Quantum spin chain dissipative mean-field dynamics

We study the emergent dynamics resulting from the infinite volume limit of the mean-field dissipative dynamics of quantum spin chains with clustering, but not time-invariant states. We focus upon three algebras of spin operators: the commutative algebra of mean-field operators, the quasi-local algebra of microscopic, local operators and the collective algebra of fluctuation operators. In the infinite volume limit, mean-field operators behave as time-dependent, commuting scalar macroscopic averages while quasi-local operators, despite the dissipative underlying dynamics, evolve unitarily in a typical non-Markovian fashion. Instead, the algebra of collective fluctuations, which is of bosonic type with time-dependent canonical commutation relations, undergoes a time-evolution that retains the dissipative character of the underlying microscopic dynamics and exhibits non-linear features. These latter disappear by extending the time-evolution to a larger algebra where it is represented by a continuous one-parameter semigroup of completely positive maps. The corresponding generator is not of Lindblad form and displays mixed quantum-classical features, thus indicating that peculiar hybrid systems may naturally emerge at the level of quantum fluctuations in many-body quantum systems endowed with non time-invariant states.


Introduction
In many physical situations concerning many-body quantum systems with N microscopic components, the relevant observables are not those referring to single constituents, rather the collective ones consisting of suitably scaled sums of microscopic operators. Among them, one usually considers macroscopic averages that scale as the inverse of N and thus lose all quantum properties in the large N limit thereby providing a description of the emerging commutative, henceforth classical, collective features of many body quantum systems.
Another class of relevant collective observables are the so-called quantum fluctuations: they account for the variations of microscopic quantities around their averages computed with respect to a chosen reference state. In analogy with classical fluctuations, they scale with the inverse square root of N so that, unlike macroscopic observables, they can retain quantum features in the large N limit [1][2][3]. Indeed, whenever the reference microscopic state presents no long-range correlations, the fluctuations behave as bosonic operators; furthermore, from the microscopic state there emerges a Gaussian state over the corresponding bosonic canonical commutation relation (CCR) algebra. These collective observables describe a mesoscopic physical scale in between the purely quantum behaviour of microscopic observables and the purely classical one of commuting macroscopic observables [4].
The dynamical structure of quantum fluctuations has been intensively studied both in the unitary [1,[3][4][5] and in the dissipative case [6][7][8]; yet, in all these examples, only time-invariant reference states have been investigated, leading to macroscopic averages not evolving in time. Here, we relax this assumption and consider the possibility, often met in actual experiments, of a non-trivial dynamics of macroscopic averages. We shall do this by focusing on dissipative, Lindbald chain dynamics of mean-field type. The model studied in the following is very general and applies to a large variety of many-body systems consisting of N microscopic finite-dimensional systems weakly interacting with their environment. We will study the large N limit of such a dissipative time-evolution (1) at the macroscopic level of mean-field observables, (2) at the microscopic scale of quasi-local observables, that is for arbitrarily large, but finite, number of chain sites, and (3) at the mesoscopic level of quantum fluctuations. These three scenarios look quite different and lead to features that, to the best of our knowledge, in particular for the cases (2) and (3), are novel in the field of many-body quantum systems.
1. Macroscopic observables: these are described by the large N-limit of mean-field observables which yields commuting scalar quantities that evolve in time according to classical macroscopic equations of motion. 2. Quasi-local observables: the emerging dynamics is generated by a Hamiltonian despite the microscopic dynamics being dissipative for each finite N. Moreover, and more interestingly, whenever macroscopic averages are not constant, such a unitary dynamics is non-Markovian, since it is implemented by a time non-local generator that always depends on the initial time. This latter is an interesting example of a unitary time-evolution manifesting memory effects. 3. Quantum fluctuations: the emerging dynamics consists of a one-parameter Gaussian family of non-linear maps. In order to make them compatible with the physical requests of linearity and complete positivity, these maps must be extended to a larger algebra, containing also classical degrees of freedom associated with the macrosocpic averages. The extended description gives raise to a dynamical hybrid system, containing both classical and quantum degrees of freedom, whose time-evolution corresponds to a semigroup of completely positive maps. Unlike in hybrid systems so far studied [9][10][11][12][13], the connection between classical and quantum degrees of freedom follows from the time-dependence of the mesoscopic commutation relations. Indeed, the commutator of two fluctuation operators is a time-evolving macroscopic average. As a consequence, the generator of the dynamics on the larger algebra contains both classical, quantum and mixed classical-quantum contributions. In particular, the dynamical maps are completely positive, even if the purely quantum contribution to the generator need not in general be characterized by a positive semi-definite Kossakowski matrix. This is the first instance where this counter-intuitive fact is reported; notice, however, that in such a hybrid context, Lindblad's theorem does not apply.
The structure of the manuscript is as follows: in section 3 we introduce mean-field and fluctuation operators for quantum spin chains and define the mesoscopic limit. In section 3, we introduce the dynamics generated by a Hamiltonian free term plus a mean field interaction and made dissipative by Lindblad type contributions of mean-field type. In section 3.1, we discuss the dynamics of macroscopic quantities and in section 3.2 the large N limit of the timeevolution of quasi-local operators. In section 4 we study the emerging mesoscopic dynamics of quantum fluctuations, discussing first the symplectic structure in section 4.1, then the timeevolution and its non-linearity in section 4.2. In section 4.3 we focus upon the extension of the non-linear maps to a semi-group of completely positive Gaussian maps on a larger algebra and on the hybrid character of its generator. Finally, section 6 contains the proofs of all results presented in the previous sections.

Quantum spin chains: macroscopic and mesoscopic descriptions
In this section, we discuss the macroscopic, respectively mesoscopic description of the collective behaviour of quantum spin chains given by classical mean-field observables, that scale with the inverse of the number of sites, N, respectively by quantum fluctuations that scale as the inverse square-root of N.
A quantum spin chain is a one-dimensional bi-infinite lattice, whose sites are indexed by integers j ∈ Z, all supporting the same d-dimensional matrix algebra A ( j) = M d (C). Its algebraic description [14,15] is by means of the quasi-local C * algebra A obtained as the inductive limit of strictly local subalgebras A [q,p] = q j=p A ( j) supported by finite intervals [q, p], with q p in Z. Namely, one considers the algebraic union q p A [q,p] and its completion with respect to the norm inherited by the local algebras. Any operator x ∈ M d (C) at site j can be embedded into A as: where 1 j−1] is the tensor product of identity matrices at each site from −∞ to j − 1, while 1 [ j+1 is the tensor product of identity matrices from site j + 1 to +∞. Quantum spin chains are naturally endowed with the translation automorphism τ : A → A such that τ (x ( j) ) = x ( j+1) . Generic states ω on the quantum spin chain are described by positive, normalised linear expectation functionals A a → ω(a) that assign mean values to all operators in A. In the following, we shall consider translation-invariant states such that At each site j ∈ Z, these states are thus locally represented by a same density matrix ρ ∈ M d (C): ω(x ( j) ) = ω(x) = Tr(ρ x), x ∈ M d (C). Furthermore, we shall focus upon translation-invariant states ω that are also spatially L 1 -clustering [1]. These are states that, for all single site operators x, y, satisfy and then the weaker clustering condition (4) Remark 1. The cluster condition (4) is often met in ground states or in thermal states associated to short-range Hamiltonians far from critical behaviours, such as phase transitions: it corresponds to the physical expectation that, in absence of long-range correlations, the farther spatially apart are observables, the closer they become to being statistically independent. On the other hand, the stronger clustering condition (3) is sufficient to ensure that fluctuations of physical observables display a Gaussian character which is again a property physically expected in systems far from phase transitions: such a condition is not strictly necessary for a system to have Gaussian fluctuations; however, it is often assumed for mathematical convenience [1]. □

Macroscopic scale: mean-field observables
In an infinite quantum spin chain, the operators belonging to strictly local subalgebras contribute to the microscopic description of the system. In order to pass to a description based on collective observables supported by infinitely many lattice sites, a proper scaling must be chosen. Most often, mean-field observables are considered; these are constructed as averages of N copies of a same single site observables x, from site j = 0 to site N − 1: In the following, operators scaling as X (N) will be referred to as mean-field operators; capital letters, like X (N) , will refer to averages over specific number of lattice sites, while small case letters, like x (k) , to operators at specific lattice sites. Given any state ω on A, the Gelfand-Naimark-Segal (GNS) construction [14] provides a representation π ω : A → π ω (A) of A on a Hilbert space H ω with a cyclic vector |ω such that the linear span of vectors of the form |Ψ a = π ω (a)|ω is dense in H ω and As shown in appendix A, given x, y ∈ M d (C), clustering yields that macroscopic averages X (N) and products of macroscopic averages X (N) Y (N) tend weakly to scalar quantities: in the sense that Moreover, in the same appendix it is shown that the L 1 -clustering condition (3) provides the following scaling: It thus follows that the weak-limits of mean-field observables are scalar quantities giving rise to a commutative (von Neumann) algebra. Mean-field observables thus describe macroscopic, classical degrees of freedom emerging from the large N limit of the microscopic quantum spin chain with no fingerprints left of its quantumness. As outlined in the Introduction, we are instead interested in studying collective observables extending over the whole spin chain that may still keep some degree of quantum behaviour; for that a less rapid scaling than 1/N is necessary.

Mesoscopic scale: quantum fluctuations
In order to disclose quantum behaviours of collective observables, one needs to look at fluctuations around average values. Indeed, fluctuations are commonly associated to an intermediate level of description in between the microscopic and the macroscopic ones, where one can hope to unveil truly mesoscopic phenomena exhibiting mixed classical-quantum features. In this section we shall review some of the known results about quantum fluctuation operators [1][2][3], introducing also the notation and relevant concepts useful to derive the results presented in the following sections.
Collective, microscopic operators of the form are quantum analogues of fluctuations in classical probability theory: we shall refer to them as 'local quantum fluctuations'. Their large N limit with respect to clustering states ω has been thoroughly investigated in [1,2] yielding a non-commutative central limit theorem and an associated quantum fluctuation algebra which turns out to be a Weyl algebra of bosonic degrees of freedom.
The scaling 1/ √ N does not guarantee convergence in the weak-operator topology.
, with respect to a clustering state ω, one has, following the same strategy used in appendix A, This means that commutators of local quantum fluctuations behave as mean-field quantities thus being, in the weak-topology, scalar multiples of the identity ω(z) 1. This latter fact clearly indicates that, in the large N limit, a non-commutative structure emerges analogous to the algebra of quantum position and momentum operators. To proceed to a formal proof of the convergence of the set of these operators to a bosonic algebra, it is convenient to work with unitary exponentials of the form e iF (N) (x) ; in the large N limit, these are expected to satisfy Weyl-like commutation relations [1].

Remark 2.
Because of the scaling 1/ √ N , quantum fluctuations provide a description level in between the microscopic (strictly local) and the macroscopic (mean-field) ones. We will refer to it as to a mesoscopic level whereby collective operators keep track of the microscopic non-commutative level they emerge from. □ In order to construct a quantum fluctuation algebra, one starts by selecting a set of p linearly independent single-site microscopic observables

and considers their local elementary fluctuations
Because of the assumption (3) on the state ω, one has that the limits are well-defined and represent the entries of a positive p × p correlation matrix C (ω) ; moreover, one chooses the elements of χ in such a way that the characteristic functions ω e itF (N) j converge to Gaussian functions of t with zero mean and covariance Σ (ω) jj , given by This can be conveniently summarized by introducing the concept of normal quantum fluctuations systems.

Definition 1.
A finite set of self-adjoint operators χ = {x j } p j=1 is said to have 'normal multivariate quantum fluctuations' with respect to a clustering state ω if the latter obeys the L 1clustering condition: and further satisfies Given a set χ as in the above definition 1, by considering all possible real linear combinations of the set elements, one introduces the real-linear span The latter set can be endowed with two real bilinear maps: the first is positive and symmetric, with The second one is, instead, anti-symmetric and defined by the real symplectic matrix σ (ω) with entries Notice that the p × p matrices introduced so far are related by the following equality For sake of compactness, because of the linearity of the map that associates an operator x with its local quantum fluctuation F (N) (x), the following notations will be used: where F (N) is the operator-valued vector with components F (N) ij ], one constructs the abstract Weyl algebra W(χ, σ (ω) ), linearly generated by the Weyl operators W( r), r ∈ R p , obeying the relations: The following theorem specifies in which sense, in the large N limit, the local exponentials W (N) ( r) yield Weyl operators W( r) [1].

Theorem 1.
Any set χ with normal fluctuations with respect to a clustering state ω admits a regular, Gaussian state Ω on W(χ, σ (ω) ) such that, for all r j ∈ R p , j = 1, 2, . . . , n, where the W( r j ) satisfy (21) and The regular and Gaussian character of Ω follows from (12). In particular, its regularity guarantees that one can write where F is the operator-valued p-dimensional vector with components F i that are collective field operators satisfying canonical commutation relations ij , or, more generically, We shall refer to the Weyl algebra W(χ, σ (ω) ) generated by the strong-closure (in the GNS representation based on Ω) of the linear span of Weyl operators as the quantum fluctuation algebra.

Mesoscopic limit
Later on, we shall focus on the effective dynamics of quantum fluctuations, emerging from the large N limit of a family of microscopic dynamical maps {Φ (N) } N∈N defined on the strictly local subalgebras A [0,N−1] . To formally state our main results, we introduce what we shall refer to as mesoscopic limit.

Definition 2 (Mesoscopic limit).
Given a discrete family of operators {X (N) } N∈N , in the quasi-local algebra A, we shall say that they posses the mesoscopic limit if and only if where Further, given a sequence of completely positive, unital maps Φ (N) : Remark 3. Notice that the right hand side of (28) is the matrix element of π Ω (Φ [W( r)]) with respect to two vector states π Ω (W( r 1,2 ))|Ω in the GNS-representation of the Weyl algebra generated by the operators W(r) based on Ω. Since these vectors are dense in the GNS-Hilbert space, the mesoscopic map Φ is defined by the matrix elements of its action on Weyl operators that arise from local quantum fluctuations. □ According to the above definition and to (22), one can then say that the Weyl operators W(x j ) are the mesoscopic limits of the local exponentials W (N) (x j ) and, by taking derivatives with respect to the parameters r j , that the operators F j are the mesocopic limits of the local quantum fluctuations F (N) j .

Mean-field dissipative dynamics
Typically, a mean-field unitary spin-dynamics emerges in the large N limit from a quadratic interaction Hamiltonian scaling as 1/N as for the case of the BCS model in the quasi-spin description [17].
In this framework, operators x ∈ A [0,N−1] pertaining to the lattice sites k = 0, 1, . . . , N − 1, evolve in time according to a group of automorphisms of with a linear and bi-linear terms, the last one scaling as 1/N: In the expressions above, the single-site operators v µ = v † µ , µ = 1, 2, . . . , d 2 , are chosen to constitute an hermitian orthonormal basis for the single-site algebra M d (C): and the coefficients µ , h µν are chosen such that In the following, we will perturb the Hamiltonian generator of the microscopic dynamics with a Lindblad type contribution [16] scaling as 1/N. We shall then study the time-evolution that emerges at the level of collective quantities from a dissipative microscopic master equa- Notice that the mean-field scaling of L (N) is that of the commutator with H (N) and is due to the scaling 1 µ . In the above expression, the coefficients C µν are chosen to form a positive semi-definite matrix C = [C µν ], known as Kossakowski matrix. Such a property of C ensures that the solution to ∂ t x t = L (N) [x t ] is a one-parameter semigroup of completely positive, unital maps γ [16,19]: Remark 4.
1. While the purely Hamiltonian mean-field dynamics studied in [1,4,5] preserve the norm, the maps γ X for all X ∈ A. Furthermore, [19] for the Hamiltonian contributions cancel and only D (N) contributes. 2. A generator as in (32) can be obtained by considering N d-level systems interacting with their environment via a Hamiltonian of the form where H (N) , H E represent the Hamiltonians of system and environment considered alone, while the coupling Hamiltonian consists of the operators V (N) α in (34) (which thus scale with 1/ √ N ) and environment operators B α = B † α . Notice that the scaling 1/ √ N of the interaction Hamiltonian is the same as in the Dicke model for light-matter interaction [20][21][22] and is the only one that, in the large N limit with respect to clustering states, can lead to a meaningful dynamics with generator as in (32). In the weak-coupling limit [23], when memory effects can be neglected, one retrieves an effective evolution of the N-body system alone, implemented by Lindblad generators of the specific type (32). The contribution D (N) describes dissipative and noisy effects due to the system-environment collective coupling in equation (37), while the Hamiltonian H (N) in (29) is an environment induced Lamb shift. □ We decompose the coefficients of the mean-field Hamiltonian in (29) µν , with the real and imaginary parts satisfying the relations Then, using (34), the mean-field Hamiltonian contribution can be written as (39) In the above expression, {x , y} = x y + y x denotes anti-commutator. At the same time, by decomposing the Kossakowski matrix C = [C µν ] in its self-adjoint symmetric and anti-symmetric components as where C tr denotes transposition, one recasts Thus, using (39) and the above expressions, the generator in (32) deomposes as a mean-field dissipator-like term plus a free Hamiltonian term: The various coefficients are conveniently regrouped into the following d 2 × d 2 matrices where A is real, but unlike A in (40), non symmetric, and B is purely imaginary, but, unlike B in (40), not anti-symmetric.

Mean-field dissipative dynamics on the quasi-local algebra
In this section we shall deal with the large N limit of the microscopic dissipative dynamics γ (N) t on the quasi-local algebra A generated by L (N) in (41)-(44); namely, we shall investigate the behaviour when N → ∞ of γ , where x ∈ A is either strictly local, that is different from the identity matrix, over an arbitrary, but fixed number of sites, or can be approximated in norm by strictly local operators.

Definition 3. An operator
The smallest such interval is the support S(O) of O ∈ A whose cardinality will be denoted by (O).
We shall consider microscopic states ω that are translation invariant and clustering, but not necessarily invariant under the large N limit of the microscopic dynamics; namely such that, in general, on strictly local x ∈ A, Thus, we shall consider the case of macroscopic averages associated with mean-field operators that may also change in time. The existence of the following macroscopic averages is first guaranteed for all t ∈ [0, R] with R defined by the norm-convergence radius of the exponential series on local and mean-field operators by corollary 1 in section 6.1, and then extended to all finite times t 0 by proposition 3.

Definition 4.
The time-dependent macroscopic averages of the commutator of single-site operators, v µ and [v µ , v ν ] ∈ M d (C), with respect to the microscopic state at any finite time t 0 will be denoted by: Using the relations (30), one writes Tr and, since the trace does not depend on the site index, one may set and Proposition 2 and corollary 3 in section 6.1 show that the macroscopic averages satisfy the following equations of motion for all times t 0: Denoting by ω t the vector with components ω α (t) and using (50), it proves convenient for later use, in particular for the derivation of the dissipative fluctuation dynamics in theorem 3, to recast the equations of motion in the following compact, matrix-like form where D( ω t ) and E have entries and D( ω t ) depends implicitly on time through the time-evolution: ω → ω t . Notice that all the scalar quantities multiplying ω γ (t) change sign under conjugation, whence the matrix D( ω t ) is real and The non-linear equations (52) with initial condition ω 0 = ω are formally solved by the matricial expression where T denotes time-ordering and the dependence of the d 2 × d 2 matrix M t ( ω) on the timeevolution ω → ω t embodies the non-linearity of the dynamics. However, this is just a formal writing, that will prove to be useful later on: the time-evolution of the macroscopic averages can be found only by directly solving the system of equations (52).
Despite the time-ordering, since there is no explicit time-dependence in the equations (52), the time-evolution of the macroscopic averages composes as a semigroup, Moreover, because of the anti-symmetry of D( ω t ) and of the fact that the macroscopic averages are real, the quantity K(t) := Furthermore, the positivity of the state ω yields the positivity of the

By expanding the matrix product
where the d 2 × d 2 matrix V α = V α µν is fixed by the chosen basis. Thus, the vectors ω of macroscopic averages {ω(v µ )} d 2 µ=1 belong to the subset S ⊂ [−1, 1] ×d 2 satisfying the constraints (57) and (58). In conclusion, the macroscopic dynamics generated by the non-linear, time-independent equations of motions (51) forms a semigroup and maps S into itself. □

Macroscopic dynamics of local observables
With the time-evolution of macroscopic averages at disposal, we are now able to derive the large N limit of the dynamics of quasi-local operators x ∈ A.
Theorem 2. Let the quasi-local algebra A be equipped with a translation-invariant, spatially L 1 -clustering state ω. In the large N limit, the local dissipative generators L (N) in (41) define on A a one-parameter family of automorphisms that depend on the state ω and are such that, for any finite t 0, with explicitly time-dependent Hamiltonian where The proof of the above theorem is given in section 6.1. Using (44), the Hamiltonian reads is hermitean. Notice that, in the large N limit, the microscopic dissipative term D (N) only contributes with a correction to the free Hamiltonian terms in (29) so that the dissipative time-evolution of local observables becomes automorphic.
Consider the dynamics of single site observables by choosing in (60) O equal to one of the orthonormal matrices at site , v where use has been made of the relations (30) and of the matrix elements (53). Notice that the expectations ω α t v ( ) γ satisfy the same equations (51) satisfied by the macroscopic observables ω γ (t); since these quantities coincide at t = 0, one has Remark 6.
1. The convergence of the mean-field dissipative dynamics γ to the automorphism α t of A occurs in the weak-operator topology associated with the GNS-representation of A based on the state ω. 2. The automorphisms α t have been derived for positive times, only. This means that, though the inverted automorphisms α −t surely exist, they cannot however arise from the underlying non-invertible microscopic dynamics. 3. The one-parameter family {α t } t 0 fails to obey the forward-in-time composition law as in (35) which is typical of time-independent generators, nor the one corresponding to two-parameter semi-groups, , 0 t 0 s t which arises from time-ordered integration of generators that depend explicitly on the running time t, but not on the initial time t 0 . Indeed, if the microscopic dynamics starts at t 0 0, then the semigroup properties ensure that, at time t t 0 , any quasi-local initial Then, adapting theorem 2 to a generic initial time t 0 0, similarly to (59), the large N limit yields a one-parameter family of automor- for all a, b ∈ A and O quasi-local. If the support of O is, for sake of simplicity, [0, S − 1], then Therefore, the time-derivative yields a generator: which depends on both the running and initial times. 4. By setting t 0 = 0 in (67), one sees that the one-parameter family {α t } t 0 is generated by a time-local master equation. However, since in general, that is for t 0 0, the generator K t−t0 depends on both the running time t and the initial time t 0 , the family of automorphisms is non-Markovian in the sense of [24]. On the other hand, if one uses lack of CP-divisibility as a criterion of non-Markovianity [25], then {α t } t 0 is Markovian. Indeed, being the dynamics unitary, there always exists a completely positive intertwining provides a time-invariant state on the quasi-local algebra A, then one recovers the one-parameter semigroup features of (35) (see also [26]). □ Example 1. We shall consider a qubit spin chain consisting of a lattice whose sites j ∈ N support the algebra M 2 (C). As a Hilbert-Schmidt orthogonal matrix basis {v µ } 4 µ=1 , we choose the spin operators s 1 , s 2 , s 3 , 1, normalized in a such way that µ , we study the following dissipative generator, with Kossa- Therefore, with respect to (45) and (46), h = 0 and E = 0, so that A and B coincide with the symmetric and anti-symmetric components of C, With respect to a translation-invariant clustering state ω, the only non-trivial macroscopic averages ω µ (t) given by (47) are ω 1,2,3 (t) while ω 4 (t) = 1 for all t 0. Since s µ 1/2, we will then consider the vector ω t = (ω 1 (t), ω 2 (t), ω 3 (t)) with components belonging to [−1/2 , 1/2]. Furthermore, from (50) and (69) one computes whence (51) and B in (71) yield the following system of differential equations: corresponding to the following matrix D( ω t ) in (52): Then, the norm is a constant of the motion; thus the third equation can readily be solved, yielding where the constant b is chosen to implement the initial condition ω 3 := ω 3 (0) = −ξ tanh (ξb).

Mean-field dynamics of quantum fluctuations
In the previous section, we studied the large N limit of the dissipative dynamics generated by (32) on (quasi) local spin operators. In this section we shall instead investigate the timeevolution of fluctuation operators scaling themselves with the inverse square-root of N.
As a set X of relevant one-site observables (see (13)), we choose the orthonormal basis of hermitian matrices {v µ } d 2 µ=1 appearing in L (N) . Accordingly, we shall focus upon the vector F (N) of local fluctuations and upon the local exponential operators in (20), As seen in section 2.2, if the matrices {v µ } d 2 µ=1 give rise to normal fluctuations with respect to the translation-invariant, clustering state ω, then In the above expression, W(r) are operators with Weyl commutation relations and Ω is a Gaussian state on the Weyl algebra W(χ, σ (ω) ) arising from the strong-closure of their linear span with respect to the GNS-representation based on Ω.
As already remarked in the previous section, the microscopic state ω need not be time- . Then, since fluctuations account for deviations of observables from their mean values that now depend on time, it is necessary to change the time-independent formulation of local quantum fluctuations given in (8) into a time-dependent one, the time-dependence occurring through the mean-values. Then, the commutator of two such local fluctuations, is a time-independent mean-field operator. However, the entries of the symplectic matrix in (17), will in general explicitly depend on time. Notice that the last two equalities follow from (48), while from (50) one derives As they depend on the initial vector ω of mean-field observables, that is of macroscopic averages, and on the time-evolution of ω into ω t , for later convenience, we shall denote by σ( ω t ) the symplectic matrix with components σ (ω) µν (t) and by σ( ω) the symplectic matrix at time t = 0 with components where we have used the assumed translation-invariance of the state ω.
Such a relation follows from (50) and (86) that yield Then, taking the time-derivative of both sides of the above equality and using (64), at t = 0. Given the local exponential operators with respect to a translation invariant, clustering state ω, in the mesoscopic limit (see definition 2 in section 2.3), they give rise to Weyl operators where the vector F has components F µ , 1 µ d 2 given by and such that

Structure of the symplectic matrix
The density matrix ρ that represents ω at each lattice site can be expanded as ρ = d 2 µ=1 r µ v µ with respect the orthonormal matrix basis. It thus turns out that the corresponding generalised is in the kernel of the symplectic matrix, is not invertible. Actually, the kernel of the symplectic matrix is at least d-dimensional for it also contains the generalized Bloch vectors corresponding to the eigenprojectors of ρ.
By an orthogonal rotation R( ω), any non-invertible σ( ω) can be brought into the form where the diagonal zero entry stands for a on the vector ω and amounts to a rotation of the hermitian matrix basis One can thus rotate the operator-valued vector F into the form so that the commutation relations (93) turn into Therefore, the first d 0 ( ω) components of G( ω) commute with all the others and among themselves and constitute a commutative set.
Definition 5. By G 0 ( ω) we will denote the d 0 ( ω)-dimensional operator-valued vector consisting of the commuting components of G( ω) and by G 1 ( ω) the vector whose components are the remaining d 1 ( ω) operators.
Then, the Weyl operators (91) split into the product of the exponentials of the commuting components of G 0 ( ω) and a quantum Weyl operators that cannot be further split: Because of (88), the matrix σ( ω t ) remains non-invertible in the course of time.

Mesoscopic dissipative dynamics
Given the local exponential operators W (N) ( r) in (90), we now study the mesoscopic limit of their dynamics at positive times t 0: We shall prove the existence of the following limit (see definition 2) where Ω is the mesoscopic state emerging from the microscopic state ω at t = 0 according to (23), W( r) = exp(i r · F) is any element of the Weyl algebra W(χ, σ (ω) ) corresponding to the matrix σ( ω) at time t = 0 with the components of F satisfying the commutation relations (93). These limits define the maps Φ ω t that describe the mesoscopic dynamics corresponding to the microscopic dissipative time-evolution γ (N) t ; their explicit form is given in the following theorem whose proof is provided in section 6.2.

Theorem 3. According to definition 2, the dynamics of quantum fluctuations is given by the mesoscopic limit
where, with T denoting time-ordering, In the above expression, is the time-dependent symplectic matrix with entries given by (85) and D( ω t ) is the matrix defined in (52).
The structure of the mesoscopic dynamics looks like that of Gaussian maps transforming Weyl operators onto Weyl operators with rotated parameters and further multiplied by a damping Gaussian factor. Indeed, the time evolution sends r into X tr t ( ω) r and the exponent r · Y t ( ω) r in the prefactor is positive since A = C+C tr 2 0 because such is the Kossakowski matrix C. However, as we shall see in the next section, the dependence on the macroscopic dynamics of mean-field quantities makes the maps Φ ω t non-linear on the Weyl algebra W(χ, σ (ω) ).

Structure of the mesoscopic dynamics
In this section we discuss in detail the properties of the mesoscopic dynamics defined by the maps Φ ω t , t 0 in (99). It turns out that they act non-linearly on products of Weyl operators. Indeed, if Φ ω t were linear, using (21), one would get Instead, the following proposition shows that the symplectic matrix in the exponent at the right hand side of the above equality is not σ( ω) at t = 0, rather σ( ω t ) at time t > 0. This is a consequence of the fact that the local operators W (N) ( r) and W (N) ( s) satisfy a Baker-Campbell-Haussdorf relation of the form Since the leading order term in the argument of the exponential function is a mean-field quanti ty, it keeps evolving in time under the action of γ (N) t in the large N limit and tends to the scalar quantity i s · (σ( ω t ) r). This result is formally derived in the proof of the following proposition given in section 6.3.

Proposition 1. The mesoscopic dynamics of the product of two Weyl operators satisfies
The non-linearity of the fluctuation dynamics conflicts with the fact that any dissipative quantum dynamics should be described by a semigroup of linear, completely positive maps. Notice that, even if systems with time-dependent macroscopic averages have already been studied [21], the puzzling result of proposition 1 had not yet emerged for, in the framework of quantum fluctuations theory only time-invariant states have been considered so far. In order to reconcile the result of proposition 1 with the desired behaviour of quantum dynamical maps, one needs to identify the proper mesoscopic algebra suited to time-evolving canonical commutation relations. One has indeed to consider quantum fluctuations obeying different algebraic rules that depend on the macroscopic averages. The proper tool is offered by an extended algebra that allows to account for the dynamics of quantum fluctuations with timevarying commutation relations. One is thus led to deal with a peculiar hybrid system, in which there appear together quantum and classical degrees of freedom, strongly connected since the commutator of two fluctuations is a classical dynamical variable. Remarkably, the need for such a mathematical setting naturally emerges from a concrete many-body quantum system as the dissipative quantum spin chain discussed above.
The maps Φ ω t can be extended to linear maps Φ ext t on a larger algebra than W(χ, σ (ω) ). Via the relations (86), the algebra W(χ, σ (ω) ) does indeed depend on the vector ω of macroscopic averages at time t = 0. We shall then denote it by W ω and by W ω ( r( ω)) its Weyl operators, where we further include the possibility that the vectors parametrizing the Weyl operators also depend on ω . We shall assume that, for all ω ∈ S, the representation of the Weyl algebra be regular so that where F( ω) is the operator-valued vector with components given by the Bosonic operators F µ ( ω), µ = 1, 2, . . . , d 2 , for each ω ∈ S so that (compare with (25)), We are thus dealing with a so-called field of von Neumann algebras {W ω } ω ∈S that can be assembled together into a direct integral von Neumann algebra [15] The most general elements of W ext are operator-valued functions of the form with f any element of the von Neumann algebra L ∞ (S) of essentially bounded functions on S with respect to the measure d ω , that is f is measurable and bounded apart from sets of zero measure, while the Weyl operators W ω ( r( ω)) ∈ W ω correspond to the operator-valued functions W 1 r evaluated at ω ; namely, W 1 r ( ω) = W ω ( r( ω)). Remark 8. Notice that the extended algebra cannot be written in a simpler tensor form; indeed, each ω determines its own Weyl algebra W ω and commutators of operators in W ω produce functions on S. Only if the algebras W ω were the same, W ω = W for all ω ∈ S, one could write W ext = L ∞ (S) ⊗ W.
States on W ext are provided by general convex combinations of the form where Ω ν is any state on the Weyl algebra W ν and ρ is any probability distribution over S. One may call Gaussian a state Ω ext on W ext if the Ω ν in (108) are all Gaussian and a specific Gaussian state Ω ω on the Weyl algebra W ω can be selected by choosing a Dirac delta distribution localised at ω ∈ S, ρ( ν ) = δ ω ( ν ). □ On the extended algebra, we can then consider the extended linear maps Φ ext t defined by their action on the building blocks W f r of W ext : Notice that Φ ext makes all parametric dependences on ω evolve in time but for the one labelling the Weyl algebra which is left fixed. Then, functions f ( ω) and vectors r( ω) are mapped into f t ( ω) := f ( ω t ), respectively r t ( ω) := r( ω t ), while, according to (99), Notice that, because of the dependence of the matrix Y t ( ω) on the whole trajectory ω → ω t , and not only on the end value ω t , the functions g r,t ( ω) = g r t ( ω) := g r ( ω t ). On the other hand, if the vector r( ω) = r does not explicitly depend on ω , then it does not evolve in time and one recovers the action (99) of the non-linear maps Φ t of which the maps Φ ext t are indeed linear extensions.
The action of the extended dynamical maps can then be recast as where it is understood that, when evaluating such an operator valued function at ω ∈ S, the matrix-valued function X t becomes X t ( ω), so that X tr t r t ( ω) = X tr t ( ω) r( ω t ). Notice that the maps Φ ext t reproduce the time-dependent algebraic relations (103). Indeed, setting E( ω) := exp(i r 1 · (σ( ω) r 2 )), with ω -independent vectors, then The expression (112) is best suited to inspect the composition law of the extended maps: When evaluated at ω , using (111), the right hand side yields The dependence on ω s of the matrix Y t ( ω s ) means that the macroscopic trajectories over which the various integral (101)-(102) are computed originates from ω s . Since the motion along a macroscopic trajectory composes in such a way that ( ω s ) t = ( ω t ) s = ω t+s for all s, t 0 (see (56)), on one hand From the first relation it follows that while the second one yields Furthermore, using (118) and (119), Together with (121), it yields In conclusion, (113) becomes whence the extended maps Φ ext t satisfy a semigroup composition law. As stated in the following proposition whose proof is given in section 6.3, the linear extended maps Φ ext t on the direct integral von Neumann algebra W ext are also completely positive. Theorem 4. The maps Φ ext t in (109) form a one parameter family of completely positive, unital, Gaussian maps on the von Neumann algebra W ext .
Since the maps Φ ext t form a semigroup on W ext , their generator L ext is obtained by taking the time-derivative of Φ ext t at t = 0 and will be of the form L ext = ⊕ S d ω L ω . The components L ω cannot be of the typical Lindblad form that is expected of the generators of Gaussian completely positive semigroups, If it were so, then , and scalar functions would remain constant in time. We will show that the generator is of hybrid form [9][10][11][12] with • a drift contribution that makes ω evolve in time as a solution to the dynamical equation (52); • mixed classical-quantum contributions; • fully quantum contributions.
Intriguingly, despite the complete positivity of the maps Φ ext t , we will show that the fully quantum terms of the generator need not be of Lindblad form.
As we shall soon see, one has to take into account the non-invertibility of the symplectic matrix σ( ω). According to section 4.1, by means of a suitable orthogonal transformation R( ω), σ( ω) can always be brought into the form (94) and the Weyl operators decomposed into a classical and quantum contribution as in (97). In the following, after rotating a given d 2 × d 2 matrix into X( ω) = R( ω) X R tr ( ω), we shall decompose it as where, as in remark 4.1,

Remark 9.
Notice that L ω contains purely classical, purely quantum and mixed classicalquantum contributions. Furthermore, the apparent Lindblad structure of the purely quantum contribution L qq ω corresponds to a Kossakowski matrix K 11 ( ω) which is in general not positive semi-definite. This is due to the correction to C 11 ( ω) = A 11 ( ω) + B 11 ( ω) 0 given by i 2 ( σ 11 ( ω)) −1 , D 11 ( ω) ; the latter matrix is traceless and cannot thus be positive semi-definite whence the positivity condition K 11 ( ω) 0 can be violated. Interestingly, despite of this, L = ⊕ S d ω L ω still generates a semigroup of completely positive maps on the extended algebra. Though the dynamics on the extended algebra consists of a semigroup of completely positive maps, the fact that its generator is not in Lindblad form with positive Kossakowski matrix is because it mixes classical and quantum terms. In order to recover the standard expression one should proceed to a fully quantum rendering of the evolution, by lifting the classical contributions to a larger non-commutative algebra in such a way that the generator in theorem 5 emerges as a restriction to a suitable commutative sub-algebra: a similar approach was proposed in a rather different context in [11]. □ Remark 10. Unlike the dissipative fluctuation time-evolution Φ ω t which is non-linear, the unitary time-evolution α t on the quasi-local algebra A given by theorem 2 is linear and does not need to be extended to a larger algebra in order to be an acceptable quantum transformation. However, if, in analogy to what has been done for Φ ω t , one introduces an extended algebra A ext whose elements are operator-valued functions O on S with values in A, ω → O ω ∈ A, unlike in remark 8, at each ω we have the same quasi-local C * algebra A, so A ext = L ∞ (S) ⊗ A. Then, the extended algebra is generated by operators of the form and O any local spin operator with finite support. We then define α ext t on A ext as follows, where ω → ω t as in (55) and α ω t given by (60) with unitary operators U (S) t ( ω) generated by Hamiltonians H s = H( ω s ) where the dependence on ω is now made explicit. It then follows that we again obtain a semigroup on A ext ; indeed, since, By taking the time-derivative of α ω t [O] at time t = 0, a time-independent generator is obtained, of the form It is a hybrid generator characterised by the absence of mixed classical-quantum contributions, by a purely classical drift part and a purely quantum contribution; explicitly, they read (compare (61) at t = 0):

Conclusions
We have considered a quantum spin chain subjected to a purely dissipative mean-field quant um dynamics. By endowing the quantum spin chain with a state not left invariant by the timeevolution, we studied the infinite volume limit of the latter on three algebras of observables. The first algebra consists of commuting macroscopic averages that behave as classical degrees of freedom obeying macroscopic equations of motions; the second algebra, build from quasilocal spin operators, despite the dissipative character of the microscopic dynamics, undergoes a unitary time-evolution with a homogeneous time-dependent Hamiltonian. Finally, the third class of observables taken into consideration represents a mesoscopic description level associated with suitable quantum fluctuations showing a collective bosonic behaviour. Due to the time-dependence of the canonical commutation relations obeyed by the fluctuations operators, the mesoscopic degrees of freedom also behave dissipatively, but their dynamics is not directly interpretable in terms of linear, completely positive maps.
We have thus extended the algebra of quantum fluctuation to accommodate the fact that macroscopic averages and quantum fluctuations are both dynamical variables. The issue is not only mathematically interesting, but also of physical relevance since in almost all experimental setups the macroscopic properties of the system actually vary in time.
On the extended algebra the non-linear fluctuations dynamics becomes linear, Gaussian and completely positive, giving rise to hybrid dynamical semigroups. Quantum fluctuations have also been experimentally investigated probing systems made of large number of atoms, and quantum effects have been reported [27][28][29]. Collective spin operators of these atomic many-body systems, once scaled by the inverse square root of the number of particles, have been observed to obey a bosonic algebra. For this reason, they have been named mechanical oscillators: they might provide a suitable concrete physical scenario where to test the theoretical results here reported.

Proofs
We first prove theorem 2 which provides the unitary dynamics of quasi-local observables and then theorem 3 which establishes the form of the dissipative dynamics of quantum fluctuations.

Dynamics of local observables
We begin with the proof of lemma 1 which provides a bound on the norm of the action of powers of the generator L (N) in (41) on products, P (N) , of mean-field and strictly local operators. Consequences of this fact are corollary 1 which asserts that the series converges uniformly in N for t 0 in a suitable finite interval of time, and corollary 2 which states that γ (N) behaves almost automorphically on products of P (N) . These two latter facts will then be used to derive firstly the time-evolution of microscopic averages in proposition 2 and then the dynamics of quasi-local operators of the quantum spin chain in theorem 2.

Lemma 1. Let P (N) ∈ A be a spin operator of the form
where O is a strictly local operator and X is a mean-field operator as in (5) for all 1 j p. Then,

where L (N) is the generator in (41), (O) is, according to definition 3, is the finite support of O and
Proof. Firstly, let us consider the action on P (N) of H (N) in (42): it consists of the sum of at most d 2 terms of the form Notice that the commutators scale as fluctuation operators since the sum is fixed by the finite support of O, while commutators of the form scale as mean-field operators further multiplied by 1/ √ N . Therefore, the action of H (N) on P (N) reduces to the sum of at most d 2 (p + 1) monomials consisting of the products of a local operator and p mean-field operators multiplied by the coefficients µ . Moreover, On the other hand, D (N) yields sums of at most d 4 terms of the form [V (136) and (137) can be used to turn the operator V (N) ν that scales as a fluctuation operator, into a mean-field one V It thus follows that the action of the generator gives rise to the sum of at most 2 d 4 ( p + 1) monomials consisting of the products of a local operator and p + 1 mean-field operators multiplied by either the coefficients A µν or B µν . With respect to the monomials contributed by H (N) , they contain one additional term, Since, by (30), v µ 1, it follows that v = max µ { v µ } 1; thus, the norms of the monomials provided by H (N) can be bounded as those provided by D. Therefore, one can estimate the norm of the action of L (N) by means of the norms of d 2 + 2 d 4 monomials containing ( p + 1) mean-field operators and a single local one. Furthermore, one sees that the monomials not containing commutators with the local operator O are bounded by Iterating this argument, L (N) h [P (N) ] will then contain at most (d 2 + 2d 4 ) h ( p+h)! p! monomials, each one with a norm that can be upper bounded as if consisted of the product of h + p mean-field quantities and strictly local operators all supported within S(O) and thus by at most (O) sites. Finally, the result follows since each of the coefficients multiplying the monomials is bounded from above by 2c and the worst case scenario is when all successive commutators act on local operators as each of them provides a factor (O) > 1. □ The previous lemma can now be used to show that γ (N) t maps mean-field quantities into infinite sums of products of mean-field quantities that converge in norm for all times t in a certain time interval [0, R]. (135); then,

Corollary 1. Let P (N) be as in
where z ∈ C and a, b ∈ A are strictly local operators. Then, for |z| < R,

Proof. Given the power series expansion
Since the bound is independent of N, the convergence is uniform in N for all |z| < R and one can exchange the infinite sum with the large N limit. □ Using the previous corollary one can show that the dissipative dynamics of products of operators of the form P (N) factorizes in the large N limit, despite the fact that, for each finite N, the time-evolution is not an automorphism of A.
Proof. The norm of the difference we want to show to vanish in the large N limit, can be recast as where, using (36), is a contraction for any N 1 (see remark 4.1), γ The result then follows by showing that In order to prove it, one can use the argument of the proof of lemma 1: operators from different sites commute, hence commutators of N−1 k=0 v (k) µ with mean-field operators yield mean-field operators, while commutators of N−1 k=0 v (k) µ with local operators yield local operators with the same or a smaller support. Therefore, the radius of norm-convergence with respect to s 0 of can be estimated by the R in the previous corollary. □ In order to proceed with the proof of theorem 2, we first derive the time evolutions of the macroscopic averages introduced in definition 4.

Proposition 2. Let ω be a translation-invariant, clustering state on the quasi-local algebra A and L (N) the dissipative Lindblad generator in (41) with R as in corollary 1. Then, for 0 t < R, the macroscopic averages in (47) evolve according to the set of non-linear equations
Proof. Consider the expression of the generator L (N) as given in (41), corollary 1 states that, for all 0 t < R, the series in (47), obtained by expanding γ (N) t , converges uniformly in N; one can then exchange the large N limit with the time-derivative obtaining: Using (48) and (50), the large N limit of the Hamiltonian contribution yields Concerning the dissipative contribution, since γ On the other hand, using (140) and corollary 2, it follows that, in norm, From corollary 1 one knows that, for 0 t < R, mean-field operators are turned into normconvergent series of mean-field operators; moreover, these latter behave as stated in (7) in the large N limit. Then, using corollary 2 together with (47) and (48) one obtains □ By means of the time-evolution of macroscopic averages, we move on to prove theorem 2: we first show that the result holds for times 0 t R, R as in corollary 1, and for strictly local operators and then relax these two constraints.

Theorem 2. Let the quasi-local algebra A be equipped with a translation-invariant, clus-
tering state ω. In the large N limit, the local dissipative generators L (N) in (41) define on A a one-parameter family of automorphisms that depend on the state ω and are such that, for all 0 t T, T 0 arbitrary, with explicitly time-dependent Hamiltonian where

Proof. Given
and study the large N limit of One finds Since H (S) t is the sum of single-site contributions, O t−s is a strictly local operator with the same support as O. Thus, as in the proof of the previous proposition, the action of A (N) of the µ (t − s) a strictly local operator with support fixed by O. Then, Therefore, one can focus upon the limit Using the Cauchy-Schwarz inequality and the Kadison inequality for completely positive , we have: Both X The result just obtained is valid for 0 t < R and for strictly local operators O. It can be extended to all times in compact subsets of the positive real line and to the whole quasi-local algebra A. While the norm-preserving maps α t , 0 t < R, can be extended by continuity to the quasi-local algebra, the extension to any finite time t 0 is obtained by the following proposition 3. □ The first extension regards the time domain and makes use of the following result [30].
The last norm is bounded uniformly in N for |y| < R; this follows by applying corollary 1, which also shows that lim N→∞ f N (z) exists for all z ∈ E = {z = t + iy : t > 0 , |z| < R}. Then, the Vitali-Porter theorem ensures that lim N→∞ f N (z) = f (z) with f (z) an analytic function, uniformly on any compact subset of Ω.
Since the Hamiltonians in (61) are sums of single-site operators that do not modify the support of the time-evolving strictly local operator O, the functions u S (z(t)) = ω a U (S) are also analytic on Ω; indeed, they are bounded: where H max := max 0 s t H (S) (z(s)) . Consider now T > R and the subset We have that f (z) and u S (z) are both analytic functions on Ω T . Moreover, due to theorem 2, f(z(t)) = u S (z(t)) for z = t ∈ [0, R); therefore, f(z(t)) = u S (z(t)) for all z(t) ∈ Ω T , so that the restriction to the real line yields the result. □ We can now conclude by extending the previous results from strictly local operators O to mean-field operators.

Corollary 3. The convergence of the microscopic dynamics
to the automorphisms α t on the quasi-local algebra A as in theorem 2 holds for operators arising as strong limits of mean-field operators.

Proof. Consider the mean-field operator X
satisfies, in the large N limit, the equation of motion of proposition 2. Furthermore, with the notations of the previous proposition, with x ∈ A strictly local. Then, as in proposition 3, lim N→∞ ω U (N) provides an analytic function on compact subsets of Ω T = {z = t + iy : 0 < t < T, |y| < R}, T R, and its restriction to t ∈ [0, T) implements the large N dynamics induced by the generator L (N) . □

Dynamics of quantum fluctuations
This section will be devoted to the proofs of the results concerning the structure and properties of the generator of the dissipative dynamics of quantum fluctuations. We start with the proof of theorem 3 which is divided into several steps, the first ones concerning the algebraic behaviour of quantum fluctuations, mean-field quantities and local exponentials in the large N limit.

Lemma 2. For all
µν ] is the mean-field operator-valued matrix with entries (84).

Proof.
Using by means of (84) we write In order to deal with Z (N) (t) = ∞ n=2 t ], notice that, since operators at different lattice sites commute, so that lim N→∞ Z (N) (t) = 0 and the result follows. Proof. Using (145) one writes Then, as in the proof of the previous lemma, the result follows from The following proposition specifies the speed with which the limit established in proposition 2 is attained, a result which will be applied in the coming estimates. (47) and (48) one has that: with E αβ the entries of the matrix E defined in (53).
Proof. Consider the time-derivative of ω (N) Since, using (47) and (50), and, as already seen in the proof of proposition 2, the action of the A (N) term of the generator on mean-field observables is in norm a O 1 N quantity, one has Using (47), (50), (83) and the fact that fluctuation have zero mean values, one rewrites ω γ The required scaling results from lemma B.2 in appendix B and the fact that the Cauchy-Schwartz inequality relative to the expectations with respect to the state ω • γ

□
The following proposition establishes the asymptotic form of the action of the generator L (N) on local exponential.
On the other hand, the second term in (156) contributes to the large N limit with where B N is a term which vanishes in norm when N → ∞. Using the matrix basis relations (30), (53) and the anti-symmetry of the operator-valued matrix T (N) , the double commutator in (160) can be recast in the form is the matrix given in (53). Analogously, the sum in (159) can be rewritten as where ω (N) t denotes the vector with d 2 real components ω (N) Observe that the time-derivative of the exponent of W where ω (N) t stands for the vector with components ω (N) (47)). Then, from the well known result of lemma B.3 reported and proved for sake of completeness in appendix B, contains an infinite sum starting from k = 3 and thus vanishes in norm when N → ∞, while the commutator yields thus, through T (N) , it exhibits a mean-field scaling when N → ∞. Finally, Setting ξ = X tr t,s ( ω) r for sake of simplicity, (162) can thus be recast as The last term does not contribute to the mesoscopic limit and proposition 5 provides the mesoscopic behaviour of the first contribution to the right hand side of the equality above. We now group together terms with the same scaling with 1/N and show that, in the mesoscopic limit, the following quantities vanish: Notice that T (N) is an operator-valued matrix with entries that scale as mean-field observables; then, we proceed by showing that, in the large N limit, in the above expressions, meanfield operators of the form M (N) α ) with respect to the large N limit of the time-evolving state ω t . Indeed, in (165) and (166) there appear terms of the type while terms of the form appear in (164) and terms as are to be found both in (163) and (166). Let us consider the latter expression and study the limit Using lemma B.1 in appendix B, and the Kadison inequality, one has Then, lemma 3 and corollary 2 yield Therefore, we have that By a similar argument, one shows that Now we consider the following quantity

The operator M
(N) α scales as a mean-field quantity; therefore, the norm of its commutator with quantities that scale as fluctuations vanishes in the large N limit. Then, because of lemma 3, we have that Finally, lemma B.1 in appendix B and the Kadison inequality applied to the term on the righthand side of the equality, yield the following bound The first term on the right-hand side is bounded by lemma B.2 in appendix B, while the second one, as already shown, vanishes in the large N limit. Therefore, Applying these considerations to the quantities (163)-(166), one thus sees that (163) vanishes in the large N limit because of proposition 4. Furthermore whence (165) vanishes in the arge N limit and analogously Finally, as regards the large N limit of (166), using (53), (101) and (45), (46), the scalar product behaves as the latter equality resulting form the fact that σ( ω t ) B + 2 h (im) σ( ω t ) is antisymmetric. □

Structure of the dissipative generator
In this section we prove various properties of the mesoscopic dynamics and its generator. We start by showing that the maps Φ ω t defined by theorem 3 cannot act linearly on the fluctuation algebra.
The function (174) can be interpreted as the expectation of the product of two Weyl operators V ω ( r i,k ( ω t )) satisfying the Weyl algebraic rules with symplectic matrix, that is real and anti-symmetric, σ t ( ω) given by (175), the expectation being defined by the functional ϕ t acting on the Weyl algebra V ω generated by the V ω ( r( ω)).
This is a positive operator in W ω ⊗ M n (C) if the expectation functional ϕ t on V is positive, namely, according to (176), if ϕ t amounts to a Gaussian state. This latter property is equivalent to having which has already been proved.
a differential operator involving the classical degrees of freedom G µ ( ω), µ = 1, 2, . . . , d 0 ( ω), Therefore, when evaluated at ω , the time-derivative of the dynamics at t = 0, together with the Using that σ tr ( ω) = −σ( ω), by means of the rotation matrix R( ω) in (94), of the decomposition (124) and with the notation of remark 4.1, s( ω) = R( ω) r , one finally gets Let us try to write the component L ω of the generator L ext in the customary Lindblad form where the d 2 × d 2 matrices H( ω) and K( ω) are both hermitian and the operators G µ ( ω) are those appearing in the decomposition (97) of the Weyl operators into classical and quantum contributions. By decomposing the matrix K( ω) and H( ω) as in (124), since the components of G 0 ( ω) commute with all the others, there are no contributions to L ω from either H 00 ( ω) or K 00 ( ω), while those from K 01 ( ω) and K 10 ( ω) can be put together with the contributions from H 01 ( ω) and H 10 ( ω) in the Hamiltonian matrix. Thus, one can, without restriction, set Then, the relations (105) yield W 1 From comparing equations (177)-(182) and (183)-(184), one finds The still unmatched terms in (178) and (181) can only be recovered by acting on the Weyl operators in a way that involves both the commuting degrees of freedom represented by the first d 0 ( ω) components of G 0 ( ω) and the remaining non-commuting ones. Then, Summing the right hand sides of the above equalities yields the classical contribution to the generator, L cc ω in (126), respectively the mixed classical-quantum one, L cq ω in (127). □

Appendix A
In the case of a clustering state ω, one can then consider the large N limit Indeed, for any integer N 0 < N one can write: While the first contribution at the rhs vanishes, concerning the second term we argue as follows. Since strictly local operators are norm dense in A, without loss of generality one can assume c to have support within [−N 0 , N 0 ], so that it commutes with N−1 k=N0+1 x (k) . Using the clustering property (4) one immediately gets the result (A.1). This means that, in the so-called weak operator topology, i.e. under the state average, the large N limit of X (N) is a scalar multiple of the identity operator: The relation (6) can be proved as follows: because of definition (5), it is equivalent to , so that ω( x (k) ) = ω( x) = 0, ω X N = 0 and similarly for y, Y N . Then, as shown in the main text for a single variable, the quasi-locality of a, b and the clustering properties of the state yield: Further, one can write: Since ω is translation-invariant, the first term vanishes as ω x y /N when N → ∞. Moreover, thanks to the clustering property (4), for any small > 0, there exists an integer N , such that for |k − | 2 > N one has: Then, using this result, one can finally write: so that, in the large N limit, the relation (6) is indeed satisfied. Notice that (6) entails that, in the GNS representation, for all a ∈ A. Namely, mean-field spin observables converge to their expectations with respect to ω in the strong operator topology on the GNS Hilbert space H ω . For what concerns (7), notice that where the last equality holds because of the translation invariance of the state ω. Now, assuming (3) one has that proves the scaling (7). Notice that, by recursion, using the norm-boundedness of the mean-field quantities and the strong-limit in (A.2), one can show that Proof. Since F (N) µ (t) 2 is a positive matrix, the following quantity: The lemma is proved if we show that G( r, t) := lim N→∞ G (N) ( r, t) is finite ∀t 0. Let us then consider where (83) and (47) (41), we first consider Since spin operators at different sites commute, the commutators read Then, one readily obtains the uniform upper bound where for later convenience we have also included B in the definition of the quantity c. Since γ (N) t is a contraction, it follows that the contribution of A (N) to (B.3) is uniformly bounded in N and t: (B.5) Let us then concentrate on the action of Concerning the contribution in(B.8), observe that V From lemma B.1 it follows that ∆ (N) (t) 2 is upper bounded by

(B.15)
The final term to consider is the Hamiltonian one contributed by the action of H N) that, by similar arguments as before, can be recast as From G (N) ( r, t) 1, it also follows that, given any uniform upper bound k to the time-derivative (B.3), one can replace it by k G (N) ( r, t), whence all upper bounds collected so far can be grouped together in an upper bound of the form K G (N) ( r, t). The only terms which escape this rule are the ones increasing with √ N in (B.11), (B.15) and (B.16). Therefore, recalling (B.3), one is left with studying the large N-limit of (B.17) First we consider the case r = 0; then, since ω γ β (t) = 0, we get I (N) (0, t) = 0. Therefore, for all N, When r = 0, we estimate Considering the time-derivative, one has: d dt ω Taking into account that ω γ and writing since the latter term is a scalar multiple of the identity, one gets I (N) ( r, t) < 4d 6 v 4 c G (N) ( r, t) ω W (N) ( r)γ (N) t m a (t)|O t N t |m b (t) = i m a (t)|Ṁ t |m b (t) e im(t) δ ab = iṁ(t) e im(t) δ ab = m a (t)|Ṅ t |m b (t) . □