be re-written as(5.26) S ( σ ) ˜Φ( x, n ) S ( σ (cid:48) ) = δ σ,σ (cid:48) S ( σ ) ˜ Q Λ( x,n ) ˜Φ( x, n ) ˜ Q Λ( x,n ) S ( σ ) . As a consequence, the bound |(cid:104) ψ, ˜Φ( x, n ) ψ (cid:105)| ≤ (cid:88) σ,σ (cid:48) |(cid:104) ψ, S ( σ ) ˜Φ( x, n ) S ( σ (cid:48) ) ψ (cid:105)| = (cid:88) σ |(cid:104) ψ, S ( σ ) ˜ Q Λ( x,n ) ˜Φ( x, n ) ˜ Q Λ( x,n ) S ( σ ) ψ (cid:105)|≤ G ( n ) (cid:88) σ (cid:107) ˜ Q Λ( x,n ) S ( σ ) ψ (cid:107) (5.27)follows from (5.15) and (5.26). Summing these orthogonal projections again, we find that (cid:88) σ (cid:107) ˜ Q Λ( x,n ) S ( σ ) ψ (cid:107) = (cid:88) σ (cid:104) ψ, ˜ Q Λ( x,n ) S ( σ ) ψ (cid:105) = (cid:104) ψ, ˜ Q Λ( x,n ) ψ (cid:105)≤ γ ( n ) (cid:104) ψ, ˜ H Λ( x,n ) ψ (cid:105) (5.28)where, for the final bound above we introduced the positive local gaps γ ( n ), used that γ ( n ) Q Λ( x,n ) ≤ H Λ( x,n ) holds in the algebra, and that the representation preserves positivity.Given the above, we conclude that for any ψ ∈ dom( H ),(5.29) |(cid:104) ψ, ˜ V in ψ (cid:105)| ≤ (cid:88) x ∈T in |(cid:104) ψ, ˜Φ( x, n ) ψ (cid:105)| ≤ G ( n ) γ ( n ) (cid:88) x ∈T in (cid:104) ψ, ˜ H Λ( x,n ) ψ (cid:105) ≤ G ( n ) γ ( n ) (cid:104) ψ, H ψ (cid:105) where we have used that Λ( x, n ) ∩ Λ( y, n ) = ∅ for x, y ∈ T in with x (cid:54) = y . Returning again to (5.21),it is now clear that(5.30) |(cid:104) ψ, ˜ V ψ (cid:105)| ≤ (cid:88) n ≥ R (cid:88) i ∈I n |(cid:104) ψ, ˜ V in ψ (cid:105)| ≤ β (cid:104) ψ, H ψ (cid:105) , with β as in (5.7), since these partitions satisfy the ( c, ζ )-polynomial growth bound. This completesthe proof. (cid:3) Proofs of Theorems 2.7 and 2.8 In Section 4 we showed in Theorem 4.1 that, for every finite Λ ⊂ Γ, the Hamiltonian H (Λ , s ) = H + sπ ( V Λ ) = H + s (cid:88) b x ( n ) ⊂ Λ π (Φ( x, n ))transforms under the spectral flow unitary as follows U (Λ , s ) ∗ H (Λ , s ) U (Λ , s ) ψ − E (Λ , s ) ψ = H ψ + (cid:88) x ∈ Γ (cid:88) m ≥ R π (Φ (2)Λ ( x, m, s )) ψ, where E (Λ , s ) is the ground state energy of H (Λ , s ) and Φ (2)Λ ( x, m, s ) is a rapidly decaying inter-action of order s . It follows from Theorem 4.2 that the new perturbation of H in the RHS is stillbounded and that it is mainly supported in a neighborhood of Λ. We now show that Theorem 5.1applies to Φ (2) ( x, m, s ) with a constant β that is independent of Λ, from which Theorem 2.7 willfollow. Proof of Theorem 2.7. Fix γ ∈ (0 , γ ) where γ is as in Assumption 2.1, and let Λ ∈ P be arbitrary.Given Theorem 4.1, it is clear that Theorem 5.1 applies and produces a non-trivial form bound(see (5.6)) so long as G (2)Λ ( m ) := sup x ∈ Γ G (2)Λ ( x, m )has a finite ( ζ + α )-moment where, as stated in Assumption 2.2, ζ is the polynomial growth of theseparating partitions and γ ( n ) ≥ γ n − α is the lower bound on the local gaps. Moreover, to showthat(6.1) s ( γ ) := inf Λ ∈P s Λ0 ( γ ) > , see (3.42), we wish to show this form bound is uniform in Λ.To this end, recall that G Λ from (4.19) is defined in terms of two functions G , G that areindependent of Λ and decay faster than any polynomial. It is then clear that for all Λ ∈ P , x ∈ Γand m ≥ R , G Λ ( x, m ) ≤ G ( m ) + G (0 , m ) =: G ( m ) , and similarly, G (1)Λ ( x, m ) ≤ G (1) ( m ) where G (1) ( m ) = (cid:80) m ≥ R G ( m ). It follows immediately thatsup Λ ∈P G (2)Λ ( m ) ≤ G (2) ( m ) := G ( m/ 2) + 2 G (1) ( m/ 2) + 2 G (1) ( R ) (cid:112) (1 + m ) ν G ( m/ G (2) has a finite ( ζ + α )-moment as long as G satisfies (2.20).Given the norm bound from Theorem 4.1, it follows form Theorem 5.1 that for all ψ ∈ dom( H ) (cid:104) ψ, W (Λ , s ) (cid:105) ≤ sβ (cid:104) ψ, H ψ (cid:105) where β = 2 c (cid:88) m ≥ R n ζ G (2) ( m ) γ ( m ) . Applying [25, Corollary 3.3] proves thatgap( H (Λ , s )) ≥ γ − sβγ . This shows that s Λ0 ( γ ) ≥ γ − γβγ , and establishes (6.1) as claimed. (cid:3) We conclude with using the uniform estimate from Theorem 2.7 to establish the claimed lowerbound estimate on the gap of the extensively perturbed system from Theorem 2.8. Proof of Theorem 2.8. Let 0 < γ < γ and consider | s | ≤ s ( γ ). Recall that for any IAS (Λ n ), thefollowing limits hold in A as n → ∞ : α Λ n s ( A ) → α s ( A ) , for all A ∈ A (6.2) δ Λ n s ( A ) → δ s ( A ) , for all A ∈ A loc , (6.3)see (2.36) and (2.37). As a consequence, ω Λ n s ( A ) = ω ◦ α Λ n s ( A ) → ω s ( A ) for all A ∈ A and,moreover, ω s is a ground state of δ s . TABILITY OF THE BULK GAP 33 Now consider A ∈ A loc for which ω s ( A ) = 0. Given Theorem 2.7, we know that along any IAS(Λ n ), the GNS Hamiltonian H (Λ n , s ) has a gap above its unique ground state lower bounded by γ for all | s | ≤ s ( γ ). Therefore, ω Λ n s satisfies (2.16) and, in particular, the inequality(6.4) ω Λ n s ( B ∗ n δ Λ n s ( B n )) ≥ γω Λ n s ( B ∗ n B n )holds for the observable B n = A − ω Λ n s ( A )1l ∈ A loc . Combining (6.3) and the locally uniformconvergence of α Λ n s , we can take the limit n → ∞ on both sides of (6.4) to obtain (2.38). Theremaining claims follow as in (2.16). 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Yosida, Functional analysis , 6th ed., Grundlehren der mathematishen Wissenschaften, Springer-Verlag, 1980. Department of Mathematics and Center for Quantum Mathematics and Physics, University ofCalifornia, Davis, Davis, CA 95616, USA Email address : [email protected] Department of Mathematics, University of Arizona, Tuscon, AZ 85721, USA Email address : [email protected] Munich Center for Quantum Science and Technology, and, Zentrum Mathematik, TU M¨unchen,85747 Garching, Germany Email address ::