A Z 2 -Topological Index for Free-Fermion Systems in Disordered Media
aa r X i v : . [ m a t h - ph ] O c t Topological Index for Free–Fermion Systems inDisordered Media
N. J. B. Aza A. F. Reyes-Lega L. A. Sequera M.October 7, 2020
Abstract
We use infinite dimensional self–dual
CAR C ∗ –algebras to study the existence of a Z –index,which classifies free–fermion systems embedded on Z d disordered lattices. Combes–Thomas es-timates are pivotal to show that the Z –index is uniform with respect to the size of the system. Weadditionally deal with the set of ground states to completely describe the mathematical structureof the underlying system. Furthermore, the weak ∗ –topology of the set of linear functionals is usedto analyze paths connecting different sets of ground states. Keywords:
Operator Algebras, Disordered fermion systems, Z –index, ground states. AMS Subject Classification:
Contents
CAR
Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Quasi–Free Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Gapped Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
A Disordered models on general graphs 29
A considerable number of mathematical results concerning gapped Hamiltonians of fermions has beenachieved in recent years. Among the most important ones are topological protection under small per-turbations and the persistence of the spectral gap for interacting fermions [Has19, DS19]. We studya Z –projection index ( Z –PI) that generalizes the one introduced long ago by Araki–Evans in theirwork where the two possible thermodynamic phases of the classical two-dimensional Ising model1re characterized using operator algebras technologies [AE83]. Here we deal with disordered free–fermion systems on the lattice within the mathematical framework of self–dual CAR C ∗ –algebras .In particular, their structure contains the information on the symmetries of free fermions embedded indisordered systems, and is also useful to study interacting fermion systems, even with superconduct-ing terms [ABPM20]. To be precise, the Z –PI is defined in terms of well–defined basis projectionsrelated to a self–adjoint operator, which typically is the Hamiltonian of the system acting on a sep-arable Hilbert space H . See Definition 2 below. Thus, the Z –PI is introduced to discriminatenonequivalent representations [EK98, BVF01].A very important problem in this context is the classification of topological matter in general. Thecurrent classification scheme can be traced back to Dyson’s [Dys62] classical work from 1962. Ofcourse that work did not contemplate topological aspects for such systems, but it provided the settingon which more recent work has been based. Indeed, a completion of this early work was made byAltland and Zirnbauer [AZ97], leading to the identification of new symmetry classes. These ideaswere generalized by Kitaev [Kit09] and led to a “periodic table” of topological insulators and su-perconductors. In that work, Kitaev showed how the classification can be achieved in terms of Bottperiodicity and K –theory . More recently an exhaustive and complete version of the classification wasmade by Ryu et al. [RSFL10]. They explore arbitrary dimensions making use again of classifyingspaces given by the Cartan symmetric spaces along with Bott periodicity in a more strong way. Thisallows them to consider disordered systems and shows the explicit relation between gapped Hamilto-nians and Anderson localization phenomena, a very important result for this kind of problem.The first iconic example of a topological fermionic system is the quantum Hall effect . The observedquantization of the conductivity was explained by Thouless et al. [TKNdN82] and led to the recogni-tion of the important role played by the
Chern number . The restrictions on the validity of this resultwhere eventually overcome by Bellissard [BvES94] and collaborators, in what was to become one ofthe main examples of applications of noncommutative geometry to physics. This was a big step to dealwith more realistic models that consider disordered media. In this line of ideas there are more recentworks, due to Carey et al [CHM +
06, PS16, BCR16], where Bellissard’s techniques are generalizedto deal with a wider class of systems.For interacting systems rigorous proofs of quantization of conductivity were provided in [GMP16,BDF18]. These studies rely on the study of families of gapped Hamiltonians , such that any two ele-ments on these families can be continuously deformed into one another. The latter was demonstratedrigorously by Bachmann, Michalakis, Nachtergaele and Sims [BMNS12] by studying spectral flowof quantum spin systems under a “quasi–adiabatic” evolution. They proved that such related systemsverify the same
Lieb–Robinson bounds and in its thermodynamic limit the spectral flow has a cocyclestructure for the automorphism in the algebra of observables. By using the dual space of the underly-ing algebras considered they also studied the ground states associated.From the point of view of physics, fundamental properties of such systems are deduced from thestudy of the set of ground states in the thermodynamic limit and zero temperature. Relevant examplesinclude electronic conduction problems (e.g. quantum Hall effect), or the study of different phases ofmatter. Nevertheless, knowledge of ground states for concrete models is a huge challenge in general.This is due to the fact that there is no general procedure to find the full set of ground states for specificsystems. As far as we know, there are very few mathematical physics results about the existence ofground states, in contrast to the theoretical point of view, see [AT85, CNN18]. Instead, one generallyverifies the existence of the ground state energy for specific physical systems .In this paper we focus on the study of Z –PI for non–interacting fermion systems. We specificallydeal with unique ground states associated to families of gapped Hamiltonians. Note that there is an Ground state energy can be understood as the states associated to the lowest energy of a physical system. For example,Giuliani and Jauslin use rigorous renormalization methods to prove the existence of the ground state energy for the bilayergraphene [GJ16]. Z –PI (26) in terms of orthogonal complex structures [BVF01, EK98]. There,the index appears naturally in the proof of the Shale–Stinespring Theorem and is related to the parity of the ground states. In [CGRL18], this approach to the Z –index was used to study ground statesfor finite Kitaev chains with different boundary conditions. More recently, for infinite translationallyinvariant fermionic chains, Bourne and Schulz-Baldes classify ground states using orthogonal com-plex structures [BSB20]. Furthermore, Matsui [Mat20] uses split–property of infinite chains and itsconnection with the Z –PI. Observe that Theorem 1 below generalizes the mentioned results in thesense that we do not require translational invariant conditions neither one–dimensional systems only.Finally, by Lemma 3 one notes that the Z –PI is uniform with respect to the size of the systems. Ina subsequent paper [AR20], we will report on results about a Z –index for interacting fermions. Ob-serve that the technical tools in that case differ from the current study and other technologies such asLieb–Robinson bounds will be required. For example, we use similar techniques as in [Oga20, BO20]for one–dimensional interacting quantum spin systems, to study a Z –index in the mentioned setting.To conclude, our main results are Theorems 1 and 2, as well as the set of Lemmata 1, 2 and 3. Fromthe mathematical point of view, Theorem 1 is reminiscent of the interacting case, however, we addi-tionally state the Z –PI result, classifying the parity of the Bogoliubov ∗ –automorphism of the infiniteself–dual CAR C ∗ –algebras. On the other hand, Theorem 2 deals with subsets of the ground states set. For instance, open spectral gap ground states are considered. As a particular case of the generalTheorem 2, we prove that in the weak ∗ –topology, paths connecting states in different components ofthe Z –PI implies the existence of a Hamiltonian having as an eigenvalue.The paper is organized as follows:• Section 2 presents the mathematical framework of CAR C ∗ –algebras. We introduce self–dual CAR C ∗ –algebras, which were introduced long ago by Araki in his elegant study of non –interacting but non –gauge invariant fermion systems. We recall pivotal properties of general CAR C ∗ –algebras.• In Section 3 we state the main Theorems, as well as some relevant definitions concerning the Z –PI and comment on the weak ∗ –topology of the set of states. In particular we discuss theconditions for a system to have pure or mixed states.• Section 4 is devoted to all technical proofs. We prove the existence of a spectral flow auto-morphism for self–dual Hilbert spaces, for families of differentiable Hamiltonians. Then, theexistence of strong limits for the dynamics, the spectral flow automorphism and the weak ∗ –convergence of ground states are proven. Well–known Combes–Thomas estimates are invokedfor families of gapped Hamiltonians, which will permit to analyze two–point correlation func-tions such that we obtain the trace class properties for relevant unitary operators.• We finally include Appendix A, providing a general framework of graphs with special attentionto disordered models. Notation 1.
A norm on the generic vector space X is denoted by k · k X and the identity map of X by X . Thespace of all bounded linear operators on ( X , k · k X ) is denoted by B ( X ) . The unit element of anyalgebra X is always denoted by , provided it exists of course. The scalar product of any Hilbert space X is denoted by h· , ·i X and Tr X represents the usual trace on B ( X ) . ␅ We introduce the mathematical framework based on Araki’s self–dual formalism [Ara68, Ara71]. Oursetting considers disorder effects, which come as is usual in physics, i.e., from impurities, crystal lat-tice defects, etc. Thus, disorder can modeled by (a) a random external potential, like in the celebrated3nderson model, (b) a random Laplacian, i.e., a self–adjoint operator defined by a next–nearest neigh-bor hopping term with random complex–valued amplitudes. In particular, random vector potentialscan also be implemented.
CAR
Algebra
If not otherwise stated, H always stands for a (complex, separable) Hilbert space. If H is finite–dimensional, we will assume it is even–dimensional, i.e., dim H ∈ N . Let Γ : H → H be a conjugation or antiunitary involution on H , i.e., an antilinear operator such that Γ = H and h Γ ϕ , Γ ϕ i H = h ϕ , ϕ i H , ϕ , ϕ ∈ H . The space H endowed with the involution Γ is named a self–dual Hilbert space and yields self–dual CAR algebra : Definition 1 (Self–dual
CAR algebra).
A self–dual
CAR algebra sCAR( H , Γ) ≡ (sCAR( H , Γ) , + , · , ∗ ) is a C ∗ –algebra generated by aunit and a family { B( ϕ ) } ϕ ∈ H of elements satisfying Conditions 1.–3.:1. The map ϕ B ( ϕ ) ∗ is (complex) linear.2. B( ϕ ) ∗ = B(Γ( ϕ )) for any ϕ ∈ H .3. The family { B( ϕ ) } ϕ ∈ H satisfies the CAR : For any ϕ , ϕ ∈ H ,(1) ␈ B( ϕ )B( ϕ ) ∗ + B( ϕ ) ∗ B( ϕ ) = h ϕ , ϕ i H . For a historic overview on self–dual
CAR algebras and some of their basic properties see [Ara68,Ara71, Ara87, Ara88, EK98]. Note that by the
CAR (1), the antilinear map ϕ B ( ϕ ) is necessarilyinjective and contractive. Therefore, H can be embedded in sCAR( H , Γ) .Conditions 1.–3. of Definition 1 only define self–dual CAR algebras up to Bogoliubov ∗ –automorphisms (see (5)). In [ABPM20], an explicit construction of ∗ –isomorphic self–dual CAR algebras from H and Γ is presented. This is done via basis projections [Ara68, Definition 3.5], which highlight therelationship between CAR algebras and their self–dual counterparts.
Definition 2 (Basis projections).
A basis projection associated with ( H , Γ) is an orthogonal projection P ∈ B ( H ) satisfying Γ P Γ = P ⊥ ≡ H − P . We denote by h P the range ran( P ) of the basis projection P . The set of all basisprojections associated with ( H , Γ) will be denoted by p ( H , Γ) . ␈ For simplicity, in the rest of this section, we will assume that H is finite–dimensional with even size: dim H ∈ N . For any P ∈ p ( H , Γ) a few remarks are in order: h P must satisfy the conditions(2) Γ( h P ) = h ⊥ P and Γ( h ⊥ P ) = h P . Then, by [Ara68, Lemma 3.3], an explicit P ∈ p ( H , Γ) can always be constructed. Moreover, ϕ (Γ ϕ ) ∗ is a unitary map from h ⊥ P to the dual space h ∗ P . In this case we can identify H with(3) H ≡ h P ⊕ h ∗ P We will assume that the inner product h· , ·i H : H × H → C associated to some Hilbert space H is a sesquilinearform on H such that is antilinear in its first component while is linear in the second one. An analogous result for
CAR algebra is, for instance, given by [BR03b, Theorem 5.2.5].
B ( ϕ ) ≡ B P ( ϕ ) . = B ( P ϕ ) + B (cid:16) Γ P ⊥ ϕ (cid:17) ∗ . Therefore, there is a natural isomorphism of C ∗ –algebras from sCAR( H , Γ) to the CAR algebra
CAR( h P ) generated by the unit and { B P ( ϕ ) } ϕ ∈ h P . In other words, a basis projection P can beused to fix so–called annihilation and creation operators. For each basis projection P associated with ( H , Γ) , by (3), h P can be seen as a one–particle Hilbert space.As shown in [Ara68, Ara71], self–dual CAR algebras naturally arise in the diagonalization of quadraticfermionic Hamiltonians (Definition 3), via Bogoliubov transformations defined as follows:For any unitary operator U ∈ B ( H ) such that U Γ = Γ U , the family of elements B( U ϕ ) ϕ ∈ H sat-isfies Conditions (a)–(c) of Definition 1 and, together with the unit , generates sCAR( H , Γ) . Likein [Ara71, Section 2], such a unitary operator U ∈ B ( H ) commuting with the antiunitary map Γ isnamed a Bogoliubov transformation , and the unique ∗ –automorphism χ U such that(5) χ U (B( ϕ )) = B( U ϕ ) , ϕ ∈ H , is called in this case a Bogoliubov ∗ –automorphism . Note that a Bogoliubov transformation U ∈ B ( H ) always satisfies(6) det ( U ) = det (Γ U Γ) = det ( U ) = ± If det ( U ) = 1 , we say that U is in the positive connected set U + . Otherwise U is said to be in the negative connected set U − . χ U (B( ϕ )) is said to be even (respectively odd ) if and only if U ∈ U + (respectively U ∈ U − ).Clearly, if P ∈ p ( H , Γ) , see Definition 2, and U ∈ B ( H ) is a Bogoliubov transformation, then P U . = U ∗ P U is another basis projection. Conversely, for any pair P , P ∈ p ( H , Γ) there is a(generally not unique) Bogoliubov transformation U such that P = U ∗ P U . See [Ara68, Lemma3.6]. In particular, Bogoliubov transformations map one–particle Hilbert spaces onto one another.Considering the Bogoliubov ∗ –automorphism (5) with U = − H , an element A ∈ sCAR( H , Γ) ,satisfying(7) χ − H ( A ) = A is called even , − A is called odd , Note that the subspace sCAR( H , Γ) + of even elements is a sub– C ∗ –algebra of sCAR( H , Γ) .It is well–known that in quantum mechanics the even elements are the ones suitable for the descrip-tion of fermion systems. For example, self–adjoint (even) elements of the CAR algebra which arequadratic in the the creation and annihilation operators are used, for instance, in the Bogoliubov ap-proximation of the celebrated (reduced) BCS model. In the context of self–dual
CAR algebra, thoseelements are called bilinear Hamiltonians and are self–adjoint bilinear elements:
Definition 3 (Bilinear elements of self–dual CAR algebra).
Given an orthonormal basis { ψ i } i ∈ I of H , we define the bilinear element associated with H ∈ B ( H ) to be h B , H B i . = X i,j ∈ I D ψ i , Hψ j E H B (cid:16) ψ j (cid:17) B ( ψ i ) ∗ . ␈ Note that h B , H B i does not depend on the particular choice of the orthonormal basis, but does dependon the choice of generators { B( ϕ ) } ϕ ∈ H of the self–dual CAR algebra sCAR( H , Γ) , and by (1),bilinear elements of sCAR( H , Γ) have adjoints equal to(8) h B , H B i ∗ = h B , H ∗ B i , H ∈ B ( H ) . ilinear Hamiltonians are then defined as bilinear elements associated with self–adjoint operators H = H ∗ ∈ B ( H ) . They include all second quantizations of one–particle Hamiltonians, but alsomodels that are not gauge invariant . Important models in condensed matter physics, like in the BCStheory of superconductivity, are bilinear Hamiltonians that are not gauge invariant.Without loss of generality (w.l.o.g.), our analysis of bilinear elements can be restricted to operators H ∈ B ( H ) satisfying H ∗ = − Γ H Γ , which, in particular, have zero trace, i.e., Tr H ( H ) = 0 . Wecall such operators self–dual operators : Definition 4 (Self–dual operators).
A self–dual operator on ( H , Γ) is an operator H ∈ B ( H ) satisfying the equality H ∗ = − Γ H Γ . If,additionally, H is self–adjoint, then we say that it is a self–dual Hamiltonian on ( H , Γ) . ␈ We say that the basis projection P (Definition 2) (block–) “diagonalizes” the self–dual operator H ∈ B ( H ) whenever(9) H = 12 (cid:16) P H P P − P ⊥ Γ H ∗ P Γ P ⊥ (cid:17) , with H P . = 2 P HP ∈ B ( h P ) . In this situation, we also say that the basis projection P diagonalizes h B , H B i , similarly to [Ara68,Definition 5.1].By the spectral theorem, for any self-dual Hamiltonian H on ( H , Γ) , there is always a basis pro-jection P diagonalizing H . In quantum physics, as discussed in Section 2.1, h P is in this case the one–particle Hilbert space and H P the one–particle Hamiltonian . Bilinear Hamiltonians are used to define so-called quasi–free dynamics: For any H = H ∗ ∈ B ( H ) ,we define the continuous group { τ t } t ∈ R of ∗ –automorphisms of sCAR( H , Γ) by(10) τ t ( A ) . = e − i t h B , H B i A e i t h B , H B i , A ∈ sCAR( H , Γ) , t ∈ R . Provided H is a self–dual Hamiltonian on ( H , Γ) (Definition 4), this group is a quasi–free dynamics,that is, a strongly continuous group of Bogoliubov ∗ –automorphisms, as defined in Equation (5).Straightforward computations using Definitions 1 and 3, together with the properties of the antiunitaryinvolution Γ , lead to show that(11) exp (cid:18) − z h B , H B i (cid:19) B ( ϕ ) ∗ exp (cid:18) z h B , H B i (cid:19) = B (cid:16) e zH ϕ (cid:17) ∗ , even for any self–dual operator H on ( H , Γ) , all z ∈ C and ϕ ∈ H .Moreover, for { τ t } t ∈ R , we define the linear subspace(12) D ( δ ) . = { A ∈ sCAR( H , Γ) : t τ t ( A ) is differentiable at t = 0 } ⊂ sCAR( H , Γ) and the linear operator (unique, generally unbounded) δ : D → sCAR( H , Γ) by(13) δ ( A ) . = d τ t ( A )d t (cid:12)(cid:12)(cid:12)(cid:12) t =0 . The operator δ is called the generator of τ and D ( δ ) is the (dense) domain of definition of δ . Herewe will assume that δ is a symmetric unbounded derivation, i.e., the domain D ( δ ) of δ is a dense ∗ –subalgebra of A and, for all A, B ∈ D ( δ ) , δ ( A ) ∗ = δ ( A ∗ ) , δ ( AB ) = δ ( A ) B + Aδ ( B ) . Note that the set of all symmetric derivations on D ( δ ) can be endowed with a real vector spacestructure. In fact, for any symmetric derivations δ and δ and all real numbers α , α , the expression ( α δ + α δ ) ( A ) . = α δ ( A ) + α δ ( A ) , A ∈ D ( δ ) , gives rise to another symmetric derivation α δ + α δ on D ( δ ) .6 .3 States A linear functional ω ∈ sCAR( H , Γ) ∗ is a “state” if it is positive and normalized, i.e., if for all A ∈ sCAR( H , Γ) , ω ( A ∗ A ) ≥ and ω ( ) = 1 . In the sequel, E ⊂ sCAR( H , Γ) ∗ will denote the set of allstates on sCAR( H , Γ) . Note that any ω ∈ E is Hermitian , i.e., for all A ∈ sCAR( H , Γ) , ω ( A ∗ ) = ω ( A ) . ω ∈ E is said to be “faithful” if A = 0 whenever A ≥ and ω ( A ) = 0 . Since sCAR( H , Γ) is a unital C ∗ –algebra, E is a weak ∗ –compact convex set, such that its extremal points coincide withthe pure states [BR03a, Theorem 2.3.15]. The latter, combined with the fact that sCAR( H , Γ) is separable allows to claim that the set of states E is metrizable in the weak ∗ –topology [Rud91,Theorem 3.16]. Note that the existence of extremal points is a consequence of the Krein–MilmanTheorem . More specifically, if E ( E ) denotes the set of extremal points of E , E = cch ( E ( E )) , where, for X a Topological Vector Space and A ⊂ X , cch( A ) refers to the closed convex hull of A . Such extremal points E ( E ) or pure states cannot be written as a linear combination of any states.As an application, we notice that extremal states can be used to write any “mixed state” ω ∈ E . By amixed state ω ∈ E we mean that there are states { ω j } mj =1 ∈ E ( E ) , m ∈ N , and positive real numbers, ≤ λ j ≤ for j ∈ { , . . . , m } , with m P j =1 λ j = 1 satisfying(14) ω = m X j =1 λ j ω j . In particular, if the state ω ∈ E is pure, ω = m P j =1 λ j ω j implies that ω = ω = · · · = ω m , and λ = · · · = λ j = m .As is usual, for the state ω ∈ E on sCAR( H , Γ) , ( H ω , π ω , Ω ω ) denotes its associated cyclic repre-sentation: H ω is the Hilbert space associated to ω , and is given by the closure of (the linear span) ofthe set { π ω ( A )Ω ω : A ∈ sCAR( H , Γ) } , H ω = π ω (sCAR( H , Γ)) Ω ω , i.e., H ω is a Hilbert space with scalar product h· , ·i H ω , π ω a representation from sCAR( H , Γ) into B ( H ω ) , the set of bounded operators acting on H ω , and Ω ω ∈ H ω is a unit cyclic vector with respectto π ω (sCAR( H , Γ)) . More specifically, for all A ∈ sCAR( H , Γ) we write ω ( A ) = h Ω ω , π ω ( A )Ω ω i H ω . ( H ω , π ω , Ω ω ) is the so–called GNS construction , which is unique up to unitary equivalence.If the state ω ∈ E is mixed, see Expression (14), its associated representation ( H ω , π ω ) is reducible,that is, it can be decomposed as a direct sum π ω = L j ∈ J π ω j on H ω = L j ∈ J H ω j . Here, { H j } j ∈ J is acountable family of orthogonal Hilbert spaces, by meaning that for two different Hilbert spaces H and H of { H j } j ∈ J , h ϕ , ϕ i H = 0 for all ϕ ∈ H and all ϕ ∈ H . The set { π ω j } j ∈ J are repre-sentations of sCAR( H , Γ) on proper subspaces of H ω . In particular if ω is pure, its representation ( H ω , π ω ) is irreducible and ω is an extremal point E ( E ) of the set of states on sCAR( H , Γ) .States ω ∈ E are said to be quasi–free when, for all N ∈ N and ϕ , . . . , ϕ N ∈ H ,(15) ω (B ( ϕ ) · · · B ( ϕ N )) = 0 , For the Topological Vector Space X , X denotes its closure. N ∈ N and ϕ , . . . , ϕ N ∈ H ,(16) ω (B ( ϕ ) · · · B ( ϕ N )) = Pf [ ω ( O k,l (B( ϕ k ) , B( ϕ l )))] Nk,l =1 , where O k,l ( A , A ) . = A A for k < l, − A A for k > l, for k = l. In Equation (16), Pf is the usual Pfaffian defined by(17) Pf [ M k,l ] Nk,l =1 . = 12 N N ! X π ∈ S N ( − π N Y j =1 M π (2 j − ,π (2 j ) for any N × N skew–symmetric matrix M ∈ Mat (2 N, C ) . Note that (16) is equivalent to thedefinition given either in [Ara71, Definition 3.1] or in [EK98, Equation (6.6.9)].Quasi–free states are therefore particular states that are uniquely defined by two–point correlationfunctions, via (15)–(16). In fact, a quasi–free state ω is uniquely defined by its so–called symbol , thatis, a positive operator S ω ∈ B ( H ) such that(18) ≤ S ω ≤ H and S ω + Γ S ω Γ = H , through the conditions(19) h ϕ , S ω ϕ i H = ω (B( ϕ )B(Γ ϕ )) , ϕ , ϕ ∈ H . Conversely, any self–adjoint operator satisfying (18) uniquely defines a quasi–free state through Equa-tion (19). In physics, S ω is called the one–particle density matrix of the system. Note that any basisprojection associated with ( H , Γ) can be seen as a symbol of a quasi–free state on sCAR( H , Γ) .Such state is pure and called a Fock state [Ara71, Lemma 4.3]. Araki shows in [Ara71, Lemmata4.5–4.6] that any quasi–free state can be seen as the restriction of a quasi–free state on sCAR( H ⊕ H , Γ ⊕ ( − Γ)) , the symbol of which is a basis projection associated with ( H ⊕ H , Γ ⊕ ( − Γ)) . Thisprocedure is called purification of the quasi–free state.Quasi–free states obviously depend on the choice of generators of the self–dual CAR algebra. An-other example of a quasi–free state is provided by the tracial state:
Definition 5 (Tracial state).
The tracial state tr ∈ E is the quasi–free state with symbol S tr . = H . ␈ The tracial state can be used to highlight the relationship between quasi–free states and bilinear Hamil-tonians. In fact, one can show, c.f. [ABPM20], that for any β ∈ (0 , ∞ ) and any self–dual Hamiltonian H on ( H , Γ) the positive operator (1 + e − βH ) − satisfies Condition (18) and is the symbol of a quasi–free state ω ( β ) H satisfying(20) ω ( β ) H ( A ) = tr (cid:16) A exp (cid:16) β h B , H B i (cid:17)(cid:17) tr (cid:16) exp (cid:16) β h B , H B i (cid:17)(cid:17) , A ∈ sCAR( H , Γ) . The state ω ( β ) H ∈ E is named the ( τ t , β ) – Gibbs state , thermal equilibrium state, or
KMS –state, asso-ciated with the self–dual (one–particle) Hamiltonian H on ( H , Γ) at fixed β ∈ (0 , ∞ ) . As is usual,we call to the parameter β ∈ (0 , ∞ ) the inverse (non–negative) temperature of a physical system.Note that, given H ∈ B ( H ) , we also can define two particular quasi–free states ω (0) H and ω ( ∞ ) H ,8hich satisfy (20) for the convergent sequence { β n } n ∈ N ⊂ R ∪ {∞} to a β ⊂ R ∪ {∞} . Theformer case is closely related with the tracial state in Definition 5, and corresponds to the infinitetemperature. Namely, the state at β = lim n →∞ β n = 0 is known as trace state or chaotic state. Thisparticular name comes from the fact that physically it corresponds to the state of maximal entropywhich occurs at infinite temperature. Its uniqueness is a well–known property. On the other hand,states at β = lim n →∞ β n = ∞ are also thermal equilibrium states. More generally, these are defined by: Definition 6 (Ground state).
Let ω ∈ E be a state on sCAR( H , Γ) and let H ∈ B ( H ) be a self–dual Hamiltonian on ( H , Γ) .We say that ω ≡ ω ( ∞ ) H is a ground state if it satisfies i ω ( A ∗ δ ( A )) ≥ , for all A ∈ D ( δ ) . Here δ is the generator with domain D ( δ ) , of the continuous group { τ t } t ∈ R of ∗ –automorphisms of sCAR( H , Γ) given by (10). ␈ From now on, we will denote by E ( β ) ∈ E the set of all KMS states at inverse temperature β ∈ R +0 ∪ {∞} associated to the self–dual Hamiltonian H on ( H , Γ) . A few of remarks regarding E ( β ) are discussed:To lighten the notation, in the sequel when we refer to the KMS state ω ( β ) H we will omit any mentionof the dependence on H , i.e., ω ( β ) H ≡ ω ( β ) . For β ∈ R + ∪ {∞} , ω ( β ) ∈ E ( β ) is τ invariant or stationary,i.e., ω ( β ) ◦ τ = ω ( β ) . See [BR03b, Propositions 5.3.3 and 5.3.19]. In contrast, the tracial case β = 0 not necessarily is. Then, for β ∈ R + ∪ {∞} , ω ≡ ω ( β ) , there is a strongly continuous one–parameterunitary group (cid:16) e i t L ω (cid:17) t ∈ R with generator L ω = L ∗ ω satisfying e i t L ω Ω ω = Ω ω such that for any t ∈ R π ω ( τ t ( A )) = e − i t L ω π ω ( A ) e i t L ω and e i t L ω ∈ π ω (sCAR( H , Γ)) ′′ . If any A ∈ D ( δ ) ⊆ sCAR( H , Γ) , π ω ( A )Ω ω ∈ D ( L ) and L ( π ω ( A )Ω ω ) = π ω ( δ ( A ))Ω ω . If ω is a ground state, then the generator satisfies L ω ≥ .For β ∈ R + , the set E ( β ) ∈ E forms a weak ∗ –compact convex set that also is a simplex , while the setof ground states or KMS states at inverse temperature ∞ , E ( ∞ ) ⊂ E , forms a face F , i.e., a subset ofa compact convex set K such that if there are finite linear combinations ω = n X j =1 λ i ω i with n X j =1 λ j = 1 of elements { ω j } nj =1 ∈ K and ω ∈ F , then { ω j } nj =1 ∈ F .Let A ∈ B ( H ) be a bounded self–dual operator on ( H , Γ) , such that E Σ ( A ) . = χ Σ ( A ) defines the spectral projection of A on the Borel set Σ ⊂ R . Here, χ Σ : Σ → { , } is the so–called characteristicfunction on Σ ⊂ R , with χ = χ Σ . For H , a self–adjoint Hamiltonian on ( H , Γ) , i.e., H = − Γ H Γ ,we denote by E , E − and E + , the restrictions of the spectral projections of H on { } , the negativereal numbers R − and the positive real numbers R + , respectively. Using functional calculus we notethat H = Z spec( H ) λ d E λ = Z ∞−∞ λ d E λ , This is true because one can show that the set of KMS E ( β ) ∈ E forms a base of the cone which is also a lattice [BR03a, Chapter 4]. spec( H ) denotes the spectrum of H . Thus, one verifies that(21) Γ E λ Γ = E − λ for all λ ∈ R and E + E − + E + = H . In particular, we have Γ E Γ = E . However, we strongly will assume throughout this paper that E = 0 so that the ground state is unique . For details see [AT85][Theorems 3 and 4]. By (21), both E + and E − are basis projections in p ( H , Γ) : Γ E ± Γ = H − E ± , i.e., ground states can be uniquelycharacterized by their spectral projections E ± . In particular, the symbol S ω in (19) can correspondsto the spectral projection E + on the positive real numbers, associated to the self–dual Hamiltonian H on ( H, Γ) in such a way that ground states are uniquely determined by the two–point correlationfunction defined by:(22) ω (B( ϕ )B(Γ ϕ )) = h ϕ , E + ϕ i H , ϕ , ϕ ∈ H . Thus, for a quasi–free system associated to some self–dual Hamiltonian H , the set of all groundstates E ( ∞ ) H ≡ E ( ∞ ) , is studied via (positive) spectral projections of H . Additionally, straightforwardcalculations show the uniqueness of ground states, even under small perturbations. See [BR03b,Chapter 5] and [Has19] for recent results on the stability of free fermion systems. More generally, fora unital C ∗ –algebra the quasi–free state for β ∈ (0 , ∞ ] is unique. We now define: Definition 7 (Quasi–free ground states).
The state ω ∈ sCAR( H , Γ) ∗ satisfying (18), (19) and (22) it will be called quasi–free ground state .The set of all quasi–free ground states it will denoted by qE ( ∞ ) ⊂ E ( ∞ ) . ␈ We consider the (possibly unbounded) self–adjoint operator h = h ∗ ∈ L ( H ) (the linear operatorson H ), for some separable Hilbert space H , whose spectrum is denoted by spec( h ) ⊂ R ∪ {∞} .Physically, we say that the system described by H has a gap if whenever we measure the spectrum ofthe associated Hamiltonian there exists a strictly positive distance γ ∈ R + between the two lowesteigenvalues E , E ∈ R such that E − E > γ , with E . = inf spec( h ) . The parameter γ , also called spectral gap , is known to be the difference between the lowest energy of the system and the energyof its first excited state. In Definition 8 below, we formally express this. On the other hand, in thecontext of fermion systems, Definition 9 is suitable for our interests. Then, introducing the notation d ( X, Y ) to denote the distance between the sets X, Y ⊂ R : d ( X, Y ) . = inf { d ( x, y ) : x ∈ X, y ∈ Y } , with d ( x, y ) . = | x − y | for x, y ∈ R , we define: Definition 8 (Gapped Hamiltonians).
Let H be a (one–particle) Hilbert space and consider h ∈ L ( H ) the (one–particle) Hamiltonian, thatis, a self–adjoint operator h = h ∗ , whose spectrum is denoted by spec( h ) ⊂ R . We will say that h is a gapped Hamiltonian if there are Σ and e Σ , nonempty and disjoint subsets of spec( h ) , such that Σ ∪ e Σ = spec( h ) and exists γ . = inf d (Σ , e Σ) > . ␈ Remark 1.
In the latter definition Σ can be though of as the Borel set in R that contains the isolatedeigenvalue E , which carries the information of the lowest energy associated to the physical system toconsider. Note that if spec( h ) is a purely point spectrum (the set of all the eigenvalues associated to h )we can define the family of elements of e Σ with indices on N \ { } as the map E : N \ { } → e Σ , such that E . = { E n } n ∈ N \{ } , the rest of eigenvalues of H , given E , belong to e Σ . ␏ H ∈ B ( H ) , where one considers a self–dual Hilbert space ( H , Γ) , with H a finite–dimensional Hilbertspace with orthonormal basis given by { ψ i } i ∈ I . Hence, for any H ∈ B ( H ) satisfying H ∗ = − Γ H Γ we have:(i) Tr H ( H ) = 0 .(ii) spec( λ H − H ) = λ − spec( H ) for λ ∈ C .Both (i) and (ii) are fundamental to study the underlying systems we are considering. On the otherhand, the physical terms we are dealing with are expressed by dΓ( h ) + dΥ( g ) = −h B , [ κ ( h ) + ˜ κ ( g )] B i + 12 Tr h ( h ) , that is, a bilinear element of a self–dual Hamiltonian (see again Definition 4) plus a constant term.This is the typical case of a free–fermion system with quasi–free dynamics provided by some bilinearHamiltonian H ∈ sCAR( H , Γ) . Instead of considering H , observe equivalently that(23) − h B , [ F + G ] B i + Tr h P ( P F P ) , gives us the description of the systems, where F and G are self–dual Hamiltonians on H , and P ∈ p ( H , Γ) is a basis projection with range ran( P ) = h P . As already mentioned F P . = 2 P HP is theso–called one–particle Hamiltonian, then, w.l.o.g. we can remove the term Tr h P ( P F P ) , by writting(23) as(24) − D B , h e F + G i B E , for e F . = F − | I | Tr h P ( P F P ) κ ( h P ) , with | I | the cardinality of the Hilbert space H , and the map κ being defined by κ ( h ) . = 12 ( P h hP h − Γ P h h ∗ P h Γ) , h ∈ B ( h ) . See also (9). Since
H . = e F + G is a self–dual Hamiltonian, we use h . = 2 P h HP h and g . = 2 P h H Γ P h , inorder to describe any quadratic Fermionic Hamiltonian. In fact, given P ∈ p ( H , Γ) with ran ( P ) = h and the self–dual Hamiltonian H ∈ B ( H ) , the bounded operators on h h . = 2 P HP and g . = 2
P HP Γ , provide all the possible free–fermion models. Further, we can add to Expression (23) or (24) aself–adjoint element W ∈ sCAR( H , Γ) which could carry interparticle interaction terms, but forsimplicity we will omit this in the sequel. The latter will be considered in a subsequent paper [AR20].Finally, based on Definition 8, items (i) and (ii), and above comments we can define the following: Definition 9 (Fermionic Gapped Hamiltonians).
Let ( H , Γ) be a self–dual Hilbert space and consider H ∈ B ( H ) be a self–dual Hamiltonian withspectrum denoted by spec( H ) ⊂ R . We will say that H is a gapped Hamiltonian if exists g ∈ R + satisfying the gap assumption g . = inf { ǫ > − ǫ, ǫ ] ∩ spec( H ) = ∅} . ␈ Following [NSY18b], we could consider unbounded one–site potentials. Thus, we would need to define
Hamiltonians on well–defined dense sets on Hilbert spaces. However, for the sake of simplicity, we will omit any mention on denselydefined self–adjoint operators. Note that in [BP16], Bru and Pedra consider unbounded one–site potentials on C ∗ –algebras. In [AR20] we will deal with unbounded one–site potentials. Σ ∈ R is a finite interval with a . = inf { Σ } and b . = sup { Σ } , e Σ is nothing but − Σ , so that − a . = sup { e Σ } and − b . = inf { e Σ } . Then, the self–dual formalism permits to consider a symmetric decomposition ofthe spectrum. Therefore Σ can be understood as a Borel set on R + related to the positive part of theenergy while e Σ ≡ − Σ its symmetric negative part: the gap g centered at zero separates these. Wefinally stress following Definition 9 that denoting by Σ and − Σ the remaining two open sets, theirclosures respectively are Σ and − Σ .Due to the above reasons, from now on we will only consider fermion systems. Thus, let usnow consider the family of self–dual Hamiltonians { H s } s ∈ C ∈ B ( H ) on ( H , Γ) , where C is thecompact set [0 , . In particular, { H s } s ∈ C will define a differentiable family of self–adjoint operatorson B ( H ) . More specifically, for any s ∈ C we will consider that the map s H s is stronglydifferentiable so that ∂ s H s ∈ B ( H ) . Thus, among the models we are taking into account, Andersonmodel is a particular case, as discussed in Appendix A. See [BPH14] and [ABPR19]. FollowingDefinition 9 we now define: Definition 10 (Phase of the Matter).
Let C ≡ [0 , and { H s } s ∈ C ∈ B ( H ) be a family of self–dual Hamiltonians on ( H , Γ) . We will saythat H s is a s –gapped Hamiltonian if the gap assumption in Definition 9 is satisfied for any s ∈ C . { H s } s ∈ C describes the same phase of the matter if there is g ∈ R + , independent of s , such that forany s ∈ C there is a uniform lower bond, i.e., inf s ∈ C g s ≥ g > . In this situation we will say that { H s } s ∈ C is in the g –phase . ␈ Observe that a difference between ground states associated to family of Hamiltonians { H s } s ∈ C in the g –phase and the general definition of ground states (Definition 6) is necessary. In fact, one can provethat if the family of Hamiltonians is gapped, then its associated ground states { ω s } s ∈ C satisfy:(25) i ω s ( A ∗ δ ( A )) ≥ g s ( ω s ( A ∗ A ) − | ω s ( A ) | ) , for any s ∈ C and A ∈ D ( δ ) , with g s ∈ R + , s ∈ C , and inf s ∈ C g s ≥ g > . For details see [Mat13]. In the sequel we will say thatstates satisfying the above inequality are gapped ground states . We study gapped Hamiltonians satisfying the following Assumption:
Assumption 1.
Take C ≡ [0 , . (a) H g . = { H s } s ∈ C ∈ B ( H ) is a differentiable family of self–dual Hamiltonians onthe g –phase such that ∂H g . = { ∂ s H s } s ∈ C ∈ B ( H ) . (b) For the infinite volume we assume that thesequences of self–dual Hamiltonians H s, L : C → B ( H ∞ ) and ∂ s H s, L : C → B ( H ∞ ) are strong andpointwise convergent, that is, lim L →∞ H s,L = H s, ∞ and lim L →∞ ∂ s H s,L = ∂ s H s, ∞ in the strong sense. ␅ Now, for any self–dual Hilbert space ( H , Γ) , take P ∈ p ( H , Γ) and P ∈ p ( H , Γ) basis projec-tions, the “ Z –projection index” ( Z –PI) σ : p ( H , Γ) × p ( H , Γ) → Z is the map defined by:(26) σ ( P , P ) . = ( − dim( P ∧ P ⊥ ) . Here, ∧ symbolizes the lower bound or intersection of the basis projections P and P in B ( H ) .Note that the Z –PI defines a topological group with two components. Then, we analyze the classof Hamiltonians described by last assumption and their connection with topological indexes. Weformally state one of the main results of the paper:12 heorem 1 ( Z –projection Index): Take C ≡ [0 , and let H g ∞ . = { H s, ∞ } s ∈ C ∈ B ( H ∞ ) be a differentiable family of self–dual Hamil-tonians on ( H ∞ , Γ ∞ ) in the g ∞ –phase, with ∂ H g ∞ . = { ∂ s H s, ∞ } s ∈ C ∈ B ( H ∞ ) , see Definition 10and Assumption 1 (b). For any s ∈ C , E + ,s, ∞ denotes the spectral projection associated to thepositive part of spec( H s, ∞ ) and consider the Z –PI given by (26) . Then:(1) For any s ∈ C , H , ∞ is unitarily equivalent to H s, ∞ via the unitary operator V s ∈ B ( H ∞ ) satisfying the differential equation (29) below.(2) For s ∈ C , the Z –PI σ ( H , ∞ , H s, ∞ ) ≡ σ ( E + , , ∞ , E + ,s, ∞ ) satisfies σ ( H , ∞ , H s, ∞ ) = , if V ( ∞ ) s ∈ U + − , if V ( ∞ ) s ∈ U − . Thus, the Bogoliubov ∗ –automorphism χ V ∗ s is inner and maintains its parity, even or odd , overthe family H g ∞ . ␄ Proof. (1) For any A ∈ B ( H ∞ ) and all s ∈ C one define the spectral flow automorphism κ s : B ( H ∞ ) → B ( H ∞ ) by κ s ( A ) . = (cid:16) V ( ∞ ) s (cid:17) ∗ AV ( ∞ ) s , where V s ∈ B ( H ∞ ) is the unitary operator satisfying V = H ∞ , and the differential equation (29).See Lemmata 1–2 and Corollary 4. In particular, since any Hamiltonian H s, ∞ in H g ∞ can be writtenas H s, ∞ = Z ∞−∞ λ d E λ,s, ∞ , with Γ E λ,s, ∞ Γ = E − λ,s, ∞ for all λ ∈ R and E − ,s, ∞ + E + ,s, ∞ = H ∞ , by Lemmata 1–2, (1) follows.(2) Concerning the Z –PI σ ( P , P ) we only need to invoke [EK98, Theo. 6.30 and Lemma 7.17]: (a) σ ( P , P ) = σ ( P , P ) , (b) If P − P is a Hilbert–Schmidt class operator, then σ ( P , P ) is continuousin P and P with respect to the norm topology in p ( H , Γ) (c) If U ∈ B ( H ) is a unitary operatorsuch that U Γ = Γ U and H − U is a trace class operator, then σ ( P, U P U ∗ ) = det U . Then we proceedto verify these statements for the family of positive spectral projections { E + ,s, ∞ } s ∈ C ∈ B ( H ∞ ) .By (21)–(22) and comments around it, any positive spectral projection in { E + ,s, ∞ } s ∈ C is a basisprojection and thus { E + ,s, ∞ } s ∈ C ⊂ p ( H ∞ , Γ ∞ ) . W.l.o.g. take E + , , ∞ and E + ,s, ∞ with s ∈ C . Wesplit the proof in three steps:(i) Part (a) follows from E + , , ∞ ∧ E ⊥ + ,s, ∞ = Γ ∞ (cid:16) E ⊥ + , , ∞ ∧ E + ,s, ∞ (cid:17) Γ ∞ . (ii) For L ∈ R +0 ∪ {∞} , we need to verify that E + , ,L − E + ,s,L ∈ B ( H L ) is Hilbert–Schmidt class.Here, ( H L , Γ L ) is the Hilbert space given by the canonical orthonormal basis { e x } x ∈ X L definedby (40) below. Since E + ,s,L and E + , ,L are self–adjoint operators on B ( H L ) by Lemma 1,there is a unitary bounded operator V ( L ) s ∈ B ( H L ) such that (cid:16) V ( L ) s (cid:17) ∗ E + ,s,L V ( L ) s = E + , ,L ,with V ( L )0 = H L . For L ∈ R +0 we have the following estimate tr H L (cid:16) ( E + , ,L − E + ,s,L ) (cid:17) = 2 X x ∈ X L (cid:26) h e x , E + , , ∞ e x i H L − D E + , ,L V ( L ) s e x , V ( L ) s E + , ,L e x E H L (cid:27) . Recall that for any separable Hilbert space H , A ∈ B ( H ) and any orthonormal basis { ψ } i ∈ I of H the trace of A , tr H ( A ) . = P i ∈ I h ψ i , Aψ i i H , does not depend of the choice of the orthonormal basis. h E + , ,L , the range ran( E + , ,L ) of E + , ,L (see Definition 2), and let { e ′ x } x ∈ X L be anorthonormal basis of h E + , ,L . Since the invariance on the choice of the orthonormal basis of thetrace we obtain for L → ∞ : lim sup L →∞ sup s ∈ C tr H L (cid:16) ( E + , ,L − E + ,s,L ) (cid:17) = lim inf L →∞ sup s ∈ C tr H L (cid:16) ( E + , ,L − E + ,s,L ) (cid:17) = 0 . It follows that E + , , ∞ − E + ,s, ∞ is Hilbert–Schmidt class.(iii) By Corollary 2, for any s ∈ C and L ∈ R +0 ∪ {∞} the unitary operator V ( L ) s ∈ B ( H L ) commutes with Γ L and by Lemma 3, for L ∈ R +0 , H L − V ( L ) s is a trace class operator, wherein particular H ∞ − V ( ∞ ) s is a trace class operator per unit volume. Then σ ( E + , , ∞ , E + ,s, ∞ ) = det (cid:16) V ( L ) s (cid:17) , L ∈ R +0 ∪ {∞} . By Lemma 3 the Bogoliubov ∗ –automorphism Υ ( L ) s on A L given Expression (36) below is inner.Additionally, if V ( L ) s ∈ B ( H L ) has some parity, we say V ( L ) s ∈ U ± (see (6)), then by [EK98,Theorem 6.15], Υ ( L ) s is even or odd and this parity holds for the family of Hamiltonians H g L .This completes the proof of the Theorem. End
Hitherto in this paper we have been interested in physical systems with open gap, which isthe case of systems of last Theorem. In fact, Theorem 1 claims that two self–dual Hamiltonians, H , ∞ , H , ∞ ∈ H g ∞ acting on H ∞ , can be connected by a path described by the spectral flowautomorphism κ s : B ( H ∞ ) → B ( H ∞ ) . As is usual, a path is nothing but a continuous map κ : [0 , → B ( H ∞ ) connecting the initial point κ = H , ∞ and the terminal point κ = H , ∞ .Equivalently, we say that H , ∞ and H , ∞ are the extremal points of the path. Observe that for any s, r ∈ C with H s, ∞ , H r, ∞ ∈ H g ∞ we can write H s, ∞ = κ s,r ( H r, ∞ ) , with κ s,r . = κ − s ◦ κ r , and then there is a path such that H s, ∞ and H r, ∞ are its extremal points, and then the family H g ∞ is arcwise connected in the strong operator topology of H ∞ , and as a consequence it is also a connectedset at the same topology.We are also interested in the possibility of having two self–dual Hamiltonians, H , ∞ and H , ∞ , actingon H ∞ but belonging to different phases of matter, in the sense of Definition 10. In this case, if H , ∞ ∈ H g ∞ while H , ∞ / ∈ H g ∞ , Theorem 2 (see Corollary 1 too) below shows that the path e κ connecting both Hamiltonians closes the gap, by meaning that there is a Hamiltonian f H ∈ B ( H ∞ ) on e κ such that is an eigenvalue of f H . Concerning the latter, observe that one can study the gapclosing in terms of the self–dual CAR , C ∗ –algebra A ∞ . = sCAR( H ∞ , Γ ∞ ) , in such a way that weassociate to H , ∞ and H , ∞ the bilinear elements h B , H , ∞ B i and h B , H , ∞ B i on A ∞ (see Expression(44) below). Instead, we can equivalently use the set of states E ( ∞ ) ∈ A ∗∞ . Since in the current workwe are dealing with the set of quasi–free ground states qE ( ∞ ) ⊂ E ( ∞ ) of Definition 7, we will analyzethe gap closing using qE ( ∞ ) .First of all, because of the properties of E ( ∞ ) provided in Section 2.3 and Theorem 4, qE ( ∞ ) is ametrizable weak ∗ –compact set. In the scope of gapped systems, for gapped quasi–free ground states Ω g ∞ ≡ { ω s } s ∈ C in the g ∞ –phase, following Theorems 1 and 5, Ω g ∞ is arcwise connected, and henceit is a connected set in the weak ∗ –topology. Here, the Bogoliubov ∗ –automorphisms Υ s in Theorem5 plays the role of implementing the path, namely, following Corollary 3 we are able to write ω s = ω ◦ Υ s , for any ω s ∈ Ω g ∞ and s ∈ C , r, s ∈ C that(27) ω r = ω s ◦ Υ s,r , with ω r , ω s ∈ Ω g ∞ , with Υ s,r . = Υ − s ◦ Υ r , which satisfies a cocycle ∗ –automorphism condition.We now invoke the following result on metric spaces:P ROPOSITION LF ˆ ANDEGA ’ S T HEOREM ).Let M ≡ ( M , d M ) be a non–empty metric space. Consider C , X ⊂ M , where C is a connected sethaving commonpointswith X and M \ X . Then, C hasapointon ∂ X , theboundaryof X . ␊ Proof.
We proceed following [Lim77, Prop. 9–Chap. 4]. We claim that x ∈ M constructed as followssatisfies the assumptions of the Theorem: On the one hand, note that exists x ∈ ∂ ( C ∩ X ) ⊂ C . Onthe other hand, for any ǫ > there are y ∈ C ∩ X ⊂ X with d M ( x, y ) < ǫ and z ∈ C \ X ⊂ M \ X with d M ( x, z ) < ǫ . End
In the context of metric spaces,
Alfˆandega’s Theorem is known as a generalization of the
Intermediatevalue theorem [Lim77]. Observe that, by definition of boundary of X , any open ball with radius r ∈ R + and center in p ∈ ∂ X , B ( p, r ) , has at least one point on X and one point on M \ X . To fixideas we desire to apply Alfˆandega’s Theorem for subsets of the metrizable weak ∗ –topology set space E associated to the self–dual CAR C ∗ –algebra A ( ∞ ) .More precisely, consider the quasi–free ground states qE ( ∞ ) ⊂ E ( ∞ ) , as well as the family of gappedquasi–free ground states Ω g ∞ ⊂ qE ( ∞ ) above defined. For s ∈ C , take ω s ∈ Ω g ∞ ⊂ qE ( ∞ ) and ω ∈ qE ( ∞ ) \ Ω g ∞ , and suppose that there is a path γ connecting ω s and ω . Note that by Expressions(18), (19) and (22), there are positive spectral (basis) projections E + ,ω s , ∞ , E + ,ω , ∞ ∈ p ( H ∞ , Γ ∞ ) on ( H ∞ , Γ ∞ ) . We state the second main result of the current paper: Theorem 2:
Let H g ∞ be the family of self–dual Hamiltonians of Theorem 1 associated to the family of gappedquasi–free ground states Ω g ∞ . Let ω ∈ qE ( ∞ ) \ Ω g ∞ be a quasi–free ground state with associatedself–dual Hamiltonian H , ∞ ∈ B ( H ∞ ) constructed from the positive spectral projection E + ,ω , ∞ .For some s ∈ C fix ω s ∈ Ω g ∞ , and suppose that there is a path γ : [0 , → qE ( ∞ ) such that γ (0) = ω s and γ (1) = ω are the extremal points of γ . Then, there is a self–dual Hamiltonian f H ∞ ∈ B ( H ∞ ) with associated ground state e ω on γ so that ∈ spec( f H ∞ ) . ␄ Proof.
Let { A n } n ∈ N be a countable set of operators on A ( ∞ ) so that k A n k A ∞ ≤ for all n ∈ N . It isa well–known fact that the metric d E ( ∞ ) ( ω , ω ) . = X n ∈ N n | ω ( A n ) − ω ( A n ) | , ω , ω ∈ E ( ∞ ) , induces the weak ∗ –topology on the set of states E ( ∞ ) . In particular, the open ball with center on ω ∈ E ( ∞ ) and radius ε ∈ R + is defined by B ( ω, ε ) . = n ω ′ ∈ E ( ∞ ) : d E ( ∞ ) ( ω, ω ′ ) < ε o ⊂ E ( ∞ ) . By the hypothesis of the Theorem, we can use Alfˆandega’s Theorem, Proposition 1, such that weknow that there is a ground state e ω on γ so that e ω ∈ ∂Ω g ∞ . Thus, for an open ball with centerin e ω ∈ ∂Ω g ∞ and radius ε ∈ R + note that there are ω in ∈ Ω g ∞ and ω out ∈ qE ( ∞ ) \ Ω g ∞ so that d E ( ∞ ) ( e ω, ω in ) < ε and d E ( ∞ ) ( e ω, ω out ) < ε . By defining ω ′ . = ( ω in + ω out ) , we can use the triangleinequality in order to obtain: d E ( ∞ ) ( e ω, ω ′ ) < ε . Because ε is arbitrary it follows that e ω is a mixedground state, that is, a convex combination of pure ground states. Finally, by [EK98, Propos. 6.37]the self–dual Hamiltonian f H ∞ ∈ B ( H ∞ ) associated to the ground state e ω has as an eigenvalue,and the proof concludes. End
15s a straightforward consequence we have the following Corollary:C
OROLLARY H g ∞ , , H g ∞ , be two families of self–dual Hamiltonians satisfying Theorem 1, for g ∞ , , g ∞ , ∈ R + as in Definition 10. Consider the Z –PI given by (26) such that σ = σ , with σ i the Z –PI associated to the family H g ∞ ,i following Theorem 1, for i ∈ { , } . For some s, r ∈ C fix H s, , H r, ∈ B ( H ∞ ) , and suppose that there is a path ˜ κ : [0 , → B ( H ∞ ) such that ˜ κ (0) = H s, and ˜ κ (1) = H r, are theextremal pointsof ˜ κ . Then, there is aself–dual Hamiltonian f H ∞ ∈ B ( H ∞ ) on ˜ κ so that ∈ spec( f H ∞ ) . ␊ Proof.
Combine Theorems 1 and 2.
End L EMMA C ≡ [0 , and let H g as in Assumption 1. For any s ∈ C , E + ,s will denote the spectralprojection associated to the positive part of spec( H s ) . Then, for the family of spectral projections { E + ,s } s ∈ C , thereexistsafamilyofautomorphisms { κ s } s ∈ C on B ( H ) satisfying κ s ( E + ,s ) = E + , . ␄ Proof.
The arguments of the proof are completely standard and we state these for the sake of com-pleteness, c.f. [Kat13, BMNS12, NSY18b]. Take C ≡ [0 , and consider { H s } s ∈ C ∈ B ( H ) bea differentiable family of self–dual Hamiltonians on the g –phase. Fix s ∈ C and let E + ,s be thespectral projection of H s on Σ s . Note that if the automorphism κ s : B ( H ) → B ( H ) satisfying κ s ( E + ,s ) = E + , exists, this implies that it is unitarily implemented by a differentiable unitary oper-ator V s ∈ B ( H ) defined by(28) κ s ( E + ,s ) . = V ∗ s E + ,s V s , with V = H , and satisfying the differential equation(29) ∂ s V s = − i D g ,s V s , where for the gap g , D g ,s : C → B ( H ) is a pointwise self–adjoint bounded operator. Here, ∂ s denotes the derivative with respect to s ∈ C . Now, for any H s , we write its spectral projection on Σ s by E + ,s = 12 π i I Γ s R ζ ( H s )d ζ, (30)where, for any s ∈ C , R ζ ( H s ) ∈ B ( H ) is the resolvent set of H s . In (30), for any s ∈ C , Γ s is a chain , that is, Γ s is a finite collection of closed rectificable curves γ s in C . In particular, Γ s surrouds Σ s and is in the complement of e Σ s . By using the second resolvent equation, i.e., R ζ ( A ) − R ζ ( B ) = R ζ ( A )( B − A ) R ζ ( B ) , for any operators A, B ∈ B ( H ) and any ζ ∈ spec( A ) ∩ spec( B ) , one can show that(31) ∂ s E + ,s = − π i I Γ s R ζ ( H s ) ( ∂ s H s ) R ζ ( H s )d ζ, ∂ s E + ,s is well–defined on C . A combination of (28)–(29) and κ s ( E + ,s ) = E + , yield us to(32) ∂ s E + ,s = − i[ D g ,s , E + ,s ] . Additionally, since for any s ∈ C , E + ,s is an orthogonal projection then E ⊥ + ,s ( ∂ s E + ,s ) E ⊥ + ,s = E + ,s ( ∂ s E + ,s ) E + ,s = 0 , where for any s ∈ C , E ⊥ + ,s denotes the orthogonal complement of E + ,s , i.e., E ⊥ + ,s . = 1 − E + ,s . Fromthe latter identity we get the following one ∂ s E + ,s = E + ,s ( ∂ s E + ,s ) E ⊥ + ,s + E ⊥ + ,s ( ∂ s E + ,s ) E + ,s , and together with (31) and the fact that E + ,s , E − ,s are basis projections, see (21), we arrive at(33) ∂ s E + ,s = − π i ℜ e (cid:18)I Γ s ( E + ,s R ζ ( H s ) ( ∂ s H s ) R ζ ( H s ) E − ,s ) d ζ (cid:19) . Here, the self–adjoint operator ℜ e ( A ) ∈ B ( H ) is the real part of A ∈ B ( H ) , given by ℜ e ( A ) . = ( A + A ∗ ) . Similarly, ℑ m ( A ) ∈ B ( H ) , the imaginary part of A , is the self–adjoint operator usuallydefined by ℑ m ( A ) . = ( A − A ∗ ) .Then, the existence of the automorphism κ s is equivalent to finding the operator D g ,s such that (32)and (33) are satisfied. This is precisely that is done in [BMNS12], and in the present context weexplicitly write ∂ s E + ,s as ∂ s E + ,s = Z Σ s Z Σ s µ + λ ℜ e (d E µ, + ,s ( ∂ s H s ) d E − λ, + ,s ) , where for any s ∈ C , E µ, + ,s is a resolution of the identity supported on the positive (negative) part of spec( H s ) , i.e.,(34) E ± ,s . = Z ± Σ s d E λ, ± ,s . The next step is to verify that the self–adjoint bounded operator(35) D g ,s . = Z R e i tH s ( ∂ s H s ) e − i tH s W g ( t )d t, for any s ∈ C , satisfies (32) and (33). Here, W g ( t ) : R → R is an odd function on L ( R ) such that its Fouriertransform, c W g : R → R is given for µ = 0 by c W g ( µ ) ≡ − √ πµ . For a complete description of the properties of W g see [BMNS12, MZ13, NSY18b]. We now notethat for any operator B ∈ B ( H ) and any orthogonal projection P ∈ B ( H ) we get − i[ B, P ] = i (
P B (1 − P ) − (1 − P ) BP ) . In particular, note that by taking B as D g ,s and P as the spectral projection of H s on Σ s , i.e., E + ,s ,we have − i[ D g ,s , E + ,s ] = i Z R Z Σ s Z − Σ s e i t ( µ − λ ) d E µ, + ,s ( ∂ s H s ) d E λ, − ,s W g ( t )d λ d µ d t − i Z R Z − Σ s Z Σ s e i t ( λ − µ ) d E λ, − ,s ( ∂ s H s ) d E µ, + ,s W g ( t )d µ d λ d t = i √ π Z Σ s Z − Σ s d E µ, + ,s ( ∂ s H s ) d E λ, − ,s c W g ( λ − µ )d λ d µ − i √ π Z − Σ s Z Σ s d E λ, − ,s ( ∂ s H s ) d E µ, + ,s c W g ( µ − λ )d µ d λ = Z Σ s Z Σ s µ + λ ℜ e (d E µ, + ,s ( ∂ s H s ) d E − λ, + ,s ) c W g is an odd function. End
In particular, the unitary operator V s satisfying the differential equation (29) commutes with the invo-lution Γ , i.e., Γ V s = V s Γ . In fact, for any s ∈ C , let C s ∈ B ( H ) be defined by C s . = [Γ , V s ] , such that C ∗ s = [ V ∗ s , Γ] . We would like to show that C s = 0 . To do this, observe that the self–adjoint boundedoperator D g ,s given by Expression (35), commutes with Γ . Using (29), after some calculations wehave ∂ s C s = i D g ,s C s and ∂ s C ∗ s = i C ∗ s D g ,s . From the left hand side equation one has ∂ s C ∗ s = − i C ∗ s D g ,s , which comparing with the right handside equation, we obtain C s = 0 . We have proven:C OROLLARY
OGOLIUBOV TRANSFORMATION ).For any s ∈ C ≡ [0 , , the unitary operator V s satisfying the differential equation (29) commuteswiththeinvolution Γ ,i.e., Γ V s = V s Γ ,then V s is aBogoliubovtransformation,see (5). ␊ One primary consequence of Lemma 1 and Corollary 2 is the existence of a strongly continuous fam-ily of one–parameter (Bogoliubov) group Υ s . = { Υ s } s ∈ C ∈ R of ∗ –automorphisms of sCAR( H , Γ) ,implemented by the Bogoliubov automorphisms V s . To be precise, for the one–parameter unitarygroup { V s } s ∈ C implementing the family of automorphism { κ s } s ∈ C of Lemma 1 over the family ofspectral projections { E + ,s } s ∈ C we are able to show that, for any s , the (Bogoliubov) ∗ –automorphism(36) Υ s (B ( ϕ )) ≡ χ V ∗ s (B ( ϕ )) = B ( V ∗ s ϕ ) , exists, for any ϕ ∈ H . The latter can be easily verified using bilinear elements, which are describedin Definition 3. More generally, for any family of self–dual Hamiltonians { H s } s ∈ C ∈ B ( H ) as inAssumption 1, we have an associated family of bilinear elements {h B , H s B i} s ∈ C ∈ sCAR( H , Γ) given by, Υ s ( h B , H s B i ) = h B , H B i , for any s ∈ C , (see Definition 3).As stressed in comments around Expression (26), for any pair P , P ∈ p ( H , Γ) there existsa Bogoliubov transformation U relating both, i.e., a unitary operator U ∈ B ( H ) so that P = U ∗ P U , with U Γ = Γ U . Thus, if det( U ) = 1 ( det( U ) = − ) we say that U is in the positive(negative) connected component. Following [EK98, Theo. 6.30 and Lemma 7.17] for the specialclass of U satisfying (i) U Γ = Γ U and (ii) − U trace class, the topological index σ ( P, U ∗ P U ) coincides with det( U ) . Note that Lemma 1 tells us about the existence of a family of unitary operators { V s } s ∈ C which implements the family of automorphisms { κ s } s ∈ C on B ( H ) . However, we need tospecify with which kind of Hamiltonians we are dealing. A wide class of fermion systems are thosesatisfying Lemma 2 and Proposition 3 below. More concretely, our results will permit to considerdisordered fermions systems in which the spectral gap does not close. Note that a suitable control ofthe properties of { V s } s ∈ C is closely related to the recently results found by Hastings in [Has19]. Then,as already mentioned we invoke [EK98, Theo. 6.30 and Lemma 7.17] in order to distinguish differentphysical systems on the same g –phase for some positive g ∈ R + (see Definition 10) and these areclassified by two components even in the interacting setting, see [NSY18a]. Additionally, Theorem 1holds for a wide family of (possibly unbounded) random Hamiltonians H g on the g –phase.From now on, we will expose some issues about quasi–free ground states for g ∈ R + and g = 0 .The former are called gapped quasi–free ground states (see Expression (25) and comments around it)and per se any information about the number of these is unknown. However, as already mentioned,mentioned, in the quasi-free setting their uniqueness is guaranteed. Since the set E of ground states ismetrizable in the weak ∗ –topology, we denote by E g ≡ ( E g , d g ) and E ≡ ( E , d ) the metric spacesin the weak ∗ –topology related to the quasi–free ground states for g ∈ R + and g = 0 respectively. In18articular, one notes that E g and E are not homeomorphic since, as we will see in Corollary 3, therepresentations associated to E g are reducible whereas those associated to E are not. This is clearfrom the fact that there is no homeomorphism between one connected metric space and another onethat is disconnected . Then the representations associated to E g and E are not physically equivalentas the intuition says. Instead, Corollary 3 below claims that any two gapped quasi–free ground statesassociated to the quasi–free dynamics of two gapped Hamiltonians on the same g –phase are unitarilyequivalent. Thus, their irreducible representations also are.In order to prove last statement, recall Expressions (18), (19) and (22), where for any s ∈ C ≡ [0 , one knows that for the positive (on Σ s ) spectral projection E + ,s ∈ B ( H ) associated to the self–dualHamiltonian H s on ( H , Γ) there is a unique quasi–free ground state ω s ∈ E g such that ω s (B ( ϕ ) B (Γ ϕ )) = h ϕ , E + ,s ϕ i H , ϕ , ϕ ∈ H . One more time, H g . = { H s } s ∈ C ∈ B ( H ) is a family of self–dual Hamiltonians in the g –phasesatisfying Assumption 1, and in this case, for any s ∈ C , ω s is also a gapped quasi–free ground state.By using the family of automorphisms { κ s } s ∈ C on B ( H ) of Lemma 1, with V s ∈ B ( H ) the unitaryoperator implementing κ s , we note that(37) ω s = ω ◦ Υ s , s ∈ C . Here, Υ s is the one–parameter (Bogoliubov) ∗ –automorphism of sCAR( H , Γ) given by Expression(36). Additionally, let ω g . = { ω s } s ∈ C be a family of gapped quasi–free ground states associated toself–dual Hamiltonians H g on some self–dual Hilbert space ( H , Γ) in the g –phase, with the sameassumptions of Lemma 1. The meaning of expression (37) in terms of representations is that theassociated (irreducible) GNS representation ( H ω g , π ω g , Ω ω g ) is unique (up to unitary equivalence):for all A ∈ sCAR( H , Γ) ω g ( A ) = D Ω ω g , π ω g ( A )Ω ω g E H ω g , where the latter notation means that for any two states ω s , ω s ∈ ω g there exists an isomorphism I s ,s from H ω s to H ω s satisfying π ω s ( A ) = I ∗ s ,s π ω s ( A ) I s ,s , i.e., π ω s and π ω s are unitarily equivalent as well as their associated cyclic vectors Ω ω s and Ω ω s .Additionally, following Definition 6 and comments around it, there is a strongly continuous one–parameter unitary group (cid:16) e i t L ω g (cid:17) t ∈ R with generator L ω g = L ∗ ω g ≥ satisfying e i t L ω g Ω ω g = Ω ω g suchthat for t ∈ R , e i t L ω g ∈ π ω g (sCAR( H , Γ)) ′′ and any A ∈ sCAR( H , Γ)e i t L ω g π ω g ( A )Ω ω g = π ω g ( τ t ( A )) Ω ω g . We summarize the latter with the following Corollary:C
OROLLARY H g ∈ B ( H ) satisfying Assumption 1 in the g –phase.Let ω g ∈ E g be a family of gapped quasi–free ground states associated to H g . Thus, the associated(irreducible)GNSrepresentation ( H ω g , π ω g , Ω ω g ) isunique(uptounitaryequivalence). Inparticular,any state ω s ∈ E g , s ∈ C , is related to ω ∈ E g by Expression (37), namely, ω s = ω ◦ Υ s . ␊ In particular E g is a weak ∗ –compact convex set metrizable in the weak ∗ –topology that can be written as E g = E g , − ∪ E g , + , for E g , − and E g , + nonempty and disjoint metrizable set in the weak ∗ –topology. Here, E g , − and E g , + areassociated to the negative and positive components of the unitary operators respectively. .2 Dynamics, ground states and spectral flow automorphism in the Thermo-dynamic limit For d ∈ N , let Z d be the Cayley graph as defined in Appendix A, see Expression (58), and let the spin set S , such that L . = Z d × S . Since we are dealing with fermions, w.l.o.g., these can be treatedas negatively charged particles. The cases of particles positively charged can be treated by exactlythe same methods. Then, in order to take the thermodynamic limit we define the Hilbert spaces H S . = ℓ ( S ) ⊕ ℓ ( S ) ∗ and H L . = ℓ (Λ L ; H S ) for all L ∈ R +0 ∪ {∞} , where Λ L for L ∈ R +0 ∪ {∞} is defined by the increasing sequence of cubic boxes(38) Λ L . = { ( x , . . . , x d ) ∈ Z d : | x | , . . . , | x d | ≤ L } ∈ P f ( Z d ) , of side length O ( L ) . Note that such a sequence is a “Van Hove net”, i.e., the volume of the boundaries ∂ Λ L ⊂ Λ L ∈ P f ( Z d ) is negligible w.r.t. the volume of Λ L for L large enough: lim L →∞ | ∂ Λ L || Λ L | = 0 .We now fix any antiunitary involution Γ S on H S . For any L ∈ R +0 ∪ {∞} , we define an antiunitaryinvolution Γ L on H L by(39) (Γ L ϕ ) ( x ) . = Γ S ( ϕ ( x )) , x ∈ Λ L , ϕ ∈ H L . Then, ( H L , Γ L ) is a local self–dual Hilbert space for any L ∈ R +0 ∪ {∞} . Note that H S and H L arefinite–dimensional, with even dimension, whenever L < ∞ : Let(40) X L . =Λ L × S × { + , −} , L ∈ R +0 ∪ {∞} . The canonical orthonormal basis { e x } x ∈ X L of H L , L ∈ R +0 ∪ {∞} , now is defined by(41) e x ( y ) . = δ x,y f s ,v , x = ( x, s , v ) ∈ X L , y ∈ Λ L , where f s , + . = Γ S f s , − ∈ H S and f s , − (t) . = δ s , t for any s , t ∈ S .Within the self–dual formalism, a lattice fermion system in infinite volume is defined by a self–dualHamiltonian H ∞ ∈ B ( H ∞ ) on ( H ∞ , Γ ∞ ) , that is, H ∞ = H ∗∞ = − Γ ∞ H ∞ Γ ∞ . See Definition 4which is here extended to the infinite–dimensional case. For a fixed basis projection P ∞ diagonalizing H ∞ , the operator P ∞ H ∞ P ∞ is the so–called one–particle Hamiltonian associated with the system.To obtain the corresponding self–dual Hamiltonians in finite volume we use the orthogonal projector P H L ∈ B ( H ∞ ) on H L and define(42) H L . = P H L H ∞ P H L , L ∈ R +0 . By construction, if H ∞ is a self–dual Hamiltonian on ( H ∞ , Γ ∞ ) , then, for any L ∈ R +0 , H L is aself–dual Hamiltonian on ( H L , Γ L ) . Note that P H L strongly converges to H ∞ as L → ∞ .For the self–dual Hilbert space ( H ∞ , Γ ∞ ) , the self–dual CAR algebra associated is denoted by A ∞ . = sCAR( H ∞ , Γ ∞ ) , with generator elements and { B( e x ) } x ∈ X ∞ satisfying CAR
Expressions ofDefinition 1. The subalgebra of even elements of A ∞ (see (7)) will be denoted by A + ∞ in the sequel.For Λ ∈ P f ( Z d ) and the finite–dimensional (one–particle) Hilbert space H Λ . = ℓ ( H S ; Λ) withinvolution given by (39), we identify the finite dimensional CAR C ∗ –algebra(43) A Λ . = sCAR( H Λ , Γ Λ ) , Λ ∈ P f ( Z d ) , with the C ∗ –subalgebra generated by the unit and { B( e x ) } x ∈ X Λ . Then, we define by(44) A (0) ∞ . = [ Λ ∈ P f ( Z d ) A Λ ⊂ A ∞ , By fixing m ≥ , the boundary ∂ Λ of any Λ ⊂ Z d is defined by ∂ Λ . = { x ∈ Λ : ∃ y ∈ Z d \ Λ with d ǫ ( x, y ) ≤ m } , where for ǫ ∈ (0 , , d ǫ ( x, y ) : Z d × Z d → [0 , ∞ ) is a well–defined pseudometric related to the distance between x, y inthe lattice Z d [BP13]. W.l.o.g. we will take the ǫ –euclidian distance d ǫ ( x, y ) . = | x − y | ǫ . ∗ –algebra of local elements, which is dense in A ∞ .From Definition 6 one notes that existence of ground states strongly relies on the existence of thedynamics in the thermodynamical limit. The latter means that the sequence { Λ L } L ∈ R +0 ∪{∞} , definedby (38), eventually will contain all the finite subsets, P f ( Z d ) of Z d as L → ∞ . In fact, for any H L = H ∗ L ∈ B ( H L ) one can associate a quasi–free dynamics (10) defining a continuous group { τ ( L ) t } t ∈ R ,L ∈ R +0 of finite volume ∗ –automorphisms of A L ≡ A Λ L by τ ( L ) t ( A ) . = e − i t h B , H L B i A e i t h B , H L B i , A ∈ A ∞ , t ∈ R . See (42) and (44). The associated finite volume generator or finite symmetric derivation is given by(13), namely,(45) δ ( L ) ( A ) = − i[ h B , H L B i , A ] , A ∈ A (0) ∞ , while, the infinite volume generator or symmetric derivation is(46) δ ( A ) = − i[ h B , H ∞ B i , A ] , A ∈ A (0) ∞ . For L ∈ R +0 and Λ L ∈ P f ( Z d ) , denote by Λ c L ≡ Z d \ Λ L the complement of Λ L . Then, A Λ c . =sCAR( H Λ c , Γ Λ c ) , will be the C ∗ –subalgebra generated by the unit and { B( e x ) } x ∈ X Λ c L . The bilinearelements associated to the (border) terms on Λ L and Λ c L are (cf. Definition 3): h B , ∂H L B i = X x , x ∈ X ∞ h e x , ∂H L e x i H ∞ B ( e x ) B ( e x ) ∗ , with H c L ≡ H Λ c L and(47) ∂H L . = P H L H ∞ P H c L + P H c L H ∞ P H L , where for any Λ L ∈ P f ( Z d ) , P H L ∈ B ( H ∞ ) is the orthogonal projector on H L , see Expression (42). Theorem 3 (Infinite volume dynamics):
Assume that the sequence { H L } L ∈ R +0 of self–dual Hamiltonians H L ∈ B ( H L ) strongly converges to H ∞ ∈ B ( H ∞ ) . For L ∈ R +0 , the continuous group { τ ( L ) t } t ∈ R with generator δ ( L ) converges stronglyto a continuous group { τ t } t ∈ R with generator δ as L → ∞ . ␄ Proof.
The proof of the statements is completely standard. We present it here for the sake of com-pleteness. We can combine Expressions (10) and (11) such that for any self–dual Hamiltonian H ∞ ∈ B ( H ∞ ) we have τ ( L ) t (B( ϕ )) = B (cid:16)(cid:16) U ( L ) t (cid:17) ∗ ϕ (cid:17) and τ t (B( ϕ )) = B ( U ∗ t ϕ ) . Here, for L ∈ R +0 , τ ( L ) t . = χ e i tHL and τ t . = χ e i tH ∞ so that n U ( L ) t . = e i tH L o t ∈ R and n U t ≡ U ( ∞ ) t . = e i tH ∞ o t ∈ R are the one–parameter unitary groups on ( H ∞ , Γ ∞ ) associated to the finite and infinite dynamicalsystems, respectively. Note that for any ϕ ∈ H ∞ , B( ϕ ) is bounded (see Definition 1). Then, usingthat k B( ϕ ) k A ∞ ≤ k ϕ k H ∞ , for any L , L ∈ R +0 , with L ≥ L , we have (cid:13)(cid:13)(cid:13) τ ( L ) t (B( ϕ )) − τ ( L ) t (B( ϕ )) (cid:13)(cid:13)(cid:13) A ∞ ≤ (cid:13)(cid:13)(cid:13) U ( L ) t − U ( L ) t (cid:13)(cid:13)(cid:13) B ( H ∞ ) k ϕ k H ∞ .
21e can write U ( L ) t − U ( L ) t = i Z t ∂ s (cid:16) U ( L ) t − s U ( L ) s (cid:17) d s = i Z t U ( L ) t − s ( H L − H L ) U ( L ) s d s, so that (cid:13)(cid:13)(cid:13) τ ( L ) t (B( ϕ )) − τ ( L ) t (B( ϕ )) (cid:13)(cid:13)(cid:13) A ∞ ≤ | t | k H L − H L k B ( H ∞ ) k ϕ k H ∞ . Since the sequence { H L } L ∈ R +0 strongly converges to H ∞ as L → ∞ , the last expression shows that itis a Cauchy sequence of self–adjoint operators. Therefore, the continuous group of ∗ –automorphisms n τ ( L ) t o t ∈ R , L ∈ R +0 , strongly converges to { τ t } t ∈ R for all t ∈ R .To show the existence of the generator, we restrict our study to bounded self–dual Hamiltonians.An extension to unbounded self–dual Hamiltonians can be found using similar arguments that in[BP16, Theorem 4.8]. With the same notation as above, note that the difference between finite volumegenerators is (see (45)) δ ( L ) ( A ) − δ ( L ) ( A ) = − i[ h B , ( H L − H L ) B i , A ] , A ∈ A (0) ∞ . We can write for L , L ∈ R +0 , with L ≥ L ,(48) H L = H L + P H L H ∞ P H L \ H L + P H L \ H L H ∞ P H L + P H L \ H L H ∞ P H L \ H L , where P H L \ H L ≡ P H L − P H L ∈ B ( H ∞ ) is the orthogonal projector on H L \ H L . Then, forany fixed Λ ⊂ Λ L Λ L (cid:13)(cid:13)(cid:13) δ ( L ) ( A ) − δ ( L ) ( A ) (cid:13)(cid:13)(cid:13) A ∞ ≤ k A k A ∞ X x , x ∈ X L \ X L (cid:12)(cid:12)(cid:12) h e x , H ∞ e x i H ∞ (cid:12)(cid:12)(cid:12) + 2 X L ,L (Λ) , with X L ,L (Λ) . = max X x ∈ X L \ X L X x ∈ X Λ (cid:12)(cid:12)(cid:12) h e x , H ∞ e x i H ∞ (cid:12)(cid:12)(cid:12) , X x ∈ X L \ X L X x ∈ X Λ (cid:12)(cid:12)(cid:12) h e x , H ∞ e x i H ∞ (cid:12)(cid:12)(cid:12) , (49)we then note that for L , L → ∞ , (cid:13)(cid:13)(cid:13) δ ( L ) ( A ) − δ ( L ) ( A ) (cid:13)(cid:13)(cid:13) A ∞ goes to zero and the sequence { δ ( L ) } L ∈ R +0 is Cauchy. In fact, it is (absolutely) convergent for any A ∈ A (0) ∞ : δ ( A ) = lim L →∞ δ ( L ) ( A ) , A ∈ A (0) ∞ . Inparticular, note that for the local element A ∈ A L k [ h B , H L B i , A ] k A ∞ ≤ | Λ L | k A k A max (cid:26) sup x ∈ X L X x ∈ X L (cid:12)(cid:12)(cid:12) h e x , H L e x i H ∞ (cid:12)(cid:12)(cid:12) , sup x ∈ X L X x ∈ X L (cid:12)(cid:12)(cid:12) h e x , H L e x i H ∞ (cid:12)(cid:12)(cid:12) (cid:27) . Finally, let us remark that the second Trotter–Kato Approximation Theorem [EBN +
06, Chap. III,Sect. 4.9] assures that δ : A ( ∞ )0 → A ( ∞ ) is the generator of { τ t } t ∈ R . For complete details see[BP16]. End
Remark 2.
Observe that for any t ∈ R , Λ ∈ P f ( Z d ) , A ∈ U Λ and L , L ∈ R +0 with Λ ⊂ Λ L Λ L wehave: τ ( L ) t ( A ) − τ ( L ) t ( A ) = Z t dd s (cid:16) τ ( L ) s (cid:16) τ ( L ) t − s ( A ) (cid:17)(cid:17) d s = − i Z t τ ( L ) s (cid:16)h h B , ( H L − H L ) B i , τ ( L ) t − s ( A ) i(cid:17) d s ecause of Expression (11) , the boundedness of the generators and { B( e x ) } x ∈ X L , and due to Λ ⊂ Λ one has: (cid:13)(cid:13)(cid:13) τ ( L ) t ( A ) − τ ( L ) t ( A ) (cid:13)(cid:13)(cid:13) A ∞ ≤ k A k A ∞ | t | X x , x ∈ X L \ X L (cid:12)(cid:12)(cid:12) h e x , H ∞ e x i H ∞ (cid:12)(cid:12)(cid:12) + 2 X L ,L (Λ) , where X L ,L (Λ) is given by (49) . Last inequality is reminiscent of Lieb–Robinson bounds (LRB) used toshow the existence of dynamics for the interacting short–range case [BP16]. ␏ In order to study quasi–free ground states at infinite volume we use:P
ROPOSITION { H L } L ∈ R +0 ∈ B ( H ∞ ) beasequenceofself–dualHamiltonianson ( H ∞ , Γ ∞ ) stronglyconvergentto H ∞ ∈ B ( H ∞ ) . Forany L ∈ R +0 ∪{∞} , E + , L willdenotethespectralprojectionon R + associatedtotheself–dualHamiltonian H L . Ifzero isnotan eigenvalueof H ∞ , then E + willbethestronglimitofthesequence { E + ,L } L ∈ R +0 , i.e., lim L →∞ E + ,L = E + . ␊ Proof.
The proof is found in [AE83, Lemma 3.3.]
End
For any L ∈ R +0 let us define the set of local quasi–free ground states by qE ( L ) ⊂ qE ( ∞ ) on A L ⊂ A ∞ . See Definition 7. To be explicit, for any self–dual Hamiltonian H ∞ ∈ B ( H ∞ ) on ( H ∞ , Γ ∞ ) and any orthogonal projection P H L ∈ B ( H ∞ ) on H L the local Hamiltonian is given by(42), namely, H L . = P H L H ∞ P H L , which has an associated local Gibbs state defined by ̺ Λ L (cid:16) B( ϕ , Λ L )B( ϕ , Λ L ) ∗ (cid:17) . = D ϕ , Λ L , E + ,L ϕ , Λ L E H L , for ϕ j, Λ L ∈ H L , j = { , } , where E + ,L denotes the sequence of spectral projections of Proposition2. Then, using Expressions (16)–(17) the local quasi–free ground state ω Λ L ∈ qE ( L ) is found to be(50) ω Λ L (cid:16) B( ϕ , Λ L )B( ϕ , Λ L ) ∗ B( ϕ , Λ c L )B( ϕ , Λ c L ) ∗ (cid:17) =2 ̺ Λ L (cid:16) B( ϕ , Λ L )B( ϕ , Λ L ) ∗ (cid:17) tr (cid:16) B( ϕ , Λ c L )B( ϕ , Λ c L ) ∗ (cid:17) where ϕ j, Λ c L ∈ H Λ c L ≡ H c L , j = { , } , and tr ∈ E is the tracial state of Definition 5, cf. [AM03,Section 4.2]. In particular, if ϕ , Λ c L = ϕ , Λ c L = H ∞ one has ω Λ L (cid:16) B( ϕ , Λ L )B( ϕ , Λ L ) ∗ (cid:17) = ̺ Λ L (cid:16) B( ϕ , Λ L )B( ϕ , Λ L ) ∗ (cid:17) . Additionally, by linearly, for any two even elements A ∈ A + L and B ∈ A + , c L , see (7), we get from(50): ω Λ L ( AB ) = 2 ̺ Λ L ( A ) tr ( B ) , see again [AM03, Section 4.2]. We now state: Theorem 4 (Quasi–free ground states):
The local quasi–free ground state ω Λ L converges to ω (B( ϕ )B( ϕ ) ∗ ) = h ϕ , E + ϕ i H ∞ , in the weak ∗ –topology, where E + ∈ B ( H ∞ ) is the spectral projection on R + associated to the self–dual Hamiltonian H ∞ ∈ B ( H ∞ ) , and ϕ , ϕ ∈ H ∞ . ␄ roof. Take L , L ∈ R +0 , with L ≥ L such that Λ L ) Λ L . Thus, we analyze the followingdifference: D ω L , ω L . = ω L (cid:16) B( ϕ , L )B( ϕ , L ) ∗ (cid:17) − ω L (cid:16) B( ϕ , L )B( ϕ , L ) ∗ (cid:17) . Here, in the way that the set of boxes Λ L was defined (38), for j = { , } we canonically iden-tify ϕ j,L ∈ H L with the element ϕ j,L ⊕ Λ L \ Λ L ∈ H L . The spectral projections on R + arerelated by E L = E L ⊕ Λ L \ Λ L ∈ B ( H L ) . Straightforward calculations yield us to note that lim L →∞ lim L →∞ D ω L , ω L equals zero. End
We are now in a position to prove the properties of the family of automorphisms κ s : B ( H ) → B ( H ) , for any s ∈ C , given by Assumption 1 and Lemma 1, which are associated to a differentiablefamily of self–dual Hamiltonians H g ∈ B ( H ∞ ) in the g –phase, see Definition 10. Observe that theexistence of such κ s is closely related to the existence of a differentiable unitary operator V s ∈ B ( H ) satisfying the non –autonomous differential equation, Expression (29): ∂ s V s = − i D g ,s V s , with V = H , where { D g ,s } s ∈ C ∈ B ( H ∞ ) is a family of self–adjoint operators that is found to be D g ,s . = Z R e i tH s ( ∂ s H s ) e − i tH s W g ( t )d t, with W g : R → R an integrable odd function the properties of which are summarized in [BMNS12,MZ13] and references therein. In the sequel, for any s ∈ C , V s , H s and ∂ s H s have to be understoodas the strong limit of the sequences { V ( L ) s } L ∈ R +0 , { H s,L } L ∈ R +0 and { ∂ s H s,L } L ∈ R +0 respectively. Weformulate:L EMMA C ≡ [0 , , fix s ∈ C and consider the family of operators satisfying Assumption 1. Then,the sequence of automorphisms { κ ( L ) s } L ∈ R +0 : B ( H L ) → B ( H L ) of Lemma 1 on the local self–dual Hilbert space ( H L , Γ L ) strongly converges uniformly on C to κ s : B ( H ∞ ) → B ( H ∞ ) . Moreprecisely,forany Λ ∈ P f ( Z d ) , B ∈ B ( H Λ ) and L ∈ R +0 suchthat Λ ⊂ Λ L wehave lim L →∞ (cid:13)(cid:13)(cid:13) κ s ( B ) − κ ( L ) s ( B ) (cid:13)(cid:13)(cid:13) B ( H ∞ ) = 0 , forany s ∈ C . ␄ Proof.
Fix Λ ∈ P f ( Z d ) and take L , L ∈ R +0 , with L ≥ L such that Λ L ) Λ L ⊃ Λ . We proceedin a similar way as in [BP16, Lemma 4.4]. Note that with a few modifications of the proof we canarrive at a result that works even in the interparticle case [AR20].For any L ∈ R +0 , let V ( L ) s ∈ B ( H L ) be the unitary operator satisfying the differential equation(29). For s, r ∈ C , one defines the unitary element(51) U L ( s, r ) . = V ( L ) s (cid:16) V ( L ) r (cid:17) ∗ , which satisfies U L ( s, s ) = H for all s ∈ C while(52) ∂ s U L ( s, r ) = − i D ( L ) g ,s U L ( s, r ) and ∂ r U L ( s, r ) = i U L ( s, r ) D ( L ) g ,r . Note that for B ∈ B ( H Λ ) one can write κ ( L ) s ( B ) − κ ( L ) s ( B ) = Z s ∂ r ( U L (0 , r ) U L ( r, s ) BU L ( s, r ) U L ( r, r. i U L (0 , r ) h(cid:16) D ( L ) g ,r − D ( L ) g ,r (cid:17) , U L ( r, s ) BU L ( s, r ) i U L ( r, with s, r ∈ C , and for L ∈ R +0 , and Λ ∈ P f ( Z d ) , Λ ⊂ Λ L .On the other hand, for any s ∈ C , t ∈ R + , Λ ∈ P f ( Z d ) and L ∈ R +0 such that Λ L ⊃ Λ , define the s –automorphism e τ ( L ) s, t : B ( H L ) → B ( H L ) by e τ ( L ) s, t ( B ) . = e i tH s, L B e − i tH s, L , with H s, L a self–dual Hamiltonian on ( H L , Γ L ) . Then, for L , L ∈ R +0 one can write the following e τ ( L ) s, t ( B ) − e τ ( L ) s, t ( B ) = Z t ∂ u (cid:16)e τ ( L ) s, u ◦ e τ ( L ) s, t − u ( B ) (cid:17) d u = i Z t e τ ( L ) u (cid:16)h H s, L − H s, L , e τ ( L ) s, t − u ( B ) i(cid:17) d u, where, for a fix s ∈ C , the difference H s, L − H s, L ∈ B ( H ∞ ) is given by Expression (48), namely, H s, L − H s, L = P H L H s, ∞ P H L \ H L + P H L \ H L H s, ∞ P H L + P H L \ H L H s, ∞ P H L \ H L , where P H L \ H L ≡ P H L − P H L ∈ B ( H ∞ ) is the orthogonal projector on H L \ H L . Here, H s, ∞ is the self–dual Hamiltonian on ( H ∞ , Γ ∞ ) at infinite volume. It follows that, (cid:13)(cid:13)(cid:13)e τ ( L ) s, t ( B ) − e τ ( L ) s, t ( B ) (cid:13)(cid:13)(cid:13) B ( H ∞ ) ≤ Z t (cid:13)(cid:13)(cid:13)h H s, L − H s, L , e τ ( L ) s, t − u ( B ) i(cid:13)(cid:13)(cid:13) d u ≤ | t | k H s, L − H s, L k B ( H ∞ ) k B k B ( H Λ ) . (54)By Assumption 1, { H s, L } L ∈ R +0 is a sequence of operators which strongly converges to H s, ∞ as L →∞ , then last expression is a Cauchy sequence of self–adjoint operators. Hence, for all t ∈ R , e τ ( L ) s, t converges strongly on B ( H L ) to e τ s, t , as L → ∞ .What is important to stress is that the difference D ( L ) g ,r − D ( L ) g ,r in Expression (53) can be written asfollows D ( L ) g ,r − D ( L ) g ,r = Z R (cid:16)e τ ( L ) r, t ( ∂ r { H r,L } ) − e τ ( L ) r, t ( ∂ r { H r,L } ) (cid:17) W g ( t )d t + Z R (cid:16)e τ ( L ) r, t ( ∂ r { H r,L } ) − e τ ( L ) r, t ( ∂ r { H r,L } ) (cid:17) W g ( t )d t. From which one has(55) (cid:13)(cid:13)(cid:13) κ ( L ) s ( B ) − κ ( L ) s ( B ) (cid:13)(cid:13)(cid:13) B ( H ∞ ) ≤ k B k B ( H ∞ ) | s | sup r ∈ C (cid:18)Z R k ∂ r { H r,L } − ∂ r { H r,L }k B ( H ∞ ) | W g ( t ) | d t + Z R (cid:13)(cid:13)(cid:13)(cid:16)e τ ( L ) r, t − e τ ( L ) r, t (cid:17) ∂ r { H r,L } (cid:13)(cid:13)(cid:13) B ( H ∞ ) | W g ( t ) | d t (cid:19) . Hence, for a fixed s ∈ C , by Assumption 1 and Inequality (54) one notes that the right hand side ofthe last inequality vanishes as L → ∞ and L → ∞ . Thus, κ ( L ) s is a pointwise Cauchy sequenceas L → ∞ and hence the family of automorphism κ ( L ) s converges strongly on B ( H L ) to κ s as L → ∞ . End
As a consequence we have: 25
OROLLARY s ∈ C , the sequence of unitary operators V ( L ) s ∈ B ( H ∞ ) stronglyconvergesto some V s ∈ B ( H ∞ ) . ␊ Proof.
As is usual, it is enough to show that the sequence V ( L ) s ∈ B ( H ∞ ) is a Cauchy sequence.Note that for any s ∈ C and L , L with L ≥ L , we can write: (cid:16) V ( L ) s (cid:17) ∗ − (cid:16) V ( L ) s (cid:17) ∗ = Z s ∂ r ( U L (0 , r ) U L ( r, s )) d r, where for any s, r ∈ C , U L ( s, r ) is the unitary element defined by (51)–(52). Straightforward calcu-lations yield to (cid:16) V ( L ) s (cid:17) ∗ − (cid:16) V ( L ) s (cid:17) ∗ = i Z s U L (0 , r ) (cid:16) D ( L ) g ,r − D ( L ) g ,r (cid:17) U L ( r, s )d r. Proceeding as in (55) we arrive at the desired result. We omit the details.
End
Fix ǫ ∈ (0 , and let ( H ∞ , Γ ∞ ) be the self–dual Hilbert space as defined in subsection 4.2. Moreover,consider the family of self–adjoint operators { A s } s ∈ C ∈ B ( H ∞ ) . Thus, for any s ∈ C we define theconstants S ( A s , µ ) . = sup x ∈ X ∞ X x ∈ X ∞ (cid:16) e µ | x − x | ǫ − (cid:17) (cid:12)(cid:12)(cid:12) h e x , A s e x i H ∞ (cid:12)(cid:12)(cid:12) ∈ R +0 ∪ {∞} , for µ ∈ R +0 and ∆( A s , z ) . = inf {| z − λ | : λ ∈ spec( A s ) } , z ∈ C , as the distance from the point z to the spectrum of A s . X ∞ is defined by (40). Here, µ is not necessarilythe same for two different operators A s , A s ∈ { A s } s ∈ C , but in the sequel w.l.o.g. we will assumethis. Since the function x (e xr − /x is increasing on R + for any fixed r ≥ , it follows that(56) S ( A s , µ ) ≤ µ µ S ( A s , µ ) , µ ≥ µ ≥ . We have the following Combes–Thomas estimates:P
ROPOSITION
OMBES –T HOMAS ).Let ǫ ∈ (0 , , C . = [0 , , the family of self–adjoint operators { A s } s ∈ C ∈ B ( H ∞ ) , µ ∈ R +0 and z ∈ C . If ∆( A s , z ) > S ( A s , µ ) then forany s ∈ C and x = ( x, s , v ) , y = ( y, t , w ) ∈ X ∞ (cid:12)(cid:12)(cid:12)(cid:12)D e x , ( z − A s ) − e y E H ∞ (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup s ∈ C ( e − µ | x − y | ǫ ∆( A s , z ) − S ( A s , µ ) ) . ␊ For a proof see [AW15, Theorem 10.5]. Some immediate consequences are summarized as follows:C
OROLLARY ǫ ∈ (0 , , C . = [0 , , the family of self–adjoint operators { A s } s ∈ C ∈ B ( H ∞ ) , µ ∈ R +0 and all x = ( x, s , v ) , y = ( y, t , w ) ∈ X ∞ . Then, 26a) Let η ∈ R + such that sup s ∈ C { S ( A s , µ ) } ≤ η/ , u ∈ R and s ∈ C , (cid:12)(cid:12)(cid:12)(cid:12)D e x , (( A s − u ) + η ) − e y E H ∞ (cid:12)(cid:12)(cid:12)(cid:12) ≤ D , ( a ) e − µ | x − y | ǫ sup s ∈ C (cid:26)D e x , (( A s − u ) + η ) − e x E / H ∞ D e y , (( A s − u ) + η ) − e y E / H ∞ (cid:27) . Moreover, forany function G ( z ) : C → C analyticon |ℑ m ( z ) | ≤ η and uniformlybounded by k G k ∞ wehave h e x , G ( A s ) e y i H ∞ ≤ D , ( b ) k G k ∞ e − µ min n , inf s ∈ C { η S ( As,µ ) } o | x − y | ǫ . (b) (Gapped Case) For z ∈ C such that inf s ∈ C ∆( A s , z ) ≥ g / > , with g as inDefinition10:(57) (cid:12)(cid:12)(cid:12)(cid:12)D e x , ( z − A s ) − e y E H ∞ (cid:12)(cid:12)(cid:12)(cid:12) ≤ g − exp − µ min ( , inf s ∈ C ( g S ( A s , µ ) )) | x − y | ǫ ! . Moreover,for η ∈ (0 , g / ,andanyfunction G ( z ) : C → C analyticon z ∈ R +0 + η + i η [ − , and uniformlybounded by k G k ∞ wehave h e x , E + G ( A s ) E + e y i H ∞ ≤ D , ( c ) k G k ∞ e − µ min n , inf s ∈ C { g S ( As,µ ) } o | x − y | ǫ . In all inequalities,thenumbers D , ( a ) , D , ( b ) , D , ( c ) ∈ R + are suitableconstants. ␊ Proof. (a) is proven as in [AG98, Theorem 3 and Lemma 3]. (b) The first part is a consequence ofTheorem 3 together Inequality (56). On the other hand, we use Cauchy’s integral formula to write,for all real E ∈ R \{ η } , χ ( η, ∞ ) G ( E ) = 12 π i Z ∞ η G ( u − i η ) u − E − i η − G ( u + i η ) u − E + i η ! d u − π Z η − η G ( η + i u ) η − E + i u d u , which yields χ ( η, ∞ ) G ( E ) = ηπ Z ∞ η G ( u − i η ) + G ( u + i η )( u − E ) + η d u − ηπ Z ∞ η G ( u )( u − E ) + 4 η d u + 12 π Z η G ( η − i u ) η − i u − E + 2i η d u + 12 π Z η G ( η + i u ) η + i u − E − η d u − π Z η − η G ( η + i u ) η − E + i u d u. By spectral calculus, together with the last equality, part (a) of this Lemma, Inequality (57) and theCauchy–Schwarz inequality, the result follows. For further details see [ABPM20, Lemma 5.12].
End
At this point it is useful to introduce the normalized trace per unit volume as Tr( · ) . = lim L →∞ H L ) tr H L ( · ) . We are able to state the following: 27
EMMA C ≡ [0 , and consider the family of operators satisfying assumptions of Corollary 5 (a) for { ∂ s H ( L ) s } s ∈ C ∈ B ( H L ) , L ∈ R +0 . Then, for the pointwise sequence V ( L ) s : C → B ( H ) , L ∈ R +0 ,of unitary operators satisfying (29), − V ( L ) s is trace class. Thus, for L ∈ R +0 , the family of one–parameter (Bogoliubov)group n Υ ( L ) s o s ∈ C ∈ R of ∗ –automorphismson A L (see (43)), given by (36), isinner . Moreover, H ∞ − V ( ∞ ) s is trace class per unit volume, where V ( ∞ ) s ∈ B ( H ) is that givenby Corollary 4. ␄ Proof.
For s ∈ C and L ∈ R +0 , let W ( L ) s ∈ B ( H L ) be the partial isometry arising from the polardecomposition of H L − V ( L ) s H L − V ( L ) s = W ( L ) s (cid:12)(cid:12)(cid:12) H L − V ( L ) s (cid:12)(cid:12)(cid:12) . From this one can calculate the trace of (cid:12)(cid:12)(cid:12) − V ( L ) s (cid:12)(cid:12)(cid:12) as follows tr H L (cid:12)(cid:12)(cid:12) H L − V ( L ) s (cid:12)(cid:12)(cid:12) = X x ∈ X L D e x , (cid:16) W ( L ) s (cid:17) ∗ (cid:16) H L − V ( L ) s (cid:17) e x E H L . Note that for the unitary bounded operator V ( L ) s on H L we can write − V ( L ) s = i R s D ( L ) g ,r V ( L ) r d r .Then, by combining the explicit form of D ( L ) g ,r given by (35), Cauchy–Schwarz inequality, Corollary5 (a), and other simple arguments we arrive at (cid:12)(cid:12)(cid:12) tr H L (cid:12)(cid:12)(cid:12) H L − V ( L ) s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ D Lem. | s | | Λ L | | S | Z R | W g ( t ) | d t X x ∈ Λ L e − µ min n , inf r ∈ C { η S ( ∂rHr,µ ) } o | x | ǫ . Then, H L − V ( L ) s is trace class, and V ( L ) s ∈ B ( H L ) is a unitary operator. It follows from [Ara87,Theorem 4.1] that the ∗ –automorphism Υ ( L ) s on A L is inner. Since there is a volume factor in lastinequality we state that H ∞ − V ( ∞ ) s is trace class per unit volume. End
A combination of Corollary 4 and Lemma 3 yields to:C
OROLLARY Υ ( L ) s on A (0) ∞ convergesuniformlyfor s ∈ C as L → ∞ totheone–parameter(Bogoliubov)group Υ s on A ∞ ,thusdefining a strongly continuousgroup on A ∞ . Moreover, (cid:16) Υ ( L ) s (cid:17) − exists and strongly converges to Υ − s . ␊ Proof.
Note that the sequence of one–parameter (Bogoliubov) group Υ ( L ) s on A (0) ∞ is Cauchy for any B ∈ A (0) ∞ . We omit the details. Existence of (cid:16) Υ ( L ) s (cid:17) − is a straight conclusion from Corollary 2, itsconvergence is immediate. We also omit the details. End
For any L ∈ R +0 , qE ( L, ∞ ) ⊂ qE ( ∞ ) denotes the local quasi–free ground states on A L ⊂ A ∞ . Wepostulate: Theorem 5 (Gapped quasi–free ground states):
Take C ≡ [0 , and consider the family of self–dual Hamiltonians satisfying Assumption 1 (b). Fix L ∈ R +0 , and let n ω ( L ) s o s ∈ C ⊂ qE ( L, ∞ ) be the family of gapped quasi–free ground states associatedto the family of Hamiltonians n H ( L ) s o s ∈ C ∈ B ( H L ) . Then, For U ∈ B ( H ) , a Bogoliubov transformation, the Bogoliubov ∗ –automorphism χ U on sCAR( H , Γ) is inner if andonly if H ∞ − U is trace class and det( U ) = ± , see [Ara87, Theorem 4.1]. Recall that A ∞ is the completeness of the normed ∗ –algebra A (0) given by (44). ω ( L ) s = ω ( L )0 ◦ Υ ( L ) s , for all s ∈ C , where Υ ( L ) s is the finite–volume Bogoliubov ∗ –automorphismon A L of Corollary 6.(2) Let ω s ∈ qE ( ∞ ) be the weak ∗ –limit of ω ( L ) s ∈ qE ( L, ∞ ) and Υ s the infinite–volume Bogoli-ubov ∗ –automorphism on A ∞ associated to the sequence Υ ( L ) s of Corollary 6. With respect tothe weak ∗ –topology, the following three statements are equivalent: (a) lim L →∞ ω ( L ) s = ω s . (b) lim L →∞ ω ( L ) s ◦ Υ s = ω s ◦ Υ s . (c) lim L →∞ ω ( L ) s ◦ Υ ( L ) s = ω s ◦ Υ s . ␄ Proof. (1) follows from Corollary 3 and Lemma 3. (2) Fix s ∈ C . Note that the existence of theweak ∗ –limit ω s is consequence of Theorem 4 while the existence of the Bogoliubov ∗ –automorphism Υ s is a consequence of Corollary 6. Now, take any A ∈ A ∞ and note that (a) ⇒ (b) because (cid:12)(cid:12)(cid:12) ω ( L ) s ◦ Υ s ( A ) − ω s ◦ Υ s ( A ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ω ( L ) s − ω s (cid:12)(cid:12)(cid:12) k A k A ∞ . (b) ⇒ (c) follows by recognizing ω ( L ) s and ω s as states and writting (cid:12)(cid:12)(cid:12) ω ( L ) s ◦ Υ ( L ) s ( A ) − ω s ◦ Υ s ( A ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ω ( L ) s − ω s (cid:12)(cid:12)(cid:12) k A k A ∞ + (cid:12)(cid:12)(cid:12) ω ( L ) s (cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13) Υ ( L ) s ( A ) − Υ s ( A ) (cid:13)(cid:13)(cid:13) A ∞ , and we have that the left hand side of last inequality is zero. Finally, we note that (cid:12)(cid:12)(cid:12) ω ( L ) s ( A ) − ω s ( A ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ω ( L ) s ◦ Υ ( L ) s − ω s ◦ Υ s (cid:12)(cid:12)(cid:12) k A k + | ω s ◦ Υ s | (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) Υ ( L ) s (cid:17) − ( A ) − Υ − s ( A ) (cid:13)(cid:13)(cid:13)(cid:13) A ∞ , and from Corollary 6, the right hand side of last inequality is zero, thus (c) ⇒ (a). End
A Disordered models on general graphs
Consider the graph G . = V × E , where V is the so–called set of vertices and E is called set of edges .A graph has the following basic properties:1. For any, v, w ∈ V , and ( v, w ) ∈ V × V , v and w are called the endpoints of ( v, w ) ∈ E .2. For v, w ∈ V , the vertices E set does not contain element of the form ( v, v ) .3. Unless otherwise indicated, the edges set is not–oriented : ( v, w ) ∈ E iff ( w, v ) ∈ E .4. For simplicity, the element g ∈ G is written as g ≡ ( v, e ) for some v ∈ V and e ∈ E .5. For ǫ ∈ (0 , and any v, w ∈ V , one can endow to G of a pseudometric d ǫ : G × G → R +0 ∪{∞} ,that is, an equivalence relation satisfying the metric properties on G , except that d ǫ ( v, w ) = 0 does not implies that v = w . d ǫ is closely related to the size of the path with the minimumnumber of edges joining the vertices v and w .6. For G , P f ( G ) ⊂ G will denote the set of all finite subsets of G .We refer the reader to [LP17] for a complete discussion about graphs.Take d ∈ N . Among the graphs that physicists consider, the d –dimensional cubic lattice or crystal Z d is taken as a subset of R d in the following way :(58) Z d . = { ( x , . . . , x d ) ∈ R d : x j ∈ Z for any ≤ j ≤ d } . Because of its spatial symmetric properties: translations, rotations.
Cayley graphs , whuch are definedvia the group V ≡ ( V , · ) generated by the subset v ≡ ( v , · ) . Then, we associate to any element of V a vertex of the Cayley graph G and the set of edges is defined by E . = n ( v, w ) ∈ V : v − w ∈ v o . In the Z d case, the group V ≡ ( V , +) is the so–called translation group.From the physical point of view, mobility or confinement of particles embedded in a graph G . = V × E will rely on the impurities of the material, crystal lattice defects (as in the Z d case), etc., which usuallyare modeled (in the simplest case) by random (one–site) external potentials on the set of vertices V as follows: We take the probability space (Ω , A Ω , a Ω ) , where Ω . = [ − , V . For any v ∈ V , Ω v is an arbitrary element of the Borel σ –algebra A v of the Borel set [ − , w.r.t. the usual metrictopology. Then, A V is the σ –algebra generated by the cylinder sets Q v ∈ V Ω v , where Ω v = [ − , forall but finitely many v ∈ V . Additionally, we assume that the distribution a Ω is an arbitrary ergodic probability measure on the measurable space (Ω , A Ω ) . I.e., it is invariant under the action ρ χ (Ω) v ( ρ ) . = χ ( V ) v ( ρ ) , v ∈ V , of the group V ≡ ( V , · ) on Ω and a Ω ( O ) ∈ { , } whenever O ∈ A Ω satisfies χ (Ω) e ( O ) = O for all v ∈ V . Here, for any ρ ∈ Ω , v ∈ V and w ∈ V χ ( E ) v ( ρ ) ( w ) . = ρ (cid:16) v − w (cid:17) . As is usual, E [ · ] denotes the expectation value associated with a Ω .For the Cayley graph G . = V × E , h . = ℓ ( G , C ) will denote a separable Hilbert space associated to G with scalar product h· , ·i h and canonical orthonormal basis denoted by { e v } v ∈ V , which is definedby e v ( w ) = δ v − w, V for all v, w ∈ v , with v the generator set of V and V the unit on V . For any, ρ ∈ Ω , one introduces the external potential V ρ ∈ B ( h ) as the self–adjoint multiplication operatoroperator V ρ : V → [ − , . On the other hand, one define for the compact set C . = [0 , , the familyof graph Laplacians { ∆ G ,s } s ∈ C defined for any s ∈ C by [∆ V ,s ( ψ )]( v ) . = deg V ( v ) ψ ( v ) − s X p ∈ V : d ǫ ( v,w )=1 ψ ( v − w ) , v ∈ V , ψ ∈ h (59)where for any ǫ ∈ (0 , , d ǫ : V × V → R +0 ∪ {∞} is a pseudometric on G . In (59), on the right handside, deg V ( v ) is the number of nearest neighbors to vertex v , or degree of v . If { deg V ( v ) } v ∈ V ∈ N isthe same for all v ∈ V , we say that the graph is regular.The random tight–binding (Anderson) model is the one–particle Hamiltonian defined by(60) h ( ρ ) V ,s . = ∆ V ,s + λV ρ , ρ ∈ Ω , λ ∈ R +0 . See [AW15] for further details. In [ABPR19], we consider a more general setting such that hoppingdisorder is present, i.e., we associate to particles a hopping probability on the non–oriented edges E . In this case, one deals with hopping amplitudes and the probability space (Ω , A Ω , a Ω ) is properlyimplemented. Acknowledgments:
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