Classical Dynamics Generated by Long-Range Interactions for Lattice Fermions and Quantum Spins
aa r X i v : . [ m a t h - ph ] S e p Classical Dynamics Generated by Long-RangeInteractions for Lattice Fermions and Quantum Spins
J.-B. Bru W. de Siqueira PedraSeptember 14, 2020
Abstract
We study the macroscopic dynamical properties of fermion and quantum-spin systems withlong-range, or mean-field, interactions. The results obtained are far beyond previous ones and re-quire the development of a mathematical framework to accommodate the macroscopic long-rangedynamics, which corresponds to an intricate combination of classical and short-range quantum dy-namics. In this paper we focus on the classical part of the long-range, or mean-field, macroscopicdynamics, but we already introduce the full framework. The quantum part of the macroscopicdynamics is studied in a subsequent paper. We show that the classical part of the macroscopicdynamics results from self-consistency equations within the (quantum) state space. As is usual,the classical dynamics is driven by Liouville’s equation.
Dedicated to V.A. Zagrebnov for his important contributions to the mathematics of quantum many-body theory.
Keywords:
Interacting fermions, self-consitency equations, quantum-spin, classical dynamics,extended quantum mechanics.
AMS Subject Classification:
Contents ∗ -Automorphisms of the CAR Algebra . . . . . . . . . . . . . . . . . . . 72.4 State Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Even States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Lattice Fermions with Long-Range Interactions 17 C ∗ -Algebra of Continuous Functions on States 21 C ∗ -Algebra of Continuous Functions on States . . . . . . . . . . . . . 215.2 Poisson Structure Associated with the State Space . . . . . . . . . . . . . . . . . . . 225.3 The Quantum C ∗ -Algebra of Continuous Functions on States . . . . . . . . . . . . 245.4 Important ∗ -Automorphisms of the Quantum C ∗ -Algebra . . . . . . . . . . . . . . . 26 C ∗ -Algebra of Functions and Dynamics . . . . . . . . . 276.3 From Quantum Dynamics to Classical Flows . . . . . . . . . . . . . . . . . . . . . 296.4 Self-Consistency Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.5 Classical Part of Long-Range Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 346.6 Quantum Part of Long-Range Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 37 More than seventy years ago, Bogoliubov proposes an ansatz, widely known as the Bogoliubov ap-proximation, which corresponds to replace, in many-boson Hamiltonians, the annihilation and cre-ation operators of zero-momentum particles with complex numbers to be determined self-consistently .See [1, Section 1.1] for more details. His motivation comes from the observation that these (un-bounded) operators almost commute with any other operator in the thermodynamic limit, leading tosome classical field in the macroscopic Bose system. In 1968, Ginibre [2] shows the exactness ofthe approximation in which concerns the thermodynamic pressure of superstable Bose gases. How-ever, even nowadays, the mathematical validity of this approximation with respect to the primordialdynamics of (stable) many-boson Hamiltonians with usual two-body interactions is an open problem.In the context of many-fermion systems, ten years after Bogoliubov’s ansatz, a similar approx-imation is used in the BCS theory of (conventional) superconductivity, as explained by Bogliubovin 1958 [3] and Haag in 1962 [4]. In 1966, this approximation is shown [8] to be exact at the levelof the thermodynamic pressure for fermion systems that are similar to the BCS model. See also theso-called approximating Hamiltonian method used on the level of the pressure of fermionic systemsin [8–11].The validity of the approximation with respect to the primordial dynamics was an open questionthat Thirring and Wehrl [5, 6] solve in 1967 for an exactly solvable permutation-invariant fermionmodel. An attempt to generalize Thirring and Wehrl’s results to a general class of fermionic models, Quantum particles carry an intrinsic form of angular momentum, the so-called spin. It is related to a spin quantumnumber s ∈ N / (depending on the nature of a given particle) which fixes a finite spin set S . = {− s , − s + 1 , . . . s − , s } .If s is a half-integer, i.e. s / ∈ N , then the corresponding particles are fermions while the case s ∈ N correspond to bosons.By the celebrated spin-statistics theorem, fermionic wave functions are antisymmetric with respect to permutations ofparticles, whereas the bosonic ones are symmetric. permutation-invariant quantum-spin systems with mean-field interactions.Even if we are rather interested in mean-field dynamics, note meanwhile that equilibrium propertiesof such quantum systems are also extensively studied in the same period, see for instance [33–35] andreferences therein.Thereafter, the mathematical research activity on this subject considerably decreases until theearly 2000s when emerges, within the mathematical physics community, a new interest in such quan-tum systems, partially because of new experiments like those on ultracold atoms (via laser and evapo-rative coolings). See, for instance, the paper [36] on mean-field dynamics, published in the year 2000.In 2005, Ginibre’s result on Bose gases at thermodynamical equilibrium is generalized [37]. Thereis also an important research activity on the mathematical foundation of the Gross-Pitaevskii (GP)or Hartree theories, starting after 1998. For more details on the GP theory and mean-field dynamicsfor indistinguishable particles (bosons), see [38–42] and references therein. In which concern lattice-fermion or quantum-spin systems with long-range , or mean-field , interactions at equilibrium, see,e.g., [43–46]. Concerning the dynamics of fermion systems in the continuum with mean-field inter-actions, see [47–56], as well as [39, Sections 6 and 7]. Such mean-field problems are even relatedto other academic disciplines, like mathematical economics, via the so-called mean-field game the-ory [57] developed since 2006 by Lasry and Lions. Mean-field theory in its extended sense is, in fact,a major research field of mathematics, even nowadays, and is still studied in physics, see, e.g., [58]and references therein. The current paper belongs to this research field, since we hereby study the dy-namical properties of fermion and quantum-spin systems with long-range, or mean-field, interactions.All mathematical studies [39, 47–56] of the 2000s on mean-field fermionic dynamics are on thecontinuum and use approximating quasi-free states (or a mixture of them) as initial states to de-rive the Hartree-Fock equation, which is originally based on the assumption that the many-fermionwavefunction is a Slater determinant. Quoting [39, p. 79]: “ Slater determinants are relevant at zerotemperature because they provide (or at least they are expected to provide) a good approximation tothe fermionic ground state of Hamilton operators like (6.1) in the mean-field limit. At positive temper-ature, equilibrium states are mixed; in the mean-field regime, they are expected to be approximatelyquasi-free mixed states, like the Gibbs state of a non-interacting gas. ” The so-called GP limit is not really a mean-field limit, but it looks similar. The fermionic nature of particles is encoded in Canonical Anti-commutation Relations (CAR) which lead, in thealgebraic formulation, to a so-called CAR algebra associated with a one-particle Hilbert space h . The term “lattice-fermion” only refers in this case to the special choice h = ℓ ( Z d × S) , where S is some finite set of spins and d ∈ N . Quantum spin systems refer to infinite tensor products of a fixed matrix algebra, enconding a spin variable at anyfixed lattice site. Their C ∗ algebras are constructed in a similar way as those for lattice fermions (in fact, these infinitetensor products are even isomorphic, as C ∗ algebras, to CAR algebras). Results of this paper hold true also in this context,via obvious modifications. By long-range interactions, we basically mean interactions that are not decaying at large interparticle distances. Notehowever that the attribute “long-range” is used in the literature also for interactions that slowly decay with respect tointerparticle distances. Our notion of “long-range” is therefore stronger, but still includes important physical models likesome of those related to conventional superconductivity. It refers to a type of mean-field interactions widely used intheoretical and mathematical physics. The attribute “mean-field” means here that the (interparticle) interaction is averaged over the number of sites of givenboxes. In some papers, only (approximating) Slater determinants are considered. , which isbased on some energy functional for the one-fermion density only, via the Hohenberg-Kohn theorems.See for instance [59].In contrast with results [39, 47–56] of the 2000s, we consider fermions on the lattice, not inthe continuum, and we meanwhile study quantum-spin systems. Our analysis of the correspondingmean-field dynamics does not require to have approximating quasi-free states (or a mixture of them)as initial states.The recent studies [39, 47–56] of the 2000s never extract effective classical dynamics within thequantum one. The reason, for most of them [39, 49, 51–54, 56], is that their mean-field scales reveal a semi-classical structure , similar to what is first done in 1981 to derive the Vlasov hierarchy in [60,61]from quantum dynamics. In order to see the intricate combination of a classical dynamics and aquantum one, one has to go back to previous results [12–32], initiated by Hepp and Lieb. In ouropinion, the most elaborated and interesting results in that context are obtained by B´ona [17,19–21] in1987-1990, who studies in detail the dynamics of a large class of permutation-invariant quantum-spinsystems with mean-field interactions. B´ona formalizes a new view point on quantum mechanics in1991 [25] and later in a mature textbook published in 2000 and revised in 2012 [62], describing whathe names “extended quantum mechanics”: It is an intricate combination of classical and quantumdynamics. B´ona’s approach does not seem to be incorporated by the physics community, until now.In contrast with the results [12–32] of the 70s-90s (1973-1992), our study does not require thepermutation invariance of quantum-spin systems. In fact, we are able to study dynamical propertiesof a very general class of lattice-fermion, or quantum-spin, systems with long-range, or mean-field, translation-invariant interactions. In this context, the initial state is only assumed to be periodicwith respect to space translations. In fact, by Proposition 2.3, the set of all initial states allowed in ourstudy is weak ∗ -dense within the set of even states, the physically relevant ones. Even for permutation-invariant systems, our results go beyond previous ones, since the class of long-range, or mean-field,interactions we are able to handle is much larger.Our study reveals an entanglement of classical and quantum dynamics, as explained in [63], whichrevisits B´ona’s approach [62] on “extended quantum mechanics”. In [63], we do not really followB´ona’s path but highlight the importance of self-consistency equations, instead. This study leads to anew theory which strongly differs from the so-called quantum-classical hybrid theories of theoreticaland mathematical physics, including the recent mathematical results [68–70]. See below Remark1.1. In this paper and in the subsequent one [64], we show that this mathematical framework isimperative to describe the dynamical properties of lattice-fermion or quantum-spin systems with long-range, or mean-field, interactions, because of the necessity of coupled classical and quantum evolutionequations.Here, we focus on the classical part of the long-range, or mean-field, dynamics of the quantum A not necessarly better, but computationally more efficient approximation. This can be easily extended to periodically invariant systems, by redefining the finite spin set. C ∗ -algebra representations, as presented, forinstance, in the monograph [65].The classical part of the dynamics of lattice-fermion, or quantum-spin, systems with long-range,or mean-field, interactions is shown to result from the solution of a self-consistency equation withinthe (quantum) state space. In general, it is a non-linear dynamics generated by a Poisson bracket. Inother words, it is driven by some Liouville’s equation, similar to what has been recently observed [42]by Ammari, Liard, and Rouffort for Bose gases in the mean-field limit. As soon as the classical part ofthis dynamics is concerned, there is no need to assume any particular property, neither on initial states,nor on local Hamiltonians. Translation invariance, or periodicity, is only required for the analysis ofthe quantum part of the dynamics. So, these particular spatial features are only pivotal in [64].It would have been interesting to add applications of our general theory, but we still have topresent (in [64]) the quantum part of such dynamics before doing that. In fact, several applicationson quasi-free models as well as permutation-invariant systems will be presented in separated papers.See, e.g., [66, 67]. We will also discuss in a forthcoming paper such mean-field dynamics within therepresentation of an arbitrary (generalized) equilibrium state of the quantum system under consid-eration. Remark 1.1 (Quantum-classical hybrid theories)
Many important models of quantum mechanics represent systems of quantum particles in interactionwith classical fields. For example, a quantum particle interacting with an external electromagneticfield is commonly studied via the magnetic Laplacian. In other words, these models implicitly combinequantum and classical mechanics. This simplification is physically justified by the huge numbers ofphotons giving origin to (effective classical) macroscopic fields, in the spirit of the correspondenceprinciple. The very recent paper [70] gives a mathematical justification of such a procedure for threeimportant quantum models: the Nelson, Pauli-Fierz and Polaron models. Mathematically, it refers toa semi-classical analysis of the bosonic degrees of freedom, leading to a new (equivalent) quantummodel interacting with a classical field. Thanks to [68, 69], the authors are able to handle the semi-classical limit for very general (possibly entangled, i.e., not factorized) states. These mathematicalresults are highly non-trivial. However, such a procedure does not entangle classical and quantumdynamics in the way it occurs in presence of long-range interactions. E.g., the relation betweenthe phase space of the classical dynamics and the state space of the C ∗ -algebra where the quantumdynamics runs is not manifest. In fact, the situation presented in this paper is essentially differentthan any quantum-classical hybrid theory. Our main results are the following: • Theorem 6.5 shows that the self-consistency equations are well-posed for any initial state. Suchequations are central to get a complete description of the macroscopic dynamics of long-rangemodels. • Proposition 6.8 states the existence of evolution groups while Corollary 6.10 shows that theirgenerators can be represented via Poisson brackets and a very natural classical energy functionon the state space of the CAR algebra. • Theorem 6.11 demonstrates the existence of non-autonomous classical dynamics on the statespace. It is a highly non-trivial mathematical statement in the theory of non-autonomous evo-lution equations [71–75]. It results from Lieb-Robinson bounds on multi-commutators of highorders derived in 2017 [76]. This class of states is studied in detail in [46]. Corollary 6.12 reduces the classical dynamics to the autonomous situation, which leads to theusual (autonomous) dynamics of classical mechanics written in terms of Poisson brackets, i.e.,
Liouville’s equation .The paper is organized as follows: Section 3 explains the well-established dynamical properties oflattice-fermion systems with short-range interactions. In this context, the celebrated Lieb-Robinsonbounds are pivotal and are also an important tool to obtain the full dynamics of systems with long-range, or mean-field, interactions. Section 4 defines what we name “long-range interactions” andwe make explicit the problem of the thermodynamic limit of their associated dynamics. From nowon, like in [46] we use the term “long-range”, only, instead of “mean-field”. In fact, we prefer thefirst rather than the second attribute, because the latter can refer to different scalings, in particularin the recent works [39, 47–56]. Note, additionally, that “Bogoliubov’s school” also uses the term“long-range” for models containing interactions that are mean-field in the sense of the present paper.In Section 6, we present the self-consistency equations as well as the classical part of long-rangedynamics. This requires the mathematical framework of [63], which has thus to be presented in thefirst subsections of Section 6. Note that we shortly explain what the quantum part of the long-rangedynamics is in Section 6.6, even if this will be done in detail in [64]. Finally, Section 7 gives theproofs of Theorems 6.5 and 6.11. Sections 8 and 9 are two appendices relating our current notion oflong-range interactions to the one of [46] and discussing possible applications in physics.In this paper, we only focus on lattice-fermion systems which are, from a technical point of view,slightly more difficult than quantum-spin systems, because of a non-commutativity issue at differ-ent lattice sites. However, all the results presented here and in the subsequent papers hold true forquantum-spin systems, via obvious modifications. Notation 1.2 (i)
A norm on a generic vector space X is denoted by k · k X and the identity mapping of X by X .The space of all bounded linear operators on ( X , k · k X ) is denoted by B ( X ) . The unit element of anyalgebra X is denoted by , provided it exists. (ii) For any topological space X and normed space ( Y , k · k Y ) , C ( X ; Y ) denotes the space of con-tinuous mappings from X to Y . If X is a locally compact topological space, then C b ( X ; Y ) denotesthe Banach space of bounded continuous mappings from X to Y along with the topology of uniformconvergence. (iii) The notion of an automorphism depends on the structure of the corresponding domain. In alge-bra, a ( ∗ -) automorphism acting on a ∗ -algebra is a bijective ∗ -homomorphism from this algebra toitself. In topology, an automorphism acting on a topological space is a self-homeomorphism, that is,a homeomorphism of the space to itself. Fix once and for all d ∈ N , the dimension of the lattice. Choose also a subset L ⊆ Z d (which willbe omitted in our notation, unless it is important to specify it). The main example we have in mindis L = Z d , as is done in [46], but it is convenient to keep this set as being any arbitrary subset of the d -dimensional lattice. Let P f ≡ P f ( L ) ⊆ L be the set of all non-empty finite subsets of L . In orderto define the thermodynamic limit, for simplicity, we use the cubic boxes Λ L . = { ( x , . . . , x d ) ∈ Z d : | x | , . . . , | x d | ≤ L } ∩ L , L ∈ N , (1) Interactions are said to be short-range when they decay sufficiently fast at large distances. This refers to a sufficientpolynomial decay such that the celebrated Lieb-Robinson bounds hold true for the corresponding dynamics.
6s a so-called van Hove sequence.
For any Λ ∈ P f , U Λ is the finite dimensional unital C ∗ -algebra generated by elements { a x, s } x ∈ Λ , s ∈ S satisfying the canonical anti-commutation relations (CAR), S being some finite set of spins: a x, s a y, t + a y, t a x, s = 0 , a x, s a ∗ y, t + a ∗ y, t a x, s = δ s , t δ x,y (2)for all x, y ∈ L and s , t ∈ S . Observe that ( U Λ L ) L ∈ N is an increasing sequence of C ∗ -algebras. Hence,the set U . = [ L ∈ N U Λ L (3)of so-called local elements is a normed ∗ -algebra with k A k U = k A k U Λ L for all A ∈ U Λ L and L ∈ N .The CAR C ∗ -algebra U ≡ U L of the full system, whose norm is denoted by k · k U , is by definition thecompletion of the normed ∗ -algebra U . It is separable, by finite dimensionality of U Λ for Λ ∈ P f .Equivalently, the C ∗ -algebra U is the universal C ∗ -algebra [77, Section II.8.3] associated with therelations (2) for all x, y ∈ L and s , t ∈ S . The (real) Banach subspace of all self-adjoint elements of U is denoted by U R U . ∗ -Automorphisms of the CAR Algebra Parity: Observe that the condition σ ( a x, s ) = − a x, s , x ∈ Λ , s ∈ S , (4)defines a unique ∗ -automorphism σ of the C ∗ -algebra U . Elements A , A ∈ U satisfying σ ( A ) = A and σ ( A ) = − A are respectively called even and odd . Note that the set U + ≡ U + L . = { A ∈ U : A = σ ( A ) } ⊆ U (5)of all even elements is a ∗ -algebra. By continuity of σ , U + is additionally norm-closed and, hence, a C ∗ -subalgebra of U . Even elements are crucial here since, for any finite subsets Λ (1) , Λ (2) ∈ P f with Λ (1) ∩ Λ (2) = ∅ , [ A , A ] . = A A − A A = 0 , A ∈ U Λ (1) ∩ U + , A ∈ U Λ (2) . (6)However, this relation is wrong in general when A is not an even element. For instance, the CAR (2)trivially yields [ a x, s , a y, t ] = 2 a x, s a y, t = 0 for any x, y ∈ L and s , t ∈ S such that ( x, s) = ( y, t) .The condition (6) is the expression of the local causality in quantum field theory. Using well-known constructions , the C ∗ -algebra U , generated by anticommuting elements, can be recoveredfrom U + . As a consequence, the C ∗ -algebra U + should thus be seen as more fundamental than U in Physics. In fact, U corresponds in this context to the so-called local field algebra. See, e.g., [78,Sections 4.8 and 6].Note that any element A ∈ U can be written as a sum of even and odd elements, respectivelydenoted by A + and A − : A = A + + A − with A ± . = 12 ( A ± σ ( A )) . (7)This directly follows from the fact that σ is an involution (i.e., σ ◦ σ = U ). More precisely, the so-called sector theory of quantum field theory. L = Z d is an important case here, because invariant states under the action of the group ( Z d , +) of lattice translations on U ≡ U Z d are pivotal in the full description of macroscopic long-range dynamics. The translations in U refer to the group homomorphism x α x from Z d to thegroup of ∗ -automorphisms of U , which is uniquely defined by the condition α x ( a y, s ) = a y + x, s , y ∈ Z d , s ∈ S . (8)This group homomorphism is used below to define the notion of (space) periodicity of states of lattice-fermion systems (Section 2.5), as well as the notion of translation-invariance of interactions (Section3.1).Permutations: Let Π be the set of all bijective mappings from L to itself which leave all but finitelymany elements invariant. It is a group with respect to the composition of mappings. The condition p π : a x, s a π ( x ) , s , x ∈ L , s ∈ S , (9)defines a group homomorphism π p π from Π to the group of ∗ -automorphisms of U . For any Λ ∈ P f , we denote by E Λ . = { ρ Λ ∈ U ∗ Λ : ρ Λ ≥ , ρ Λ ( ) = 1 } = { ρ Λ ∈ U ∗ Λ : k ρ Λ k U ∗ Λ = ρ Λ ( ) = 1 } (10)the set of all states on U Λ . By finite dimensionality of U Λ for Λ ∈ P f , the set E Λ is a norm-compactconvex subset of the dual space U ∗ Λ . It is not a simplex, being affinely equivalent to the set of stateson a matrix algebra. It does not have of course a dense set of extreme points. In comparison, thestructure of the set of states for infinite systems is more subtle.The state space associated with U is defined by E . = { ρ ∈ U ∗ : ρ ≥ , ρ ( ) = 1 } = { ρ ∈ U ∗ : k ρ k U ∗ = ρ ( ) = 1 } . (11)In particular, if L ∈ P f then E = E L . In any case, for U is a separable Banach space, E is ametrizable and weak ∗ -compact convex subset of the dual space U ∗ , by [91, Theorems 3.15 and 3.16].It is also the state space of the classical dynamics [63, Definition 2.1] introduced later on.By the Krein-Milman theorem [91, Theorem 3.23], E is the weak ∗ -closure of the convex hull ofthe (non-empty) set E ( E ) of its extreme points: E = co E ( E ) . (12)When L is an infinite set, U is antiliminal and simple, because it is a UHF (uniformly hyperfinite)algebra. See, e.g., [63, Section 8]. Therefore, by [63, Lemma 8.5], the state space E has a weak ∗ -dense subset of extreme points: E = E ( E ) . (13)This fact is well-known and already discussed in [81, p. 226]. As a matter of fact, the property of aconvex weak ∗ -compact set having a weak ∗ -dense set of extreme points is not accidental, but generic in infinite dimension, by [63, Theorem 2.4]. More precisely, by [63, Theorem 2.4], the set of all convex weak ∗ -compact subsets of the dual X ∗ of an infinite-dimensional separable Banach space X is a weak ∗ -Hausdorff-dense G δ subset of the space of convex weak ∗ -compactsubsets of X ∗ . Λ ∈ P f , the restriction of a state ρ ∈ E to U Λ yields a state in E Λ , always denoted by ρ Λ ∈ E Λ . Conversely, for any Λ ∈ P f , a state ρ Λ ∈ E Λ can be seen as a state ˜ ρ on U by setting ˜ ρ ( AB ) = ρ Λ ( A )tr( B ) , A ∈ U Λ , B ∈ U Z , Z ∈ P f , Z ∩
Λ = ∅ , (14)where tr is the unique tracial state on U , also named normalized trace, see [82, Section 4.2]. Inparticular, similar to (3), the set E . = [ L ∈ N E Λ L can be seen as a weak ∗ -dense subset of E , by density of U ⊆ U . As explained below Equation (5), recall that the C ∗ -algebra U + should thus be seen as more funda-mental than U in Physics, because of the local causality in quantum field theory. As a consequence,states on the C ∗ -algebra U + should be seen as being the physically relevant ones. As it is explicitlyshown in the proof of Proposition 2.1, the set of states on U + is canonically identified with the set of even states on U , defined by E + . = { ρ ∈ E : ρ ◦ σ = ρ } , (15) σ being the unique ∗ -automorphism of U satisfying (4). E + is again a metrizable and weak ∗ -compactconvex set and has a (non-empty) set E ( E + ) of extreme points such that E + = co E ( E + ) , by the Krein-Milman theorem [91, Theorem 3.23]. Like for the space E of all states (cf. (13)), if L isan infinite set then E ( E + ) is a dense subset of the set of all even states: Proposition 2.1 (Weak ∗ -density of extreme even states) Let L ⊆ Z d be an infinite set. The set E ( E + ) of extreme points of E + is a weak ∗ -dense subset of E + ,i.e., E + = E ( E + ) . Proof.
Let ˜ E + be the set of all states on U + . Obviously, E + ⊆ ˜ E + by seeing any state on E as a stateon U + , by restriction. Conversely, for any state ˜ ρ ∈ ˜ E + , we define the linear functional ρ . = ˜ ρ ◦ (cid:18) σ + U (cid:19) ∈ U ∗ . Note that k σ k B ( U ∗ ) = 1 because σ is a ∗ -automorphism of a C ∗ -algebra. Since ρ ( ) = 1 and k ρ k U ∗ ≤ , we deduce from [79, Proposition 2.3.11] that ρ ∈ E . By construction, the restriction of ρ to U + is ˜ ρ while ρ ( A ) = 0 for all odd elements A ∈ U . In other words, ρ ∈ E + and one can identify ˜ E + with E + . It easy to check that the mapping ˜ ρ ρ is an affine weak ∗ homeomorphism from ˜ E + to E + .By [80], if L is an infinite set then U + is ∗ -isomorphic to U . Thus, by (13), the assertion follows. Remark 2.2 (States of quantum-spin systems)
As already mentioned, all results presented here can be extended to quantum-spin systems on a lattice,the corresponding C ∗ -algebra being an infinite tensor product of finite-dimensional matrix algebrasattached to each lattice site x ∈ L = Z d . In this case, there is no even property to consider, whichcorresponds to take E + = E in all the discussions. (14) is well defined because of [82, Theorem 11.2], the tracial state tr being even. L = Z d . Consider thesub-groups ( Z d~ℓ , +) ⊆ ( Z d , +) , ~ℓ ∈ N d , where Z d~ℓ . = ℓ Z × · · · × ℓ d Z . Any state ρ ∈ E satisfying ρ ◦ α x = ρ for all x ∈ Z d~ℓ is called Z d~ℓ -invariant on U or ~ℓ -periodic , α x being the unique ∗ -automorphism of U satisfying (8). Translation-invariant states refer to (1 , · · · , -periodic states. The set of all periodic states is denoted by E p . = [ ~ℓ ∈ N d E ~ℓ , (16)where, for any ~ℓ ∈ N d , E ~ℓ . = (cid:8) ρ ∈ E : ρ ◦ α x = ρ for all x ∈ Z d~ℓ (cid:9) . (17)By [46, Lemma 1.8], periodic states are even and form a weak ∗ -dense subset of even states: Proposition 2.3 (Weak ∗ -density of periodic states) Let L = Z d . The set E p of periodic states is a weak ∗ -dense set of E + , i.e., E + = E p . Proof.
For any ρ ∈ E + and n ∈ N , we define the state ˜ ρ n to be some (2 n +1 , . . . , n +1) -periodic statefor which (˜ ρ n ) Λ n = ρ Λ n in the cubic box Λ n (1). Such a state always exists because of [82, Theorem11.2], since ρ is, by definition, even. Clearly, { ˜ ρ n } n ∈ N converges, as n → ∞ , towards ρ ∈ E + withrespect to the weak ∗ -topology, by density of U ⊆ U .The sets E ~ℓ , ~ℓ ∈ N d , of ~ℓ -periodic states all share the same peculiar geometrical structure: Like theset E of all states for L = Z d , they are metrizable and weak ∗ -compact convex sets with a weak ∗ -denseset E ( E ~ℓ ) of extreme points: E ~ℓ = co E ( E ~ℓ ) = E ( E ~ℓ ) , ~ℓ ∈ N d . (18)Compare with Equations (12)-(13). In fact, up to an affine homeomorphism, for any ~ℓ ∈ N d , E ~ℓ is theso-called Poulsen simplex [46, Theorem 1.12]. This property is well-known and also holds true forquantum-spin systems [79, p. 405-406, 464]. The fact that all E ~ℓ , ~ℓ ∈ N d , have the same topologicalstructure is not so surprising since, for any fixed ~ℓ ∈ N d , we can redefine the spin set S = S ~ℓ and, asa consequence, the CAR algebra U = U ~ℓ to see any ~ℓ -periodic state ρ ∈ E ~ℓ as a translation-invariantstate on the new CAR algebra U ~ℓ . By [63, Proposition 6.14], E + can also be seen as the weak ∗ -Hausdorff limit of the increasing sequence { E ~ℓ } ~ℓ ∈ N d of weak ∗ -compact simplices with weak ∗ -denseset of extreme points.By [46, Theorem 1.16] note that the set of all ( ~ℓ -) ergodic states, as defined by [46, Definition1.15] for any ~ℓ ∈ N d , is equal to E p . = [ ~ℓ ∈ N d E (cid:0) E ~ℓ (cid:1) ⊆ E p ⊆ E + . (19)By Proposition 2.3, E p is also a weak ∗ -dense set of E + .The set E p is important because, in all cyclic representations of U associated with a state ρ ∈ E p ,the infinite-volume dynamics of interacting lattice fermions with long-range interactions exists. Theergodicity of states of E p plays a crucial rˆole in this context.Instead of the set E p of periodic states, previous studies extracting classical dynamics for lattice-fermion and quantum-spin systems with long-range interactions (cf. [5, 6, 12–32]) use classical flowswithin the subset E Π . = { ρ ∈ E : ρ ◦ p π = ρ for all π ∈ Π } (20)10f permutation-invariant states, p π being the unique ∗ -automorphism of U satisfying (9). This set hasa much simpler structure than the set E p of periodic states: E Π ⊆ \ ~ℓ ∈ N d E ~ℓ ⊆ E p . Compare this assertion with (16). E Π is a closed metrizable face of E ~ℓ for all ~ℓ ∈ N d and a Bauersimplex, i.e., a (Choquet) simplex whose set of extreme points is closed. Indeed, by Størmer’s theorem[46, Theorem 5.2], extreme points of E Π are so-called product states . See [46, Section 5.1] for moredetails on permutation-invariant states. A (complex) interaction is a mapping
Φ : P f → U + such that Φ Λ ∈ U Λ for any Λ ∈ P f . The set ofall interactions can be naturally endowed with the structure of a complex vector space as follows: (Φ + ˜Φ) Λ . = Φ Λ + ˜Φ Λ and ( λ Φ) Λ . = λ Φ Λ (21)for all interactions Φ , ˜Φ and λ ∈ C . The mapping Φ Φ ∗ . = (Φ ∗ Λ ) Λ ∈P f (22)is a natural involution on the vector space of interactions. Self-adjoint interactions are interactions Φ satisfying Φ = Φ ∗ . The set of all self-adjoint interactions forms a real subspace of the space ofall interactions. Any interaction Φ can be decomposed into its real and imaginary parts, which areself-adjoint interactions respectively defined by Re { Φ } . = 12 (Φ ∗ + Φ) and Im { Φ } . = 12 i (Φ − Φ ∗ ) . (23)We now define a Banach space W of short-range interactions by introducing a norm for inter-actions that take into account their spacial decay. To this end, we use a positive-valued symmetricfunction F : L → (0 , with maximum value F ( x, x ) = 1 for all x ∈ L . Like for instance in [76],we impose the following conditions on F : • Summability on L . k F k , L . = sup y ∈ L X x ∈ L F ( x, y ) ∈ [1 , ∞ ) . (24) • Bounded convolution constant. D . = sup x,y ∈ L X z ∈ L F ( x, z ) F ( z, y ) F ( x, y ) < ∞ . (25)Examples of functions F : L → (0 , satisfying (24)-(25) for any lattice L ⊆ Z d ( d ∈ N ) are givenby F ( x, y ) = (1 + | x − y | ) − ( d + ǫ ) or F ( x, y ) = e − ς | x − y | (1 + | x − y | ) − ( d + ǫ ) (26)for every ς, ǫ ∈ R + . In all the paper, (24)-(25) are assumed to be satisfied.11hen, a norm for interactions Φ is defined by k Φ k W . = sup x,y ∈ L X Λ ∈P f , Λ ⊇{ x,y } k Φ Λ k U F ( x, y ) (27)and W ≡ W ( L ) ≡ ( W , k·k W ) denotes the separable Banach space of interactions Φ satisfying k Φ k W < ∞ . Elements Φ ∈ W arenamed short-range interactions on L ⊆ Z d . The (real) Banach subspace of all self-adjoint interactionsis denoted by W R W , similar to U R U .By definition, an interaction Φ on L = Z d is translation-invariant if, for all x ∈ Z d and Λ ∈ P f , Φ Λ+ x = α x (Φ Λ ) , where Λ + x . = (cid:8) y + x ∈ Z d : y ∈ Λ (cid:9) . (28)Recall that { α x } x ∈ Z d is the family of (translation) ∗ -automorphisms of U defined by (8). We denoteby W W the (separable) Banach subspace of translation-invariant, short-range interactions on L = Z d .For any Φ ∈ W and ~ℓ ∈ N d , we define the even observable e Φ ,~ℓ . = 1 ℓ · · · ℓ d X x =( x ,...,x d ) , x i ∈{ ,...,ℓ i − } X Z∈P f , Z∋ x Φ Z |Z| . (29)From (24) and (27), note that k e Φ ,~ℓ k U ≤ k F k , L k Φ k W , Φ ∈ W , ~ℓ ∈ N d . (30)For any self-adjoint, translation-invariant interaction Φ , they refer to the energy density observablesof [46, Eq. (1.16)]. See also Proposition 3.2 below. Additionally, for any Λ ∈ P f , we define theclosed subspaces W Λ . = { Φ ∈ W : Φ Z = 0 whenever Z * Λ , Z ∋ } (31)of interactions that are both finite-range and translation-invariant. Note that, for any Λ ∈ P f and ~ℓ ∈ N d , W Λ ⊆ n Φ ∈ W : e Φ ,~ℓ ∈ U Λ ( ~ℓ ) o ⊆ W , (32)where Λ ( ~ℓ ) . = ∪ { Λ + x : x = ( x , . . . , x d ) , x i ∈ { , . . . , ℓ i − }} ∈ P f . (33)Recall that, for Λ ∈ P f , U Λ ⊆ U is the finite-dimensional unital C ∗ -algebra generated by elements { a x, s } x ∈ Λ , s ∈ S satisfying the CAR (2). Similar to (3), W . = [ L ∈ N W Λ L ⊆ W (34)is a dense subspace of W . Elements of W are finite-range , translation-invariant interactions. This follows from the continuity and linearity of the mappings Φ Φ Z for all Z ∈ P f . .2 Local Energy Elements We define a sequence of local elements associated with any complex interaction Φ ∈ W as follows: Definition 3.1 (Local energy)
The local energy elements associated with a complex interaction Φ ∈ W are given by U Φ L . = X Λ ⊆ Λ L Φ Λ ∈ U Λ L ∩ U + , L ∈ N . If Φ ∈ W R , then ( U Φ L ) L ∈ N ∈ U R and so, Definition 3.1 yields a sequence of local Hamiltonians,which are used to generate finite-volume dynamics.By straightforward estimates using (24) and (27), as in (30), note that, for any complex interactions Φ , Ψ ∈ W , (cid:13)(cid:13) U Φ L − U Ψ L (cid:13)(cid:13) U = (cid:13)(cid:13) U Φ − Ψ L (cid:13)(cid:13) U ≤ | Λ L | k F k , L k Φ − Ψ k W , L ∈ N . (35)In addition, similar to [46, Lemma 1.32], local energy elements yield energy densities for translation-invariant short-range interactions: Proposition 3.2 (Energy density of periodic states)
For any ~ℓ ∈ N d , ~ℓ -periodic state ρ ∈ E ~ℓ (17) and translation-invariant complex interaction Φ ∈ W , lim L →∞ ρ (cid:0) U Φ L (cid:1) | Λ L | = ρ ( e Φ ,~ℓ ) with e Φ ,~ℓ being the even observable defined by (29). Proof.
All arguments of the proof can be found in [46, Lemma 4.17].By Proposition 3.2, the energy density e Φ ( ρ ) . = lim L →∞ ρ (cid:0) U Φ L (cid:1) | Λ L | , Φ ∈ W , exists for all periodic states, which form a weak ∗ -dense subset E p of the set E + of even states, byProposition 2.3. Note that there are infinitely many (uncountable) other states with this property.Examples of non-periodic states for which e Φ ( ρ ) exists can be constructed by using KMS statesassociated with random interactions together with the Akcoglu-Krengel ergodic theorem [84]. See,e.g., [85]. For all ρ ∈ E such that e Φ ( ρ ) exists, observe from Inequality (35) that | e Φ ( ρ ) − e Ψ ( ρ ) | ≤ k F k , L k Φ − Ψ k W , Φ , Ψ ∈ W . (36) For any short-range interaction Φ ∈ W , the elements of Definition 3.1 determine a sequence ofbounded operators on U : Definition 3.3 (Derivations on the CAR algebra for short-range interactions)
The derivations { δ Φ L } L ∈ N ⊆ B ( U ) associated with any interaction Φ ∈ W are defined by δ Φ L ( A ) . = i (cid:2) U Φ L , A (cid:3) . = i (cid:0) U Φ L A − AU Φ L (cid:1) , A ∈ U , L ∈ N . U since, for any Φ ∈ W and L ∈ N , δ Φ L ( AB ) = δ Φ L ( A ) B + Aδ Φ L ( B ) , A, B ∈ U . (37)They are symmetric (or ∗ -derivations) when Φ ∈ W R : δ Φ L ( A ) ∗ = δ Φ L ( A ∗ ) , A ∈ U , L ∈ N . (38)In the thermodynamic limit L → ∞ , this sequence of derivations leads to a limit derivationdefined on the (dense) subset U (3) of local elements: Proposition 3.4 (Strong convergence of finite-volume derivations)
For any Φ ∈ W , Λ ∈ P f , A ∈ U Λ and L ≥ L ≥ , (cid:13)(cid:13) δ Φ L ( A ) − δ Φ L ( A ) (cid:13)(cid:13) U ≤ | Λ | k A k U k Φ k W sup y ∈ Λ X x ∈ Λ cL F ( x, y ) and sup L ∈ N (cid:13)(cid:13) δ Φ L ( A ) (cid:13)(cid:13) U ≤ | Λ | k A k U k Φ k W k F k , L . Proof.
Fixing all parameters of the proposition, we straightforwardly get the estimate (cid:13)(cid:13) δ Φ L ( A ) − δ Φ L ( A ) (cid:13)(cid:13) U ≤ X y ∈ Λ X x ∈ Λ cL X Z∈P f , Z⊇{ x,y } k [Φ Z , A ] k U ≤ k A k U k Φ k W X y ∈ Λ X x ∈ Λ cL F ( x, y ) , which implies the first assertion. The second assertion is even simpler to prove. We omit the details. Corollary 3.5 (Generators of infinite-volume short-range dynamics)
For any Φ ∈ W and A ∈ U , the limit δ Φ ( A ) = lim L →∞ δ Φ L ( A ) exists and defines a (densely defined) derivation δ Φ from U ⊆ U to U satisfying the following bound: (cid:13)(cid:13) δ Φ ( A ) (cid:13)(cid:13) U ≤ | Λ | k A k U k Φ k W k F k , L , A ∈ U Λ , Λ ∈ P f . Additionally, δ Φ is symmetric (or a ∗ -derivation) whenever Φ ∈ W R . Proof.
Combine Proposition 3.4 with Equations (37)-(38) and the completeness of U . Remark 3.6 (Closure of limit derivations) If Φ ∈ W R then the symmetric derivation δ Φ is (norm-) closable [76, Lemma 4.6]. It is proven fromits dissipativity [86, Definition 1.4.6, Proposition 1.4.7], which is, in turn, deduced from [86, Theorem1.4.9] because A ∈ U and A ≥ implies A / ∈ U . Moreover, its closure generates a stronglycontinuous group of ∗ -automorphisms of U , by [76, Theorem 4.8]. See also Proposition 3.7 below. .4 Dynamics Generated by Short-Range Interactions We now consider time-dependent interactions. Let Ψ ∈ C ( R ; W ) be a continuous function from R tothe Banach space W of interactions on L ⊆ Z d . Then, for any L ∈ N , there is a unique (fundamental)solution ( τ ( L, Ψ) t,s ) s,t ∈ R in B ( U ) to the (finite-volume) non-autonomous evolution equations ∀ s, t ∈ R : ∂ s τ ( L, Ψ) t,s = − δ Ψ( s ) L ◦ τ ( L, Ψ) t,s , τ ( L, Ψ) t,t = U , (39)and ∀ s, t ∈ R : ∂ t τ ( L, Ψ) t,s = τ ( L, Ψ) t,s ◦ δ Ψ( t ) L , τ ( L, Ψ) s,s = U . (40)In these two equations, U refers to the identity mapping of U . Note also that, for any L ∈ N and Ψ ∈ C ( R ; W ) , ( τ ( L, Ψ) t,s ) s,t ∈ R is a continuous two-parameter family of bounded operators that satisfiesthe (reverse) cocycle property ∀ s, r, t ∈ R : τ ( L, Ψ) t,s = τ ( L, Ψ) r,s τ ( L, Ψ) t,r . If Ψ ∈ C ( R ; W R ) then δ Ψ( t ) L is always a symmetric derivation (or ∗ -derivation) and thus, in thiscase, τ ( L, Ψ) t,s is a ∗ -automorphism of U for all lengths L ∈ N and times s, t ∈ R . Moreover, in thethermodynamic limit L → ∞ , the family ( τ ( L, Ψ) t,s ) s,t ∈ R strongly converges to a strongly continuoustwo-parameter family of ∗ -automorphisms of U , associated with the family { δ Ψ( t ) } t ∈ R of limit sym-metric derivations of Corollary 3.5: Proposition 3.7 (Infinite-volume short-range dynamics)
For any Ψ ∈ C ( R ; W R ) , as L → ∞ , ( τ ( L, Ψ) t,s ) s,t ∈ R converges strongly, uniformly for s, t on com-pacta, to a strongly continuous two-parameter family ( τ Ψ t,s ) s,t ∈ R of ∗ -automorphisms of U , which isthe unique solution in B ( U ) to the non-autonomous evolutions equation ∀ s, t ∈ R : ∂ t τ Ψ t,s = τ Ψ t,s ◦ δ Ψ( t ) , τ Ψ s,s = U , (41) in the strong sense on the dense subspace U ⊆ U . In particular, it satisfies the reverse cocycleproperty: ∀ s, r, t ∈ R : τ Ψ t,s = τ Ψ r,s τ Ψ t,r . (42) Proof.
See [76, Corollary 5.2].It is convenient to introduce at this point the notation ∂ Ψ Λ . = { x ∈ Λ : ∃Z ∈ P f with x ∈ Z , Ψ Z = 0 and Z ∩ Λ = 0 , Z ∩ Λ c = 0 } for any interaction Ψ and any finite subset Λ ∈ P f with complement Λ c . = L \ Λ . For any s, t ∈ R , t ∧ s and t ∨ s stand, respectively, for the minimum and maximum of the set { s, t } . We are nowin a position to give additional estimates on the limit dynamics, like the celebrated Lieb-Robinsonbounds: Proposition 3.8 (Estimates on short-range dynamics)
For any Ψ ∈ C ( R ; W R ) , ( τ Ψ t,s ) s,t ∈ R satisfies the following bounds: (i) Lieb-Robinson bounds (cf. (6)). For any s, t ∈ R , sets Λ (1) , Λ (2) ∈ P f with Λ (1) ∩ Λ (2) = ∅ , everyeven element A ∈ U + ∩ U Λ (1) and all A ∈ U Λ (2) , (cid:13)(cid:13) [ τ Ψ t,s ( A ) , A ] (cid:13)(cid:13) U ≤ D − k A k U k A k U (cid:16) e D R t ∨ st ∧ s k Ψ( α ) k W d α − (cid:17) X x ∈ ∂ Ψ Λ (1) X y ∈ Λ (2) F ( x, y ) . Rate of convergence. For any s, t ∈ R , Λ ∈ P f , A ∈ U Λ and L ∈ N such that Λ ⊆ Λ L , (cid:13)(cid:13)(cid:13) τ Ψ t,s ( A ) − τ ( L, Ψ) t,s ( A ) (cid:13)(cid:13)(cid:13) U ≤ k A k U Z t ∨ st ∧ s (cid:16) k Ψ ( α ) k W e D R α ∨ sα ∧ s k Ψ( α ) k W d α (cid:17) d α X y ∈ L \ Λ L X x ∈ Λ F ( x, y ) . (iii) Lipschitz continuity with respect to Ψ . For any s, t ∈ R , Ψ , ˜Ψ ∈ C (cid:0) R ; W R (cid:1) , Λ ∈ P f and A ∈ U Λ , (cid:13)(cid:13)(cid:13) τ Ψ t,s ( A ) − τ ˜Ψ t,s ( A ) (cid:13)(cid:13)(cid:13) U ≤ | Λ | k A k U k F k , L Z t ∨ st ∧ s e D R t ∨ αt ∧ α k Ψ( α ) k W d α (cid:13)(cid:13)(cid:13) ˜Ψ ( α ) − Ψ ( α ) (cid:13)(cid:13)(cid:13) W d α . (iv) Uniform continuity with respect to times. For any s , s , t , t ∈ R , Ψ ∈ C ( R ; W R ) , Λ ∈ P f and A ∈ U Λ , (cid:13)(cid:13) τ Ψ t ,s ( A ) − τ Ψ t ,s ( A ) (cid:13)(cid:13) U ≤ | Λ | k A k U k F k , L (cid:18)Z t ∨ t t ∧ t k Ψ ( α ) k W d α + Z s ∨ s s ∧ s e D R t ∨ αt ∧ α k Ψ( α ) k W d α k Ψ ( α ) k W d α (cid:19) . Proof.
The proof of Assertion (i) is almost done in [76, Theorem 5.1, Corollary 5.2 (ii)]. However,the bound there refers to the supremum with respect to α of the norm k Ψ ( α ) k W . Here, we need aslightly more accurate estimate (a point-wise estimate). In fact, by [76, equation after Eq. (5.4)] andsimilar arguments as in [76, Eqs. (4.16)-(4.18)], we get Assertion (i). Then, Assertion (ii) is provenexactly like in the proof of [76, Theorem 5.1 (ii)], by replacing [76, Theorem 5.1 (i)] with Assertion(i). It remains to prove Assertions (iii) and (iv). We start with (iii):For any s, t ∈ R , Ψ , ˜Ψ ∈ C ( R ; W R ) , Λ ∈ P f , A ∈ U Λ and any sufficiently large L ∈ N such that U Λ ⊆ U Λ L , by (39) and Proposition 3.7, observe that τ ˜Ψ t,s ( A ) − τ ( L, Ψ) t,s ( A ) = Z ts τ ˜Ψ α,s ◦ (cid:16) δ ˜Ψ( α ) − δ Ψ( α ) (cid:17) ◦ τ ( L, Ψ) t,α ( A ) d α (43) + Z ts τ ˜Ψ α,s ◦ (cid:16) δ Ψ( α ) − δ ( L, Ψ( α )) (cid:17) ◦ τ ( L, Ψ) t,α ( A ) d α . By Definitions 3.1, 3.3 and Corollary 3.5, it follows that (cid:13)(cid:13)(cid:13) τ ˜Ψ t,s ( A ) − τ ( L, Ψ) t,s ( A ) (cid:13)(cid:13)(cid:13) U ≤ Z t ∨ st ∧ s X Z∈P f (cid:13)(cid:13)(cid:13)h(cid:16) ˜Ψ ( α ) − Ψ ( α ) (cid:17) Z , τ ( L, Ψ) t,α ( A ) i(cid:13)(cid:13)(cid:13) U d α (44) + Z t ∨ st ∧ s X Z∈P f , Z∩ Λ cL = ∅ (cid:13)(cid:13)(cid:13)h Ψ ( α ) Z , τ ( L, Ψ) t,α ( A ) i(cid:13)(cid:13)(cid:13) U d α , where Λ cL . = L \ Λ L is the complement of the cubic box Λ L (1). Now, by using Assertion (i) for Ψ L ∈ C ( R ; W R ) defined , for L ∈ N , by Ψ L ( t ) Z = Ψ ( t ) Z [ Z ⊆ Λ L ] , Z ∈ P f , t ∈ R , together with k Ψ L ( t ) k W ≤ k Ψ ( t ) k W , (24)-(25) and Equation (27), we get that X Z∈P f , Z∩ Λ cL = ∅ (cid:13)(cid:13)(cid:13)h Ψ ( α ) Z , τ ( L, Ψ) t,α ( A ) i(cid:13)(cid:13)(cid:13) U (45) ≤ k A k U (cid:16) e D R t ∨ αt ∧ α k Ψ( α ) k W d α (cid:17) k Ψ ( α ) k W X x ∈ Λ X y ∈ Λ cL F ( x, y ) [ p ] = 1 when the proposition p is true and , else. X Z∈P f (cid:13)(cid:13)(cid:13)h(cid:16) ˜Ψ ( α ) − Ψ ( α ) (cid:17) Z , τ ( L, Ψ) t,α ( A ) i(cid:13)(cid:13)(cid:13) U (46) ≤ | Λ | k A k U k F k , L e D R t ∨ αt ∧ α k Ψ( α ) k W d α (cid:13)(cid:13)(cid:13) ˜Ψ ( α ) − Ψ ( α ) (cid:13)(cid:13)(cid:13) W . To prove these two inequalities, see [76, Eqs. (4.25)-(4.25)]. Since lim L →∞ X x ∈ Λ X y ∈ Λ cL F ( x, y ) = 0 , (47)because of (24), Assertion (iii) follows by combining (44)-(47) with Assertion (ii).Finally, to get Assertion (iv), note first that Corollary 3.5 directly implies that (cid:13)(cid:13) τ Ψ t ,s ( A ) − τ Ψ t ,s ( A ) (cid:13)(cid:13) U ≤ | Λ | k A k U k F k , L Z t ∨ t t ∧ t k Ψ ( α ) k W d α (48)for any s, t , t ∈ R , Ψ ∈ C ( R ; W R ) , Λ ∈ P f and A ∈ U Λ . Meanwhile, fix s , s , t ∈ R , Ψ ∈ C ( R ; W R ) , Λ ∈ P f and A ∈ U Λ . By Assertion (ii), for any ε ∈ R + there is L ∈ R such that (cid:13)(cid:13) τ Ψ t,s ( A ) − τ Ψ t,s ( A ) (cid:13)(cid:13) U ≤ (cid:13)(cid:13)(cid:13) τ ( L, Ψ) t,s ( A ) − τ ( L, Ψ) t,s ( A ) (cid:13)(cid:13)(cid:13) U + ε , which, by Equation (39), implies that (cid:13)(cid:13) τ Ψ t,s ( A ) − τ Ψ t,s ( A ) (cid:13)(cid:13) U ≤ Z s ∨ s s ∧ s X Z∈P f (cid:13)(cid:13)(cid:13)h Ψ ( α ) Z , τ ( L, Ψ) t,α ( A ) i(cid:13)(cid:13)(cid:13) U d α + ε . (49)Similar to (46), it follows that (cid:13)(cid:13) τ Ψ t,s ( A ) − τ Ψ t,s ( A ) (cid:13)(cid:13) U ≤ | Λ | k A k U k F k , L Z s ∨ s s ∧ s e D R t ∨ αt ∧ α k Ψ( α ) k W d α k Ψ ( α ) k W d α . (50)Assertion (iv) is a combination of (48) and (50). Fix now L = Z d , d ∈ N . Let S be the unit sphere of the Banach space W of translation-invariant(complex) interactions. Observe that any finite signed Borel measure a on S defines an interaction Z S Ψ a (dΨ) ∈ W (51)by (cid:18)Z S Ψ a (dΨ) (cid:19) Λ . = Z S Ψ Λ a (dΨ) , Λ ∈ P f . (52)This last integral is well-defined because, for each Λ ∈ P f , the integrand is an absolutely integrablefunction taking values in a finite-dimensional normed space, which is U Λ . Below, we extend thisobservation to define long-range, or mean-field, models. Note that (51) can also be seen as a Bochner17ntegral because the measure a is finite and W is a separable Banach space. See, e.g., [94, Theorems1.1 and 1.2].For any n ∈ N and any finite signed Borel measure a on the Cartesian product S n (endowed withits product topology), we define the finite signed Borel measure a ∗ to be the pushforward of a throughthe automorphism (cid:0) Ψ (1) , . . . , Ψ ( n ) (cid:1) ((Ψ ( n ) ) ∗ , . . . , (Ψ (1) ) ∗ ) ∈ S n (53)of S n as a topological space. A finite signed Borel measure a on S n is, by definition, self-adjoint whenever a ∗ = a .For any n ∈ N , we denote the space of self-adjoint , finite, signed Borel measures on S n by S ( S n ) ,which is a real Banach space with the norm of the total variation k a k S ( S n ) . = | a | ( S n ) , n ∈ N . (54)The set of all sequences a ≡ ( a n ) n ∈ N of self-adjoint, finite, signed Borel measures a n ∈ S ( S n ) is areal vector space, where ( a + ˜ a ) n . = a n + ˜ a n and ( λ a ) n . = λ a n , n ∈ N , for any sequence a ≡ ( a n ) n ∈ N , ˜ a ≡ (˜ a n ) n ∈ N and all λ ∈ R . We define the (real) space S to be the setof all sequences a ≡ ( a n ) n ∈ N of self-adjoint, finite signed Borel measures a n ∈ S ( S n ) with k a k S . = X n ∈ N n k F k n − , Z d k a n k S ( S n ) < ∞ , (55)where we recall that F : Z d × Z d → (0 , is the decay function with maximum value F ( x, x ) = 1 for x ∈ Z d and satisfying Conditions (24)-(25). See also (27). Observe that ( S , k · k S ) is a real Banachspace. We are now in a position to define long-range models: Definition 4.1 (Long-range models)
The (real) Banach space of long-range models is the space M . = W R × S along with the norm k m k M . = k Φ k W + k a k S , m . = (Φ , a ) ∈ M . Note that W R and S are canonically seen as subspaces of M , i.e., W R ⊆ M and S ⊆ M . We emphasize that long-range models are not necessarily translation-invariant since, obviously, M . = (cid:0) W ∩ W R (cid:1) × S M . (56)Similar to (29)-(31), we define the subsets M Λ . = W R × S Λ ⊆ M , Λ ∈ P f , (57)where, for any Λ ∈ P f , S Λ . = { ( a n ) n ∈ N ∈ S : ∀ n ∈ N , | a n | ( S n ) = | a n | (( S ∩ W Λ ) n ) } . (58)Note that the short-range part of models in M Λ is not necessarily finite-range, but their long-rangeinteractions are built from finite-range interactions. Similar to (34) we can define a dense subspace M . = [ L ∈ N M Λ L ⊆ M . (59)18 .2 Local Hamiltonians and Derivations on the CAR Algebra Similar to Definition 3.1, we define a sequence of local Hamiltonians for any model m ∈ M : At anyfixed n ∈ N and L ∈ N , the mapping (cid:0) Ψ (1) , . . . , Ψ ( n ) (cid:1) U Ψ (1) L · · · U Ψ ( n ) L from S n to U is continuous (see (35)), and so, for any long-range model m ∈ M , we can define thefollowing self-adjoint element of U : Definition 4.2 (Hamiltonians)
The local Hamiltonians of any model m ∈ M are U m L . = U Φ L + X n ∈ N | Λ L | n − Z S n U Ψ (1) L · · · U Ψ ( n ) L a n (cid:0) dΨ (1) , . . . , dΨ ( n ) (cid:1) , L ∈ N . Note that U m L ∈ U Λ L ∩ U R ∩ U + , L ∈ N , and straightforward estimates using Equations (35), (54)-(55) and Definition 4.1 yield the bound k U m L k U ≤ | Λ L | k F k , L k m k M , L ∈ N . (60)(This upper bound is relatively coarse, in general.)All well-established Hamiltonians for lattice fermions in condensed matter physics can be writtenas U m L via some model m ∈ M . In Section 9, we shortly explain an important example related tothe BCS theory. Other examples can also be found in [46, Section 2.2] (for instance, in relation withthe forward scattering approximation) as well as in [67] which refers to the dynamics of the strong-coupling BCS-Hubbard model. The latter is an interesting model because it predicts the existenceof a superconductor-Mott insulator phase transition, like in cuprates which must be doped to becomesuperconductors. See [44] for more details.Note that the long-range character of m ∈ M with local internal energy U m L – as compared tothe usual models defined from short-range interactions Φ ∈ W only – can be understood as follows:Take m = (Φ , a ) ∈ M . For each fixed ǫ ∈ (0 , , we define the long-range truncation of U Φ L , theshort-range component of the total internal energy (Definition 3.1), by U Φ L,ǫ . = X Λ ⊆ Λ L , ø (Λ) >ǫL Φ Λ , where the function ø (Λ) is the diameter of Λ ∈ P f . Analogously, the long-range truncation of theinternal energy ( U m L − U Φ L ) associated with the long-range part of m is by definition equal to U m L,ǫ . = X n ∈ N | Λ L | n − Z S n X Λ ⊆ Λ L , ø (Λ (1) ∪···∪ Λ ( n ) ) >ǫL Ψ (1)Λ (1) · · · Ψ ( n )Λ ( n ) a n (cid:0) dΨ (1) , . . . , dΨ ( n ) (cid:1) . If m = (Φ , a ) ∈ M\W , for any ǫ ∈ (0 , one generally has that lim L →∞ k U Φ L,ǫ k = 0 and lim L →∞ k U m L,ǫ k > . In particular, the long-range part ( U m L − U Φ L ) of the internal energy U m L generally dominates the inter-action at large distances, in the infinite volume limit.19or any translation-invariant model m . = (Φ , a ) ∈ M (cf. (56)), observe that U m L . = U ˜Φ L + ∞ X n =2 | Λ L | n − Z S n U Ψ (1) L · · · U Ψ ( n ) L a n (cid:0) dΨ (1) , . . . , dΨ ( n ) (cid:1) , L ∈ N , where ˜Φ . = Φ + Z S Ψ a (dΨ) ∈ W , the last integral being defined by (52). If the model is short-range and translation-invariant, i.e., Φ , ˜Φ ∈ W , then the interaction ˜Φ can be encoded in some self-adjoint, finite, signed Borel measure ˜ a , leading to the definition of a new model ˜ m . = (0 , ˜ a ) ∈ M such that U m L = U ˜ m L . In other words,if one is only interested in models that are short-range and translation-invariant, then one can onlyconsider the Banach space S . Finally, for any translation-invariant model m = (Φ , (0 , a , , . . . )) ∈M , remark that U m L . = U Φ L + 1 | Λ L | Z S U Ψ (1) L U Ψ (2) L a (cid:0) dΨ (1) , dΨ (2) (cid:1) , (61)which can be seen as a local Hamiltonian of a long-range model in the sense of [46], as explained inSection 8. Explicit examples of such models are given in [46, Section 2.2].Like in Definition 3.3, any model m ∈ M yields a sequence of bounded derivations: Definition 4.3 (Derivations on the CAR algebra for long-range interactions)
The (symmetric) derivations { δ m L } L ∈ N ⊆ B ( U ) associated with any model m ∈ M are defined by δ m L ( A ) . = i [ U m L , A ] . = i ( U m L A − AU m L ) , A ∈ U , L ∈ N . For any long-range model m ∈ M , the finite-volume dynamics are always well-defined: For all L ∈ N there is a strongly continuous one-parameter group ( τ ( L, m ) t ) t ∈ R of ∗ -automorphisms of U generated by δ m L ∈ B ( U ) : τ ( L, m ) t ( A ) . = e itU m L A e − itU m L , A ∈ U . (62)Compare with (39)-(40) for short-range interactions in the autonomous situation.Nevertheless, in contrast with short-range interactions (cf. Corollary 3.5 and Proposition 3.7), inthe thermodynamic limit L → ∞ , the finite-volume dynamics does not generally converge withinthe C ∗ -algebra U . To see this, consider the following elementary example: Choose a model m =(0 , (0 , a , , . . . )) ∈ M such that U m L = 12 | Λ L | N L with N L . = X x ∈ Λ L , s ∈ S a ∗ x, s a x, s . Take A = a , s ∈ U for some fixed spin s ∈ S . Observe that τ ( L, m ) t ( a , s ) = e it (2 | Λ L | ) − e it | Λ L | − a ∗ , s a , s N L a , s e − it | Λ L | − a ∗ , s a , s N L . Therefore, for any t ∈ R , e − it (2 | Λ L | ) − τ ( L, m ) t ( a , s ) = a , s + it | Λ L | − (cid:2) a ∗ , s a , s N L , a , s (cid:3) + R L ( t ) (63)with k R L ( t ) k U ≤ t . Note that | Λ L | − (cid:2) a ∗ , s a , s N L , a , s (cid:3) = − | Λ L | − a , s N L U , as L → ∞ . ByEquation (63), at least at small times | t | > , ( τ ( L, m ) t ( A )) L ∈ N ⊆ U does not converge, as L → ∞ .The non-convergence property is generic: For any integer n ≥ , Ψ (1) , . . . , Ψ ( n ) ∈ W , A ∈ U andall L ∈ N , | Λ L | n − h U Ψ (1) L · · · U Ψ ( n ) L , A i = h U Ψ (1) L , A i U Ψ (2) L | Λ L | · · · U Ψ ( n ) L | Λ L | + U Ψ (1) L | Λ L | · · · U Ψ ( n − L | Λ L | h U Ψ ( n ) L , A i + n − X m =2 U Ψ (1) L | Λ L | · · · U Ψ ( m − L | Λ L | h U Ψ ( m ) L , A i U Ψ ( m +1) L | Λ L | · · · U Ψ ( n ) L | Λ L | . (64)(Compare with Definitions 4.2 and 4.3.) Note that the element (64) of U is uniformly bounded withrespect to L ∈ N , since, by Corollary 3.5, the commutators in the right-hand side of this last equationhave a limit in U for any A ∈ U , as L → ∞ . However, | Λ L | − U Ψ L does not generally converge in thenorm sense of U : Since lim L →∞ | Λ L | (cid:2) U Ψ L , A (cid:3) = 0 , A ∈ U , if the sequence ( | Λ L | − U Ψ L ) L ∈ N would converge in U , as L → ∞ , then its limit would be an elementof the center of U . For U is a simple algebra (cf. [79, Corollary 2.6.19]), its center is trivial. Therefore, ( | Λ L | − U Ψ L ) L ∈ N would converge to c for some c ∈ C . In particular, by Proposition 3.2, for Ψ ∈ W , (cid:16) e Ψ ,~ℓ − c (cid:17) ∈ \ ρ ∈ E ~ℓ ker ρ , which is clearly wrong, in general. This observation is well-known. See, e.g., [19, p. 2225].By contrast, taking Ψ ∈ W , ~ℓ ∈ N d and any cyclic representation ( H ρ , π ρ , Ω ρ ) of an extreme (orergodic) ~ℓ -periodic state ρ ∈ E ~ℓ (17), one has that the sequence ( | Λ L | − π ρ ( U Ψ L )) L ∈ N does stronglyconverge to π ρ ( c ) for some c = c ρ ∈ C . This is basically Haag’s argument [4] proposed in 1962 inorder to give a mathematical meaning to the dynamics of the BCS model. In the more general case ofa not necessarily extreme state ρ ∈ E ~ℓ , the strong operator limit of the sequence is an element of the(possibly non-trivial) center of the von Neumann algebra π ρ ( U ) ′′ . These facts compel us to considera more general setting, in particular the notion of state-dependent observables and interactions. C ∗ -Algebra of Continuous Functions on States In this section, we define the extended quantum framework introduced in [63], starting with the clas-sical C ∗ -algebra. Note that we consider here the C ∗ -algebra U , with state space E , as the primordial C ∗ -algebra X of [63]. Nevertheless, from the point of view of physics, only the subalgebra U + ⊆ U of even elements is relevant, as already explained in Section 2.3. In this case, the physical state spaceis the set E + ⊆ E of all even states. C ∗ -Algebra of Continuous Functions on States Recall that E stands for the metrizable, weak ∗ -compact and convex set of states on U , as defined by(11). It is the state space of the classical dynamics defined on the space C ( E ; C ) of complex-valuedweak ∗ -continuous functions on E :Classical algebra: Endowed with the point-wise operations and complex conjugation, C ( E ; C ) be-comes a unital commutative C ∗ -algebra denoted by C . = (cid:16) C ( E ; C ) , + , · C , × , ( · ) , k·k C (cid:17) , (65)21here the corresponding C ∗ -norm is k f k C . = max ρ ∈ E | f ( ρ ) | , f ∈ C . (66)Note that the “ max ” in the definition of the norm is well-defined because of the continuity of f together with the compactness of E . C is the classical C ∗ -algebra of weak ∗ -continuous complex-valued functions on states. The (real) Banach subspace of all real-valued functions is denoted by C R C . The C ∗ -algebra C is separable, E being metrizable and compact.Gelfand transform: Elements of the (separable and unital) C ∗ -algebra U naturally define continuousand affine functions ˆ A ∈ C by ˆ A ( ρ ) . = ρ ( A ) , ρ ∈ E, A ∈ U . (67)This is the well-known Gelfand transform. Note that A = B implies ˆ A = ˆ B , as states separateselements of U . Since k A k U = max ρ ∈ E | ρ ( A ) | , A ∈ U R , (68)the mapping A ˆ A defines a linear isometry from the Banach space U R of all self-adjoint elementsto the space C R of all real-valued functions on E .Dense classical subalgebra: Recall that U is the normed ∗ -algebra of local elements of U defined by(3). We denote respectively by C U . = C [ { ˆ A : A ∈ U } ] and C U . = C [ { ˆ A : A ∈ U } ] (69)the subalgebras of polynomials in the elements of { ˆ A : A ∈ U } and { ˆ A : A ∈ U } , with complexcoefficients. The unit ˆ ∈ C mapping any state to belongs to C U , by definition. Since U is, byconstruction, dense in U , the subalgebra C U separates states. Therefore, by the Stone-Weierstrasstheorem, C U ⊆ C U is dense in C , i.e., C = C U . We define A ( E ; C ) ⊆ C to be the closed subspace of all affine , weak ∗ -continuous complex-valuedfunctions over E . By [63, Definition 3.7], the convex Gˆateaux derivative of f ∈ C at a fixed state ρ ∈ E is an affine weak ∗ -continuous complex-valued function over E , defined as follows: Definition 5.1 (Convex weak ∗ -continuous G ˆateaux derivative) For any f ∈ C and ρ ∈ E , we say that d f ( ρ ) : E → C is the (unique) convex weak ∗ -continuousGˆateaux derivative of f at ρ ∈ E if d f ( ρ ) ∈ A ( E ; C ) and lim λ → + λ − ( f ((1 − λ ) ρ + λυ ) − f ( ρ )) = [d f ( ρ )] ( υ ) , ρ, υ ∈ E .
A function f ∈ C such that d f ( ρ ) exists for all ρ ∈ E is called (convex-) differentiable and weuse the notation d f ≡ (d f ( ρ )) ρ ∈ E : E → A ( E ; C ) . If f ∈ A ( E ; C ) then d f ( ρ ) = f − f ( ρ ) , ρ ∈ E , which means that affine functions of C are continuously (convex-)differentiable, as expected.22e define the (non-empty) subspace of continuously differentiable complex-valued functions overthe convex and weak ∗ -compact set E by Y ≡ Y ( E ) . = { f ∈ C : d f ∈ C ( E ; C ) } . (70)We endow this vector space with the norm k f k Y . = k f k C + max ρ ∈ E k d f ( ρ ) k C < ∞ , f ∈ Y , (71)in order to obtain a Banach space, again denoted by Y . The “ max ” in the definition of the norm iswell-defined because of the continuity of f and d f together with the compactness of E . The normedvector space Y is complete, see [63, Section 3.4].The (real) Banach subspace of all continuously differentiable real-valued functions is denoted by Y R Y . By [63, Proposition 3.8] together with Equations (66)-(68) and (70), for any f ∈ Y R C R ,there is a unique D f ∈ C ( E ; U R ) such that d f ( ρ ) = \ D f ( ρ ) , ρ ∈ E . (72)By (66) and (68), note that k D f ( ρ ) k U = k d f ( ρ ) k C , ρ ∈ E . (73)Since the convex weak ∗ -continuous Gˆateaux derivative is linear and because any function f ∈ Y canbe decomposed into real and imaginary parts, it follows that, for any f ∈ Y C , there is a unique D f ∈ C ( E ; U ) satisfying (72). For instance, D ˆ A ( ρ ) = A − ρ ( A ) , A ∈ U . Therefore, using [63, Definition 3.9] and the standard notation [ A, B ] . = AB − BA , A, B ∈ U ,for the commutator, we can define a Poisson bracket for continuously differentiable complex-valuedfunctions on the state space:
Definition 5.2 (Poisson bracket)
We define the mapping {· , ·} : Y × Y → C by { f, g } ( ρ ) . = ρ ( i [D f ( ρ ) , D g ( ρ )]) , f, g ∈ Y . By [63, Proposition 3.10 and discussions afterwards], this mapping is a Poisson bracket on C U , that is,for complex-valued polynomials. In other words, it is a skew-symmetric biderivation on C U satisfyingthe Jacobi identity.In fact, by generalizing the well-known construction of a Poisson bracket for the polynomial func-tions on the dual space of finite dimensional Lie groups [83, Section 7.1], we define in [63, Section3.2] a Poisson bracket for the polynomial functions on the hermitian continuous functional (like thestates) on any C ∗ -algebra. Then, in [63, Section 3.3], the Poisson bracket is localized on the state andphase spaces associated with this algebra, by taking quotients with respect to conveniently chosenPoisson ideals. In particular, this leads, in an elegant way, to a Poisson bracket for polynomial func-tions of the classical C ∗ -algebra C . Definitions 5.1-5.2 just yield a very convenient explicit expressionof this Poisson bracket for functions on the state space. The phase space in [63, Definition 2.2] is the weak ∗ closure of the subset E ( E ) of extreme points of the state space E . For general C ∗ -algebras, E and the weak ∗ closure of E ( E ) are not necessarly the same. But here, if L is infinite then E = E ( E ) , i.e., phase and state spaces coincide. .3 The Quantum C ∗ -Algebra of Continuous Functions on States The long-range dynamics takes place in the space C ( E ; U ) of weak ∗ -continuous U -valued functionson the metrizable compact space E , which can be endowed with a C ∗ -algebra structure:Quantum algebra: Endowed with the point-wise ∗ -algebra operations inherited from U , C ( E ; U ) is aunital non-commutative C ∗ -algebra denoted by U ≡ U L . = ( C ( E ; U ) , + , · C , × , ∗ , k·k U ) . (74)The unique C ∗ -norm k·k U is the supremum norm for functions on E taking values in the normedspace U , i.e., k f k U . = max ρ ∈ E k f ( ρ ) k U , f ∈ U . (75)Recall that U R U is the (real) Banach subspace of all self-adjoint elements of U . The (real) Banachsubspace of all U R -valued functions of U is similarly denoted by U R U .We identify the primordial C ∗ -algebra U , on which the quantum dynamics is usually defined,with the subalgebra of constant functions of U . Meanwhile, the classical dynamics appears in thealgebra C of complex-valued weak ∗ -continuous functions on E . This unital commutative C ∗ -algebrais identified with the subalgebra of functions of U whose values are multiples of the unit ∈ U . Inother words, we have the canonical inclusions U ⊆ U and C ⊆ U . (76)See [63, Eq. (67)].Dense subalgebras: Similar to (3), we define the ∗ -subalgebras U Λ . = { f ∈ U : f ( E ) ⊆ U Λ } , Λ ∈ P f , (77)and U . = [ L ∈ N U Λ L ⊆ U . (78)The union C U ∪ U generates the C ∗ -algebra U and U = span { C U } ⊇ span { C U U } (79)are dense ∗ -subalgebras of U . To see this, use the density of U ⊆ U and C U , as well as the com-pactness of E together with the existence of partitions of unity subordinated to any open cover of themetrizable (weak ∗ -compact) space E (paracompactness and metrizability of E ).Positivity of elements of U : The positivity of elements of U is equivalent to their point-wise positivity.This is a direct consequence of the following lemma: Lemma 5.3 (Spectrum)
For any f ∈ U , its spectrum equals spec ( f ) = [ ρ ∈ E spec ( f ( ρ )) . roof. Fix f ∈ U and take any z ∈ C in the resolvent set of f . Then, clearly, z also belongs to theresolvent set of f ( ρ ) ∈ U for all ρ ∈ E . It follows that [ ρ ∈ E spec ( f ( ρ )) ⊆ spec ( f ) . (80)Take now z ∈ C in the resolvent set of f ( ρ ) ∈ U for all ρ ∈ E . Using the Neumann series ( A − B − z U ) − = ( z U − A ) − ∞ X n =1 (cid:0) B ( z U − A ) − (cid:1) n for all A, B ∈ U with k B ( z U − A ) − k U < , one sees that the mapping ρ ( z U − f ( ρ )) − from E to U is weak ∗ -continuous. In particular, this mapping is an element of U which, by construc-tion, is the resolvent of f at z ∈ C . It follows that spec ( f ) ⊆ [ ρ ∈ E spec ( f ( ρ )) . (81)By (80)-(81), the assertion follows. Corollary 5.4 (Positivity)
Any element f ∈ U R is positive iff f ( ρ ) ∈ U R is positive for all ρ ∈ E . Functional calculus for elements of U : It turns out that the continuous functional calculus in U coin-cides with the point-wise continuous functional calculus in U : Lemma 5.5 (Continuous functional calculus)
For any self-adjoint f ∈ U R with spectrum spec( f ) and any continuous function ϕ ∈ C (spec( f ); C ) , ϕ ( f ) = ϕ ◦ f , where ϕ ◦ f ( ρ ) . = ϕ ( f ( ρ )) , ρ ∈ E .
Note that ϕ ( f ( ρ )) is well-defined because spec ( f ( ρ )) ⊆ spec( f ) for all ρ ∈ E , by Lemma 5.3. Proof.
Note first that, for all self-adjoint f ∈ U R and any polynomial function ϕ ∈ C (spec( f ); C ) , ϕ ( f ) = ϕ ◦ f ∈ U . Observe next that, for any ϕ , ϕ ∈ C (spec( f ); C ) and ρ ∈ E , k ϕ ◦ f ( ρ ) − ϕ ◦ f ( ρ ) k U ≤ sup ρ ∈ spec( f ( ρ )) | ϕ ( ρ ) − ϕ ( ρ ) | ≤ k ϕ − ϕ k ∞ , (82)by Lemma 5.3. If ϕ ∈ C (spec( f ); C ) is a general continuous function then take any sequence ofpolynomial functions ϕ n ∈ C (spec( f ); C ) converging uniformly to ϕ . Such a sequence alwaysexists, by the Stone-Weierstrass theorem and the compactness of spec( f ) . Therefore, we infer from(82) that ϕ ◦ f is the uniform limit of the sequence of weak ∗ -continuous function ( ϕ n ◦ f ) n ∈ N ⊆ U .In particular, ϕ ◦ f is weak ∗ -continuous, i.e., ϕ ◦ f ∈ U . It is easy to check that the mapping ϕ ϕ ◦ f is a ∗ -homomorphism from C (spec( f ); C ) to U with ◦ f = being the unit of U andid spec( f ) ◦ f = f . By the uniqueness of the continuous functional calculus, ϕ ◦ f = ϕ ( f ) for all f ∈ U R and ϕ ∈ C (spec( f ); C ) . 25 .4 Important ∗ -Automorphisms of the Quantum C ∗ -Algebra Parity: The ∗ -automorphism σ of U uniquely defined by the condition (4) naturally induces a ∗ -automorphism Ξ of U defined by [Ξ ( f )] ( ρ ) . = σ ( f ( ρ )) , ρ ∈ E, f ∈ U . (83)Elements f , f ∈ U satisfying Ξ( f ) = f and Ξ( f ) = − f are respectively called even and odd .The set U + ≡ U + L . = { f ∈ U : f = Ξ( f ) } = (cid:8) f ∈ U : f ( E ) ⊆ U + (cid:9) ⊆ U (84)of all even weak ∗ -continuous U -valued functions on states is a C ∗ -subalgebra. Compare with (5).Translations: Let L = Z d . The ∗ -automorphisms α x , x ∈ Z d , of U uniquely defined by the condition(8) naturally induce a group homomorphism x A x from Z d to the group of ∗ -automorphisms of U ,defined by [A x ( f )] ( ρ ) . = α x ( f ( ρ )) , ρ ∈ E, f ∈ U , x ∈ Z d . (85)These ∗ -automorphisms represent the translation group in U .Permutations: In the same way, we can define a group homomorphism π P π from the set Π ofall bijective mappings from L into itself, which leave all but finitely many elements invariant, to thegroup of ∗ -automorphisms of U by using (9) and the definition [ P π ( f )] ( ρ ) . = p π ( f ( ρ )) , ρ ∈ E, f ∈ U , π ∈ Π . The ∗ -automorphisms representing the permutation group in U are not used here and are only givenfor completeness. Similar to what is done in Section 3.1, a state-dependent (complex) interaction is defined to be amapping Φ : P f → U + such that Φ Λ ∈ U Λ for any Λ ∈ P f . See Equations (77) and (84). Similar to(21) and (22), the set of all state-dependent interactions is naturally endowed with the structure of acomplex vector space on which the natural involution Φ Φ ∗ . = ( Φ ∗ Λ ) Λ ∈P f (86)is defined. Self-adjoint state-dependent interactions Φ are, by definition, those satisfying Φ = Φ ∗ . Astate-dependent interaction Φ can be identified with a mapping ρ Φ ( ρ ) from E to the vector spaceof usual interactions (of Section 3.1) via the definition Φ ( ρ ) Λ . = Φ Λ ( ρ ) , Λ ∈ P f . In this paper, we only consider a particular space of state-dependent short-range interactions:Using the Banach space W of (usual) short-range interactions, we define W . = ( C ( E ; W ) , + , · C , ∗ , k·k W ) to be the Banach space of weak ∗ -continuous, state-dependent short-range interactions, along with thesupremum norm k Φ k W . = max ρ ∈ E k Φ ( ρ ) k W , Φ ∈ W , (87)26here k · k W is defined by (27). Note that (86) is an isometric antilinear involution on W . Recall that W R W is the (real) Banach subspace of all self-adjoint interactions and the (real) Banach subspaceof all self-adjoint state-dependent interactions is similarly denoted by W R ≡ (cid:0) C (cid:0) E ; W R (cid:1) , + , · R , k·k W (cid:1) W . To simplify notation, for any Ψ ∈ C ( R ; W ) and ρ ∈ E , Ψ ( ρ ) ∈ C ( R ; W ) stands for the time-dependent interaction defined by Ψ ( ρ ) ( t ) . = Ψ ( t ; ρ ) , ρ ∈ E, t ∈ R . (88)When L = Z d , Φ ∈ W is, by definition, translation-invariant if, for all x ∈ Z d and Λ ∈ P f , Φ Λ+ x = A x ( Φ Λ ) , see (28). Recall that { A x } x ∈ Z d is the family of (translation) ∗ -automorphisms on U defined by (85). Similar to the (separable) Banach subspace W W of translation-invariant, short-range interactions on L = Z d , we denote by W W the Banach subspace of translation-invariant,state-dependent, short-range interactions on L = Z d . C ∗ -Algebra of Functions and Dynamics For any state-dependent (short-range) interaction Φ ∈ W , we can naturally define limit derivations inthe quantum C ∗ -algebra U : Definition 6.1 (Derivations for state-dependant interactions)
The symmetric derivations δ Φ associated with any Φ ∈ W is defined on the dense subset U =span { C U } (see (79)) by (cid:2) δ Φ ( f A ) (cid:3) ( ρ ) . = f ( ρ ) δ Φ ( ρ ) ( A ) , ρ ∈ E, f ∈ C , A ∈ U . The right-hand side of the last equation defines an element of U , by Corollary 3.5. Similar to Remark3.6, observe also that, if Φ ∈ W R , then the symmetric derivation δ Φ is (norm-) closable: By Corollary5.4 and Lemma 5.5, for all f ∈ U , f ≥ implies f / ∈ U . It follows from [86, Theorem 1.4.9]that δ Φ is dissipative [86, Definition 1.4.6], and, by [86, Proposition 1.4.7], it thus norm-closable andits closure is also dissipative.Any Ψ ∈ C ( R ; W R ) determines a two-parameter family T Ψ ≡ ( T Ψ t,s ) s,t ∈ R of ∗ -automorphisms ofthe quantum C ∗ -algebra U defined by (74): (cid:2) T Ψ t,s ( f ) (cid:3) ( ρ ) . = τ Ψ ( ρ ) t,s ( f ( ρ )) , ρ ∈ E, f ∈ U , s, t ∈ R . (89)For all f ∈ U , the right -hand side of (89) defines an element of U , by Proposition 3.8 (iii) andLebesgue’s dominated convergence theorem. By density of U ⊆ U and the fact that τ Ψ ( ρ ) t,s is a con-traction, it follows that the right-hand side of (89) defines an element of U for all f ∈ U ! U . Similarto Proposition 3.7, this family satisfies a non-autonomous evolution equation with infinitesimal gen-erator δ Ψ ( t ) for t ∈ R : Proposition 6.2 (Infinite-volume state-dependent short-range dynamics)
For any Ψ ∈ C ( R ; W R ) , T Ψ ≡ ( T Ψ t,s ) s,t ∈ R is a strongly continuous two-parameter family of ∗ -auto-morphisms of U , which is the unique solution in B ( U ) to the non-autonomous evolution equation ∀ s, t ∈ R : ∂ t T Ψ t,s = T Ψ t,s ◦ δ Ψ ( t ) , T Ψ s,s = U , (90) in the strong sense on the dense subspace U ⊆ U , U being the identity mapping of U . In particular,it satisfies the reverse cocycle property: ∀ s, r, t ∈ R : T Ψ t,s = T Ψ r,s T Ψ t,r . (91)27 roof. Fix Ψ ∈ C ( R ; W R ) . The fact that ( T Ψ t,s ) s,t ∈ R is a family of ∗ -automorphisms of U satisfying(91) is a direct consequence of (89), since ( τ Ψ ( ρ ) t,s ) s,t ∈ R is a family of ∗ -automorphisms of U satisfying(42) for any ρ ∈ E .We now prove that the family ( T Ψ t,s ) s,t ∈ R is strongly continuous: By (89) and [63, Lemma 5.1(iii)], it suffice to prove that ( τ Ψ ( ρ ) t,s ) ( ρ,s,t ) ∈ E × R is a strongly continuous family. To this end, take threesequences ( s n ) n ∈ N , ( t n ) n ∈ N ⊆ R and ( ρ n ) n ∈ N ⊆ E converging respectively to s, t ∈ R and ρ ∈ E .For any A ∈ U , (cid:13)(cid:13)(cid:13) τ Ψ ( ρ n ) t n ,s n ( A ) − τ Ψ ( ρ ) t,s ( A ) (cid:13)(cid:13)(cid:13) U ≤ (cid:13)(cid:13)(cid:13) τ Ψ ( ρ n ) t n ,s n ( A ) − τ Ψ ( ρ ) t n ,s n ( A ) (cid:13)(cid:13)(cid:13) U + (cid:13)(cid:13)(cid:13) τ Ψ ( ρ ) t n ,s n ( A ) − τ Ψ ( ρ ) t,s ( A ) (cid:13)(cid:13)(cid:13) U . (92)By Proposition 3.7, lim n →∞ (cid:13)(cid:13)(cid:13) τ Ψ ( ρ ) t n ,s n ( A ) − τ Ψ ( ρ ) t,s ( A ) (cid:13)(cid:13)(cid:13) U = 0 , A ∈ U . (93)Because Ψ ∈ C (cid:0) R ; W R (cid:1) , observe from (87) that, for T ∈ R + , sup α ∈ [ − T,T ] sup ρ ∈ E k Ψ ( α ; ρ ) k W = sup α ∈ [ − T,T ] k Ψ ( α ) k W < ∞ . (94)Therefore, we infer from Proposition 3.8 (iii) and Lebesgue’s dominated convergence theorem that lim n →∞ (cid:13)(cid:13)(cid:13) τ Ψ ( ρ n ) t n ,s n ( A ) − τ Ψ ( ρ ) t n ,s n ( A ) (cid:13)(cid:13)(cid:13) U = 0 , A ∈ U . By Combining (92)-(93) with (89), [63, Lemma 5.1 (iii)], the density of U in U and the fact that τ Ψ ( ρ ) t,s is a contraction for any s, t ∈ R , we deduce that ( T Ψ t,s ) s,t ∈ R is a strongly continuous two-parameterfamily.We next prove the non-autonomous evolution equation for ( T Ψ t,s ) s,t ∈ R : By Equation (89), ∀ s, t ∈ R , f ∈ C ⊆ U : ∂ t T Ψ t,s ( f ) = 0 , T Ψ s,s = U . (95)For any s, t ∈ R , A ∈ U ⊆ U and h ∈ R \{ } , observe additionally that (cid:13)(cid:13)(cid:13) h − (cid:0) T Ψ t + h,s ( A ) − T Ψ t,s ( A ) (cid:1) − T Ψ t,s ◦ δ Ψ ( t ) ( A ) (cid:13)(cid:13)(cid:13) U (96) = sup ρ ∈ E (cid:13)(cid:13)(cid:13) h − (cid:16) τ Ψ ( ρ ) t + h,t ( A ) − A (cid:17) − δ Ψ ( t ; ρ ) ( A ) (cid:13)(cid:13)(cid:13) U , using Proposition 3.7. Now, by contradiction, assume the existence of a zero sequence ( h n ) n ∈ N , asequence ( ρ n ) n ∈ N ⊆ E and a positive constant D > such that inf n ∈ N (cid:13)(cid:13)(cid:13) h − n (cid:16) τ Ψ ( ρ n ) t + h n ,t ( A ) − A (cid:17) − δ Ψ ( t ; ρ n ) ( A ) (cid:13)(cid:13)(cid:13) U ≥ D > . By weak ∗ -compactness of E , we can assume without loss of generality that ( ρ n ) n ∈ N converges in theweak ∗ topology to some ρ ∈ E , as n → ∞ . From Corollary 3.5, it follows that lim inf n →∞ (cid:13)(cid:13)(cid:13) h − n (cid:16) τ Ψ ( ρ n ) t + h n ,t ( A ) − A (cid:17) − δ Ψ ( t ; ρ ) ( A ) (cid:13)(cid:13)(cid:13) U ≥ D > . (97)Meanwhile, using Proposition 3.8 (iii), for any ε ∈ R + , there is n ∈ N such that, for all n ≥ n , h − n (cid:16) τ Ψ ( ρ n ) t + h n ,t − τ Ψ ( ρ ) t + h n ,t (cid:17) ( A ) ≤ | Λ | k A k U k F k , L e D R ε − ε k Ψ( ρ ; t + α ) k W d α (98) × max α ∈ [ − ε,ε ] k Ψ ( t + α ; ρ n ) − Ψ ( t + α ; ρ ) k W . ( t, ρ ) Ψ ( t ; ρ ) is (jointly) continuous on R × E , by definition of W , and weinfer from Inequalities (97)-(98) that lim inf n →∞ (cid:13)(cid:13)(cid:13) h − n (cid:16) τ Ψ ( ρ ) t + h n ,t ( A ) − A (cid:17) − δ Ψ ( t ; ρ ) ( A ) (cid:13)(cid:13)(cid:13) U ≥ D > , which contradicts (41) for Ψ = Ψ ( ρ ) . By Equation (96), it follows that ∀ s, t ∈ R , A ∈ U ⊆ U : ∂ t T Ψ t,s ( A ) = T Ψ t,s ◦ δ Ψ ( t ) ( A ) . (99)By using that δ Ψ ( t ) , t ∈ R , are derivations and T Ψ t,s , s, t ∈ R , are ∗ -automorphisms of U , we deduce(90) on U , from (95) and (99). Recall that U = span { C U } , by (79).Finally, in order to prove the uniqueness of the solution to (90), assume that ( ˜ T t,s ) s,t ∈ R ⊆ B ( U ) is a two-parameter family satisfying (90) on U . Since δ Ψ ( t ) ( C ) = { } , C is a subspace of the fixedpoint algebra of ( ˜ T t,s ) s,t ∈ R . In particular, by [63, Lemma 5.2], it comes from a strongly continuousfamily (˜ τ ρt,s ) ( ρ,s,t ) ∈ E × R defined by ˜ τ ρt,s ( A ) . = [ ˜ T t,s ( A )] ( ρ ) , ρ ∈ E, A ∈ U ⊆ U , s, t ∈ R . Through (90) (cf. (96)), for each ρ ∈ E , ( τ Ψ ( ρ ) t,s ) s,t ∈ R and (˜ τ ρt,s ) s,t ∈ R are both solution in B ( U ) to thenon-autonomous evolution equation (41) on U for Ψ = Ψ ( ρ ) . Therefore, by Proposition 3.7, thesolution to the non-autonomous evolution equation (90) on U is also unique. Proposition 6.2 means that, for any Ψ ∈ C ( R ; W R ) , ( U , T Ψ ) is a state-dependent C ∗ -dynamicalsystem, as defined in [63, Definition 5.3]. Therefore, as explained in [63, Section 5.2], for any Ψ ∈ C ( R ; W R ) , ( U , T Ψ ) induces a Feller dynamics within the classical C ∗ -algebra C defined by (65)-(66):State-space trajectories: Let C ( E ; E ) be the set of weak ∗ -continuous functions from the state space E to itself endowed with the topology of uniform convergence. In other words, any net ( f j ) j ∈ J ⊆ C ( E ; E ) converges to f ∈ C ( E ; E ) whenever lim j ∈ J max ρ ∈ E | f j ( ρ )( A ) − f ( ρ )( A ) | = 0 , for all A ∈ U . (100)We denote by Aut ( E ) C ( E ; E ) the subspace of all automorphisms of E , i.e., element of C ( E ; E ) with weak ∗ -continuous inverse. Equivalently, Aut ( E ) is the set of all bijective mappings in C ( E ; E ) ,because E is a compact Hausdorff space. From the family T Ψ ≡ ( T Ψ t,s ) s,t ∈ R , we define a continuousfamily ( φ Ψ t,s ) s,t ∈ R ⊆ Aut ( E ) by φ Ψ t,s ( ρ ) . = ρ ◦ τ Ψ ( ρ ) t,s , ρ ∈ E, s, t ∈ R , (101)where ( τ Ψ ( ρ ) t,s ) ( ρ,s,t ) ∈ E × R is the unique strongly continuous family of ∗ -automorphisms of U satisfying(89), see also [63, Lemma 5.2].Classical flows as Feller evolution systems: The state-space trajectories, in turn, yield a strongly con-tinuous two-parameter family ( V Ψ t,s ) s,t ∈ R of ∗ -automorphisms of the classical C ∗ -algebra C , definedby V Ψ t,s f . = f ◦ φ Ψ t,s , f ∈ C , s, t ∈ R . (102)29his classical dynamics is a Feller evolution system in the following sense: As a ∗ -automorphism, V Ψ t,s is self-adjointness- and positivity-preserving while k V Ψ t,s k B ( C R ) = 1 ; ( V Ψ t,s ) s,t ∈ R is a strongly con-tinuous two-parameter family satisfying ∀ s, r, t ∈ R : V Ψ t,s = V Ψ r,s ◦ V Ψ t,r , (103)by (91). Therefore, as explained in [63, Section 4.4], the classical dynamics defined as the restrictionof ( V Ψ t,s ) s,t ∈ R to the real space C R can be associated in this case with Feller processes in probabilitytheory: By the Riesz-Markov representation theorem and the monotone convergence theorem, thereis a unique two-parameter group ( p Ψ t,s ) s,t ∈ R of Markov transition kernels p Ψ t,s ( · , · ) on E such that V Ψ t,s f ( ρ ) = Z E f (ˆ ρ ) p Ψ t,s ( ρ, dˆ ρ ) , f ∈ C R . The right-hand side of the above identity makes sense for bounded measurable functions from E to R . In fact, one can naturally extend ( V Ψ t,s ) s,t ∈ R to this more general class of functions on E .The notion of Feller evolution system, which is only an extension of Feller semigroups to non-autonomous two-parameter families, has been introduced (at least) in 2014 [87].Parity: For any Ψ ∈ C ( R ; W R ) , the family ( T Ψ t,s ) s,t ∈ R is parity-preserving, i.e., Ξ ◦ T Ψ t,s = T Ψ t,s ◦ Ξ , s, t ∈ R , where Ξ is the ∗ -automorphism of U defined by (83). It follows that φ Ψ t,s (cid:0) E + (cid:1) ⊆ E + , φ Ψ t,s (cid:0) E \ E + (cid:1) ⊆ E \ E + , s, t ∈ R , (104)which in turn implies that V Ψ t,s can be seen as a mapping on either C ( E + , C ) or C ( E \ E + , C ) : V Ψ t,s ( f | E + ) . = (cid:0) V Ψ t,s f (cid:1) | E + , V Ψ t,s (cid:0) f | E \ E + (cid:1) . = (cid:0) V Ψ t,s f (cid:1) | E \ E + , f ∈ C , s, t ∈ R . (105)Recall that E + is the weak ∗ -compact convex set of even states defined by (15), which is the physicalstate space. Similar to [63, Corollary 4.3], the set E ( E + ) of extreme points of E + is also conservedby the flow.Translations: Let L = Z d . For any continuous mapping Ψ ∈ C ( R ; W ∩ W R ) from R to the space W ∩ W R of translation-invariant, self-adjoint and state-dependent interactions, the mapping x A x from Z d to the group of ∗ -automorphisms of U , defined by (85), is a symmetry group of the state-dependent C ∗ -dynamical system ( U , T Ψ ) , i.e., A x ◦ T Ψ t,s = T Ψ t,s ◦ A x , s, t ∈ R , x ∈ Z d . As a consequence, for any ~ℓ ∈ N d , φ Ψ t,s (cid:0) E ~ℓ (cid:1) ⊆ E ~ℓ , φ Ψ t,s (cid:0) E \ E ~ℓ (cid:1) ⊆ E \ E ~ℓ , s, t ∈ R , (106)which in turn implies that V Ψ t,s can be seen as a mapping on either C ( E ~ℓ , C ) or C ( E \ E ~ℓ , C ) : V Ψ t,s ( f | E ~ℓ ) . = ( V Ψ t,s f ) | E ~ℓ , V Ψ t,s ( f | E \ E ~ℓ ) . = ( V Ψ t,s f ) | E \ E ~ℓ , f ∈ C , s, t ∈ R . (107)Recall that E ~ℓ ⊆ E + is the weak ∗ -compact convex set of ~ℓ -periodic states defined by (17) for any ~ℓ ∈ N d . Similar to [63, Corollary 4.3], for every ~ℓ ∈ N d , the set E ( E ~ℓ ) of extreme points of E ~ℓ is, inthis case, also conserved by the flow. The positivity and norm-preserving property are reminiscent of Markov semigroups. .4 Self-Consistency Equations By using Equation (64), Proposition 3.2 and Corollary 3.5 together with the linearity of the mapping Φ δ Φ , for every integer n ≥ , any translation-invariant interactions Ψ (1) , . . . , Ψ ( n ) ∈ W , eachlocal element A ∈ U , ~ℓ ∈ N d and every extreme ~ℓ -periodic state ρ ∈ E ( E ~ℓ ) ⊆ E ~ℓ , one can prove that lim L →∞ | Λ L | n − ρ (cid:16) i h U Ψ (1) L · · · U Ψ ( n ) L , A i(cid:17) = ρ ◦ δ Ψ ( ρ ) ( A ) , (108)where, in this case, the state-dependent interaction Ψ ∈ W equals Ψ ( ρ ) = (cid:4) ρ ; Ψ (1) , . . . , Ψ ( n ) (cid:5) ~ℓ . = n X m =1 Ψ ( m ) Y j ∈{ ,...,n } ,j = m ρ ( e Ψ ( j ) ,~ℓ ) ∈ W , ρ ∈ E . (109)Compare (108) with Definitions 4.2 and 4.3.The proof of (108) uses the ergodicity of extreme periodic states in a crucial way. It is non-trivial and will be performed in detail in [64]. One central argument in this proof, like in Haag’sapproach [4] to mean-field theories, is that, for any translation-invariant interaction Φ ∈ W andany cyclic representation ( H ρ , π ρ , Ω ρ ) of an extreme state ρ ∈ E ( E ~ℓ ) at fixed ~ℓ ∈ N d , the uniformlybounded family ( π ρ (cid:0) U Φ L (cid:1) | Λ L | ) L ∈ N ⊆ B ( H ρ ) converges to the operator ρ ( e Φ ,~ℓ ) H ρ , in the sense of the strong operator topology. See [64, Section5.3] for more details. Compare with Proposition 3.2.Having in mind the discussions of Section 4.3 and the linearity of the mapping Φ δ Φ , we givehere the limit (108) in order to convince the reader that appropriate approximating (state-dependent,short-range) interactions naturally appears in the description of the infinite-volume dynamics of lattice-fermion (or quantum-spin) systems with long-range interactions. To define them, recall that the inte-gral of interactions is defined by (51)-(52). Definition 6.3 (Non-autonomous approximating interactions)
For ~ℓ ∈ N d and any continuous functions m = (Φ( t ) , a ( t )) t ∈ R ∈ C ( R ; M ) , ξ ∈ C ( R ; E ) , we definethe mapping Φ ( m ,ξ ) from R to W R by Φ ( m ,ξ ) ( t ) . = Φ ( t ) + X n ∈ N Z S n (cid:4) ξ ( t ) ; Ψ (1) , . . . , Ψ ( n ) (cid:5) ~ℓ a ( t ) n (cid:0) dΨ (1) , . . . , dΨ ( n ) (cid:1) , t ∈ R , with ⌊ ρ ; Ψ ⌋ ~ℓ . = Ψ . If ξ ∈ C ( R ; Aut ( E )) , then a mapping Φ ( m , ξ ) from R to W R is defined, for any ρ ∈ E and t ∈ R , by Φ ( m , ξ ) ( t ; ρ ) . = Φ ( t ) + X n ∈ N Z S n (cid:4) ξ ( t ; ρ ) ; Ψ (1) , . . . , Ψ ( n ) (cid:5) ~ℓ a ( t ) n (cid:0) dΨ (1) , . . . , dΨ ( n ) (cid:1) . Such approximating interactions can be used to define, via Proposition 3.7 or Proposition 6.2, ∗ -automorphisms of U or U , because they are always bounded continuous functions: Lemma 6.4 (Continuity of approximating interactions)
Let ~ℓ ∈ N d and m ∈ C ( R ; M ) . For any ξ ∈ C ( R ; E ) , Φ ( m ,ξ ) ∈ C ( R ; W R ) and, for any ξ ∈ C ( R ; Aut ( E )) , Φ ( m , ξ ) ∈ C ( R ; W R ) with (cid:13)(cid:13) Φ ( m ,ξ ) ( t ) (cid:13)(cid:13) W ≤ k m ( t ) k M and (cid:13)(cid:13) Φ ( m , ξ ) ( t ) (cid:13)(cid:13) W ≤ k m ( t ) k M , t ∈ R . (110)31 roof. Inequalities (110) are direct consequences of Equations (30), (54)-(55), Definition 4.1 and thetriangle inequality, recalling that S n is the n -fold Cartesian product of the unit sphere S of the Banachspace W . Using the same arguments, note also that, for any m ∈ C ( R ; M ) , ξ ∈ C ( R ; E ) and t , t ∈ R , (cid:13)(cid:13) Φ ( m ,ξ ) ( t ) − Φ ( m ,ξ ) ( t ) (cid:13)(cid:13) W (111) ≤ k Φ ( t ) − Φ ( t ) k W + X n ∈ N n k F k n − , L k a ( t ) n − a ( t ) n k S ( S n ) + X n ∈ N k F k n − , L Z S n n X j =1 (cid:12)(cid:12)(cid:12) ( ξ ( t ) − ξ ( t )) ( e Ψ ( j ) ,~ℓ ) (cid:12)(cid:12)(cid:12) | a ( t ) n | (cid:0) dΨ (1) , . . . , dΨ ( n ) (cid:1) . Since m ∈ C ( R ; M ) and ξ ∈ C ( R ; E ) , we invoke Lebesgue’s dominated convergence theorem todeduce from the last inequality that Φ ( m ,ξ ) ∈ C ( R ; W R ) . The difference of Φ ( m , ξ ) for two differ-ent times at a fixed state ρ satisfies the same inequality as (111), ξ ( t ) , ξ ( t ) being replaced with ξ ( t ; ρ ) , ξ ( t ; ρ ) , respectively. Since Aut ( E ) C ( E ; E ) is the subspace of all automorphisms of E endowed with the topology of uniform convergence, as stated in Equation (100), one also infers fromLebesgue’s dominated convergence theorem that Φ ( m , ξ ) ∈ C ( R ; W R ) , provided m ∈ C ( R ; M ) and ξ ∈ C ( R ; Aut ( E )) .As is usual, we do the following identifications for the subspaces of constant functions: E ⊆ C ( R ; E ) and Aut ( E ) ⊆ C ( R ; Aut ( E )) . (112)When ξ ∈ E and m = (Φ , (0 , a , , . . . )) ∈ M , the first part of Definition 6.3 corresponds tothe (autonomous) approximating interactions first introduced in [46, Definition 2.31], see also Sec-tion 8. They are used there to characterize (generalized equilibrium) states minimizing the free-energy density through self-consistency equations (gap equations), whose solutions are related tonon-cooperative equilibria of a two-person zero-sum game (thermodynamic game).More generally, by Proposition 6.2 and Lemma 6.4, for any ξ ∈ C ( R ; Aut ( E )) , there is astrongly continuous two-parameter family ( T Φ ( m , ξ ) t,s ) s,t ∈ R of ∗ -automorphisms of U satisfying the re-verse cocycle property. This family satisfies a non-autonomous evolution equation, similar to Equa-tion (41). It is used to construct the infinite-volume limit of the non-autonomous dynamics of time-dependent long-range models of M within a cyclic representation associated with an arbitrary pe-riodic state. We show that, generically, a long-range (or mean-field) dynamics is equivalent to anintricate combination of a classical and short-range quantum dynamics. Both dynamics will be(non-trivial) consequences of the well-posedness of self-consistency equations , which are reminis-cent of [63, Theorem 4.1].To present these equations, recall that, for any Λ ∈ P f , M Λ belongs to the dense subset M ⊆ M of models with an arbitrary short-range part, while the long-range interactions are polynomials ofinteractions that are finite-range and translation-invariant. See (57)-(59) and Definition 4.2. Theorem 6.5 (Self-consistency equations)
Fix Λ ∈ P f and m ∈ C b ( R ; M Λ ) . There is a unique ̟ m ∈ C ( R ; Aut ( E )) such that ̟ m ( s, t ) = φ Φ ( m , ̟ m ( α, · )) t,s | α = s , s, t ∈ R , where ( φ Φ ( m , ξ ) t,s ) s,t ∈ R is the continuous family of automorphisms of E defined by (101) with the state-dependent interaction Ψ = Φ ( m , ξ ) of Definition 6.3 for ξ ∈ C ( R ; Aut ( E )) . Proof.
The theorem is a consequence of Lemmata 7.2 and 7.8.32 emark 6.6
Section 7 proves stronger results than Theorem 6.5. See, in particular, Lemma 7.4.
Remark 6.7
Self-consistency equations have been, very recently, also highlighted in [92] by studying particlesgoverned by the Vlasov equation and interacting with an oscillatory environment. In this case, self-consistency equations turn out to be essential to describe the behavior of this system at large times,giving an innovative use of such equations in EDP.
At fixed Λ ∈ P f and long-range model m ∈ C b ( R ; M Λ ) , Theorem 6.5 means that, for any s, t ∈ R , ρ ∈ E and A ∈ U , ρ s,t ( A ) = ρ ◦ τ Φ ( m , ̟ m ( α, · )) ( ρ ) t,s ( A ) | α = s with ρ s,t . = ̟ m ( s, t ; ρ ) . = ̟ m ( s, t ) ( ρ ) ∈ E . (113)See Equations (88), (89) and (101). Let ( C n ) n ∈ N be an arbitrary family of closed sets C n ⊆ ( S ∩ W Λ ) n , n ∈ N , such that, for t ∈ R , | a ( t ) n | ( C n ) = | a ( t ) n | ( S n ) with m = (Φ( t ) , a ( t )) t ∈ R ∈ C b ( R ; M Λ ) (cf. (31) and (57)-(58)). Then, (113) for the time-dependent expectation ρ s,t ( A ) ∈ C , s, t ∈ R , of elements A ∈ V m . = n e Ψ ,~ℓ : Ψ ∈ C n , n ∈ N o leads to a systems of non-autonomous, coupled and non-linear equations, in general. These self-consistency equations are strongly related to the self-consistency equations (gap equations) explainedin [46, Section 2.8] for the special case of (generalized) equilibrium states. By contrast, Equations(113) for elements A ∈ U \V m are not coupled to each other.Last but not least, for any m ∈ C b ( R ; M Λ ) , observe from (104) that ̟ m (cid:0) s, t ; E + (cid:1) ⊆ E + , ̟ m (cid:0) s, t ; E \ E + (cid:1) ⊆ E \ E + , s, t ∈ R . Recall that E + is the weak ∗ -compact convex set of even states defined by (15). When long-rangemodels are translation-invariant, that is, if m ∈ C b ( R ; M ∩ M Λ ) (see (56)), the non-autonomousapproximating interactions of Definition 6.3 are also translation-invariant. By (106), it follows, in thiscase, that ̟ m ( s, t ) maps periodic states to periodic states: for any ~ℓ ∈ N d , ̟ m (cid:0) s, t ; E ~ℓ (cid:1) ⊆ E ~ℓ , ̟ m (cid:0) s, t ; E \ E ~ℓ (cid:1) ⊆ E \ E ~ℓ , s, t ∈ R . (114)Recall that E ~ℓ ⊆ E is the weak ∗ -compact convex subset of ~ℓ -periodic states defined by (17). Like E , ithas a dense set of extreme points ( ~ℓ -ergodic states). Additionally, in this case, similar to [63, Corollary4.3], for every ~ℓ ∈ N d , the set E ( E ~ℓ ) of extreme points of E ~ℓ is conserved by the flow.33 .5 Classical Part of Long-Range Dynamics Similar to (102), the continuous family ̟ m of Theorem 6.5 yields a family ( V m t,s ) s,t ∈ R of ∗ -auto-morphisms of C defined by V m t,s ( f ) . = f ◦ ̟ m ( s, t ) , f ∈ C , s, t ∈ R . (115)It is a strongly continuous two-parameter family defining a classical dynamics on the commutative C ∗ -algebra C of continuous complex-valued functions on states defined by (65)-(66): Proposition 6.8 (Classical dynamics as Feller evolution system)
Fix Λ ∈ P f and m ∈ C b ( R ; M Λ ) . Then, ( V m t,s ) s,t ∈ R is a strongly continuous two-parameter family of ∗ -automorphisms of C satisfying the reverse cocycle property: ∀ s, r, t ∈ R : V m t,s = V m r,s ◦ V m t,r . If m ∈ M Λ C b ( R ; M Λ ) , i.e., m is constant in time, then V m t,s = V m t − s, for any s, t ∈ R and ( V m t, ) t ∈ R is a C -group of ∗ -automorphisms of C . Proof.
In order to prove these assertions, one simply adapts the argument used to prove [63, Proposi-tion 3.4], having in mind results of Section 7. We omit the details.Like (102), ( V m t,s ) s,t ∈ R can be associated with a Feller process in probability theory. If m ∈ C b ( R ; M Λ ∩ M ) , note that the classical flow conserves the Poulsen simplex E ~ℓ , ~ℓ ∈ N d , and V m t,s canbe seen as either a mapping from C ( E ~ℓ ; C ) to itself or from C ( E \ E ~ℓ ; C ) to itself: V m t,s ( f | E ~ℓ ) . = ( V m t,s f ) | E ~ℓ , V m t,s ( f | E \ E ~ℓ ) . = ( V m t,s f ) | E \ E ~ℓ , f ∈ C , s, t ∈ R , (116)using (114). Compare with Equation (107). For all m ∈ C b ( R ; M Λ ) , the same holds true for theweak ∗ -compact convex set E + of even states.For any constant function m ∈ M Λ C b ( R ; M Λ ) , ( V m t, ) t ∈ R is a C -group of ∗ -automorphismsof C and we denote by k m its (well-defined) generator. By [88, Chap. II, Sect. 3.11], it is a closed(linear) operator densely defined in C . Since V m t, , t ∈ R , are ∗ -automorphisms, we infer from theNelson theorem [86, Theorem 1.5.4], or the Lumer-Phillips theorem [79, Theorem 3.1.16], that ± k m are dissipative operators, i.e., k m is conservative. The ∗ -homomorphism property of V m t, , t ∈ R , isreflected by the fact that k m has to be a symmetric derivation of C . In fact, similar to [63, Theorem4.5], k m is directly related to a Poissonian symmetric derivation.In order to understand this fact, we need to figure out the appropriate classical energy functions ,which correspond in [63, Theorem 4.5] to the function h . Recall that ˆ A ∈ C is the continuous andaffine function defined by (67) for any A ∈ U , while {· , ·} is the Poisson bracket of Definition 5.2.Now, having in mind Definition 4.2, it is natural to define the following family of classical energyfunctions of C : Definition 6.9 (Classical energy functions of long-range dynamics)
For any m ∈ M , we define the functions h m L . = c U Φ L + X n ∈ N | Λ L | n − Z S n [ U Ψ (1) L · · · [ U Ψ ( n ) L a ( t ) n (cid:0) dΨ (1) , . . . , dΨ ( n ) (cid:1) ∈ C R , L ∈ N , which we name the local classical energy functions associated with m . h m L = c U m L , in general. Local classical energyfunctions are continuously differentiable real-valued functions over the convex and weak ∗ -compactset E , i.e., { h m L } L ∈ N ⊆ Y R , the subspace Y R ⊆ C R being defined by (70): For any m ∈ M , bystraightforward estimates using Equations (35), (54)-(55) and Definition 4.1 together with Lebesgue’sdominated convergence theorem, one checks that, for any L ∈ N , h m L is continuously differentiableand Dh m L = U Φ L − c U Φ L + X n ∈ N n X m =1 Z S n (cid:18) U Ψ ( m ) L − \ U Ψ ( m ) L (cid:19) Y j ∈{ ,...,n } ,j = m [ U Ψ ( j ) L | Λ L | a ( t ) n (cid:0) dΨ (1) , . . . , dΨ ( n ) (cid:1) , (117)where Dh m L ∈ C ( E ; U R ) = U R is the function defined by (72) for f = h m L . Moreover, by (35), (68),(71) and (73), for any m ∈ M , k h m L k Y = k h m L ( ρ ) k C + max ρ ∈ E k Dh m L ( ρ ) k U ≤ | Λ L | k F k , L k m k M , L ∈ N , which is similar to Inequality (60) that bounds the norm of the Hamiltonians U m L , L ∈ N .Last but not least, it is very instructive to compare (117) with the local Hamiltonians associatedwith approximating interactions of Definition 6.3, in the light of Proposition 3.2: All terms of theform c U Ψ L ( ρ ) | Λ L | = ρ (cid:0) U Ψ L (cid:1) | Λ L | , Ψ ∈ S , ρ ∈ E, in (117) should converge, as L → ∞ . By Proposition 3.2, we know this holds true whenever ρ is aperiodic state, the limit being ρ ( e Φ ,~ℓ ) for some ~ℓ ∈ N d . The periodicity of states is therefore a veryuseful property in this context. Recall that periodic states form a weak ∗ -dense set E p (16) of thephysically relevant set E + (15) of even states, by Proposition 2.3.So, restricting our study to periodic states, we are now in a position to link the generator k m to aPoissonian symmetric derivation: Corollary 6.10 (Classical evolutions via Poisson brackets)
For any Λ ∈ P f and m ∈ M Λ , the dense ∗ -subalgebra C U (69) belongs to the domain of k m and k m ( f ) | E p = lim L →∞ { h m L , f } | E p , f ∈ C U , where the limit has to be understood point-wise on the weak ∗ -dense subspace E p ⊆ E + of all periodicstates. Proof.
Fix Λ ∈ P f and m ∈ M Λ . Comparing Definitions 3.3 and 6.3 with Definition 5.2 and(117) and using Proposition 3.2 and Corollary 3.5 together with Lebesgue’s dominated convergencetheorem, one computes that lim L →∞ { h m L , f } ( ρ ) = ρ ◦ δ Φ ( m ,ρ ) (D f ( ρ )) , f ∈ C U , ρ ∈ E p . (118)Note that one can interchange the local quantum derivation (which is a commutator) in the left-handside of (118) with every integral over ( S ∩ W Λ ) n , n ∈ N , by finite dimensionality of the space U Λ L forall L ∈ N . Here, we use (112) to define the approximating interaction Φ ( m ,ρ ) , as well as the fact that D f ( ρ ) ∈ U whenever f ∈ C U . Now, the rest of the proof is very similar to the one of [63, Theorem4.5]: By Lemma 7.4, one verifies that, for any A ∈ U and t ∈ R , ∂ t V m t, ( ˆ A ) ( ρ ) = V m t, ◦ k m ( ˆ A ) ( ρ ) = ̟ m (0 , t ; ρ ) ◦ δ Φ ( m , ̟ m (0 ,t ; ρ )) ( A ) . (119)35ince k m and d h m L , L ∈ N , are symmetric derivations, the assertion follows by combining (118) and(119).Equation (119) also holds true for any time-dependent model m ∈ C b ( R ; M Λ ) and Λ ∈ P f , with k m being replaced with k m ( t ) in (119). Therefore, in the non-autonomous situation, for any s, t ∈ R , ρ ∈ E and polynomial function f ∈ C U , ∂ t V m t,s ( f ) ( ρ ) = V m t,s ◦ k m ( t ) ( f ) ( ρ ) = ̟ m ( s, t ; ρ ) ◦ δ Φ ( m , ̟ m ( s,t ; ρ )) ( f ( ρ )) . (120)Similar (point-wise) identities for ∂ s V m t,s like ∂ s V m t,s ( f ) = − k m ( s ) ◦ V m t,s ( f ) (121)are not at all obvious. In fact, no unified theory of non-autonomous evolution equations that gives acomplete characterization of the existence of fundamental solutions in terms of properties of genera-tors, analogously to the Hille-Yosida generation theorems for the autonomous case, is available. See,e.g., [71–75] and references therein.An important, highly non-trivial , result in that direction is proven in the next theorem, whichdepends in a crucial way on Lieb-Robinson bounds for multi-commutators of [76, Theorems 4.11,5.4]: Theorem 6.11 (Non-autonomous classical dynamics)
Fix Λ ∈ P f , ς, ǫ ∈ R + and take F ( x, y ) = e − ς | x − y | (1 + | x − y | ) − ( d + ǫ ) , x, y ∈ L , as the decay function, see (26). Then, ( V m t,s ) s,t ∈ R is a strongly continuous two-parameter family of ∗ -automorphisms of C satisfying, on the dense ∗ -subalgebra C U (69), ∀ s, t ∈ R : ∂ t V m t,s ( f ) | E p = lim L →∞ V m t,s (cid:16) { h m ( t ) L , f } (cid:17) | E p , V m s,s = C , (122) for any m ∈ C b ( R ; M Λ ∩ M ) while, for any m ∈ C b ( R ; M Λ ) , s, t ∈ R and f ∈ C U , V m t,s ( f ) ∈ Y with ∀ s, t ∈ R : ∂ s V m t,s ( f ) | E p = − lim L →∞ { h m ( s ) L , V m t,s ( f ) }| E p , V m t,t = C . (123) All limits have to be understood point-wise on the weak ∗ -dense subspace E p ⊆ E + of all periodicstates. Proof.
Fix all parameters of the theorem. Equation (122) results from (114), (118), (120), Lemma3.2 and the fact that m ∈ C b ( R ; M Λ ∩ M ) (cf. (116)). In order to prove V m t,s ( f ) ∈ Y and (123), itsuffices to invoke Lemma 7.13, which says that ∂ s V m t,s ( ˆ A ) | E p = − lim L →∞ { h m ( s ) L , V m t,s ( ˆ A ) }| E p for any s, t ∈ R and A ∈ U . Since ( V m t,s ) s,t ∈ R is a family of ∗ -automorphisms of C , by usingthe (bi)linearity and Leibniz’s rule satisfied by the derivatives and the bracket {· , ·} , we deduce that V m t,s ( f ) ∈ Y and (123) for all polynomial functions f ∈ C U and times s, t ∈ R .By Corollary 6.10 and Theorem 6.11, Equation (121) seems to hold true, but it cannot be directlydeduced from Theorem 6.11. We refrain from doing such a study in this paper, since Theorem 6.11already shows that we have a non-autonomous classical dynamics in the usual sense. Note that, inthe case L = Z d , we have to consider the limit L → ∞ . This is the classical counterpart of thethermodynamic limit L → ∞ in the derivations of Corollary 3.5.In the autonomous situation, as already suggested by Corollary 6.10, we obtain from Theorem6.11 the usual (autonomous) dynamics of classical mechanics written in terms of Poisson brackets(see, e.g., [89, Proposition 10.2.3]), i.e., Liouville’s equation :36 orollary 6.12 (Liouville’s equation)
Under conditions of Theorem 6.11, for any t ∈ R and f ∈ C U , ∂ t V m t, ( f ) = V m t, ◦ k m ( f ) = V m t, (cid:16) lim L →∞ { h m L , f } (cid:17) = lim L →∞ (cid:8) h m L , V m t, ( f ) (cid:9) = k m ◦ V m t, ( f ) , where all limits have to be understood point-wise on the weak ∗ -dense subspace E p ⊆ E + of allperiodic states. Proof.
Combine Corollary 6.10 with Theorem 6.11.Writing the classical dynamics in terms of Liouville’s equation, as in Corollary 6.12, is concep-tually illuminating and also very useful from a purely mathematical point of view. For instance, theGross-Pitaevskii and Hartree hierarchies mathematically derived from Bose gases with mean-field interactions are infinite systems of coupled PDEs and therefore, a direct proof of the uniquenessof its solutions is technically quite demanding, usually involving Feynman graphs, multilinear es-timates, etc. In [42], the authors show that a solution to these hierarchies is basically a family oftime-dependent correlation functions associated with a certain positive measure on the unit ball of a L -space, whose dynamical evolution is driven by Liouville’s equation, similarly to Corollary 6.12.Uniqueness of a solution to such Liouville’s equation can be proven in a general setting, as shown in2018 [42, 90], implying the uniqueness of a solution to the Gross-Pitaevskii and Hartree hierarchieswithout highly technical issues depending on the particularities of the hierarchies under consideration. The classical part of the dynamics of lattice-fermion systems with long-range interactions, whichis defined within the classical C ∗ -algebras C of continuous complex-valued functions on states, isshown to result from self-consistency equations, as explained in Theorem 6.5. Since C can be seen asa subalgebra (76) of the quantum C ∗ -algebras U of continuous U -valued functions on states, definedby (74)-(75), there is a natural extension of the classical dynamics on C : The continuous family ̟ m of Theorem 6.5 yields a family ( V m t,s ) s,t ∈ R of ∗ -automorphisms of U defined by V m t,s ( f ) . = f ◦ ̟ m ( s, t ) , f ∈ U , s, t ∈ R . In particular, by (115), V m t,s | C = V m t,s for any s, t ∈ R . However, it is not what we have in mind here : Emphasizing rather the inclusion
U ⊆ U , in thelong-range dynamics, the classical algebra C becomes a subalgebra of the fixed-point algebra of thestate-dependent long-range dynamics on U . In [64] we describe in detail the quantum part of thelong-range dynamics, which is defined in a representation of the C ∗ -algebra U . We give below a fewkey points of this study:(i): As soon as the classical part of the long-range dynamics is concerned, there is no need to imposeany additional property on initial states to define it. By contrast, for the quantum part, periodicityof initial states is needed. Note, however, that the set E p of all periodic states is still a weak ∗ -densesubset of the physically relevant set E + of all even states, by Proposition 2.3.(ii): From now on, fix ~ℓ ∈ N d and consider the set E ~ℓ of ~ℓ -periodic states defined by (17). Note thatany ~ℓ -periodic state ρ ∈ E ~ℓ ⊆ U ∗ naturally extends to a state on U : There is a natural conditionalexpectation Ξ from U to C ⊆ U defined by Ξ ( f ) ( ρ ) = ρ ( f ( ρ )) , ρ ∈ E .
Then, since E ~ℓ is a Choquet simplex, any state ρ ∈ E ~ℓ can be uniquely identified with its Choquetmeasure µ ρ which, in turn, is a state of C , canonically viewed as a measure on E . The state of U ρ ∈ E ~ℓ is the state µ ρ ◦ Ξ . Moreover, if ( H ρ , π ρ , Ω ρ ) is a cyclic representation of ρ ∈ E ~ℓ ,then, by [64, Proposition 4.2], there is a representation Π ρ : U → B ( H ρ ) such that [Π ρ ( U )] ′′ = [Π ρ ( U )] ′′ and Π ρ ( A ) = π ρ ( A ) for any A ∈ U ⊆ U . In particular, [Π ρ ( C )] ′′ ⊆ [ π ρ ( U )] ′ ∩ [ π ρ ( U )] ′′ . In fact, for ρ ∈ E ~ℓ , ( H ρ , Π ρ , Ω ρ ) is a cyclic representation of the state µ ρ ◦ Ξ ∈ U ∗ . The existence ofsuch an extension of π ρ strongly depends on the orthogonality of the ( ~ℓ -) ergodic decomposition of ~ℓ -periodic states, as it is explained in [64]. The ergodicity property of extreme states of E ~ℓ is pivotalin order to get the limit dynamics stated in the third point.(iii): Let m ∈ C b ( R ; M Λ ∩ M ) for some Λ ∈ P f . Assume that the state at initial time s ∈ R is ρ ∈ E ~ℓ . Then, by taking the above cyclic representation ( H ρ , Π ρ , Ω ρ ) of ρ , seen as the state µ ρ ◦ Ξ on U , we show in [64, Theorem 4.3] that, for any t ∈ R and A ∈ U ⊆ U , in the thermodynamic limit L → ∞ , Π ρ (cid:16) T Φ ( m , ̟ m ( α, · )) t,s ( A ) | α = s − τ ( L, m ) t,s ( A ) (cid:17) (124)converges to in the σ -weak topology within B ( H ρ ) . In particular, the restriction to U of the state ρ ◦ T Φ ( m , ̟ m ( α, · )) t,s | α = s (125)can be seen as the state of the system at any time t ∈ R when ρ is the (initial) state at time t = s .Here, ̟ m ∈ C (cid:0) R ; Aut ( E ) (cid:1) results from Theorem 6.5, Φ ( m , ξ ) is the state-dependent interaction of Definition 6.3 for ξ ∈ C ( R ; Aut ( E )) , ( T Ψ t,s ) s,t ∈ R is the strongly continuous two-parameter family of ∗ -automorphisms of U of Proposition6.2 for Ψ ∈ C ( R ; W R ) , and ( τ ( L, m ) t ) t ∈ R is the strongly continuous one-parameter group of ∗ -auto-morphisms of U defined by (62). Note that, even if the local dynamics is autonomous, the limitdynamics can still be non-autonomous .A similar result holds true for non-autonomous long-range dynamics, i.e., for all time-dependentmodels m ∈ C b ( R ; M Λ ∩ M ) , Λ ∈ P f . In this case, one replaces the autonomous local dynamics in (124) with the non-autonomous localone, similar to (39)-(40).
The aim of this section is to prove Theorems 6.5 and 6.11. In fact, we prove here stronger results thanthese theorems. The proof of Theorem 6.5 is done in five lemmata and two corollaries. The proofof Theorem 6.11 is a direct consequence of Lemma 7.13. Note that those proofs are a much moreinvolved version of the ones performed in [63, Section 7] to prove [63, Theorems 4.1 and 4.6].38e start with preliminary definitions: In all the present section, fix once and for all ~ℓ ∈ N d , Λ ∈ P f and a time-dependent model m = ((Φ ( t ) , a ( t ) )) t ∈ R ∈ C b ( R ; M Λ ) , with M Λ . = W R × S Λ ⊆ M , (see (57)), S Λ ⊆ S being defined by (58). By (32)-(33), we can assume without loss of generality that e Φ ,~ℓ ∈ U Λ . In order to simplify mathematical expressions, we use the standard notation k m k ∞ ≡ k m k C b ( R ; M ) . = sup t ∈ R k m ( t ) k M , m ∈ C b ( R ; M Λ ) . (126)Recall that E Λ ⊆ U ∗ Λ , which is defined by (10), is the norm-compact set of states on the finite-dimensional C ∗ -algebra U Λ . For every continuous function ζ ∈ C ( R ; E Λ ) , let Ψ ζ ( t ) . = Φ ( t ) + X n ∈ N Z ( S ∩W Λ ) n (cid:4) ζ ( t ) ; Ψ (1) , . . . , Ψ ( n ) (cid:5) ~ℓ a ( t ) n (cid:0) dΨ (1) , . . . , dΨ ( n ) (cid:1) , t ∈ R , (127)with W Λ ⊆ W being defined by (31). In particular, by Definition 6.3 and Equation (58), for allcontinuous functions ξ ∈ C ( R ; E ) , Ψ ξ | U Λ ( t ) = Φ ( m ,ξ ) ( t ) , t ∈ R . (128)Note that such approximating interactions can be used to define a strongly continuous two-parameterfamily ( τ Ψ ζ t,s ) s,t ∈ R of ∗ -automorphisms of U for any ζ ∈ C ( R ; E Λ ) , by Proposition 3.7 and Lemma6.4. Such approximating dynamics satisfy the following estimate: Lemma 7.1 (Estimates on approximating dynamics)
For any s, t ∈ R , ˜Λ ∈ P f , A ∈ U ˜Λ and ζ , ζ ∈ C ( R ; E Λ ) ⊆ C ( R ; U ∗ Λ ) , (cid:13)(cid:13)(cid:13)(cid:16) τ Ψ ζ t,s − τ Ψ ζ t,s (cid:17) ( A ) (cid:13)(cid:13)(cid:13) U ≤ | ˜Λ | k A k U k F k , L k m k ∞ e D k m k ∞ | t − s | Z t ∨ st ∧ s k ζ ( α ) − ζ ( α ) k U ∗ Λ d α . Here, U ∗ Λ is endowed with the usual norm for continuous linear functionals. Proof.
Similar to Inequality (111), by Equations (30), (54)-(55) and (128), note that, for any ζ , ζ ∈ C ( R ; E Λ ) , (cid:13)(cid:13) Ψ ζ ( α ) − Ψ ζ ( α ) (cid:13)(cid:13) W ≤ k m ( α ) k M k ζ ( α ) − ζ ( α ) k U ∗ Λ , α ∈ R . Combining this inequality with Proposition 3.8 (iii) and Lemma 6.4, we obtain the assertion. Notethat C ( R ; E Λ ) ⊆ C ( R ; U ∗ Λ ) , because the norm and weak ∗ topologies of U ∗ Λ are the same, by finitedimensionality of U Λ for Λ ∈ P f .We now show the existence and uniqueness of the solution to the self-consistency equation: Lemma 7.2 (Self-consistency equations)
For any s ∈ R and ρ ∈ E , there is a unique solution ̟ ρ,s to the following equation in ξ ∈ C ( R ; E ) : ∀ t ∈ R : ξ ( t ) = ρ ◦ τ Ψ ξ |U Λ t,s . (129) Moreover, ̟ ρ,s ( t ) = ̟ ̟ ρ,s ( r ) ,r ( t ) for any r, s, t ∈ R . roof. The proof is similar to the one of [63, Lemma 7.3]: Fix the initial time s ∈ R and state ρ ∈ E .The existence and uniqueness of a solution ̟ ρ,s to (129) is proven via the Banach fixed point theorem:Step 1: Fix T ∈ R + and observe that C ([ s − T, s + T ]; E Λ ) is a closed bounded subset of the Ba-nach space C ([ s − T, s + T ]; U ∗ Λ ) , with U ∗ Λ being endowed with the usual norm for continuous linearfunctionals and k ζ k C ([ s − T,s + T ]; U ∗ Λ ) . = sup t ∈ [ s − T,s + T ] k ζ ( t ) k U ∗ Λ , ζ ∈ C ([ s − T, s + T ]; U ∗ Λ ) . Define the mapping F from C ([ s − T, s + T ]; E Λ ) to itself by F ( ζ ) ( t ) . = ρ ◦ τ Ψ ζ t,s (cid:12)(cid:12)(cid:12) U Λ , t ∈ [ s − T, s + T ] . (130)The existence of such a ∗ -automorphism τ Ψ ζ t,s for any ζ ∈ C ([ s − T, s + T ]; E Λ ) and t ∈ [ s − T, s + T ] follows from Proposition 3.7 and Lemma 6.4. By Proposition 3.7 (or Proposition 3.8 (iv)), (130)defines a mapping from C ([ s − T, s + T ]; E Λ ) to itself. Moreover, by (127) and (130), we infer fromLemma 7.1 that, for any ζ , ζ ∈ C ([ s − T, s + T ]; E Λ ) , k F ( ζ ) ( t ) − F ( ζ ) ( t ) k C ([ s − T,s + T ]; U ∗ Λ ) (131) ≤ T | Λ | k F k , L k m k ∞ e D T k m k ∞ k ζ − ζ k C ([ s − T,s + T ]; U ∗ Λ ) . Therefore, by fixing the time parameter T ∈ R + such that T ≤ e − D T k m k ∞ | Λ | k F k , L k m k ∞ , (132)the function F is a contraction. Hence, we obtain a unique solution ג ρ,s ∈ C ([ s − T, s + T ]; E Λ ) toEquation (129) with ξ | U Λ = ג ρ,s at fixed s ∈ R and ρ ∈ E .Step 2: By (128), the restriction of any solution ̟ ρ,s ∈ C ([ s − T, s + T ]; E ) to (129) to the subspace U Λ ⊆ U must equal ג ρ,s ∈ C ([ s − T, s + T ]; E Λ ) and Φ ( m ,̟ ρ,s ( t )) = Ψ ג ρ,s ( t ) . With this observation,we see that ̟ ρ,s ( t ) . = ρ ◦ τ Ψ ג ρ,s t,s , t ∈ [ s − T, s + T ] , is the unique solution in C ([ s − T, s + T ]; E ) to (129) at fixed initial time s ∈ R and state ρ ∈ E .Step 3: In the same way we prove the existence and uniqueness of a solution to (129) at fixed s ∈ R and ρ ∈ E , one shows that, for each r ∈ [ s − T, s + T ] , the self-consistency equation ∀ t ∈ [ r − ˜ T , r + ˜ T ] : ξ ( t ) = ̟ ρ,s ( r ) ◦ τ Ψ ξ |U Λ t,r , (133)has also a unique solution ̟ ̟ ρ,s ( r ) ,r in C ([ r − ˜ T , r + ˜ T ]; E ) for any ˜ T ∈ (0 , T ] . By the reversecocycle property (42), at fixed s ∈ R and ρ ∈ E , ̟ ρ,s solves the self-consistency equation (133) forany r ∈ ( s − T, s + T ) and t ∈ [ s − ˜ T , s + ˜ T ] with ˜ T = T − | s − r | ∈ R + , whence ̟ ρ,s ( t ) = ̟ ̟ ρ,s ( r ) ,r ( t ) for any r ∈ ( s − T, s + T ) and t ∈ [ s − ˜ T , s + ˜ T ] .Step 4: Assume the existence and uniqueness of a solution ̟ ρ,s in C ([ s − T , s + T ]; E ) to Equation(129) for some parameter T ∈ R + . Take r ∈ ( s − T , s − T + T ) ∪ ( s + T − T, s + T ) .
40y combining the existence and uniqueness of a solution ̟ ̟ ρ,s ( r ) ,r to (133) in C ([ r − ˜ T , r + ˜ T ]; E ) for any ˜ T ∈ (0 , T ] together with the reverse cocycle property (42), we deduce that ̟ ρ,s ( t ) = ̟ ̟ ρ,s ( r ) ,r ( t ) , t ∈ ( s − T , s + T ) , as well as the existence of a unique solution ̟ ρ,s to (129) in C ([ s − T − T, s + T + T ]; E ) . As aconsequence, one can infer from a contradiction argument the existence and uniqueness of a solutionin ξ ∈ C ( R ; E ) to the self-consistency equation (129). Moreover, this solution must satisfy theequality ̟ ρ,s ( t ) = ̟ ̟ ρ,s ( r ) ,r ( t ) for any r, s, t ∈ R . Corollary 7.3 (Bijectivity of the solution to the self-consistency equation)
For any s, t ∈ R , ̟ s ( t ) ≡ ( ̟ ρ,s ( t )) ρ ∈ E is a bijective mapping from E to itself. Proof.
The assertion is a direct consequence of Lemma 7.2, in particular the equality ̟ ρ,s ( t ) = ̟ ̟ ρ,s ( r ) ,r ( t ) for r, s, t ∈ R .By combining Lemma 7.2 with Proposition 3.7, note that, for any s ∈ R , ρ ∈ E and A ∈ U , ∂ t { ̟ ρ,s ( t ) ( A ) } = ρ ◦ τ Ψ ג t,s ◦ δ Ψ ג ( t ) ( A ) , t ∈ R , with ג . = ̟ ρ,s | U Λ . This property can be strengthened within the local C ∗ -algebra U Λ : Lemma 7.4 (Differentiability of the solution – I)
For any s ∈ R and ρ ∈ E , ג . = ̟ ρ,s | U Λ ∈ C ( R ; U ∗ Λ ) with derivative given by ∂ t { ̟ ρ,s ( t ) | U Λ } = ρ ◦ τ Ψ ג t,s ◦ δ Ψ ג ( t ) | U Λ , t ∈ R . Here, U ∗ Λ is endowed with the usual norm for continuous linear functionals. Proof.
To prove that ג . = ̟ ρ,s | U Λ ∈ C ( R ; U ∗ Λ ) at fixed s ∈ R and ρ ∈ E , we first remark that, forany A ∈ U Λ and h ∈ R \{ } , (cid:12)(cid:12)(cid:12) h − (cid:16) ρ ◦ τ Ψ ג t + h,s ( A ) − ρ ◦ τ Ψ ג t,s ( A ) (cid:17) − ρ ◦ τ Ψ ג t,s ◦ δ Ψ ג ( t ) ( A ) (cid:12)(cid:12)(cid:12) (134) ≤ max α ∈ [ t − h,t + h ] (cid:13)(cid:13)(cid:13) δ Ψ ג ( α ) − Ψ ג ( t ) ( A ) (cid:13)(cid:13)(cid:13) U + max α ∈ [ t − h,t + h ] (cid:13)(cid:13)(cid:13)(cid:16) τ Ψ ג α,s − τ Ψ ג t,s (cid:17) ◦ δ Ψ ג ( t ) ( A ) (cid:13)(cid:13)(cid:13) U . Since ג ∈ C ( R ; E Λ ) , by Corollary 3.5, for any A ∈ U Λ with k A k U = 1 , max α ∈ [ t − h,t + h ] (cid:13)(cid:13)(cid:13) δ Ψ ג ( α ) − Ψ ג ( t ) ( A ) (cid:13)(cid:13)(cid:13) U ≤ | Λ | k F k , L max α ∈ [ t − h,t + h ] (cid:13)(cid:13) Ψ ג ( α ) − Ψ ג ( t ) (cid:13)(cid:13) W and hence, by using Lemma 6.4, we arrive at lim h → sup A ∈U Λ , k A k U =1 max α ∈ [ t − h,t + h ] (cid:13)(cid:13)(cid:13) δ Ψ ג ( α ) − Ψ ג ( t ) ( A ) (cid:13)(cid:13)(cid:13) U = 0 . (135)Meanwhile, by Proposition 3.4, for any A ∈ U Λ satisfying k A k U = 1 , (cid:13)(cid:13)(cid:13)(cid:16) τ Ψ ג α,s − τ Ψ ג t,s (cid:17) ◦ δ Ψ ג ( t ) ( A ) (cid:13)(cid:13)(cid:13) U ≤ (cid:13)(cid:13)(cid:13)(cid:16) τ Ψ ג α,s − τ Ψ ג t,s (cid:17) ◦ δ Ψ ג ( t ) L ( A ) (cid:13)(cid:13)(cid:13) U + 2 | Λ | (cid:13)(cid:13) Ψ ג ( t ) (cid:13)(cid:13) W sup y ∈ Λ X x ∈ Λ cL F ( x, y ) . ε ∈ R + , there is L ∈ N such that, for any A ∈ U Λ with k A k U = 1 , max α ∈ [ t − h,t + h ] (cid:13)(cid:13)(cid:13)(cid:16) τ Ψ ג α,s − τ Ψ ג t,s (cid:17) ◦ δ Ψ ג ( t ) ( A ) (cid:13)(cid:13)(cid:13) U ≤ max α ∈ [ t − h,t + h ] (cid:13)(cid:13)(cid:13)(cid:16) τ Ψ ג α,s − τ Ψ ג t,s (cid:17) ◦ δ Ψ ג ( t ) L ( A ) (cid:13)(cid:13)(cid:13) U + ε , while we obtain from Proposition 3.8 (iv) that lim h → sup A ∈U Λ , k A k U =1 max α ∈ [ t − h,t + h ] (cid:13)(cid:13)(cid:13)(cid:16) τ Ψ ג α,s − τ Ψ ג t,s (cid:17) ◦ δ Ψ ג ( t ) L ( A ) (cid:13)(cid:13)(cid:13) U = 0 . It follows that lim h → sup A ∈U Λ , k A k U =1 max α ∈ [ t − h,t + h ] (cid:13)(cid:13)(cid:13)(cid:16) τ Ψ ג α,s − τ Ψ ג t,s (cid:17) ◦ δ Ψ ג ( t ) ( A ) (cid:13)(cid:13)(cid:13) U = 0 . We finally combine the last limit with (134)-(135) to deduce that ג ∈ C ( R ; U ∗ Λ ) with derivative givenby ∂ t { ̟ ρ,s ( t ) | U Λ } = ∂ t ג ( t ) = ρ ◦ τ Ψ ג t,s ◦ δ Ψ ג ( t ) | U Λ , t ∈ R , at any fixed s ∈ R and ρ ∈ E . Lemma 7.5 (Continuity with respect to the initial condition)
For any s, t ∈ R , ̟ s ( t ) ≡ ( ̟ ρ,s ( t )) ρ ∈ E ∈ C ( E ; E ) . Proof.
Take s ∈ R and two states ρ , ρ ∈ E . Then, define the quantity X ( T ) . = max t ∈ [ s − T,s + T ] (cid:13)(cid:13)(cid:0) ̟ ρ ,s ( t ) − ̟ ρ ,s ( t ) (cid:1) | U Λ (cid:13)(cid:13) U ∗ Λ , T ∈ R + . (136)By Proposition 3.8 (ii) and Lemma 6.4 together with Equations (24) and (128), for any T, ε ∈ R + ,there is L ∈ N such that, for any A ∈ U Λ with k A k U = 1 , sup t ∈ [ s − T,s + T ] sup ζ ∈ C ( R ; E Λ ) (cid:13)(cid:13)(cid:13) τ Ψ ζ t,s ( A ) − τ ( L, Ψ ζ ) t,s ( A ) (cid:13)(cid:13)(cid:13) U ≤ ε . (137)By combining Lemmata 7.1 (for L = Λ L ) and 7.2 with (137), we thus obtain the bound X ( T ) ≤ ε + (cid:13)(cid:13)(cid:13) ( ρ − ρ ) | U Λ L (cid:13)(cid:13)(cid:13) U ∗ Λ L + 2 | Λ | k F k , L k m k ∞ e D T k m k ∞ Z T X ( α ) d α , where k m k ∞ is defined by (126). By Gr¨onwall’s inequality, X ( T ) ≤ ε + (cid:13)(cid:13)(cid:13) ( ρ − ρ ) | U Λ L (cid:13)(cid:13)(cid:13) U ∗ Λ L ! (cid:16) | Λ | k F k , L k m k ∞ e D T k m k ∞ Y ( T ) (cid:17) (138)with Y ( T ) . = Z T exp n | Λ | k F k , L k m k ∞ e D T k m k ∞ α o d α . (139)By finite dimensionality of U Λ L , the norm and weak ∗ topologies of U ∗ Λ L are the same and, by theweak ∗ continuity property of ̟ s ( t ) , we infer from (136) and (138)-(139) that ( ̟ ρ,s ( t ) | U Λ ) ρ ∈ E ∈ C ( E ; U ∗ Λ ) , s, t ∈ R . (140)The continuity is even uniform for times t in compact sets. Now, by Lemma 7.2, for any s, t ∈ R , ρ , ρ ∈ E and A ∈ U , one obviously gets from the triangle inequality that (cid:12)(cid:12)(cid:0) ̟ ρ ,s ( t ) − ̟ ρ ,s ( t ) (cid:1) ( A ) (cid:12)(cid:12) (141) ≤ (cid:12)(cid:12)(cid:12) ( ρ − ρ ) ◦ τ Ψ ̟ρ ,s |U Λ t,s ( A ) (cid:12)(cid:12)(cid:12) + (cid:13)(cid:13)(cid:13)(cid:16) τ Ψ ̟ρ ,s |U Λ t,s − τ Ψ ̟ρ ,s |U Λ t,s (cid:17) ( A ) (cid:13)(cid:13)(cid:13) U . ( ̟ ρ,s ( t ) ( A )) ρ ∈ E ∈ C ( E ; C ) , A ∈ U . By density of U ⊆ U and the fact that any state ρ ∈ E satisfies k ρ k U ∗ = 1 , the assertion follows. Corollary 7.6 (Solution to the self-consistency equation as self-homeomorphisms)
At fixed s, t ∈ R , ̟ s ( t ) ≡ ( ̟ ρ,s ( t )) ρ ∈ E ∈ Aut ( E ) , i.e., ̟ s ( t ) is an automorphism of the state space E . Moreover, it satisfies a cocycle property: ∀ s, r, t ∈ R : ̟ s ( t ) = ̟ r ( t ) ◦ ̟ s ( r ) . (142) Proof.
The proof is the same as the one of [63, Corollary 7.7]: By Corollary 7.3, for any s, t ∈ R , ̟ s ( t ) is a weak ∗ -continuous bijective mapping from E to itself. Since E is weak ∗ -compact, itsinverse is also weak ∗ -continuous. The cocycle property is a rewriting of the equality ̟ ρ,s ( t ) = ̟ ̟ ρ,s ( r ) ,r ( t ) of Lemma 7.2.Before stating the next lemma, recall that the topology used in the subspace Aut ( E ) C ( E ; E ) of all automorphisms of E is the one of uniform convergence of weak ∗ -continuous functions, as statedin Equation (100). Lemma 7.7 (Well-posedness of the self-consistency equation)
For any s ∈ R , ̟ m s ≡ ( ̟ s ( t )) t ∈ R ≡ (( ̟ ρ,s ( t )) ρ ∈ E ) t ∈ R ∈ C ( R ; Aut ( E )) . At fixed s ∈ R , the mapping t ̟ s ( t ) is uniformly continuous for times t in compact sets, i.e., forany T, ε ∈ R + and A ∈ U there is η ∈ R + such that, for all t , t ∈ [ − T, T ] with | t − t | ≤ η , max ρ ∈ E | ̟ s ( t ) ( A ) − ̟ s ( t ) ( A ) | ≤ ε . Proof.
The proof is not exactly the same as the one of [63, Lemma 7.8], but it is similar: Takeany sequence ( t n ) n ∈ N ⊆ R converging to t ∈ R . Assume that ̟ s ( t n ) does not converge to ̟ s ( t ) ,uniformly. So, by density of U , there are ( ρ n ) n ∈ N ⊆ E , k ∈ N , L ∈ N , ε , . . . , ε k ∈ R + and A , . . . , A k ∈ U Λ L such that lim inf n →∞ (cid:12)(cid:12)(cid:2) ̟ ρ n ,s ( t n ) − ̟ ρ n ,s ( t ) (cid:3) ( A j ) (cid:12)(cid:12) ≥ ε j > , j ∈ { , . . . , k } . (143)By weak ∗ -compactness and metrizability of E , we can assume without loss of generality that thesequence ( ρ n ) n ∈ N weak ∗ -converges to some ρ ∈ E . By Lemma 7.2 and Equation (143), this in turnimplies that lim inf n →∞ (cid:12)(cid:12)(cid:12)h ρ n ◦ τ Ψ ̟ρn,s |U Λ t n ,s − ̟ ρ,s ( t ) i ( A j ) (cid:12)(cid:12)(cid:12) ≥ ε j > , j ∈ { , . . . , k } , (144)since ̟ s ( t ) ∈ C ( E ; E ) . Using Lemma 7.1, Lebesgue’s dominated convergence theorem and Equa-tion (140), we obtain from (144) that lim inf n →∞ (cid:12)(cid:12)(cid:12)h ρ n ◦ τ Ψ ̟ρ,s |U Λ t n ,s − ̟ ρ,s ( t ) i ( A j ) (cid:12)(cid:12)(cid:12) ≥ ε j > , j ∈ { , . . . , k } . (145)But this is a contradiction because ( τ Ψ t,s ) s,t ∈ R is a strongly continuous two-parameter family (Propo-sition 3.7) and hence, lim n →∞ ρ n ◦ τ Ψ ̟ρ,s |U Λ t n ,s ( A j ) = ρ ◦ τ Ψ ̟ρ,s |U Λ t,s ( A j ) = [ ̟ ρ,s ( t )] ( A j ) j ∈ { , . . . , k } . By (136) and (138)-(139) taken for some arbitrarily large (but finite) cubic box Λ L ⊇ Λ , note that ( ̟ ρ,s ( t ) | U Λ ) ρ ∈ E ∈ C (cid:0) E ; U ∗ Λ L (cid:1) is uniformly continuous for times t in compact setsand, as n → ∞ , ( τ Ψ ̟ρn,s |U Λ t,s − τ Ψ ̟ρ,s |U Λ t,s ) ( A ) , A ∈ U , converges to ∈ U uniformly with respect to times t in compact sets, by Lemma 7.1. It means thatthe mapping t ̟ s ( t ) is in fact uniformly continuous for times t in compact sets, at fixed s ∈ R . Lemma 7.8 (Joint continuity with respect to initial and final times)
The solution to the self-consistency equation is jointly continuous with respect to initial and finaltimes: ̟ m ≡ ( ̟ m s ) s ∈ R ≡ ( ̟ s ( t )) s,t ∈ R ≡ (( ̟ ρ,s ( t )) ρ ∈ E ) s,t ∈ R ∈ C (cid:0) R ; Aut ( E ) (cid:1) . Proof.
The proof is not the same as the one of [63, Lemma 7.9], but it is similar: Fix ρ ∈ E , s ∈ R and T ∈ R + and remark that C ([ s − T, s + T ] ; E Λ ) is a closed bounded subset of the Banach space C ([ s − T, s + T ] ; U ∗ Λ ) , where k ζ k C ( [ s − T,s + T ] ; U ∗ Λ ) . = sup α,t ∈ [ s − T,s + T ] k ζ ( α, t ) k U ∗ Λ , ζ ∈ C (cid:0) [ s − T, s + T ] ; U ∗ Λ (cid:1) . Similar to (130), we define the mapping F from C ([ s − T, s + T ] ; E Λ ) to itself by F ( ζ ) ( α, t ) . = ρ ◦ τ Ψ ζ t,α | U Λ , α, t ∈ [ s − T, s + T ] . See Proposition 3.7 and Lemma 6.4. By Inequality (131), F is also a contraction when the time T ∈ R + satisfies (132). Hence, in this case, for any ρ ∈ E and s ∈ R , there is a unique ˜ ג ∈ C (cid:0) [ s − T, s + T ] ; E Λ (cid:1) (146)such that ∀ α, t ∈ [ s − T, s + T ] : ˜ ג ( α, t ) = ρ ◦ τ Ψ ˜ ג ( α, · ) t,α | U Λ . Now, by uniqueness of the solution to (129), ̟ ρ,α ( t ) | U Λ = ˜ ג ( α, t ) = ρ ◦ τ Ψ ˜ ג ( α, · ) t,α | U Λ (147)for any α, t ∈ [ s − T, s + T ] . Similar to (141), for any A ∈ U and α , α , t , t ∈ [ s − T, s + T ] ,note that | ( ̟ ρ,α ( t ) − ̟ ρ,α ( t )) ( A ) | ≤ (cid:13)(cid:13)(cid:13)(cid:16) τ Ψ ˜ ג ( α , · ) t ,α − τ Ψ ˜ ג ( α , · ) t ,α (cid:17) ( A ) (cid:13)(cid:13)(cid:13) U + (cid:13)(cid:13)(cid:13)(cid:16) τ Ψ ˜ ג ( α , · ) t ,α − τ Ψ ˜ ג ( α , · ) t ,α (cid:17) ( A ) (cid:13)(cid:13)(cid:13) U + (cid:13)(cid:13)(cid:13)(cid:16) τ Ψ ˜ ג ( α , · ) t ,α − τ Ψ ˜ ג ( α , · ) t ,α (cid:17) ( A ) (cid:13)(cid:13)(cid:13) U . Using this elementary inequality together with Proposition 3.8 (iv), Lemmata 6.4, 7.1 and Equations(146)-(147), one gets that, for any local element A ∈ U , the mapping ( α, t ) ̟ ρ,α ( t ) ( A ) from [ s − T, s + T ] to C is continuous. By density of U U , we deduce that ( ̟ ρ,α ( t )) α,t ∈ R ∈ C (cid:0) [ s − T, s + T ] ; E (cid:1) , ρ ∈ E, s ∈ R , the parameter T ∈ R + satisfying (132). Via Corollary 7.6 and Lemma 7.7, we then deduce that ( ̟ ρ,s ( t )) s,t ∈ R ∈ C (cid:0) R ; E (cid:1) , ρ ∈ E . (148)44o get the assertion, it only remains to reproduce the compactness argument performed in theproof of Lemma 7.7: Take two sequences ( s n ) n ∈ N , ( t n ) n ∈ N ⊆ R converging to s, t ∈ R , respectively.Assume that ̟ s n ( t n ) does not converge to ̟ s ( t ) , uniformly, which corresponds to have Equation(144), the term τ Ψ ̟ρn,s |U Λ t n ,s being replaced with τ Ψ ̟ρn,sn |U Λ t n ,s n . Thanks to the triangle inequality andLemma 7.1, for any L ∈ N , A ∈ U Λ L , sufficiently large T ∈ R + and n ∈ N such that s, t ∈ ( − T, T ) and s n , t n ∈ [ − T, T ] , (cid:13)(cid:13)(cid:13)(cid:16) τ Ψ ̟ρn,sn |U Λ t n ,s n − τ Ψ ̟ρ,s |U Λ t n ,s n (cid:17) ( A ) (cid:13)(cid:13)(cid:13) U ≤ | Λ L | k A k U k F k , L k m k ∞ e D T k m k ∞ × Z T − T (cid:16)(cid:13)(cid:13)(cid:0) ̟ ρ n ,s n ( α ) − ̟ ρ,s n ( α ) (cid:1) | U Λ (cid:13)(cid:13) U ∗ Λ + k ( ̟ ρ,s n ( α ) − ̟ ρ,s ( α )) | U Λ k U ∗ Λ (cid:17) d α . As a consequence, by using (138)-(139) and (148) together with Lebesgue’s dominated convergencetheorem, one arrives from (144) at Equation (145), τ Ψ ̟ρ,s |U Λ t n ,s being replaced with τ Ψ ̟ρ,s |U Λ t n ,s n . This isnot possible because ( τ Ψ t,s ) s,t ∈ R is a strongly continuous two-parameter family, by Proposition 3.7.The proof of Theorem 6.5, being a consequence of Lemmata 7.2 and 7.8, is finished. In orderto prove Theorem 6.11, we give now several technical assertions. Concerning the first one, recallDefinition 5.1: For any f ∈ C and ρ ∈ E , d f ( ρ ) : E → C is the (unique) convex weak ∗ -continuousGˆateaux derivative of f at ρ ∈ E if d f ( ρ ) ∈ A ( E ; C ) and lim λ → + λ − ( f ((1 − λ ) ρ + λυ ) − f ( ρ )) = [d f ( ρ )] ( υ ) , υ ∈ E .
By Equation (72) extended to the complex case, for any f ∈ Y , there is a unique D f ∈ C ( E ; U ) suchthat d f ( ρ ) ( υ ) = \ D f ( ρ ) ( υ ) = υ (D f ( ρ )) , ρ, υ ∈ E .
In the next lemma, we compute the convex weak ∗ -continuous Gˆateaux derivative of the classicalevolution (115) of elementary functions defined by (67) for local elements of U : Lemma 7.9 (Differentiability of the solution – II)
For any s, t ∈ R and A ∈ U , ( ̟ ρ,s ( t ) ( A )) ρ ∈ E ≡ ( ̟ ρ,s ( t, A )) ρ ∈ E ∈ C ( E ; C ) (149) and, for any υ ∈ E , [d ̟ ρ,s ( t, A )] ( υ ) = υ (D ̟ ρ,s ( t, A )) = ( υ − ρ ) ◦ τ Ψ ג t,s ( A ) + z A [d ̟ ρ,s ( · , · ) ( υ )] , where ג ≡ ג ρ,s . = ̟ ρ,s | U Λ and, for any continuous function ξ : R × U Λ → C , z A [ ξ ] . = X n ∈ N X Z∈P f Z ts d α Z ( S ∩W Λ ) n a ( α ) n (cid:0) dΨ (1) , . . . , dΨ ( n ) (cid:1) n X m ,m =1 ,m = m ξ ( α, e Ψ ( m ,~ℓ ) ρ (cid:16) i h τ Ψ ג α,s (cid:16) Ψ ( m ) Z (cid:17) , τ Ψ ג t,s ( A ) i(cid:17)Y j ∈{ ,...,n }\{ m ,m } ̟ ρ,s (cid:16) α, e Ψ ( j ) ,~ℓ (cid:17) . (150) The above series is absolutely summable. Moreover, for any s ∈ R , ρ ∈ E and ˜Λ ∈ P f , the mapping ( t, A ) D ̟ ρ,s ( t, A ) from R × U ˜Λ to U is continuous. roof. We start with a preliminary observation: By Proposition 3.7 and (39) together with Lieb-Robinson bounds for multi-commutators (cf. Proposition 3.8 (i) and [76, Theorems 4.11, 5.4]) andLebesgue’s dominated convergence theorem, observe that (cid:0) τ Ψ t,s − τ Ψ t,s (cid:1) ( A ) = X Z∈P f Z ts d ατ Ψ α,s (cid:0) i (cid:2) (Ψ ( α ) − Ψ ( α )) Z , τ Ψ t,α ( A ) (cid:3)(cid:1) (151)for any s, t ∈ R , A ∈ U and Ψ , Ψ ∈ C ( R ; W R ∩ W Λ ) . (Note that, in this case, Ψ , Ψ are finite-range interactions.) The above series is absolutely summable, because of Lieb-Robinson bounds, asstated in Proposition 3.8 (i). In order to arrive at (151), we also use the equality (cid:0) τ Ψ t,s − τ Ψ t,s (cid:1) | U ˜Λ = lim L →∞ Z ts τ Ψ α,s ◦ (cid:16) δ Ψ ( α ) − δ Ψ ( α ) L (cid:17) ◦ τ (Ψ ,L ) t,α | U ˜Λ d α for any ˜Λ ∈ P f , s, t ∈ R and Ψ , Ψ ∈ C ( R ; W R ) , deduced from (39) and Proposition 3.7. Seealso Definitions 3.1, 3.3 and Corollary 3.5. In particular, to get (151), in the light of Equations(43)-(44), we have to estimate multi-commutators of order three . This is done by the extensionto multicommutators of the Lieb-Robinson bounds, contributed in [76, Theorems 4.11, 5.4]. Lieb-Robinson bounds for multi-commutators of order three stated in [76, Theorems 4.11, 5.4] requiresufficient polynomial decays of interactions. An obvious sufficient condition for such decays is totake Ψ ( t ) , Ψ ( t ) ∈ W Λ for all t ∈ R , meaning that they are all finite-range. See (31).Fix now all parameters of the lemma. For any s, t ∈ R , ρ, υ ∈ E , h ∈ (0 , and A ∈ U , i ( h, t, A ; υ ) . = h − (cid:0) ̟ (1 − h ) ρ + hυ,s ( t, A ) − ̟ ρ,s ( t, A ) (cid:1) = ( υ − ρ ) ◦ τ Ψ ג ρ,s t,s ( A ) + h − ((1 − h ) ρ + hυ ) ◦ (cid:16) τ Ψ ג (1 − h ) ρ + hυ,s t,s − τ Ψ ג ρ,s t,s (cid:17) ( A ) . Using now (109), (127) and (151) we deduce that, for any υ ∈ E and A ∈ U , i ( h, t, A ; υ ) = ( υ − ρ ) ◦ τ Ψ ג ρ,s t,s ( A ) (152) + X n ∈ N X Z∈P f Z ts d α Z ( S ∩W Λ ) n a ( α ) n (cid:0) dΨ (1) , . . . , dΨ ( n ) (cid:1) n X m ,m =1 ,m = m i (cid:16) h, α, e Ψ ( m ,~ℓ ; υ (cid:17) × ((1 − h ) ρ + hυ ) ◦ τ Ψ ג (1 − h ) ρ + hυ,s α,s (cid:16) i h Ψ ( m ) Z , τ Ψ ג ρ,s t,α ( A ) i(cid:17) × Y j ∈{ ,...,m − }\{ m } ̟ ρ,s (cid:16) α, e Ψ ( j ) ,~ℓ (cid:17) × Y j ∈{ m +1 ,...,n }\{ m } ̟ (1 − h ) ρ + hυ,s (cid:16) α, e Ψ ( j ) ,~ℓ (cid:17) , where the two products over j are, by definition, equal to when j ranges over the empty set . Notethat (151) can be used here because m ∈ C b ( R ; M Λ ) , implying that Ψ ג ∈ C b ( R ; W R ∩ W Λ ) . SeeLemma 6.4 and Equation (128).From Equation (152), one sees that i ( λ, t, A ; υ ) is given by a Dyson-type series which is ab-solutely summable, uniformly with respect to h ∈ (0 , . To show that, use m ∈ C b ( R ; M Λ ) (see(55) and (57)-(58)), Lemma 6.4 together with Equation (128), the fact that ( τ Ψ ξ t,s ) s,t ∈ R is a family of This happens when m ∈ { , n } . -automorphisms of U for any ξ ∈ C ( R ; E ) , Inequality (30) and the (usual) Lieb-Robinson bounds(Proposition 3.8 (i)). Note that, for any ˜Λ ∈ P f , the mapping R × U ˜Λ to C defined by ( t, A ) ( υ − ρ ) ◦ τ Ψ ג ρ,s t,s ( A ) is continuous. By Lemma 7.1 and Lebesgue’s dominated convergence theorem, lim h → + ((1 − h ) ρ + hυ ) ◦ τ Ψ ג (1 − h ) ρ + hυ,s α,s (cid:16) i h Ψ ( m ) Z , τ Ψ ג ρ,s t,α ( A ) i(cid:17) = ρ ◦ τ Ψ ג ρ,s α,s (cid:16) i h Ψ ( m ) Z , τ Ψ ג ρ,s t,α ( A ) i(cid:17) , uniformly for α in a compact set and, by Equations (136) and (138)-(139), lim h → + max α ∈ [ s − T,s + T ] (cid:13)(cid:13)(cid:0) ̟ (1 − h ) ρ + hυ,s,s ( α ) − ̟ ρ,s ( α ) (cid:1) | U Λ (cid:13)(cid:13) U ∗ Λ = 0 , T ∈ R + . Hence, using again Lebesgue’s dominated convergence theorem, we deduce that i (0 , t, A ; υ ) . = lim h → + i ( h, t, A ; υ ) = lim h → + h − (cid:0) ̟ (1 − h ) ρ + hυ,s ( t, A ) − ̟ ρ,s ( t, A ) (cid:1) (153)exists for all s, t ∈ R , ρ, υ ∈ E and A ∈ U , as given by a Dyson-type series. In particular, for any υ ∈ E and ˜Λ ∈ P f , the complex-valued function ( t, A ) i (0 , t, A, υ ) on R × U ˜Λ is the uniquesolution in ξ ∈ C ( R × U ˜Λ ; C ) to the equation ξ ( t, A ) = ( υ − ρ ) ◦ τ Ψ ג t,s ( A ) + z A [ ξ ] (154)with z A defined by (150). Compare with (152) taken at h = 0 .The uniqueness of the solution in ξ ∈ C ( R × U ˜Λ ; C ) to (154) is a consequence of the fact thatone can iterate Equation (154) in order to prove that ξ is the Dyson-type series i (0 , t, A, υ ) : First,it suffices to show this fact for ˜Λ = Λ and A ∈ U Λ with k A k U ≤ k F k , L because this case fixesall ξ ( t, e Ψ ,~ℓ ) for all Ψ ∈ S ∩ W Λ and t ∈ R (cf. (30)). Secondly, observe that any function ξ ∈ C ( R × U Λ ; C ) is bounded on compact sets of R × U Λ and any norm-closed ball of U Λ is compact,by finite dimensionality of U Λ . Using these observations together with (24)-(25), (30), (55), Lieb-Robinson bounds (Proposition 3.8 (i)), Lemma 6.4 and tedious computations, one checks that, forany A ∈ U Λ satisfying k A k U ≤ k F k , L , arbitrary time T ∈ R + and t ∈ [ s − T, s + T ] , | z A [ ξ ] | ≤ k F k , L e D T k m k ∞ | Λ | k m k ∞ sup t ∈ [ s − T,s + T ] sup B ∈U Λ : k B k U ≤k F k , L | ξ ( t, B ) | (cid:18)Z ts d α (cid:19) . (155)More generally, we can reconstruct from (154) the k th first terms of the Dyson-type series i (0 , t, A, υ ) and, in the same way one obtains (155) (i.e., the (0 + 1) th remaining term), the ( k + 1) th remainingterm is bounded by sup t ∈ [ s − T,s + T ] sup B ∈U Λ : k B k U ≤k F k , L | ξ ( t, B ) | (cid:16) k F k , L e D T k m k ∞ | Λ | k m k ∞ (cid:17) k +1 Z ts d α · · · Z α k − s d α k , and thus vanishes in the limit k → ∞ . So, by Lebesgue’s dominated convergence theorem, anysolution ξ ∈ C ( R × U ˜Λ ; C ) to (154) is equal to i (0 , t, A, υ ) .Similar to (152), using in particular the Lieb-Robinson bounds (Proposition 3.8 (i)), note that the47ntegral equation D ( t, A ) = τ Ψ ג t,s ( A ) − ρ ◦ τ Ψ ג t,s ( A ) (156) + X n ∈ N X Z∈P f Z ts d α Z ( S ∩W Λ ) n a ( α ) n (cid:0) dΨ (1) , . . . , dΨ ( n ) (cid:1) n X m ,m =1 ,m = m D (cid:16) α, e Ψ ( m ,~ℓ (cid:17) × ρ ◦ τ Ψ ג α,s (cid:16) i h Ψ ( m ) Z , τ Ψ ג t,α ( A ) i(cid:17) × Y j ∈{ ,...,n }\{ m ,m } ̟ ρ,s (cid:16) α, e Ψ ( j ) ,~ℓ (cid:17) . uniquely determines, by absolutely summable (in U ) Dyson-type series, a continuous mapping ( t, A ) D ( t, A ) from R × U ˜Λ to U for any ˜Λ ∈ P f , which, via (154), satisfies υ ( D ( t, A )) = i (0 , t, A ; υ ) . = lim h → + h − (cid:0) ̟ (1 − h ) ρ + hυ,s ( t, A ) − ̟ ρ,s ( t, A ) (cid:1) (157)for all s, t ∈ R , ρ, υ ∈ E and A ∈ U . Observe that the integrals in the corresponding Dyson-typeseries are well-defined as Bochner integrals: For any ˜Λ ∈ P f , the mapping from R × U ˜Λ to U definedby ( t, A ) τ Ψ ג t,s ( A ) − ρ ◦ τ Ψ ג t,s ( A ) (158)is continuous. Since the measures a ( α ) n , n ∈ N , are finite and U is a separable Banach space,by [94, Theorems 1.1 and 1.2], all terms appearing in the arguments of the integrals are Bochner-integrable. By Definition 5.1, the assertion follows.For the last assertions, we assume that the decay function equals F ( x, y ) = e − ς | x − y | (1 + | x − y | ) − ( d + ǫ ) , x, y ∈ L , (159)for some fixed ς , ǫ ∈ R + . See (26). Lemma 7.10 (Graph norm continuity of dynamics on local elements)
Assume (159). For any Ψ ∈ C ( R ; W R ) , Φ ∈ W , s, t ∈ R and A ∈ U , τ Ψ t,s ( A ) ∈ dom( δ Φ ) and δ Φ ◦ τ Ψ t,s ( A ) = lim L →∞ δ Φ ◦ τ ( L, Ψ) t,s ( A ) = lim L →∞ δ Φ L ◦ τ Ψ t,s ( A ) , uniformly for s, t on compacta. Additionally, for any s ∈ R and ˜Λ ∈ P f , the mapping ( t, A ) δ Φ ◦ τ Ψ t,s ( A ) from R × U ˜Λ to U is continuous. Proof.
For any Ψ ∈ C ( R ; W R ) , Φ ∈ W , s, t ∈ R and A ∈ U , a simple adaptation of [76, Equation(5.47), m = 1 ] (using Lieb-Robinson bounds for multi-commutators of order three [76, Theorems4.11, 5.4]) implies that lim L →∞ sup L ∈ N (cid:13)(cid:13)(cid:13) δ Φ L ◦ (cid:16) τ Ψ t,s − τ ( L , Ψ) t,s (cid:17) ( A ) (cid:13)(cid:13)(cid:13) U = 0 . (160)Using again a similar argument together with Corollary 3.5 and the closedness of δ Φ , we also deducethat δ Φ ◦ τ Ψ t,s ( A ) = lim L →∞ δ Φ ◦ τ ( L, Ψ) t,s ( A ) and τ Ψ t,s ( A ) ∈ dom( δ Φ ) . (161)48ote that the limits in Equations (160)-(161) are uniform for s, t on compacta (cf. [76, Theorem 5.6(i)]). For any Ψ ∈ C ( R ; W R ) , Φ ∈ W , L, L ∈ N , s, t ∈ R and A ∈ U , observe meanwhile that (cid:13)(cid:13) δ Φ ◦ τ Ψ t,s ( A ) − δ Φ L ◦ τ Ψ t,s ( A ) (cid:13)(cid:13) U ≤ (cid:13)(cid:13)(cid:13) δ Φ ◦ (cid:16) τ Ψ t,s − τ ( L , Ψ) t,s (cid:17) ( A ) (cid:13)(cid:13)(cid:13) U + (cid:13)(cid:13)(cid:13) δ Φ L ◦ (cid:16) τ Ψ t,s − τ ( L , Ψ) t,s (cid:17) ( A ) (cid:13)(cid:13)(cid:13) U + (cid:13)(cid:13)(cid:13)(cid:0) δ Φ − δ Φ L (cid:1) ◦ τ ( L , Ψ) t,s ( A ) (cid:13)(cid:13)(cid:13) U . By (160)-(161) and Corollary 3.5, it follows that, for any Φ ∈ W , s, t ∈ R and A ∈ U , δ Φ ◦ τ Ψ t,s ( A ) = lim L →∞ δ Φ L ◦ τ Ψ t,s ( A ) , uniformly for s, t on compacta.For any L ∈ R , the mapping from R × U to U defined by ( t, A ) δ Φ ◦ τ ( L, Ψ) t,s ( A ) (162)is continuous, by Corollary 3.5. By (161), as L → ∞ , this mapping on any fixed A ∈ U convergesuniformly for s, t on compacta to the mapping from R × U to U defined by ( t, A ) δ Φ ◦ τ Ψ t,s ( A ) , (163)which is thus continuous with respect to t ∈ R . By finite dimensionality of U ˜Λ for ˜Λ ∈ P f andlinearity of the mapping (163) with respect to A ∈ U , the function (163) is continuous on R × U ˜Λ . Proposition 7.11 (Graph norm continuity of convex derivatives)
Assume (159). For any Φ ∈ W , s, t ∈ R , ρ ∈ E and A ∈ U , D ̟ ρ,s ( t, A ) ∈ dom( δ Φ ) and δ Φ (D ̟ ρ,s ( t, A )) = lim L →∞ δ Φ L (D ̟ ρ,s ( t, A )) . Additionally, for any s ∈ R , ρ ∈ E and ˜Λ ∈ P f , the mapping ( t, A ) δ Φ (D ̟ ρ,s ( t, A )) from R × U ˜Λ to U is continuous. Proof.
Fix all parameters of the proposition. Let G . = (cid:0) dom( δ Φ ) , k·k G (cid:1) be the Banach space obtained by endowing the domain of δ Φ with its graph norm. By [76, Theorem4.8 (ii)], U is a core of the derivation δ Φ and hence, G is a separable Banach space.By using Lemma 7.10, the closedness of δ Φ and the fact that all terms appearing in the argu-ments of the integrals in the Dyson-type series of D ( t, A ) are Bochner-integrable, one checks that theDyson-type series deduced from (156) yields an element D ( t, A ) ∈ G with ( t, A ) δ Φ ( D ( t, A )) on R × U ˜Λ ( ˜Λ ∈ P f ) being the unique solution in E ∈ C ( R × U ˜Λ ; U ) to the equation E ( t, A ) = δ Φ ◦ τ Ψ ג t,s ( A ) (164) + X n ∈ N X Z∈P f Z ts d α Z ( S ∩W Λ ) n a ( α ) n (cid:0) dΨ (1) , . . . , dΨ ( n ) (cid:1) n X m ,m =1 ,m = m E (cid:16) α, e Ψ ( m ,~ℓ (cid:17) × ρ ◦ τ Ψ ג α,s (cid:16) i h Ψ ( m ) Z , τ Ψ ג t,α ( A ) i(cid:17) × Y j ∈{ ,...,n }\{ m ,m } ̟ ρ,s (cid:16) α, e Ψ ( j ) ,~ℓ (cid:17) t ∈ R , A ∈ U ˜Λ and ˜Λ ∈ P f . Compare this equation with (156). The integrals in (164) are againwell-defined as Bochner integrals: Combine Lemma 7.10 and [94, Theorems 1.1 and 1.2] with thefact that the measure a ( α ) n , n ∈ N , are finite and U is separable.Additionally, for any s ∈ R , ρ ∈ E and A ∈ U , the mapping t δ Φ (D ̟ ρ,s ( t, A )) from R to U is continuous, again by Lemma 7.10, the last Dyson-type series and the identity D ̟ ρ,s ( t, A ) = D ( t, A ) . (165)By linearity with respect to A ∈ U and finite dimensionality of U ˜Λ for ˜Λ ∈ P f , it follows that themapping ( t, A ) δ Φ (D ̟ ρ,s ( t, A )) from R × U ˜Λ to U is continuous.By combining Lemma 7.10 with Lebesgue’s dominated convergence theorem and the fact that theunique solution E = δ Φ ◦ D to (164) is given by a Dyson-type series, we arrive at δ Φ ( D ( t, A )) = lim L →∞ δ Φ L ( D ( t, A )) (166)in U . Note that one can interchange the local derivation δ Φ L (which is a commutator) in the right-handside of (166) with every integral over ( S ∩ W Λ ) n for any n ∈ N , because it is a bounded operator on U . By (165), this concludes the proof of the proposition. Lemma 7.12 (Differentiability of the solution – III)
Assume (159). Then, for any t ∈ R , ρ ∈ E and A ∈ U , ( ̟ ρ,s ( t ) ( A )) s ∈ R ≡ ( ̟ ρ,s ( t, A )) s ∈ R ∈ C ( R ; C ) with derivative satisfying, for any A ∈ U , ∂ s ̟ ρ,s ( t, A ) = − ρ ◦ δ Ψ ̟ρ,s |U Λ ( s ) ◦ τ Ψ ̟ρ,s |U Λ t,s ( A ) + z A [ ∂ s ̟ ρ,s ] . (167) Here, z A is defined by (150). Additionally, for any ˜Λ ∈ P f , ( t, A ) ∂ s ̟ ρ,s ( t, A ) is a continuousfunction on R × U ˜Λ . Proof.
Fix all parameters of the Lemma. By Lemma 7.2, for any ρ ∈ E , s, t ∈ R , A ∈ U and ε ∈ R \{ } , ˜ i ( ε, t, A ) . = ε − ( ̟ ρ,s + ε ( t, A ) − ̟ ρ,s ( t, A ))= ε − ρ ◦ ( τ Ψ ג ρ,s t,s + ε − τ Ψ ג ρ,s t,s ) ( A ) + ε − ρ ◦ ( τ Ψ ג ρ,s + ε t,s + ε − τ Ψ ג ρ,s t,s + ε ) ( A ) with ג ≡ ג ρ,s . = ̟ ρ,s | U Λ . Similar to (152), via Equations (109), (127) and (151) (keeping in mind that m ∈ C b ( R ; M Λ ) ), we deduce that ˜ i ( ε, t, A ) = ε − ρ ◦ ( τ Ψ ג ρ,s t,s + ε − τ Ψ ג ρ,s t,s ) ( A ) (168) + X n ∈ N X Z∈P f Z ts + ε d α Z ( S ∩W Λ ) n a ( α ) n (cid:0) dΨ (1) , . . . , dΨ ( n ) (cid:1) n X m ,m =1 ,m = m ˜ i (cid:16) ε, α, e Ψ ( m ,~ℓ (cid:17) × ρ ◦ τ Ψ ג ρ,s + ε α,s (cid:16) i h Ψ ( m ) Z , τ Ψ ג ρ,s t,α ( A ) i(cid:17) × Y j ∈{ ,...,m − }\{ m } ̟ ρ,s (cid:16) α, e Ψ ( j ) ,~ℓ (cid:17) × Y j ∈{ m +1 ,...,n }\{ m } ̟ ρ,s + ε (cid:16) α, e Ψ ( j ) ,~ℓ (cid:17) , j are, by definition, equal to when j ranges over the empty set . Again,using the same arguments as for i ( h, t, A ; υ ) in (152), one sees from Equation (168) that ˜ i ( ε, t, A ) isgiven by a Dyson-type series which is absolutely summable, uniformly with respect to ε in a boundedset when m ∈ C b ( R ; M Λ ) .Now, assuming (159), we can apply [76, Theorem 5.5] to the interaction Ψ ג ρ,s : ∀ s, r, t ∈ R : ∂ r τ Ψ ג ρ,s t,r = − δ Ψ ג ρ,s ( r ) ◦ τ Ψ ג ρ,s t,r , (169)in the strong sense on the dense set U . It is a highly non-trivial outcome resulting again from Lieb-Robinson bounds for multi-commutators of order three given in [76, Theorems 4.11, 5.4]. Similar to(153), by Lemmata 7.1 and 7.8 together with Lebesgue’s dominated convergence theorem, we deducefrom the Dyson-type series coming from (168) that ∂ s ̟ ρ,s ( t, A ) . = lim ε → ˜ i ( ε, t, A ) = lim ε → ε − ( ̟ ρ,s + ε ( t, A ) − ̟ ρ,s ( t, A )) exists, for all s, t ∈ R , ρ ∈ E and A ∈ U , and is also given by a Dyson-type series. Note that theDyson-type series are well-defined because, for any ˜Λ ∈ P f , the mapping from R × U ˜Λ to C definedby ( t, A ) ρ ◦ δ Ψ ג ρ,s ( s ) ◦ τ Ψ ג ρ,s t,s ( A ) (170)is continuous, by Lemma 7.10.By Proposition 3.8 (i), for any ˜Λ ∈ P f , the complex-valued function ( t, A ) ∂ s ̟ ρ,s ( t, A ) on R × U ˜Λ is the unique solution in ξ ∈ C ( R × U ˜Λ ; C ) to the equation ξ ( t, A ) = − ρ ◦ δ Ψ ג ρ,s ( s ) ◦ τ Ψ ג ρ,s t,s ( A ) + z A [ ξ ] (171)with z A defined by (150). Compare with (168) taken at ε = 0 . To prove the uniqueness of a solutionin ξ ∈ C ( R × U ˜Λ ; C ) to (171), we use the same arguments than for (154), keeping in mind Lemma7.10. This is again a consequence of Lieb-Robinson bounds for multi-commutators of order threegiven in [76, Theorems 4.11, 5.4].We conclude this section with the derivation of Liouville’s equation for (elementary) continuousand affine functions defined by (67), from which Theorem 6.11 is deduced. Lemma 7.13 (Liouville’s equation for affine functions)
Assume (159). Then, ∂ s V m t,s ( ˆ A ) ( ρ ) = − lim L →∞ { h m ( s ) L , V m t,s ( ˆ A ) } ( ρ ) , s, t ∈ R , A ∈ U , ~ℓ ∈ N d , ρ ∈ E ~ℓ , with ˆ A ∈ C being defined by (67). Proof.
Fix s ∈ R and ρ ∈ E . By (67) and (115), note that V m t,s ( ˆ A ) = ̟ ρ,s ( t ) ( A ) ≡ ̟ ρ,s ( t, A ) , t ∈ R , A ∈ U . Therefore, by Lemma 7.9 and (156), D V m t,s ( ˆ A ) ( ρ ) = D ̟ ρ,s ( t ; A ) = D ( t, A ) , t ∈ R , A ∈ U . See also Definition 5.1 and Equation (72). By Proposition 7.11 and (164), the continuous complex-valued function ( t, A )
7→ − ρ ◦ δ Ψ ̟ρ,s |U Λ ( s ) (cid:16) D V m t,s ( ˆ A ) ( ρ ) (cid:17) This happens when m ∈ { , n } . R × U ˜Λ solves Equation (171), like the well-defined continuous mapping ( t, A ) ∂ s V m t,s ( ˆ A ) ( ρ ) = ∂ s ̟ ρ,s ( t ) ( A ) ≡ ∂ s ̟ ρ,s ( t, A ) from R × U ˜Λ to C (Lemma 7.12), at any fixed ˜Λ ∈ P f . By uniqueness of the solution to (171), ∂ s V ht,s ( ˆ A ) ( ρ ) = − ρ ◦ δ Ψ ̟ρ,s |U Λ ( s ) (cid:16) D V m t,s ( ˆ A ) ( ρ ) (cid:17) , s, t ∈ R , A ∈ U . By tedious computations using Definitions 3.3, 5.2, Corollary 3.5, Lemma 3.2, Proposition 7.11,Lebesgue’s dominated convergence theorem and Equations (109), (117) and (127), one meanwhilechecks that, for any ~ℓ ∈ N d and ρ ∈ E ~ℓ , ρ ◦ δ Ψ ̟ρ,s |U Λ ( s ) (cid:16) D V m t,s ( ˆ A ) ( ρ ) (cid:17) = lim L →∞ { h m ( s ) L , V m t,s ( ˆ A ) } ( ρ ) , s, t ∈ R , A ∈ U . Recall that L = Z d , see Section 4.1. In [46, Definition 2.1], we give a definition of translation-invariant long-range models that differs from Equation (56). It turns out that any model of M (2)1 . = (cid:8) (Φ , a ) ≡ (Φ , ( a n ) n ∈ N ) ∈ W R × S : ∀ n ∈ N \{ } , a n = 0 (cid:9) , where W R . = W ∩ W R , can be identified with a long-range model in the sense of [46, Definition2.1], and vice-versa. (The notation W in [46] corresponds here to W R .) This identification can bedone in a such a way that the sequences of local Hamiltonians associated with each long-range modelare the same in both cases:(i): We start with a preliminary observation which simplifies the arguments. At L ∈ N , the localHamiltonian of any model m = (Φ , (0 , a , , . . . )) ∈ M (2)1 of Definition 4.2 is equal to U m L = U Φ L + 12 | Λ L | Z S (cid:16) U Ψ (1) L U Ψ (2) L + ( U Ψ (2) L ) ∗ ( U Ψ (1) L ) ∗ (cid:17) a (cid:0) dΨ (1) , dΨ (2) (cid:1) , because a is, by definition, self-adjoint, meaning that it equals its pushforward through the homeo-morphism (53) for n = 2 . Since, for any A, B ∈ U , AB + B ∗ A ∗ = 12 (( A ∗ + B ) ∗ ( A ∗ + B ) − ( A ∗ − B ) ∗ ( A ∗ − B )) , observe that U m L = U Φ L + 1 | Λ L | Z S (cid:16) | U Ψ (1) ∗ +Ψ (2) L | − | U Ψ (1) ∗ − Ψ (2) L | (cid:17) a (cid:0) dΨ (1) , dΨ (2) (cid:1) with | C | . = C ∗ C for C ∈ U . Let B ⊇ S be the unit closed ball of the Banach space W of translation-invariant (complex) interactions and define the continuous functions F ± : S → B by F ± (cid:0) Ψ (1) , Ψ (2) (cid:1) . = 12 (cid:0) Ψ (1) ∗ ± Ψ (2) (cid:1) , Ψ (1) , Ψ (2) ∈ S . F ±∗ ( a ) the two pushforwards of the measure a through the continuous functions F ± ,we arrive at the equality U m L = U Φ L + 1 | Λ L | Z B | U Ψ L | a (dΨ) , a . = F + ∗ ( a ) − F −∗ ( a ) . (172)Then, any model of M (2)1 can be identified with a long-range model in the sense of [46, Definition2.1]: • The measure space of [46, Definition 2.1] is ( B , Σ , | a | ) , with Σ ≡ Σ B being the Borel σ -algebra associated with B . Σ is countably generated, by separability of W ⊇ B . Ergo, by [93,Proposition 3.4.5], the space L ( B ; C ) . = L ( B , | a | ; C ) of square-integrable complex-valuedfunctions on B is a separable Hilbert space, i.e., ( B , Σ , | a | ) is a separable measure space. • The corresponding L -functions (Φ Ψ ) Ψ ∈ B , (Φ ′ Ψ ) Ψ ∈ B of [46, Definition 2.1] are defined by Φ Ψ . = Re { Ψ } and Φ ′ Ψ . = Im { Ψ } with Ψ ∈ B . See (23) for the definition of real and imaginary parts of interactions. These two functions areBochner measurable, by [94, Theorems 1.1 and 1.2], for they are continuous functions and W is a separable Banach space. Additionally, they are L -functions because there are bounded ona space of finite measure. • The corresponding measurable function γ Ψ ∈ {− , } of [46, Definition 2.3] is obtained fromany Hahn decomposition P a , N a ∈ Σ of the signed measure a (172) by γ Ψ = [Ψ ∈ P a ] − [Ψ ∈ N a ] , Ψ ∈ B , where P a and N a are respectively positive and negative sets for a .(ii): Conversely, let ( A , A , a ) be a separable measure space with A and a : A → R +0 being respec-tively some σ -algebra on A and some measure on A . Fix a measurable function a γ a ∈ {− , } on A (see [46, Definition 2.3]) and a model (Φ , (Φ a ) a ∈A , (Φ ′ a ) a ∈A ) ∈ W R × L (cid:0) A , W R (cid:1) × L (cid:0) A , W R (cid:1) (173)in the sense of [46, Definition 2.1]. ( W in [46] refers here to W R .)Define the set A Φ , Φ ′ . = { a ∈ A : Φ a + i Φ ′ a = 0 } . The mapping a Φ a + i Φ ′ a from A Φ , Φ ′ to R + is measurable, since the vector space operations in W are jointly continuous. Hence, A Φ , Φ ′ a , the preimage of W \{ } by this mapping, is an element of the σ -algebra A . As a consequence,by [46, Definition 2.3], at L ∈ N , the local Hamiltonian associated with a model of [46, Definition2.1] equals U L = U Φ L + 1 | Λ L | Z A Φ , Φ ′ γ a | U Φ a + i Φ ′ a Λ L | a (d a ) , (174)recalling that | C | . = C ∗ C for C ∈ U .Define the functions G : A Φ , Φ ′ → S by G ( a ) . = g ( a ) (Φ a + i Φ ′ a ) , a ∈ A Φ , Φ ′ , g ( a ) . = k Φ a + i Φ ′ a k − W , a ∈ A Φ , Φ ′ . The mapping g from A Φ , Φ ′ to R + is measurable, as a composition of a measurable function from A Φ , Φ ′ to W \{ } with a continuous one from W \{ } to R + . Again by the joint continuity of thevector space operations in W , G is a measurable function. As a consequence, at L ∈ N , by (174), thelocal Hamiltonian associated with a model of [46, Definition 2.1] equals U L = U Φ L + 1 | Λ L | Z S | U ΨΛ L | a (dΨ) , a . = G ∗ ( γg − a ) , where G ∗ ( γg − a ) is the pushforward of the signed measure γg − a through the measurable function G . Note that a is a finite measure on S because Z S | a | (dΨ) = Z S k Φ a + i Φ ′ a k W a (d a ) ≤ Z S k Φ a k W a (d a ) + 2 Z S k Φ ′ a k W a (d a ) < ∞ . Now, using the continuous mapping K from S to S × S defined by K (Ψ) . = (Ψ ∗ , Ψ) , Ψ ∈ S , we define a . = 12 ( K ∗ ( a ) + K ∗ ( a ) ∗ ) to be the real part of the pushforward of the signed measure a through the measurable function K .Then, by construction, a = a ∗ is self-adjoint and U L = U Φ L + 12 | Λ L | Z S (cid:16) U Ψ (1) L U Ψ (2) L + ( U Ψ (2) L ) ∗ ( U Ψ (1) L ) ∗ (cid:17) a (cid:0) dΨ (1) , dΨ (2) (cid:1) for any L ∈ N . The most general form of a translation invariant model for fermions with two-body interactions in acubic box Λ L . = { Z ∩ [ − L, L ] } d (see (1)) of volume | Λ L | , L ∈ N , is given in momentum space by H F ullL = X k ∈ Λ ∗ L , s ∈ S ( ε k − µ ) ˜ a ∗ k ˜ a k + 1 | Λ L | X k,k ′ ,q ∈ Λ ∗ L s , s , s , s ∈ S g s , s , s , s ( k, k ′ , q ) ˜ a ∗ k + q, s ˜ a ∗ k ′ − q, s ˜ a k ′ , s ˜ a k, s . (175)See [95, Eq. (2.1)]. Recall that S is some finite (spin) set representing the internal degrees of freedomof quantum particles. The set Λ ∗ l . = 2 π (2 l + 1) Λ l ⊆ [ − π, π ] d is the reciprocal lattice of quasi-momenta (periodic boundary conditions) associated with Λ L and theoperator ˜ a ∗ k, s . = 1 | Λ L | / X x ∈ Λ L e − ik · x a ∗ x, s (respectively ˜ a k, s ) creates (respectively annihilates) a fermion with spin s ∈ S and (quasi-) momentum k ∈ Λ ∗ L . The function ε k represents the kinetic energy of a fermion with (quasi-) momentum k . Inphysics, it is usually the Fourier transform of the discrete Laplacian. The real number µ is the chemical54otential. The last term of (175) corresponds to a translation-invariant two-body interaction writtenin the momentum space.One important example of a lattice-fermion system with long-range interactions is given in thescope of the celebrated BCS theory – proposed in the late 1950s (1957) to explain conventional typeI superconductors. The lattice version of this theory is obtained from (175) by taking S . = {↑ , ↓} andimposing g s , s , s , s ( k, k ′ , q ) = δ k, − k ′ δ s , ↑ δ s , ↓ δ s , ↓ δ s , ↑ f ( k, − k, q ) for some function f : It corresponds to the so-called (reduced) BCS Hamiltonian H BCSL . = X k ∈ Λ ∗ L , s ∈ S ( ε k − µ ) ˜ a ∗ k, s ˜ a k, s − | Λ L | X k,q ∈ Λ ∗ L γ k,q ˜ a ∗ k, ↑ ˜ a ∗− k, ↓ ˜ a − q, ↓ ˜ a q, ↑ , (176)where γ k,q is a positive function. Because of the term δ k, − k ′ , the interaction of this model has along-range character, in position space. In physics, one usually takes γ k,q . = (cid:26) γ ≥
00 for | k − q | ≤ Cfor | k − q | > C with C ∈ (0 , ∞ ] . The simple choice C = ∞ , i.e., γ k,q = γ > in (176), is still physically very inter-esting since, even when ε k = 0 , the BCS Hamiltonian qualitatively displays most of basic propertiesof real conventional type I superconductors. See, e.g. [96, Chapter VII, Section 4].The BCS Hamiltonian H BCSL with γ k,q = γ > can be rewritten in the x –space as H BCSL = X x,y ∈ Λ L , s ∈ S h ( x − y ) a ∗ x, s a y, s − µ X x ∈ Λ L , s ∈ S a ∗ x, s a x, s − γ | Λ L | X x,y ∈ Λ N a ∗ x, ↑ a ∗ x, ↓ a y, ↓ a y, ↑ (177)for some appropriate function h : Z d → C representing the (kernel of) one-particle operator (orthe Fourier transform of the function k ε k − µ ). Note that the long-range character of the BCSinteraction in (177) is in this case clear since it can be seen as an hopping at any distance of (Cooper)pairs of fermions with spins ↓ and ↑ , respectively. It is also a mean-field term since it can be seen asa space average of interactions: − | Λ L | X x,y ∈ Λ L a ∗ x, ↑ a ∗ x, ↓ a y, ↓ a y, ↑ = − X y ∈ Λ L | Λ L | X x ∈ Λ L a ∗ x, ↑ a ∗ x, ↓ ! a y, ↓ a y, ↑ . Therefore, the BCS Hamiltonian H BCSL with γ k,q = γ > can be explicitly expressed as the localHamiltonian U m L of a long-range model m ∈ M : Let Ψ ( h ) be defined by Ψ ( h )Λ . = h ( x − y ) X s ∈ S a ∗ x, s a y, s + (1 − δ x,y ) h ( x − y ) X s ∈ S a ∗ y, s a x, s whenever Λ = { x, y } for x, y ∈ Z d , and Ψ ( h )Λ . = 0 otherwise. The interaction Ψ ( h ) belongs to W R assoon as sup x,y ∈ Z d | h ( x − y ) | F ( x, y ) < ∞ . (178)See Equation (27). Such a condition is trivially satisfied for all kinetic terms used in condensedmatter physics for many-fermions on lattices, by taking F as in Equation (26). Let ˜Ψ ∈ S be defined The positivity of γ k,q imposes constraints on the choice of the function f . ˜Ψ { x } . = a x, ↓ a x, ↑ for x ∈ Z d and ˜Ψ Λ . = 0 otherwise. Let ˜ a be defined, for all Borel subset B ⊆ S ,by ˜ a ( B ) = − γ [ ˜Ψ ∈ B ] . If Inequality (178) holds true, then m = (Ψ ( h ) , ˜ a ) ∈ M and, by Definition 4.2, U m L = H BCSL , L ∈ N . As a consequence, the dynamics at infinite volume of the BCS Hamiltonian H BCSL with γ k,q = γ > can be explicitly computed from results of this paper, in which concerns its classical part, and from[64] for its quantum part.Other examples can also be found in [46, Section 2.2] (in relation with the forward scattering ap-proximation) as well as in [66, 67], which explain the dynamics of the strong-coupling BCS-Hubbardmodel. The latter is an interesting model because it predicts the existence of a superconductor-Mottinsulator phase transition, like in cuprates which must be doped to become superconductors. See [44]for more details. Acknowledgments:
This work is supported by CNPq (308337/2017-4), FAPESP (2017/22340-9), aswell as by the Basque Government through the grant IT641-13 and the BERC 2018-2021 program,and by the Spanish Ministry of Science, Innovation and Universities: BCAM Severo Ochoa accred-itation SEV-2017-0718, MTM2017-82160-C2-2-P. The formulation of long-range models as pairs (Φ , a ) ∈ W R × S ( S ) with associated Hamiltonians defined by (172), instead of [46, Definitions 2.1and 2.3], has been proposed to us by S. Breteaux. We thank him very much for this hint. References [1] V.A. Zagrebnov and J.-B. Bru, The Bogoliubov Model of Weakly Imperfect Bose Gas. Phys.Rep. (2001) 291-434.[2] J. Ginibre, On the Asymptotic Exactness of the Bogoliubov Approximation for many BosonsSystems. Commun. Math. Phys. (1968) 26-51.[3] N. N. Bogoliubov. On a new method in the theory of superconductivity. Nuovo Cimento (1958)794-805.[4] R. Haag. The mathematical structure of the Bardeen-Cooper-Schrieffer model. Nuovo Cimento, (1962) 287-298.[5] W. Thirring and A. Wehrl, On the mathematical structure of the B.C.S.-model, Comm. Math.Phys. (1967) 303-314.[6] W. Thirring, On the mathematical structure of the B.C.S.-model. II, Comm. Math. Phys. (1968)181-189.[7] L. van Hemmen, Linear Fermion Systems, Molecular Field Models, and the KMS Condition.Bortachritte der Physik (1978) 397-439; Section III.[8] N.N. Bogoliubov Jr., On model dynamical systems in statistical mechanics. Physica (1966)933. 569] N.N. Bogoliubov Jr., J.G. Brankov, V.A. Zagrebnov, A.M. Kurbatov and N.S. Tonchev, Metodapproksimiruyushchego gamil’toniana v statisticheskoi fizike . Sofia: Izdat. Bulgar. Akad.Nauk , 1981[10] N.N. Bogoliubov Jr., J.G. Brankov, V.A. Zagrebnov, A.M. Kurbatov and N.S. Tonchev, Someclasses of exactly soluble models of problems in Quantum Statistical Mechanics: the method ofthe approximating Hamiltonian. Russ. Math. Surv. (1984) 1-50.[11] J.G. Brankov, D.M. Danchev and N.S. Tonchev, Theory of Critical Phenomena in Finite–sizeSystems: Scaling and Quantum Effects.
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