Vojkan Jaksic

Yan Pautrat

Claude-Alain Pillet

V Jakšić

C.-A Pillet

Central limit theorem for locally interacting Fermi gas

published or not. The documents may come

Introduction

This paper and its companion [START_REF] Aschbacher | Fluctuations in quasi-free fermionic systems[END_REF] are first in a series of papers dealing with fluctuation theory of non-equilibrium steady states in quantum statistical mechanics. They are part of a wider program initiated in [Ru2, Ru3, JP1, JP2, JP4] which deals with the development of a mathematical theory of non-equilibrium statistical mechanics in the framework of algebraic quantum statistical mechanics [START_REF] Bratteli | Operator Algebras and Quantum Statistical Mechanics 1[END_REF][START_REF] Bratteli | Operator Algebras and Quantum Statistical Mechanics 2[END_REF][START_REF] Pillet | Quantum dynamical systems[END_REF]. For additional information about this program we refer the reader to the reviews [START_REF] Ruelle | Topics in quantum statistical mechanics and operator algebras[END_REF][START_REF] Jakšić | Mathematical theory of non-equilibrium quantum statistical mechanics[END_REF][START_REF] Aschbacher | Topics in non-equilibrium quantum statistical mechanics[END_REF].

In this paper we study the same model as in [START_REF] Jakšić | The Green-Kubo formula for locally interacting fermionic open systems[END_REF]: A free Fermi gas in a quasi-free state perturbed by a sufficiently regular local interaction. It is well-known that under the influence of such a perturbation this system approaches, as time t → +∞, a steady state commonly called the natural non-equilibrium steady state (NESS) [BM1, AM, BM2, FMU, JOP4]. Our main result is that under very general conditions the Quantum Central Limit Theorem (QCLT) holds for this NESS. Combined with the results of [START_REF] Jakšić | The Green-Kubo formula for locally interacting fermionic open systems[END_REF], the QCLT completes the proof of the near-equilibrium Fluctuation-Dissipation Theorem and the development of linear response theory for this class of models.

The rest of this introduction is organized as follows. In Subsection 1.1 for notational purposes we review a few basic concepts of algebraic quantum statistical mechanics. In this subsection the reader can find the definition of QCLT for quantum dynamical systems and a brief review of related literature. Our main result is stated in Subsection 1.2. In Subsection 1.3 we discuss our results in the context of linear response theory.

Acknowledgment. A part of this work has been done during Y.P.'s stay at McGill University and the C.R.M. as ISM Postdoctoral Fellow, during his visit to McGill University funded by NSERC and his visit to Erwin Schrödinger Institut. The research of V.J. was partly supported by NSERC. We wish to thank Manfred Salmhofer for useful discussions.

Central limit theorem for quantum dynamical systems

Let O be a C * -algebra with identity 1l and let τ t , t ∈ R, be a strongly continuous group of * -automorphisms of O. We denote by (Hω, πω, Ωω) the GNS-representation of the C * -algebra O associated to the state ω. The state ω is called modular if Ωω is a separating vector for the enveloping von Neumann algebra πω(O) ′′ . The states of thermal equilibrium are described by the (τ, β)-KMS condition where β > 0 is the inverse temperature. Any (τ, β)-KMS state on O is τ -invariant and modular.

Let f be a bounded Borel function on R and A ∈ O self . With a slight abuse of notation in the sequel we will often denote f (πω(A)) by f (A) and write ω(f (A)) = (Ωω, f (πω(A))Ωω). With this convention, 1 [a,b] (A) denotes the spectral projection on the interval [a, b] of πω(A). We shall use the same convention for the products f1(πω(A1)) • • • fn(πω(An)), etc.

An involutive skew * -automorphism

Θ of O is called time-reversal if Θ • τ t = τ -t • Θ. A state η on O is called time-reversal invariant if η • Θ(A) = η(A * ) holds for all A ∈ O.
For any subset A ⊂ O we denote by A self = {A ∈ A | A = A * } the set of self-adjoint elements of A. We say that a subset

A ⊂ O is L 1 -asymptotically abelian for τ if for all A, B ∈ A, Z ∞ -∞ ' ' [A, τ t (B)] ' ' dt < ∞.
Throughout the paper we shall use the shorthand

Ãt ≡ 1 √ t Z t 0 (τ s (A) -ω(A)) ds.
Let C be a * -vector subspace of O. We say that C is CLT-admissible if for all A, B ∈ C, Z ∞ -∞ ˛ω(τ t (A)B)ω(A)ω(B) ˛dt < ∞.

For A, B ∈ C we set

L(A, B) ≡ Z ∞ -∞ ω `(τ t (A) -ω(A))(B -ω(B)) ´dt = Z ∞ -∞ `ω `τ t (A)B ´-ω(A)ω(B) ´dt, ς(A, B) ≡ 1 2i Z ∞ -∞ ω `[τ t (A), B] ´dt = 1 2i (L(A, B) -L(B, A)) .
The functional (A, B) → L(A, B) is obviously bilinear. Other properties of this functional are summarized in:

Proposition 1.1 Suppose that C is CLT-admissible and let A, B ∈ C. Then:

(i) L(A * , A) ≥ 0.

(ii) L(A, B) = L(B * , A * ). In particular, if A and B are self-adjoint, then ς(A, B) = Im L(A, B).

(iii) |L(A * , B)| 2 ≤ L(A * , A)L(B * , B).
(iv) (A, B) → ς(A, B) is a symplectic form on the real vector space C self .

(v) If ω is a mixing (τ, β)-KMS state, then ς = 0.

(vi) Suppose that ς = 0, that C is dense in O and L 1 -asymptotically abelian for τ , and that ω is either a factor state or 3-fold mixing: For all A1, A2, A3 ∈ O,

lim min i =j |t i -t j |→∞
ω `τ t 1 (A1)τ t 2 (A2)τ t 3 (A3) ´= ω(A1)ω(A2)ω(A3).

Then ω is a (τ, β)-KMS state for some β ∈ R ∪ {±∞}.

Proof. Note that

0 ≤ ω " Ã * t Ãt " = Z t -t " 1 - |s| t « ω `(τ t (A * ) -ω(A * ))(A -ω(A)) ´ds.
This identity and the dominated convergence theorem yield We shall say that the Quantum Central Limit Theorem (QCLT) holds for C if for all n and all A1,

L(A * , A) = lim
• • • , An in C self , lim t→∞ ω " e i Ã1t • • • e i Ãnt " = exp 0 @ - 1 2 X 1≤j,k≤n L (A k , Aj) -i X 1≤j<k≤n ς(Aj, A k ) 1 A . (1.1)
The SQCLT is obviously a special case of the QCLT. Under sufficient ergodic assumptions, however, the QCLT can be deduced from the SQCLT.

Theorem 1.3 Suppose that C is CLT-admissible and L 1 -asymptotically abelian for τ . Suppose also that the system (O, τ, ω) is ergodic and that the state ω is modular. If the SQCLT holds for C w.r.t. (O, τ, ω) then the QCLT also holds for C.

We shall prove Theorem 1.3 in Section 2 following the ideas of [GV].

The SQCLT has the same probabilistic interpretation as the classical central limit theorem. The probability of measuring a value of A in [a, b] when the system is in the state ω is given by

Probω{A ∈ [a, b]} = ω(1 [a,b] (A)).
If SQCLT holds for A, then

lim t→∞ Probω  1 t Z t 0 τ s (A) ds ∈ ω(A) + » a √ t , b √ t -ff = 1 p 2πL(A, A) Z b a e -x 2 /2L(A,A) 2 dx. (1.2)
Except in trivial cases, the QCLT does not have a classical probabilistic interpretation. In this case the relevant concept is the CCR algebra over the symplectic space (C self , ς), often called the fluctuation algebra [START_REF] Goderis | Noncommutative central limits[END_REF]. The mathematical structure of the fluctuation algebra is discussed in many places in the literature, see e.g. [GVV1]- [START_REF] Goderis | Dynamics of fluctuations for quantum lattice systems[END_REF] and [START_REF] Manuceau | The smallest C * -algebra for canonical commutations relations[END_REF][START_REF] Bratteli | Operator Algebras and Quantum Statistical Mechanics 2[END_REF][START_REF] Petz | An invitation to the algebra of canonical commutation relations[END_REF][START_REF] Ohya | Quantum Entropy and its Use[END_REF][START_REF] Dereziński | Introduction to representations of canonical commutation and anticommutation relations[END_REF] for general results about CCR algebras. For notational and reference purposes we recall a few basic facts. Let W be the C * -algebra generated by the elements

{W (A) | A ∈ C self } such that for all A, B in C self W (-A) = W (A) * , W (A)W (B) = e -iς(A,B)/2 W (A + B),
equipped with the minimal regular norm. The map

ωL(W (A)) ≡ e -L(A,A)/2 ,
uniquely extends to a quasi-free state on W and (1.1) can be written as

lim t→∞ ω " e i Ã1t • • • e i Ãnt " = ωL(W (A1) • • • W (An)). (1.3)
The pair (W, ωL) describes the fluctuations of C w.r.t. the quantum dynamical system (O, τ, ω). Let (HL, πL, ΩL) be the GNS representation of W associated to ωL. We shall also denote by ωL the induced state on the enveloping von Neumann algebra πL(W) ′′ . Since for all A ∈ C self the map

R ∋ x → ωL(W (xA)),
extends to an entire analytic function on C, there exist self-adjoint operators ϕL(A) on HL such that

πL(W (A)) = e iϕ L (A) .
Moreover, the operators ϕL(A), A ∈ C self have a common dense set of analytic vectors A ⊂ HL and on this set

[ϕL(A), ϕL(B)] = iς(A, B)1l.
The operators ϕL(A) are the Bose fields associated by QCLT to (O, τ, ω). For any n and A1,

• • • , An ∈ C self , ΩL is in the domain of ϕL(A1) • • • ϕL(An)
and , as usual, we denote

ωL(ϕL(A1) • • • ϕL(An)) ≡ (ΩL, ϕL(A1) • • • ϕL(An)ΩL).
(1.4)

In particular ωL(ϕL(A1)ϕL(A2)) = L(A1, A2). For any integer n we denote Pn the set of all permutations π of {1, . . . , 2n} such that π(2j -1) < π(2j), and π(2j -1) < π(2j + 1), (1.5) for every j ∈ {1, . . . , n}. The cardinality of Pn is (2n)!/(2 n n!). Then

ωL(ϕL(A1) • • • ϕL(An)) = 8 > > < > > : X π∈P n/2 n/2 Y j=1 ωL(ϕL(A π(2j-1) )ϕL(A π(2j) )), if n is even; 0,
if n is odd.

(1.6)

With these preliminaries, we can formulate:

lim t→∞ ω(f1( Ã1t) • • • fn( Ãnt)) = ωL(f1(ϕL(A1)) • • • fn(ϕL(An))).
(1.7)

If n = 1, then Theorem 1.4 is an immediate consequence of the classical Lévy-Cramér Continuity Theorem. The proof in the case n > 1 is given in Subsection 2.2. For a probabilistic interpretation of Theorem 1.4 in the context of repeated quantummechanical measurements we refer the reader to Section 2 in [START_REF] Davies | Quantum Theory of Open Systems[END_REF].

The QCLT does not imply that

lim t→∞ ω( Ã1t • • • Ãnt) = ωL(ϕL(A1) • • • ϕL(An)), (1.8)
and in principle the convergence of moments has to be established separately. In our model, the proof of (1.8) is an intermediate step in the proof of the QCLT.

To define Bose annihilation and creation operators associated with fields ϕL(A), we need to assume that the symplectic form ς is non-degenerate (this implies that C self is either even-or infinite-dimensional). In this case there exists a complex structure J on C self satisfying ς(JA, JB) = ς(A, B), and one can define the operators aL(A)/a * L (A) on A by

aL(A) ≡ 1 √ 2 (ϕL(A) + iϕL(JA)) , a * L (A) ≡ 1 √ 2 (ϕL(A) -iϕL(JA)) .
(1.9)

These operators are closable and satisfy

[aL(A), a * L (B)] = i (ς(A, B) -iς(A, JB)) , on A.
We expect that in typical physical examples the symplectic form ς will be degenerate in which case the Bose annihilation and creation operators (1.9) cannot be defined globally. Let us assume, for simplicity, that L is non-degenerate (in the general case one has to further factor out the kernel of L). Consider first the extreme case ς = 0 (this will hold, for example, if ω is a mixing (τ, β)-KMS state). Let Ĉself be the group of all characters of the discrete Abelian group C self . The dual group Ĉself endowed with the topology of pointwise convergence is a compact topological group and the algebra W is isomorphic to the C * -algebra of all continuous functions on Ĉself . The state ωL is identified with the Gaussian measure on Ĉself uniquely determined by Z χ(A) dµL(χ) = e -L(A,A)/2 .

More generally, let C

(1) self = {A | ς(A, B) = 0 for all B ∈ C self } , and suppose that there exist

C (2) self such that C self = C (1) self ⊕ C (2) self and L = L (1) ⊕ L (2) where L (j) denotes the restriction of L to C (j)
self . This is certainly the case if C self is finite dimensional, i.e., if we consider QCLT with respect to finitely many observables. The restriction of ς to C

(2) self is non-degenerate, and if W (j) , ω (j) L , j = 1, 2 denote the respective CCR algebras and quasi-free states, then

W = W (1) ⊗ W (2) , ωL = ω (1) L ⊗ ω (2)
L . In particular, annihilation and creation operators can be associated to the elements of W (2) .

Besides QCLT one may consider the related and more general existence problem for the quantum hydrodynamic limit (QHL). For ǫ > 0 and t > 0, let

Âǫ(t) ≡ ǫ Z t/ǫ 2 0 (τ s (A) -ω(A)) ds.
We say that C has QHL w.r.t.

(O, τ, ω) if for all A1, • • • An ∈ C self , and all t1 > 0, • • • , tn > 0, lim ǫ↓0 ω " e i Â1ǫ (t 1 ) • • • e i Ânǫ(tn) " = ωL(W (χ [0,t 1 ] ⊗ A1) • • • W (χ [0,tn] ⊗ An)). (1.10)
where χI is the characteristic function of the interval I and, in the definition of the Weyl algebra, the bilinear form L must be replaced by

LQHL(χ [0,s] ⊗ A, χ [0,t] ⊗ B) = min(s, t) L(A, B).
The special case where all tj's are equal corresponds to QCLT. The QHL is interpreted as the weak convergence of the quantum stochastic process Âǫ(t) to a quantum Brownian motion. With the obvious reformulation, Theorem 1.4 holds for QHL. Convergence of moments

lim ǫ↓0 ω( Â1ǫ(t1) • • • Ânǫ(tn)) = ωL(ϕL(χ [0,t 1 ] ⊗ A1) • • • ϕL(χ [0,tn] ⊗ An), (1.11)
is of independent interest. Even more generally, one may associate to a class F of real valued integrable functions on R the observables

Âǫ(f ) ≡ ǫ -1 Z ∞ 0 f (ǫ 2 t) `τ t (A) -ω(A) ´dt, with f ∈ F, A ∈ C and study the limit ǫ ↓ 0 of ω " e i Â1ǫ (f 1 ) • • • e i Ânǫ(fn)
" .

(1.12)

Note that QHL corresponds to the choice

F = {χ [0,t] | t > 0}.
For reasons of space and notational simplicity we will focus in the paper on the QCLT for locally interacting fermionic systems. With only notational changes our proofs can be extended to establish QHL and (1.11). It is likely that the proofs can be extended to a much larger class of functions F, but we shall not pursue this question here (see [START_REF] Dereziński | Boson free fields as a limit of fields of a more general type[END_REF] for a related discussion).

We finish this section with a few remarks about earlier quantum central limit type results.

First notice that since the law of a single observable is well-defined, the description of the limiting law of a family { Ãt | t > 0} of observables as the parameter t → ∞, is covered by the classical Lévy-Cramèr theorem. Several results of interest exist, which are only of quantum nature insofar as the computation of limt→∞ ω(e iα Ãt ) is made more complicated by the quantum setting.

Truly quantum central limit theorems attempt to describe the joint limiting behavior, as t → ∞, of a family of observables {( Ã1t, . . . , Ãnt) | t > 0}. The earliest results of this type were quantum probabilistic versions of classical theorems concerning sums of independent, identically distributed random variables. Physically interesting applications of these results include the study of spatial fluctuations of local observables of quantum spin systems in a translation-invariant product state. The generality of the framework and the formulation of the limit vary. We mention in particular [GvW] which apply to general *-algebras but where only the convergence of moments is proved; [Kup] which works for general C*-algebras and where convergence in distribution (to a classical Gaussian family) is proved, but only with respect to a tracial state . We also mention [CH] which, although not a central limit theorem, is a first attempt to characterize the convergence in distribution of a family of non-commuting operators in terms of a (pseudo)-characteristic function.

The series of papers [GVV1]- [START_REF] Goderis | Dynamics of fluctuations for quantum lattice systems[END_REF], [GV] is more directly oriented towards quantum statistical mechanics. Following ideas of [START_REF] Hepp | Phase transitions in reservoir driven open systems with applications to lasers and superconductivity[END_REF][START_REF] Hepp | On the superradiant phase transition for molecules in a quantized radiation field: the Dicke maser model[END_REF], [W], a fluctuation algebra is associated to the spatial fluctuations of local observables for a quantum spin system in a translation-invariant state. That state does not have to be a product state but it must have very strong ergodic properties. It is therefore very difficult to apply these results beyond the product state case. Nevertheless, these works laid a solid conceptual ground and our construction owes much to them. The papers [Ma1]- [START_REF] Matsui | On the algebra of fluctuation in quantum spin chains[END_REF] are similar in spirit but require less stringent ergodic conditions. Their results apply, for example, to spatial fluctuations of local observables in XY -chains at thermal equilibrium.

A distinct feature of our work is that we study QCLT with respect to the group τ t describing the microscopic dynamics of the system. There is a number of technical and conceptual aspects of QCLT which are specific to the dynamical group. For example, the ergodic properties of the system (laws of large numbers), which have to be established prior to the study of fluctuations, are typically much harder to prove for the dynamical group than for the lattice translation group. As for the conceptual differences, we mention that if ω is a (τ, β)-KMS state, then by Proposition 1.1 (v), ς = 0 and the CCR algebra of fluctuations W is commutative (Part (vi) provides a partial converse to this statement). This is in sharp contrast with QCLT w.r.t. the translation group, where even in the simple example of product states of spin systems the fluctuation algebra is non-commutative.

The CLT for classical dynamical systems is discussed in [Li]. For a review of results on dynamical CLT for interacting particle systems in classical statistical mechanics we refer the reader to [Sp] and [KL]. The CLT for classical spin systems is discussed in Section V.7 of [E].

After this paper was completed, we have learned of the work [START_REF] Dereziński | Boson free fields as a limit of fields of a more general type[END_REF] which is technically and conceptually related to ours. We shall comment on Dereziński's result at the end of Subsection 3.3.

QCLT for locally interacting fermions

A free Fermi gas is described by the C * -dynamical system (O, τ0) where:

(i) O = CAR(h) is the CAR algebra over the single particle Hilbert space h;

(ii) τ t 0 is the group of Bogoliubov * -automorphisms generated by the single particle Hamiltonian h0,

τ t 0 (a # (f )) = a # (e ith 0 f ),
where a * (f )/a(f ) are the Fermi creation/annihilation operators associated to f ∈ h and a # stands for either a or a * . We denote by δ0 the generator of τ0.

Let O be the τ0-invariant C * -subalgebra of O generated by {a * (f )a(g) | f, g ∈ h} and 1l. Physical observables are gauge invariant and hence belong to O.

Let v be a vector subspace of h and let O(v) be the collection of the elements of the form

A = K X k=1 n k Y j=1 a * (f kj )a(g kj ), (

Our main assumption is :

(A) There exists a dense vector subspace d ⊂ h such that the functions

R ∋ t → (f, e ith 0 g), are in L 1 (R, dt) for all f, g ∈ d.
This assumption implies that h0 has purely absolutely continuous spectrum. Specific physical models which satisfy this assumption are discussed at the end of this subsection.

Let V ∈ O(d) self be a self-adjoint perturbation. We shall always assume that nV ≥ 2. The special case nV = 1 leads to quasi-free perturbed dynamics and is discussed in detail in the companion paper [START_REF] Aschbacher | Fluctuations in quasi-free fermionic systems[END_REF], see also [AJPP1, AJPP2, JKP] and Remark after Theorem 1.6 below.

Let λ ∈ R be a coupling constant and let τ λ be the C * -dynamics generated by δ λ = δ0 + iλ[V, • ]. By rescaling λ, without loss of generality we may assume that max

f ∈F (V ) f = 1. (1.14)
We shall consider the locally interacting fermionic system described by (O, τ λ ). Note that τ λ preserves O and that the pair

(O, τ λ ) is also a C * -dynamical system. Let λV ≡ 1 2nV KV ℓV (2nV -2) 2n V -2 (2nV -1) 2n V -1 , (1.15) where ℓV ≡ Z ∞ -∞ sup f,g∈F (V )
|(f, e ith 0 g)|dt.

(1.16)

The following result was proven in [START_REF] Jakšić | The Green-Kubo formula for locally interacting fermionic open systems[END_REF] (see also [START_REF] Botvich | Unitary equivalence of temperature dynamics for ideal and locally perturbed Fermi gas[END_REF][START_REF] Aizenstadt | Spin interaction with an ideal Fermi gas[END_REF][START_REF] Botvich | Asymptotic completeness and all that for an infinite number of fermions[END_REF][START_REF] Fröhlich | Dissipative transport: contacts and tunneling junctions[END_REF]).

Theorem 1.5 Suppose that (A) holds. Then:

1. For all A ∈ O(d) and any monomial B = a # (f1) • • • a # (fm) with {f1, . . . , fm} ⊂ d, one has

sup |λ|≤λ V Z R ' ' [τ t λ (A), B] ' ' dt < ∞.
2. For |λ| ≤ λV the Møller morphisms

γ + λ ≡ s -lim t→∞ τ -t 0 • τ t λ ,
exist and are * -automorphisms of O.

In what follows we shall assume that (A) holds. Let T be a self-adjoint operator on h satisfying 0 ≤ T ≤ I and [T, e ith 0 ] = 0 for all t, and let ω0 be the gauge invariant quasi-free state on O associated to T . We will sometimes call T the density operator. The state ω0 is τ0-invariant and is the initial (reference) state of our fermionic system. The quantum dynamical system (O, τ0, ω0) is mixing. We denote by N0 the set of all ω-normal states on O. Theorem 1.5 yields that any state η ∈ N0 evolves to the limiting state

ω + λ = ω0 • γ + λ , i.e., for A ∈ O and |λ| ≤ λV , lim t→∞ η(τ t λ (A)) = ω + λ (A),
see, e.g., [START_REF] Robinson | Return to equilibrium[END_REF][START_REF] Aschbacher | Topics in non-equilibrium quantum statistical mechanics[END_REF]. The state ω + λ is the NESS (non-equilibrium steady state) of (O, τ λ ) associated to the initial state ω0. Clearly, ω + λ is τ λ -invariant and γ + λ is an isomorphism of the quantum dynamical systems (O, τ0, ω0) and (O, τ λ , ω + λ ). In particular, the system (O, τ λ , ω + λ ) is mixing. In what follows we shall always assume that Ker T = Ker (I -T ) = {0}. This assumption ensures that the states ω0 and ω + λ are modular.

Let c ⊂ d be a vector subspace such that the functions

R ∋ t → (f, e ith 0 T g), are in L 1 (R, dt) for all f, g ∈ c.
In general, it may happen that c = {0}, and so the existence of a non-trivial c is a dynamical regularity property of the pair (T, h0). If T = F (h0), where F ∈ L 1 (R, dx) is such that its Fourier transform

F (t) = 1 √ 2π Z ∞ -∞ e itx F (x)dx, is also in L 1 (R, dt), then one can take c = d. Let λV ≡ 2 -8(n V -1) λV , (1.17) and C ≡ O(c).
The main result of this paper is: Remark. If nV = 1, then Theorem 1.5 holds for any 0 < λV < (2KV ℓV ) -1 , see [START_REF] Jakšić | The Green-Kubo formula for locally interacting fermionic open systems[END_REF]. With this change, Theorem 1.6 holds with λV = λV . The case nV = 1 is however very special.

Theorem 1.6 Suppose that (A) holds, that V ∈ C self ,
If V = P k a * (f k )a(g k ), then τ λ is quasi-free dynamics generated by h λ = h0 + λ P k (g k , •
)f k and Theorem 1.5 can be derived from the scattering theory of the pair (h λ , h0), see [START_REF] Robinson | Return to equilibrium[END_REF][START_REF] Aschbacher | Topics in non-equilibrium quantum statistical mechanics[END_REF]. This alternative approach is technically simpler, yields better constants, and can be also used to prove a Large Deviation Principle and to discuss additional topics like Landauer-Büttiker formula which cannot be handled by the method of [START_REF] Jakšić | The Green-Kubo formula for locally interacting fermionic open systems[END_REF] and this paper. For this reason, we shall discuss this special case separately in the companion paper [START_REF] Aschbacher | Fluctuations in quasi-free fermionic systems[END_REF].

As we have already remarked, our proof of Theorem 1.6 also yields the convergence of moments (see Theorem 3.2), and is easily extended to the proof of existence of QHL for locally interacting fermionic systems (recall (1.10), (1.11)).

We finish this subsection with some concrete models to which Theorem 1.6 applies. The models on graphs are the same as in [START_REF] Jakšić | The Green-Kubo formula for locally interacting fermionic open systems[END_REF]. Let G be the set of vertices of a connected graph of bounded degree, ∆G the discrete Laplacian acting on l 2 (G), and δx the Kronecker delta function at x ∈ G. We shall call a graph G admissible if there exists γ > 1 such that for all x, y ∈ G,

|(δx, e -it∆ G δy)| = O(|t| -γ ), (1.18) as t → ∞. Examples of admissible graphs are G = Z d for d ≥ 3, G = Z+ × Z d-1 where Z+ = {0, 1, • • • } and d ≥ 1,
tubular graphs of the type Z+ × Γ, where Γ ⊂ Z d-1 is finite, a rooted Bethe lattice, etc. Assumption (A) holds and Theorem 1.6 holds with c = d if:

(i) G is an admissible graph;

(ii) h = ℓ 2 (G) (or more generally ℓ 2 (G) ⊗ C L ) and h0 = -∆G;
(iii) d is the subspace of finitely supported elements of h;

(iv) T = F (h0) where F ∈ L 1 (R, dt) and 0 < F (x) < 1 for x ∈ sp(h0).
The continuous examples are similar. Let D ⊂ R d be a domain and let ∆D be the Dirichlet Laplacian on L 2 (D, dx). We shall say that a domain D is admissible if there exists γ > 1 such that

|(f, e -it∆ D g)| = O(|t| -γ ),
for all bounded f and g with compact support. Examples of admissible domains are (i) D is an admissible domain;

D = R d for d ≥ 3, D = R+ × R d-1 for d ≥ 1,
(ii) h = L 2 (D, dx) (or more generally L 2 (D, dx) ⊗ C L ) and h0 = -∆D;
(iii) d is the subspace of bounded compactly supported elements of h;

(iv) T = F (h0) where F ∈ L 1 (R, dt) and 0 < F (x) < 1 for x ∈ sp(h0).

QCLT and linear response

In addition to the assumptions of the previous subsection, we assume that h, h0, T have the composite structure

h = M M j=1 hj, h0 = M M j=1 hj, T = M M j=1 1 1 + e β j (h j -µ j ) , (1.19)
where hj's are bounded from below self-adjoint operators on the Hilbert subspaces hj, βj > 0, and µj ∈ R. We denote by pj the orthogonal projections onto hj. The subalgebras Oj = CAR(hj) describe Fermi gas reservoirs Rj which are initially in equilibrium at inverse temperatures βj and chemical potentials µj. The perturbation λV describes the interaction between the reservoirs and allows for the flow of heat and charges within the system.

The non-equilibrium statistical mechanics of this class of models has been studied recently in [START_REF] Jakšić | The Green-Kubo formula for locally interacting fermionic open systems[END_REF] (see also [FMU] for related models and results). We briefly recall the results we need.

Suppose that pjF (V ) ⊂ Dom (hj) for all j. The entropy production observable of (O, τ λ ) associated to the reference state ω0 is

σ λ ≡ - M X j=1 βj(Φj -µjJj),
where

Φj ≡ iλ[dΓ(hjpj), V ] and Jj ≡ iλ[dΓ(pj), V ]. Explicitly, Φj = λ K V X k=1 n k X l=1 l-1 Y i=1 a * (f ki )a(g ki ) ! {a * (ihjpjf kl )a(g kl ) + a * (f kl )a(ihjpjg kl )} n k Y i=l+1 a * (f ki )a(g ki ) ! , Jj = λ K V X k=1 n k X l=1 l-1 Y i=1 a * (f ki )a(g ki ) ! {a * (ipjf kl )a(g kl ) + a * (f kl )a(ipjg kl )} n k Y i=l+1 a * (f ki )a(g ki ) ! .
The observable Φj/Jj describes the heat/charge flux out of the reservoir Rj (note that Φj, Jj ∈ O). The conservation laws

M X j=1 ω + λ (Φj) = 0, M X j=1 ω + λ (Jj) = 0,
hold. By the general result of [JP1, Ru2, JP4], the entropy production of the NESS ω + λ is non-negative,

Ep(ω + λ ) ≡ ω + λ (σ λ ) = - M X j=1 βj(ω + λ (Φj) -µjω + λ (Jj)) ≥ 0.
If all βj's and µj's are equal, i.e.

β1 = • • • = βM = β and µ1 = • • • = µM = µ, then ω0 ↾ O is a (τ0, β)-KMS state
and so the reference state is a thermal equilibrium state of the unperturbed system. Then

ω + λ ↾ O is a (τ λ , β)-KMS state, ω + λ (Φj) = ω + λ (Jj)
= 0 for all j, and in particular Ep(ω + λ ) = 0, see [START_REF] Jakšić | Linear response theory for thermally driven quantum open systems[END_REF]. On physical grounds, vanishing of the fluxes and the entropy production in thermal equilibrium is certainly an expected result. It is also expected that if either βj's or µj's are not all equal, then Ep(ω + λ ) > 0. For specific interactions V one can compute ω + λ (σ λ ) to the first non-trivial order in λ and hence establish the strict positivity of entropy production by a perturbative calculation (see [START_REF] Fröhlich | Dissipative transport: contacts and tunneling junctions[END_REF][START_REF] Jakšić | [END_REF] and [START_REF] Jakšić | Mathematical theory of non-equilibrium quantum statistical mechanics[END_REF] for a related results). The strict positivity of the entropy production for a generic perturbation λV has been established in [START_REF] Jakšić | On the strict positivity of entropy production[END_REF].

To establish QCLT for the flux observables in addition to the Assumption (A) we need:

(B) For all j, hjpjd ⊂ d.
This assumption and the specific form of density operator ensure that one may take c = d and that if V ∈ C self , then {Φj, Jj} ⊂ C self . Hence, for |λ| ≤ λV the QCLT holds for the flux observables.

We finish with a discussion of linear response theory (for references and additional information about linear response theory in algebraic formalism of quantum statistical mechanics we refer the reader to [START_REF] Aschbacher | Topics in non-equilibrium quantum statistical mechanics[END_REF] and [JOP1]- [START_REF] Jakšić | The Green-Kubo formula for locally interacting fermionic open systems[END_REF]). We will need the following two assumptions:

(C) The operators hj are bounded.

(D) There exists a complex conjugation c on h which commutes with all hj and satisfies cf = f for all f ∈ F (V ).

Assumption (C) is of technical nature and can be relaxed. Assumption (D) ensures that the system (O, τ λ , ω0) is time-reversal invariant. Time-reversal invariance is of central importance in linear response theory.

Let βeq > 0 and µeq ∈ R be given equilibrium values of the inverse temperature and chemical potential. We denote

β = (β1, • • • , βM ), µ = (µ1, • • • , µM ), βeq = (βeq, • • • , βeq), µeq = (µeq, • • • , µeq),
(Φj) = ω + λ, βeq, µeq (Jj) = 0, L λ, βeq, µeq (A, B) = Z ∞ -∞ ω + λ, βeq, µeq `Aτ t λ (B) ´dt, for A, B ∈ {Φj, Jj | 1 ≤ j ≤ M }.
Assuming the existence of derivatives, the kinetic transport coefficients are defined by

L kj λhh ≡ -∂ β j ω + λ, β, µ (Φ k ) ˛ β= βeq, µ= µeq , L kj λhc ≡ βeq∂µ j ω + λ, β, µ (Φ k ) ˛ β= βeq, µ= µeq , L kj λch ≡ -∂ β j ω + λ, β, µ (J k ) ˛ β= βeq, µ= µeq , L kj λcc ≡ βeq∂µ j ω + λ, β, µ (J k ) ˛ β= βeq, µ= µeq , (1.20)
where the indices h/c stand for heat/charge. We then have Theorem 1.7 Suppose that Assumptions (A)-(D) hold. Then, for any |λ| < λV , the functions

( β, µ) → ω + λ, β, µ (Φj), ( β, µ) → ω + λ, β, µ (Jj),
are analytic in a neighborhood of ( βeq, µeq). Moreover, (1) The Green-Kubo formulas hold:

L kj λhh = 1 2 L λ, βeq, µeq (Φ k , Φj), L kj λhc = 1 2 L λ, βeq, µeq (Φ k , Jj), L kj λch = 1 2 L λ, βeq, µeq (J k , Φj), L kj λcc = 1 2 L λ, βeq, µeq (J k , Jj).
(1.21)

(2) The Onsager reciprocity relations hold:

L kj λhh = L jk λhh , L kj λcc = L jk λcc , L kj λhc = L jk λch .
(

V = V hop + V int where V hop = X x,y t(x, y) (a * (δx)a(δy) + a * (δy)a(δx)) ,
and t : G × G → R is a finitely supported function (G = ∪jGj), and

V int = X x,y v(x, y)a * (δx)a * (δy)a(δy)a(δx),
where v : G × G → R is finitely supported. V hop describes tunneling junctions between the reservoir and V int is a local pair interaction.

2 General aspects of CLT

Proof of Theorem 1.3

Our argument follows the ideas of [GV]. We set

D(A, B) ≡ e iA e iB -e i(A+B) e -1 2 [A,B] .
The first ingredient of the proof is:

Proposition 2.1 If {A, B} ⊂ O self is L 1 -
D(A + a, B + b) ≤ D(A, B) + 4 ` a 3 + b 3 ´+ [[A, B], [a, b]] + (2 + a + b ) X X∈{A,B} y∈{a,b} [X, y] .
Proof. We decompose D(A + a, B + b) = P 9 j=1 Dj according to the following table and get an upper bound of the norm of each term using the elementary estimates 5

' ' 'e i(x+y) -e ix ' ' ' ≤ y , ' ' 'e i(x+y) -e ix e iy ' ' ' ≤ 1 2 [x, y] , ' ' 'e ix e iy -e iy e ix ' ' ' ≤ [x, y] .
" e ia e ib -e i(a+b) e -1 2 [a,b] " e i(A+B) e -1 2 [A,B] D(a, b
)

6 e i(a+b) " e -1 2 [a,b] e i(A+B) -e i(A+B) e -1 2 [a,b] " e -1 2 [A,B] 1 2 [A + B, [a, b]] 7 e i(a+b) e i(A+B) " e -1 2 [a,b] e -1 2 [A,B] -e -1 2 [A,B]-1 2 [a,b] " 1 8 [[A, B], [a, b]] 8 e i(a+b) e i(A+B) " e -1 2 [A,B]-1 2 [a,b] -e -1 2 [A+a,B+b] " 1 2 ( [A, b] + [B, a] ) 9 " e i(a+b) e i(A+B) -e i(A+B+a+b) " e -1 2 [A+a,B+b] 1 2 ( [A, a] + [A, b] + [B, a] + [B, b] )
From the BCH estimate we further get

D5 ≤ D(a, b) ≤ [a, [a, b]] + [b, [a, b]] ≤ 4( a 3 + b 3 ),
and the Jacobi identity yields

D6 ≤ a ( [A, b] + [B, b] ) + b ( [A, a] + [B, a] ).
The result follows. 2

Proof of Proposition 2.1. For t > 0 and j ∈ N set p(t) ≡ log(1 + t) and Ij(t) ≡ [jp(t), (j + 1)p(t)[.

For X ∈ O self define X (j) t ≡ t -1/2 Z I j (t)∩[0,t] τ s (X) ds, X (<k) t ≡ X 0≤j<k X (j) t .
If N (t) denotes the integer such that N (t)p(t) ≤ t < (N (t) + 1)p(t) then repeated use of Lemma 2.2 yields

D( Ãt, Bt) ≤4 N (t) X j=0 " A (j) t 3 + B (j) t 3 " + N (t) X j=0 " 2 + A (j) t + B (j) t " X X,Y ∈{A,B} [X (<j) t , Y (j) t ]
+

N (t) X j=0 [[A (<j) t , B (<j) t ], [A (j) t , B (j)
t ]] .

(2.24)

We now estimate the right hand side of this inequality. We first note that

X (j) t ≤ X t -1/2 p(t) ≤ X , (2.25)
and hence

N (t) X j=0 " A (j) t 3 + B (j) t 3 " ≤ ` A 3 + B 3 ´(N (t) + 1)t -3/2 p(t) 3 ≤ 2 ` A 3 + B 3 ´t-1/2 p(t) 2 → 0, as t → ∞. Next consider N (t) X j=0 j-1 X k=0 [X (k) t , Y (j) t ] ≤ 1 t N (t) X j=0 Z jp(t) 0 Z (j+1)p(t) jp(t) [X, τ v-u (Y )] dvdu.
The change of variables ξ = vjp(t), η = vu, leads to

N (t) X j=0 j-1 X k=0 [X (k) t , Y (j) t ] ≤ 1 t N (t) X j=0 Z p(t) 0 Z ξ+jp(t) ξ [X, τ η (Y )] dηdξ ≤ N (t) + 1 t Z p(t) 0 Z ∞ ξ [X, τ η (Y )] dηdξ.
Since (N (t) + 1)/t ≤ 2/p(t) we obtain, for X, Y ∈ {A, B},

lim t→∞ N (t) X j=0 j-1 X k=0 [X (k) t , Y (j) t ] ≤ lim p→∞ 2 p Z p 0 »Z ∞ ξ [X, τ η (Y )] dη - dξ = 0. (2.26)
Combining this with (2.25) we get

N (t) X j=0 " 2 + A (j) t + B (j) t " X X,Y ∈{A,B} [X (<j) t , Y (j) t ] ≤ (2 + A + B ) X X,Y ∈{A,B} N (t) X j=0 j-1 X k=0 [X (k) t , Y (j)
t ] → 0, as t → ∞. To estimate the last term on the right hand side of (2.24) we write

N (t) X j=0 [[A (<j) t , B (<j) t ], [A (j) t , B (j) t ]] ≤ N (t) X j=0 j-1 X k=0 j-1 X l=0 [[A (k) t , B (l) t ], [A (j) t , B (j)
t ]] = Z1 + Z2 + Z3, where

Z1 ≡ N (t) X j=0 j-1 X k=0 [[A (k) t , B (k) t ], [A (j) t , B (j)
t ]] ,

and

Z2 ≡ N (t) X j=0 j-1 X k=0 k-1 X l=0 [[A (k) t , B (l) t ], [A (j) t , B (j)
t ]] , Z3 ≡

N (t) X j=0 j-1 X l=0 l-1 X k=0 [[A (k) t , B (l) t ], [A (j) t , B (j)
t ]] .

Combined with (2.25) and (2.26), the identity

[[A k , B k ], [Aj, Bj]] = [[[A k , Aj], B k ], Bj] + [[[Aj, B k ], A k ], Bj] + [[[B k , Bj], A k ], Aj] + [[[Bj, A k ], B k ], Aj], yields Z1 ≤ 4( A 2 + B 2 ) X X,Y ∈{A,B} N (t) X j=0 j-1 X k=0 [X (k) t , Y (j)
t ] → 0, as t → ∞. The estimate

N (t) X j=0 [A (j) t , B (j) t ] ≤ 1 t N (t) X j=0 Z I j (t) Z I j (t) [A, τ u-v (B)] dudv ≤ 1 t N (t) X j=0 Z p(t) 0 Z p(t)-v -v [A, τ u (B)] dudv ≤ N (t) + 1 t p(t) Z p(t) -p(t) [A, τ u (B)] du ≤ 2 Z R [A, τ u (B)] du, together with (2.26) yield Z2 ≤ 2 N (t) X j=0 [A (j) t , B (j)
t ]

N (t) X k=0 k-1 X l=0 [A (k) t , B (l)
t ] → 0, as t → ∞. The same argument applies to Z3 and completes the proof.2

Let (Hω, πω, Ωω) be the GNS-representation of the algebra O associated to the state ω. The second ingredient of the proof of Theorem 1.3 is:

Proposition 2.3 Suppose that (O, τ, ω
) is an ergodic quantum dynamical system and that ω is a modular state. If {A, B} is an L 1 -asymptotically abelian pair for τ , then

s-lim t→∞ πω " [ Ãt, Bt] " = Z ∞ -∞ ω([τ s (A), B]) ds.
Proof. We shall first prove that

lim t→∞ πω " [ Ãt, Bt] " Ωω = "Z ∞ -∞ ω([τ s (A), B]) ds « Ωω. (2.27) Writing πω " [ Ãt, Bt] " = 1 t Z t 0 Z t 0 πω `τ s 1 ([τ s 2 -s 1 (A), B]) ´ds1ds2, the change of variable u = s1, v = s2 -s1 yields that πω " [ Ãt, Bt] " = Z t -t ft(v) dv, (2.28)
where

ft(v) = 1 t Z min(t,t-v) max(-v,0) πω (τ u ([τ v (A), B])) du. Clearly, ft(v) ≤ [τ v (A), B]) ∈ L 1 (R, dv), (2.29)
and so, by the dominated convergence theorem, it suffices to show that

lim t→∞ ft(v)Ωω = ω ([τ v (A), B]) Ωω, (2.30)
for all v ∈ R to prove (2.27). Let Lω be the standard Liouvillean associated to ω. We recall that Lω is the unique self-adjoint operator on Hω such that πω(τ t (A)) = e itLω πω(A)e -itLω , LωΩω = 0.

Then πω(τ u ([τ v (A), B]))Ωω = e iuLω πω([τ v (A), B])Ωω, implies ft(v)Ωω = 1 t Z min(t,t-v) max(-v,0) e iuLω πω ([τ v (A), B]) Ωω du.
Since (O, τ, ω) is ergodic, zero is a simple eigenvalue of Lω, and von Neumann's mean ergodic theorem yields

s-lim t→∞ 1 t Z min(t,t-v) max(-v,0)
e iuLω du = s-lim

t→∞ 1 t Z t 0 e iuLω du = Ωω(Ωω| • ),
for all v ∈ R. This implies (2.30) and (2.27) follows.

To finish the proof note that for any

X ∈ πω(O) ′ one has πω " [ Ãt, Bt] " XΩω = Xπω " [ Ãt, Bt] " Ωω,
and so for all

Ψ ∈ πω(O) ′ Ωω lim t→∞ πω " [ Ãt, Bt] " Ψ = "Z ∞ -∞ ω([τ s (A), B]) ds « Ψ.
(2.31)

Since ω is modular πω(O) ′ Ωω is dense in Hω and it follows from the estimate

sup t>0 [ Ãt, Bt] ≤ Z R [τ s (A), B] ds < ∞,
that (2.31) extends to all Ψ ∈ Hω. 2

We are now ready to complete:

Proof of Theorem 1.3. Let {A1, • • • , An} ∈ C self . For j = 1, . . . , n -1, we set Ujt = exp " - 1 2 [ Ãjt, Ã(j+1)t + • • • + Ãnt] « ,
and

Ut = U1t • • • U (n-1)t .
Clearly, the Ujt's are unitary and repeated use of Proposition 2.1 yields that

lim t→∞ ' ' 'e i Ã1t • • • e i Ãnt -e i( Ã1t +•••+ Ãnt ) Ut ' ' ' = 0,
and hence, lim

t→∞ ˛ω " e i Ã1t • • • e i Ãnt " -ω " e i( Ã1t +•••+ Ãnt ) Ut "˛= 0. (2.32) Proposition 2.3 implies that s-lim t→∞ πω(Ujt) = exp 0 @ -i n X k=j+1 ς(Aj, A k ) 1 A ,
and so

s-lim t→∞ πω(Ut) = exp 0 @ -i X 1≤j<k≤n ς(Aj, A k ) 1 A .
(2.33)

Since SQCLT holds, Relations (2.32) and (2.33) yield

lim t→∞ ω " e i Ã1t • • • e i Ãnt " = lim t→∞ ω " e i( Ã1t +•••+ Ãnt ) Ut " = lim t→∞ ω " e i( Ã1t +•••+ Ãnt ) " exp 0 @ -i X 1≤j<k≤n ς(Aj, A k ) 1 A = exp - 1 2 L n X k=1 A k , n X j=1 Aj !! exp 0 @ -i X 1≤j<k≤n ς(Aj, A k ) 1 A ,
and the theorem follows. 2

Proof of Theorem 1.4

Let f1, . . . , fn be as in the theorem and set M = sup 1≤j≤n,x∈R |fj(x)|. Then, for each j = 1, . . . , n, there exists a sequence (g jk ) k∈N of compactly supported continuous functions such that

sup k∈N,x∈R |g jk (x)| ≤ M and lim k→∞ g jk (x) = fj(x), (2.34)
for all x ∈ R. For η > 0 set

g jkη (x) = Z R e -(x-y) 2 /η g jk (y) dy √ πη .
ĝjkη denotes the Fourier transform of g jkη . Note that ĝjkη ∈ L 1 (R) and

Lemma 2.4

lim

k 1 →∞ lim η 1 →0 • • • lim kn→∞ lim ηn→0 lim t→∞ ω(g 1k 1 η 1 ( Ã1t) • • • g nknηn ( Ãnt)) = ωL(f1(ϕL(A1)) • • • fn(ϕL(An))). Proof. Write ω(g 1k 1 η 1 ( Ã1t) • • • g nknηn ( Ãnt)) = Z R n ĝ1k 1 η 1 (ξ1) • • • ĝnknηn (ξn)ω " e -iξ 1 Ã1t • • • e -iξn Ãnt " dξ1 • • • dξn (2π) n .
The QCLT implies that

lim t→∞ ω " e -iξ 1 Ã1t • • • e -iξn Ãnt " = ωL(W (-ξ1A1) • • • W (-ξnAn)),
and the dominated convergence theorem yields

lim t→∞ ω(g 1k 1 η 1 ( Ã1t) • • • g nknηn ( Ãnt)) = Z R n ĝ1k 1 η 1 (ξ1) • • • ĝnknηn (ξn)ωL(W (-ξ1A1) • • • W (-ξnAn)) dξ1 • • • dξn (2π) n = ωL(g 1k 1 η 1 (ϕL(A1)) • • • g nknηn (ϕL(An))).
By the functional calculus for self-adjoint operators (2.34) and (2.35) imply

lim k 1 →∞ lim η 1 →0 • • • lim kn→∞ lim ηn→0 ωL(g 1k 1 η 1 (ϕL(A1)) • • • g nknηn (ϕL(An))) = ωL(f1(ϕL(A1)) • • • fn(ϕL(An))),
and the statement follows. 2 Lemma 2.5 For any given ξ1, . . . , ξj-1 ∈ R,

lim k→∞ lim η→0 lim t→∞ ω " e -iξ 1 Ã1t • • • e -iξ j-1 Ã(j-1)t ˛fj( Ãjt) -g jkη ( Ãjt) ˛2 e iξ j-1 Ã(j-1)t • • • e iξ 1 Ã1t « = 0.
Proof. Let µt be the spectral measure for πω( Ãjt) and the unit vector πω(e iξ j-1 Ã(j-1)t • • • e iξ 1 Ã1t )Ωω. Then

ω " e -iξ 1 Ã1t • • • e -iξ j-1 Ã(j-1)t ˛fj( Ãjt) -g jkη ( Ãjt) ˛2 e iξ j-1 Ã(j-1)t • • • e iξ 1 Ã1t « = Z R |fj(x) -g jkη (x)| 2 dµt(x).
The QCLT yields that for any α ∈ R,

lim t→∞ Z R e iαx dµt(x) = lim t→∞ ω " e -iξ 1 Ã1t • • • e -iξ j-1 Ã(j-1)t e iα Ãjt e iξ j-1 Ã(j-1)t • • • e iξ 1 Ã1t " = exp 0 @ - α 2 2 L(Aj, Aj) -2iας 0 @ Aj, X 1≤k≤j-1 ξ k A k 1 A 1 A = Z R e iαx dµ(x),
where µ is a Gaussian measure with variance L(Aj, Aj) and expectation -2ς

" Aj, P j-1 k=1 ξ k A k "
(note that if L(Aj, Aj) = 0 then µ is the δ-measure at 0). This fact and the Lévy-Cramér Continuity Theorem yield that

lim t→∞ Z R |fj(x) -g jkη (x)| 2 dµt(x) = Z R |fj(x) -g jkη (x)| 2 dµ(x),
and the statement follows from (2.34), (2.35) and the dominated convergence theorem. 2

We are now ready to complete:

Proof of Theorem 1.4. Write ω(f1( Ã1t) • • • fn( Ãnt)) -ω(g 1k 1 η 1 ( Ã1t) • • • g nknηn ( Ãnt)) = n X j=1 Tj,
where

Tj = ω " g 1k 1 η 1 ( Ã1t) • • • g (j-1)k j-1 η j-1 ( Ã(j-1)t ) h fj( Ãjt) -g jk j η j ( Ãjt) i fj+1( Ã(j+1)t ) • • • fn( Ãnt) " = Z R j-1 ĝ1k 1 η 1 (ξ1) • • • ĝ(j-1)k j-1 η j-1 (ξj-1)Dj(ξ1, . . . , ξj-1) dξ1 • • • dξj-1 (2π) j-1 , with Dj(ξ1, . . . , ξj-1) = ω " e -iξ 1 Ã1t • • • e -iξ j-1 Ã(j-1)t h fj( Ãjt) -g jk j η j ( Ãjt) i fj+1( Ã(j+1)t ) • • • fn( Ãnt)
" .

The Cauchy-Schwartz inequality yields that |Dj(ξ1, . . . , ξj-1)| ≤ M n-j

˛ω " e -iξ 1 Ã1t • • • e -iξ j-1 Ã(j-1)t ˛fj( Ãjt)g jk j η j ( Ãjt) ˛2 e iξ j-1 Ã(j-1)t • • • e iξ 1 Ã1t «˛1 /2 , and so Lemma 2.5 and the dominated convergence theorem gives lim sup

k 1 →∞ lim sup η 1 →0 • • • lim sup k j →∞ lim sup η j →0
lim sup t→∞ |Tj| = 0, for all j. Hence, lim sup

k 1 →∞ lim sup η 1 →0 • • • lim sup kn→∞ lim sup ηn→0 lim sup t→∞ |ω(f1( Ã1t) • • • fn( Ãnt)) -ω(g 1k 1 η 1 ( Ã1t) • • • g nknηn ( Ãnt))| = 0,
and the statement follows from Lemma 2.4. 2

Norm localization

For ǫ > 0 we denote Dǫ = {z ∈ C | |z| < ǫ}.

Proposition 2.6 Let

A ∈ O self be such that Z ∞ -∞ |ω(Aτ t (A)) -ω(A) 2 | dt < ∞.
Suppose that there exists ǫ > 0 such that lim t→∞ ω(e iα Ãt ) = e -L(A,A)α 2 /2 , (2.36)

for α ∈ Dǫ. Then (2.36) holds for all α ∈ R.

Proof. This follows from well-known results in classical probability. See section 30 in [Bil]. 2

Locally interacting fermions

In this section we describe the strategy of the proof of our main result, Theorem 1.6, and establish a number of preliminary results needed for the proof. In particular, we shall reduce the proof of Theorem 1.6 to the proof of Theorem 3.5 (stated in Subsection 3.3 and proven in Section 4). Theorem 3.5, which is the main technical result of our paper, concerns only the unperturbed system (O, τ0, ω0).

Strategy

Suppose that the assumptions of Theorem 1.6 hold and let

A = K A X k=1 n k Y j=1 a * (f kj )a(g kj ),
be an element of C. Clearly,

ω + λ " ( Ãt) n " = t -n/2 Z [0,t] n ω + λ n Y j=1 " τ t j λ (A) -ω + λ (A) " ! dt1 • • • dtn.
The first ingredient of the proof of Theorem 1.6 is:

Theorem 3.1 There exists a finite constant CV,A such that for all n,

sup |λ|≤ λV ,t>0 t -n/2 Z [0,t] n ˛ω+ λ n Y j=1 " τ t j λ (A) -ω + λ (A) " !˛d t1 • • • dtn ≤ C n V,A n!.
(3.37)

Remark 1. Our proof also gives an explicit estimate on the constant CV,A, see Formula (3.51) below. Remark 2. In the special case n = 2, Theorem 3.1 yields that for all t > 0 and |λ| ≤ λV ,

Z t -t " 1 - |s| t « ˛ω+ λ `(τ s λ (A) -ω + λ (A))(A -ω + λ (A)) ´˛d s ≤ 2C 2 V,A . As t → ∞ the monotone convergence theorem yields Z ∞ -∞ ˛ω+ λ `(τ s λ (A) -ω + λ (A))(A -ω + λ (A)) ´˛d s ≤ 2C 2 V,A .
In particular, we derive that C is CLT-admissible.

The second ingredient of the proof of Theorem 1.6 is:

Theorem 3.2 For |λ| ≤ λV and all n ≥ 1,

lim t→∞ ω + λ " ( Ãt) n " = 8 > < > : n! 2 n/2 (n/2)! L(A, A) n/2 if n is even, 0 if n is odd.
Remark. With only notational changes the proof of Theorem 3.2 yields that for all A1, • • • , An ∈ C,

lim t→∞ ω + λ " Ã1t • • • Ãnt " = ωL (ϕL(A1) • • • ϕL(An)) ,
where the r.h.s. is defined by (1.6).

Given Theorems 3.1 and 3.2, we can complete:

Proof of Theorem 1.6. Let A ∈ C self . For α ∈ C one has Proposition 2.6 yields that (3.39) holds for all α ∈ R, and so SQCLT holds for C w.r.t. (O, τ λ , ω + λ ). Our standing assumption Ker (T ) = Ker (I -T ) = {0} ensures that the state ω0 is modular, and since ω + λ = ω0 • γ + λ , the state ω + λ is also modular. By Theorem 1.5, if |λ| ≤ λV , then C is L 1 -asymptotically Abelian for τ λ and it follows from Theorem 1.3 that the QCLT also holds.2 Notice that in the initial step of the proof we did not use the assumption that A is self-adjoint, and so the following weak form of QCLT holds for any A ∈ C: Corollary 3.3 For any A ∈ C there exists ǫ > 0 such that for |λ| ≤ λV and |α| < ǫ,

ω + λ " e iα Ãt " = X n≥0 (iα) n n! ω + λ " ( Ãt)
lim t→∞ ω + λ " e iα Ãt " = e -L(A,A)α 2 /2 .
In the rest of this section we shall describe the strategy of the proof of Theorems 3.1 and 3.2.

The commutator estimate

We shall need the following result Theorem 3.4 Suppose that Assumption (A) holds. Let V ∈ O(d) self be a perturbation such that nV ≥ 2 and

max f ∈F (V ) f = 1. Let A = a # (f1) • • • a # (fm) be a monomial such that F (A) = {f1, • • • , fm} ⊂ d, and let C (n) A (s1, . . . , sn) = [τ sn 0 (V ), [• • • , [τ s 1 0 (V ), A] • • • ]].
Then for all n ≥ 0 there exist a finite index set Qn(A), monomials F (n)

A,q ∈ O, and scalar functions G (n)

A,q such that

C (n) A (s1, . . . , sn) = X q∈Qn(A) G (n)
A,q (s1, . . . , sn)F

(n)

A,q (s1, . . . , sn).

(3.40)

Moreover,

The order of the monomial F (n)

A,q does not exceed 2n(nV -1) + m.

If m is even then the order of F (n)

A,q is also even.

The factors of

F (n) A,q are from n a # (e ish 0 g) ˛g ∈ F (V ), s ∈ {s1, . . . , sn} o ∪ n a # (g) ˛g ∈ F (A) o ,
The number of factors from the first set does not exceed n(2nV -1) while the number of factors from the second set does not exceed m -1. In particular, F

A,q ≤ max(1, max f ∈F (A) f m-1 ).

Let λV be given by (1.15). Then

WV,A ≡ ∞ X n=1 |λV | n X q∈Qn(A) Z -∞<sn≤•••≤s 1 ≤0 ˛G(n) A,q (s1, . . . , sn) ˛ds1 • • • dsn < ∞.
(3.41)

The proof of Theorem 3.4 is identical to the proof of Theorem 1.1 in [START_REF] Jakšić | The Green-Kubo formula for locally interacting fermionic open systems[END_REF]. Parts 1-3 are simple and are stated for reference purposes. The Part 4 is a relatively straightforward consequence of the fundamental Botvich-Guta-Maassen integral estimate [BGM] which also gives an explicit estimate on WV,A. A pedagogical exposition of the Botvich-Guta-Maassen estimate can be found in [START_REF] Jakšić | [END_REF].

If A is as in Theorem 3.4 then

γ + λ (A) = lim t→∞ τ -t 0 • τ t λ (A),
can be expanded in a power series in λ which converges for |λ| ≤ λV . Indeed, it follows from the Araki-Dyson expansion that

τ -t 0 • τ t λ (A) = A + ∞ X n=1 (iλ) n Z -t≤sn≤•••≤s 1 ≤0 [τ sn 0 (V ), [• • • , [τ s 1 0 (V ), A] • • • ]] ds1 • • • dsn.
Hence, for |λ| ≤ λV ,

γ + λ (A) = A + ∞ X n=1 (iλ) n X q∈Qn(A) Z -∞<sn≤•••≤s 1 ≤0 G (n) A,q (s1, . . . , sn)F (n) A,q (s1, . . . , sn) ds1 • • • dsn, (3.42)
where the series on the right-hand side is norm convergent by Parts 3 and 4 of Theorem 3.4. This expansion will be used in the proof of Theorems 3.1 and 3.2.

Quasi-free correlations

Let O, τ0 and ω0 be as in Subsection 1.2. We denote by

ϕ(f ) = 1 √ 2 (a(f ) + a * (f )) ,
the Fermi field operator associated to f ∈ h. The Fermi field operators satisfy the commutation relation

ϕ(f )ϕ(g) + ϕ(g)ϕ(f ) = Re(f, g)1l,
and the CAR algebra O is generated by {ϕ(f

) | f ∈ h}. Clearly, a(f ) = 1 √ 2 (ϕ(f ) + iϕ(if )) , a * (f ) = 1 √ 2 (ϕ(f ) -iϕ(if )) . (3.43)
We recall that ω0, the gauge invariant quasi-free state associated to the density operator T , is uniquely specified by

ω0(a * (fn) • • • a * (f1)a(g1) • • • a(gm)) = δn,mdet{(gi, T fj)}.
Alternatively, ω0 can be described by its action on the Fermi field operators. Let Pn be the set of all permutations π of {1, . . . , 2n} in Subsection 1.1 (recall (1.5)). Denote by ǫ(π) the signature of π ∈ Pn. ω0 is the unique state on O such that ω0(ϕ(f1)ϕ(f2

)) = 1 2 (f1, f2) -i Im(f1, T f2), and
ω0(ϕ(f1) • • • ϕ(fn)) = 8 > > > > < > > > > : X π∈P n/2 ǫ(π) n/2 Y j=1 ωT `ϕ(f π(2j-1) ), ϕ(f π(2j) ) ´if n is even; 0 if n is odd.
For any bounded subset f ⊂ h we set

M f = sup f ∈f f , and
C f = max " 1, sup f,g∈f 2 f g Z ∞ -∞ ˛ω0 `ϕ(f )τ t 0 (ϕ(g)) ´˛d t « ,
and we denote by M(f) the set of monomials with factors from {ϕ(f )|f ∈ f}. We further say that A ∈ M(f) is of degree at most k if, for some f1, . . . ,

f k ∈ f, one can write A = ϕ(f1) • • • ϕ(f k ).
Theorem 3.5 Suppose that C f < ∞. Then for any A1, . . . , An ∈ M(f) of degrees at most k1, . . . , kn the following holds:

1.

sup t>0 t -n/2 Z [0,t] n ˛ω0 n Y i=1 `τ t i 0 (Ai) -ω0(Ai) ´!˛d t1 • • • dtn ≤ " 2 7/2 M f " P i k i C n f n!.

If n is odd,

lim t→∞ t -n/2 Z [0,t] n ω0 n Y i=1 `τ t i 0 (Ai) -ω0(Ai) ´! dt1 • • • dtn = 0.

If n is even,

lim t→∞ t -n/2 Z [0,t] n ω0 n Y i=1 `τ t i 0 (Ai) -ω0(Ai) ´! dt1 • • • dtn = X π∈P n/2 n/2 Y j=1 L0(A π(2j-1) , A π(2j) ), where L0(Ai, Aj) = Z ∞ -∞ ω0 `(τ t 0 (Ai) -ω0(Ai))(Aj -ω0(Aj)) ´dt. (3.44)
Remark. As in Remark 2 after Theorem 3.1, Part 1 of the previous theorem with

n = 2 implies that Z ∞ -∞ ˛ω0 `(τ t 0 (Ai) -ω0(Ai))(Aj -ω0(Aj)) ´˛d t < ∞,
and so L0(Ai, Aj) is well defined.

Theorem 3.5 is in essence the main technical result of our paper. Its proof is given in Section 4.

We have formulated Theorem 3.5 in terms of field operators since that allows for a combinatorially natural approach to its proof. Using the identities (3.43) one effortlessly gets the following reformulation which is more convenient for our application.

Denote by M(f) the set of monomials with factors from {a

# (f )|f ∈ f}. A ∈ M(f) is of degree at most k if, for some f1, . . . , f k ∈ f, one can write A = a # (f1) • • • a # (f k ). Let D f = max " 1, sup f,g∈f∪if 2 f g Z ∞ -∞ ˛ω0 `ϕ(f )τ t 0 (ϕ(g)) ´˛d t « ,
Corollary 3.6 Suppose that D f < ∞. Then for any A1, . . . , An ∈ M(f) of degrees at most k1, . . . , kn the following holds:

1.

sup t>0 t -n/2 Z [0,t] n ˛ω0 n Y i=1 `τ t i 0 (Ai) -ω0(Ai) ´!˛d t1 • • • dtn ≤ `24 M f ´Pi k i D n f n!.

If n is odd,

lim t→∞ t -n/2 Z [0,t] n ω0 n Y i=1 `τ t i 0 (Ai) -ω0(Ai) ´! dt1 • • • dtn = 0.
3. If n is even,

lim t→∞ t -n/2 Z [0,t] n ω0 n Y i=1 `τ t i 0 (Ai) -ω0(Ai) ´! dt1 • • • dtn = X π∈P n/2 n/2 Y j=1 L0(A π(2j-1) , A π(2j) ),
where L0(Ai, A k ) is defined by (3.44).

Note that if c is as in Subsection 1.2 and f is a finite subset of c, then C f < ∞ and D f < ∞.

After this paper was completed we have learned of a beautiful paper [START_REF] Dereziński | Boson free fields as a limit of fields of a more general type[END_REF] which is perhaps deepest among early works on quantum central limit theorems (Dereziński's work was motivated by [Ha1, Ha2, Ru1, HL1, HL2, HL3, Da2]). In relation to our work, in [De1] Theorem 3.5 was proven in the special case k1 = • • • = kn = 2 of quadratic interactions. This suffices for the proof of SQCLT for quasi-free dynamics and for observables which are polynomials in Fermi fields. The proofs of Parts (2) and (3) of Theorem 3.5 are not that much different in the general case kj ≥ 2. The key difference is in Part (1) which in the quadratic case follows easily from Stirling's formula. To prove Part (1) for any kj ≥ 2 is much more difficult and the bulk of the proof of Theorem 3.5 in Section 4 is devoted to this estimate. The proof of QCLT for locally interacting fermionic systems critically depends on this result.

Proofs of Theorems 3.1 and 3.2

In this subsection we shall show that Theorems 3.4 and 3.5 imply Theorems 3.1 and 3.2, thereby reducing the proof of Theorem 1.6 to the proof of Theorem 3.5.

If η is a state, we shall denote

ηT(A1, . . . , An) ≡ η n Y i=1 (Ai -η(Ai)) ! . (3.45) Let A = K A X k=1 A k , A k = n k Y j=1 a * (f kj )a(g kj ),
be an element of C. Without loss of generality we may assume that

max f ∈F (A) f = 1. With f = n e ish 0 f ˛f ∈ F (V ) ∪ F (A), s ∈ R o , DV,A = max " 1, max f,g∈F (V )∪F (A) 1 f g Z ∞ -∞ " 2 -1 |(f, e ith 0 g)| + |(f, e ith 0 T g)| " dt « ,
we clearly have M f = 1 and D f ≤ DV,A.

Proof of Theorem 3.1. For |λ| ≤ λV ,

ω + λT `τ t 1 λ (A), . . . , τ tn λ (A) ´= K A X k 1 ,...,kn=1 ω0T `τ t 1 0 • γ + λ (A k 1 ), . . . , τ tn 0 • γ + λ (A kn ) ´, (3.46)
and the expansion (3.42) yields that

τ t 0 • γ + λ (A k ) -ω0 • γ + λ (A k ) = X j≥0 (iλ) j X q∈Q j (A k ) Z ∆ j G (j)
A k ,q (s)

" τ t 0 " F (j)
A k ,q (s)

" -ω0 " F (j)
A k ,q (s)

"" ds, (3.47)
where ∆j denotes the simplex {s = (s1, . . . , sj

) ∈ R j | -∞ < sj < • • • < s1 < 0}.
We have adopted the convention that

Q0(A k ) is a singleton, that G (0)
A k ,q = 1 and that F

A k ,q = A k . Moreover, integration over the empty simplex ∆0 is interpreted as the identity map. Applying Fubini's theorem we get

t -n/2 Z [0,t] n ω0T `τ t 1 0 • γ + λ (A k 1 ), . . . , τ tn 0 • γ + λ (A kn ) ´dt1 • • • dtn = X j 1 ,...,jn≥0 (iλ)
j 1 +•••+jn X q 1 ∈Q j 1 (A k 1 ),...,qn∈Q jn (A kn ) Z ∆ j 1 ds1 • • • Z ∆ jn dsn n Y l=1 G (j l ) A k l q l (s l ) ! Ct(j, q, s; A k 1 , . . . , A kn ), (3.48)
where we have set

Ct(j, q, s; A k 1 , . . . , A kn ) = t -n/2 Z [0,t] n ω0T " τ t 1 0 " F (j 1 ) A k 1 q 1 (s1) " , . . . , τ tn 0 " F (jn)
A kn qn (sn)

"" dt1 • • • dtn.
We derive from Corollary 3.6 and Theorem 3.4 that

|Ct(j, q, s; A k 1 , . . . , A kn )| ≤ 2 8(n V -1) P n l=1 j l " 2 8n A D f " n n!, (3.49)
holds for t > 0. Using this bound we further get from (3.48)

sup t>0 t -n/2 Z [0,t] n ˛ω0T `τ t 1 0 • γ + λ (A k 1 ), . . . , τ tn 0 • γ + λ (A kn ) ´˛d t1 • • • dtn ≤ n Y l=1 0 B @2 8n A D f X j l ≥0 |2 8(n V -1) λ| j l X q l ∈Q j l (A k l ) Z ∆ l ˛G(j l ) A k l q l (s l ) ˛ds l 1 C A n!. (3.50)
For |λ| ≤ λV we have (recall Definitions (1.17) and (3.41)),

X j l ≥0 |2 8(n V -1) λ| j l X q l ∈Q j l (A k l ) Z ∆ l ˛G(j l ) A k l q l (s l ) ˛ds l ≤ 1 + WV,A k l .
By Theorem 3.4, the right hand side of this inequality is finite. Combining this bound with (3.46) and (3.50) we finally obtain

sup |λ|< λV ,t>0 t -n/2 Z [0,t] n ˛ω+ λT `τ t 1 λ (A), . . . , τ tn λ (A) ´˛d t1 • • • dtn ≤ 2 8n A D f K A X k=1 (1 + WV,A k ) !n n!,
which concludes the proof.2

The above proof gives that in Theorem 3.1 one may take

CV,A = 2 8n A DV,A K A X j=1 (1 + WV,A k ) . (3.51)
For an explicit estimate on WV,A k we refer the reader to [START_REF] Jakšić | The Green-Kubo formula for locally interacting fermionic open systems[END_REF].

Proof of Theorem 3.2. Note that

ω + λ " ( Ãt) n " = K A X k 1 ,...,kn=1 t -n/2 Z [0,t] n ω0T `τ t 1 0 • γ + λ (A k 1 ), . . . , τ tn 0 • γ + λ (A kn ) ´dt1 • • • dtn. (3.52)
In the proof of Theorem 3.1 we have established that the power series (3.48) converges uniformly for |λ| ≤ λV and t > 0.

Suppose first that n is odd. Corollary 3.6 yields that lim t→∞ Ct(j, q, s; A k 1 , . . . , A kn ) = 0.

(3.53) By (3.49) and Part 3 of Theorem 3.4 we can apply the dominated convergence theorem to the s-integration in (3.48) to conclude that each term of this power series vanishes as t → ∞, and so

lim t→∞ ω + λ " ( Ãt) n
" = 0, for |λ| ≤ λV .
If n is even, Corollary 3.6 yields

lim t→∞ Ct(j, q, s; A k 1 , . . . , A kn ) = X π∈P n/2 n/2 Y i=1 L0 " F (j π(2i-1) ) A k π(2i-1)
,q π(2i-1) (s π(2i-1) ), F

(j π(2i) ) A k π(2i) ,q π(2i) (s π(2i) ) « = X π∈P n/2 Z R n/2 n/2 Y i=1 ω0T " τ t i 0 " F (j π(2i-1) ) A k π(2i-1) ,q π(2i-1) (s π(2i-1) ) « , F (j π(2i) ) A k π(2i) ,q π(2i) (s π(2i) ) « dt1 • • • dt n/2 .
The estimate (3.49) (applied in the case n = 2) yields that

Z R ˛ω0T " τ t 0 " F (j)
A k ,q (s)

" , F (j ′ ) A k ′ ,q ′ (s ′ ) "˛d t ≤ " 2 8n A +1/2 D f " 2 2 8(n V -1)(j+j ′ ) ,
from which we obtain ˛lim t→∞ Ct(j, q, s; A k 1 , . . . , A kn )

˛≤ "

2 8n A +1/2 D f " n 2 8(n V -1) P i j i .
Arguing as in the previous case we get, for |λ| ≤ λV , the expansion

lim t→∞ t -n/2 Z [0,t] n ω0T `τ t 1 0 • γ + λ (A k 1 ), . . . , τ tn 0 • γ + λ (A kn ) ´dt1 • • • dt n/2 = X j 1 ,...,jn≥0 (iλ) j 1 +•••+jn X q 1 ∈Q j 1 (A k 1 ),••• ,qn∈Q jn (A kn ) Z ∆ j 1 ds1 • • • Z ∆ jn dsn n Y l=1 G (j l ) A k l ,q l (s l ) ! (3.54) X π∈P n/2 Z R n/2 n/2 Y i=1 ω0T " τ t i 0 " F (j π(2i-1) ) A k π(2i-1) ,q π(2i-1) (s π(2i-1) ) « , F (j π(2i) ) A k π(2i) ,q π(2i) (s π(2i) ) « dt1 • • • dt n/2 .
By Fubini's theorem, this can be rewritten as

X π∈P n/2 Z R n/2 2 6 4 X j 1 ,...,jn≥0 (iλ) j 1 +•••+jn X q 1 ∈Q j 1 (A k 1 ),••• ,qn∈Q jn (A kn ) Z ∆ j 1 ds1 • • • Z ∆ jn dsn n Y l=1 G (j l ) A k l ,q l (s l ) ! n/2 Y i=1 ω0T " τ t i 0 " F (j π(2i-1) ) A k π(2i-1)
,q π(2i-1) (s π(2i-1) )

« , F (j π(2i) ) A k π(2i) ,q π(2i) (s π(2i) ) « 3 5 dt1 • • • dt n/2 .
By Expansion (3.42), the expression inside the square brackets is

n/2 Y i=1 ω0T " τ t i 0 • γ + λ " A k π(2i-1) " , γ + λ (A k π(2i) ) " = n/2 Y i=1 ω + λT " τ t i " A k π(2i-1) " , A k π(2i)
" , so that, by (3.52),

lim t→∞ ω + λ " ( Ãt) n " = K A X k 1 ,...,kn=1 X π∈P n/2 n/2 Y i=1 "Z R ω + λT " τ t " A k π(2i-1) " , A k π(2i) " dt « = X π∈P n/2 n/2 Y i=1 "Z R ω + λT `τ t (A) , A ´dt « = n! 2 n/2 (n/2)! L(A, A) n/2 .
2 4 Proof of Theorem 3.5

For notational simplicity throughout this section we shall drop the subscript 0 and write h for h0, τ for τ0, ω for ω0. We shall also use the shorthand (3.45).

Graphs, pairings and Pfaffians

An graph is a pair of sets g = (V, E) where E is a set of 2-elements subsets of V . The elements of V are called points or vertices of g, those of E are its edges. Abusing notation, we shall write v ∈ g for vertices of g and e ∈ g for its edges. If v ∈ e ∈ g we say that the edge e is incident to the vertex v. If the edge e is incident to the vertices u and v we write e = uv and say that the edge e connects u to v. degree of a vertex v ∈ g is the number of distinct edges e ∈ g incident to v. A graph is k-regular if all its vertices share the same degree k. A vertex v ∈ g of degree said to be isolated. A path on g is a sequence (v0, e1, v1, e2, . . . , en, vn) where vi ∈ V , ei ∈ E and ei = vi-1vi. We say that such a path connects the vertices v0 and vn. If v0 = vn the path is closed and is called a loop. The graph g connected if, given any pair v, v ′ ∈ V there is a path on g which connects v and v ′ . A connected graph without loops is a tree.

A graph g ′ = (V ′ , E ′ ) is a subgraph of the graph g = (V, E) if V ′ ⊂ V and E ′ ⊂ E. A subgraph g ′ of g is said to be spanning g if V ′ = V .
A connected graph g has a spanning tree i.e., a subgraph which is spanning and is a tree.

Let g = (V, E) be a graph. To a subset W ⊂ V associate a subgraph g |W = (W, E |W ) of g by setting

E |W = {e = uv ∈ E | u, v ∈ W }.
Given two graphs g1 = (V1, E1) and g2 = (V2, E2) such that V1 and V2 are disjoint we denote by g1 ∨ g2 the joint graph (V1 ∪ V2, E1 ∪ E2).

Let g = (V, E) be a graph and Π = {V1, . . . , Vn} a partition of V . The set

E/Π = {ViVj | there are u ∈ Vi, v ∈ Vj such that uv ∈ E}. v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 ε(p) = (-1) 2 = +1
p = 1 2 3 4 5 6 7 8 1 4 2 6 3 5 7 8

Figure 1: Diagrammatic representation of a pairing p defines a graph g/Π = (Π, E/Π). We say that g/Π is the Π-skeleton of g.

A graph g = (V, E) is said to be (V1, V2)-bipartite if there is a partition V = V1 ∪ V2 such that all edges e ∈ E connect a vertex of V1 to a vertex of V2.

A pairing on a set V is a graph p = (V, E) such that every vertex v ∈ V belongs to exactly one edge e ∈ E. Equivalently, p = (V, E) is a pairing if E is a partition of V or if it is 1-regular. We denote by P(V ) the set of all pairings V . Clearly, only sets V of even parity |V | = 2n admit pairings and in this case one has

|P(V )| = (2n)! 2 n n! = (2n -1)!!.
If the set V = {v1, . . . , v2n} is completely ordered, v1 < v2 < • • • < v2n, writing E = {π(v1)π(v2), π(v3)π(v4), . . . , π(v2n-1)π(v2n)} , sets a one-to-one correspondence between pairings p = (V, E) and permutations π ∈ SV such that π(v2i-1) < π(v2i) and π(v2i-1) < π(v2i+1) for i = 1, . . . , n (compare with (1.5)). In the sequel we will identify the two pictures and denote by p the permutation of V associated to the pairing p. In particular, the signature ε(p) of a pairing p is given by the signature of the corresponding permutation. A diagrammatic representation of a pairing p ∈ P(V ) is obtained by drawing the vertices v1, . . . , v2n as 2n consecutive points on a line. Each edge e ∈ p is drawn as an arc connecting the corresponding points above this line (see Figure 1). It is well known that the signature of is then given by ε(p) = (-1) k where k is the total number of intersection points of these arcs.

If V = V1 ∪ V2 is a partition of V into two equipotent subsets we denote by P(V1, V2) ⊂ P(V ) the corresponding set of (V1, V2)-bipartite pairings and note that |P(V1, V2)| = n!.

If V1 = {v1, . . . , vn} and V2 = {vn+1, . . . , v2n} are completely ordered by v1 < • • • < vn < • • • < v2n then p(v2i-1) = vi and σ(vn+i) = p(v2i) for 1 ≤ i ≤ n defines a one-to-one correspondence between bipartite pairings p ∈ P(V1, V2) and permutations σ ∈ SV 2 A simple calculation shows that ε(p) = (-1) n(n-1)/2 ε(σ).

In the special case V = {1, . . . , 2n}, V1 = {1, . . . , n} and V2 = {n + 1, . . . , 2n} we shall set P(V ) = Pn and P(V1, V2) = e Pn.

The Pfaffian of a 2n × 2n skew-symmetric matrix M is defined by

Pf(M ) = X p∈Pn ε(p) n Y i=1 M p(2i-1)p(2i) .
If B is a n × n matrix and = (-1) n(n-1)/2 det(B).

M = » 0 B -B T 0 - , then

Truncating quasi-free expectations

Let V ⊂ h be finite and totally ordered. To any subset W ⊂ V we assign the monomial

Φ(W ) ≡ Y u∈W ϕ(u),
where the product is ordered from left to right in increasing order of the index u.

Let ω be a gauge invariant quasi-free state on CAR(h). We define a |V | × |V | skew-symmetric matrix Ω by setting

Ωuv ≡ ω(ϕ(u)ϕ(v)),
for u, v ∈ V and u < v. We also denote by Ω W the sub-matrix of Ω whose row and column indices belong to W . Then we have The following simple lemma is our fundamental tool when dealing with such expansions.

ω(Φ(W )) =  Pf(Ω W ) if |W | is even, 0 otherwise. ( 4
Lemma 4.1 Let W1 = {u1, . . . , ur} and W2 = {v1, . . . , vs} be disjoint even subsets of V such that u1 < u2 < • • • < ur and v1 < v2 < • • • < vs. Denote by ε(W1, W2) the signature of the permutation of W1 ∪ W2 which "orders" the sequence W1W2 i.e., which maps the sequence u1, u2, • • • ur, v1, v2, • • • , vs into the ordered sequence of elements of W1 ∪ W2. Then, for any p1 ∈ P(W1) and p2 ∈ P(W2) one has

Ω(p1 ∨ p2) = Ω(p1)Ω(p2), ε(p1 ∨ p2) = ε(W1, W2)ε(p1)ε(p2). (4.58)
Proof. The statement about Ω(p1 ∨ p2) is obvious. To prove the statement about signatures we draw the following diagram (see Figure 2). Draw two parallel lines and on the top one the two diagrams corresponding to the pairings p1 and p2, one next to the other. On the bottom line draw the diagram representing the pairing p1 ∨ p2 but with the edges drawn below the baseline. Now draw segments connecting each point of the top line with its representant on the bottom line. These segments represent the permutation referred to in the Lemma. Thus, if there are q intersection points of these segments then ε(W1, W2) = (-1) q . Denote by j the number of intersection points in our diagram lying above the top line and by j ′ the number of those intersections points lying below the bottom line. Then, we have ε(p1)ε(p2) = (-1) j and ε(p1 ∨ p2) = (-1) j ′ Now observe that our diagram is a disjoint union of closed loops. Thus, it has an even number of intersection points i.e., Iterating Equ. (4.58) we obtain, for disjoint even subsets W1, . . . , W k ⊂ V and arbitrary pairings pi ∈ P(Wi), the formulas

(-1) j+q+j ′ = 1, u 2 u 4 v 1 v 2 v 3 v 4 u 3 u 1 u 2 u 3 u 4 v 3 v 4 u 1 ε(p 1 ) = -1 ε(p 1 ∨ p 2 ) = +1 ε(p 2 ) = +1 ε(J 1 , J 2 ) = -1 v 1 v 2
Ω k _ i=1 pi ! = k Y i=1 Ω(pi), ε k _ i=1 pi ! = ε(W1, . . . , W k ) k Y i=1 ε(pi),
where ε(W1, . . . , W k ) denotes the signature of the permutation which "orders" the sequence W1, . . . , W k . Moreover, the recurrence relation ε

(W1, . . . , W k ) = ε(W1 ∪ • • • ∪ W k-1 , W k )ε(W1, . . . , W k-1 ), holds.
If X, Y are subsets of V we write X < Y whenever max(X) < min(Y ).

Remark. where P(Π) denotes the set of pairings p ∈ P(V ) which have a Π-skeleton p/Π without isolated vertex. The result follows from the fact that the second sum in the last expression vanishes unless Is(p) is empty.2

If W1 < W2 < • • • < W k it
V 4 V 5 V 1 V 2 V 3

Resummation

The setup in this subsection is the same as in the previous one. We consider a fixed ordered partition Π = (V1, . . . , Vn) of V by even subsets as in Lemma 4.2 and fix our attention on the expansion (4.59) of the truncated correlation.

Consider a fixed term in this expansion i.e., a pairing p ∈ P(Π). Since its skeleton p/Π has no isolated point, for each i ∈ I = {1, . . . , n} the set of edges of p connect a vertex in Vi to a vertex outside Vi is not empty. We call exit edge of p from Vi the element of this set which contains the smallest vertex in Vi. The set of all exit edges of p defines a subgraph of p which we denote by ex(p) (see Figure 3). We also denote by Ex(Π) = {ex(p) | p ∈ P(Π)} the set of all exit graphs. We can rewrite expansion (4.59) as ωT (A1, . . . , An) = X g∈Ex(Π) X p∈ex -1 ({g})

ε(p)Ω(p).

A given exit graph g can be seen as a pairing on X(g) = {u ∈ V | uv ∈ g for some v ∈ V }. Setting V (g) = V \ X(g) and applying Lemma 4.1 we get ωT (A1, . . . , An) = X g∈Ex(Π) Ω(g)S(g), (4.61) where S(g) ≡ X

p∈ex -1 ({g}) ε(p)Ω(p |V (g) ).

(4.62)

Our next result is a partial resummation formula for S(g).

Define the exit point from Vi by xi(g) ≡ min(X(g) ∩ Vi). We say that

θ (X, L, M, M ′ , R), is a g-admissible partition of V if X, L, M, M ′ and R are disjoint subsets V such that X = X(g), V = X ∪ L ∪ M ∪ M ′ ∪ R,
and which, for all i ∈ I, satisfy the following conditions

(1) max((L ∪ M ) ∩ Vi) < xi(g);

(2) min((R ∪ M ′ ) ∩ Vi) > xi(g);

(3) |L ∩ Vi| is even;

(4) |M ∩ Vi| = |M ′ ∩ Vi|.
If X, Y are two subsets of V denote by Ω X,Y the sub-matrix of Ω with row (resp. column) indices in (resp. Y ).

Lemma 4.3 For

g ∈ Ex(Π) one has S(g) = X θ=(X,L,M,M ′ ,R)∈Θ(g) ε(θ)ω(Φ(R)) Y i∈I " ω(Φ(L ∩ Vi)) det(Ω M ∩V i ,M ′ ∩V i ) « , (4.63)
where Θ(g) denotes the set of g-admissible partitions of V and

ε(θ) ≡ ε(X, L ∩ V1, . . . , L ∩ Vn, (M ∪ M ′ ) ∩ V1, . . . , (M ∪ M ′ ) ∩ Vn, R)ε(g |X ) Y i∈I (-1) |M ∩V i |(|M ∩V i |-1)/2 .
Proof. Let us have a closer look at a pairing p whose exit graph is g. What happens in Xi(g) ≡ Vi ∩ X(g) is completely determined by g. However, the structure of p |V i (g) where Vi(g) ≡ Vi ∩ V (g) depends on finer details of p. Edges of p which incident to a vertex in Vi(g) located to the left of the exit point xi(g) must connect this vertex to another vertex in Vi(g). These edges split in two categories: the ones which connect two vertices on the left of the exit point and the ones which connect a vertex on the left to a vertex on the right. We denote by Li(p) the set of vertices which belong to an edge of the first category, and by Mi(p) the vertices located to the left of xi(g) and belonging to an edge of the second one. By M ′ i (p) we denote the set of vertices which are connected to elements of Mi(p). This subset of Vi(g) is located on the right of the exit point. We group the remaining vertices of Vi(g), which are all on the right of the exit point, into a fourth set Ri(p). Elements of this set connect among themselves or with elements of Rj(p) for some j = i (see Figure 4). Setting we obtain a map Ψ from ex -1 ({g}) to the set

L(p) ≡ [ i∈I Li(p), M (p) ≡ [ i∈I Mi(p), M ′ (p) ≡ [ i∈I M ′ i (p), R(p) ≡ [ i∈I Ri(p), we obtain a partition θ(p) ≡ (X(g), L(p), M (p), M ′ (p), R(p)), of V which is clearly g-admissible. Moreover, setting li(p) ≡ p |L(p)∩V i ∈ P(L(p) ∩ Vi), mi(p) ≡ p |(M (p)∪M ′ (p))∩V i ∈ P(M (p) ∩ Vi, M ′ (p) ∩ Vi), r(p) ≡ p |R(p) ∈ P(R(p)). X i (g) V i R i (p) M ′ i (p) M i (p) L i (p) x i (g)
[ θ=(X,L,M,M ′ ,R)∈Θ(g) " {θ} × Y i∈I P(L ∩ Vi) ! × Y i∈I P(M ∩ Vi, M ′ ∩ Vi) ! × P(R) # . Since p = g ∨ _ i∈I li(p) ! ∨ _ i∈I mi(p) ! ∨ r(p),
Ψ is injective. For any g-admissible partition θ = (X, L, M, M ′ , R) and any We conclude that Ψ is bijective. Thus, using Lemma 4.1, we can rewrite the sum S(g) as

li ∈ P(L ∩ Vi), mi ∈ P(M ∩ Vi, M ′ ∩ Vi), r ∈ P(R), (4.64
X θ=(X,L,M,M ′ ,R)∈Θ(g) ε(g |X )ε(X, L ∩ V1, . . . , L ∩ Vn, (M ∪ M ′ ) ∩ V1, . . . , (M ∪ M ′ ) ∩ Vn, R) Y i∈I 0 @ X l i ∈P(L∩V i ) ε(li)Ω(li) 1 A Y i∈I 0 @ X m i ∈P(M ∩V i ,M ′ ∩V i ) ε(mi)Ω(mi) 1 A X r∈P(R) ε(r)Ω(r).
The result now follows from Equ. (4.55) and (4.57). 2

Estimating truncated expectations

Apart from the entropic factor |Θ(g)|, the following Lemma controls the partial sum S(g). Combining this estimate with the following Lemma, the result is an immediate consequence of Formula (4.63). 2 Lemma 4.5 Let B be the k × k matrix defined by Bij = ω(ϕ(ui)ϕ(vj)). Then, the estimate

| det(B)| ≤ 2 -k k Y i=1 " ui vi « , holds.
Proof. Let • be a complex conjugation on h. The real-linear map The result follows from the facts that the binomial coefficient is bounded by 2 2N and ki ≤ 2 k i .

Q : h → h ⊕ h f → (1 -T ) 1/2 f ⊕ T 1/2 f ,

Counting exit graphs and their admissible partitions

A g-admissible partition is a partition of V (g) into four sets. Since there are 4 |V (g)| such partitions the second estimate follows.2

Proof of Theorem 3.5

To prove Theorem 3.5 we set Ai = ϕ(e it i h fi1) • • • ϕ(e it i h f ik i ) and apply Lemma 4.7 to the case Vi ≡ {e it i h fi1, . . . , e it i h f ik i }, i ∈ I ≡ {1, . . . , n}.

We set 2N = |V | = P i ki and obtain Z Proof. Assume first that the skeleton g/Π is connected. Then it has a spanning tree (Π, T ). Fix a root Vr in T and for j ∈ I \ {r} let V l(j) be the parent of Vj in T . Let π ∈ Sn be a relabeling of the vertices of T such that π(r) = 1 and π(l(j)) < π(j) for j ∈ I \ {r}. Define new variables by sj = tjt l(j) for j ∈ I \ {r} and sr = tr. The corresponding Jacobian matrix is Jij = δij -(1δir)δ l(i)j . By our choice of the relabeling the reordered matrix

J ′ ij = J π -1 (i)π -1 (j) = δij -(1 -δi1)δ π(l(π -1 (i)))j ,
is lower triangular with ones on the diagonal. Thus the Jacobian determinant is given by | det J| = | det J ′ | = 1.

V π(1) V π(2) V π(3) V π(4)

Figure 5: The pairing π induced by a maximally disconnected pairing p.

For each edge VjV l(j) ∈ T there is a corresponding edge ej = ujvj ∈ g with uj = e it j h fj,a j ∈ Vj and vj = e it l(j) h f l(j)b j ∈ V l(j) and therefore a factor j (sj) = 2 fja j f l(j)b j To prove part 2 it suffices to notice that if n is odd then the skeleton of an exit graph can have at most (n -1)/2 connected components.

To prove part 3, we go back to Formula (4.59) and write Thus, the pairings p ∈ P(Π) which contribute to the limit t → ∞ are maximally disconnected in the sense that their skeleton have exactly n/2 connected components. The skeleton p/Π of such a pairing induces a pairing π ∈ P n/2 such that p = p1 ∨ • • • ∨ p n/2 , pj ∈ P0(V π(2j-1) , V π(2j) ),

where P0(Vi, Vj) denotes the set of pairings on Vi ∪ Vj whose skeleton w.r.t. the partition (Vi, Vj) has no isolated vertex (see Figure 5). Since the map p → (π, p1, . . . , p n/2 ) is clearly bijective we can, for the purpose of computing the limit of (4.68) as t → ∞, replace ωT (A1, . . . , An) X and by the remark following it ε(V π(1) , . . . , V π(n) ) = 1. Thus, the last expression can be rewritten as

X π∈P n/2
n/2 Y j=1 0 @ X p j ∈P 0 (V π(2j-1) ,V π(2j) )

ε(pj)Ω(pj)

1 A
Finally observe that, by Lemma 4.2, X p∈P 0 (V i ,V j ) ε(p)Ω(p) = ωT (Ai, Aj).

One easily concludes the proof by the remark following Theorem 3.5 and the dominated convergence theorem.

  The pair (O, τ ) is called a C * -dynamical system. A positive normalized element of the dual O * is called a state on O. In what follows ω is a given τ -invariant state on O. The triple (O, τ, ω) is called a quantum dynamical system. The system (O, τ, ω) is called ergodic if lim * τ s (A)B) ds = ω(B * B)ω(A), and mixing if lim |t|→∞ ω `B * τ t (A)B ´= ω(B * B)ω(A), for all A, B ∈ O.

  follows. Parts (ii) and (iv) are obvious. (i) and (ii) imply the Cauchy-Schwartz inequality (iii). Part (v) follows from Proposition 5.4.12 in [BR2]. Part (vi) is the celebrated stability result of Bratteli, Kishimoto and Robinson [BKR, BR2]. 2 Definition 1.2 Let C be CLT-admissible. We shall say that the Simple Quantum Central Limit Theorem (SQCLT) holds for C w.r.t. (O, τ, ω) if for all A ∈ C self ,

  and that |λ| ≤ λV . Then C is CLT-admissible and the QCLT holds for C w.r.t. (O, τ λ , ω + λ ).

  tubular domains of the type R+ × Γ, where Γ ⊂ R d-1 is a bounded domain, etc. Assumption (A) holds and Theorem 1.6 holds with c = d if:

j Dj upper bound on Dj 1 "

 1 e i(A+a)e ia e iA " e i(B+b) 1 2 [A, a] 2 e ia e iA " e i(B+b)e ib e iB " 1 2 [B, b] 3 e ia " e iA e ibe ib e iA " e iB [A, b] 4 e ia e ib " e iA e iBe i(A+B) e -1 2 [A,B] " D(A, B)

  sup k∈N,η>0,x∈R |g jkη (x)| ≤ M and lim η→0 g jkη (x) = g jk (x), (2.35) for all x ∈ R.

  ǫ = 1/(2CV,A) and suppose that |λ| ≤ λV . Theorems 3.1 and 3.2 yield that sup L(A,A) α 2 /2 .(3.39)

  only bipartite pairings p ∈ e Pn contribute to the Pfaffian of M which reduces to

  .56) If |W | is even, assigning to any pairing p ∈ P(W ) the weight

Figure 2 :

 2 Figure 2: Proof of Lemma 4.1

  immediately follows from the fact that the Wi are even that ε(W π(1) , . . . , W π(k) ) = 1 for any permutation π ∈ S k . Lemma 4.2 Let Π = (V1, . . . , Vn) be an ordered partition of V by even subsets i.e.,V1 < V2 < • • • < Vn, |Vi| even.and set Ai = Φ(Vi). Then one has ωT (A1, . . . , An)

Figure 3 :

 3 Figure 3: The exit graph ex(p) (solid lines) for p ∈ P(Π)

Figure 4 :

 4 Figure 4: The partition of V i induced by a pairing p. Solid lines belong to the exit graph ex(p).

  ) = g, θ(p) = θ, li(p) = li, mi(p) = mi, r(p) = r.

Lemma 4. 4

 4 For g ∈ Ex(Π) one has |S(g)| ≤ 2 -|V (g)|/2 | Θ(g)| Y v∈V (g)we have, for any X ⊂ V , the simple bound|ω(Φ(X))| ≤ 2 -|X|/2 Y v∈X v .

.

  is isometric and such that ω(ϕ(ui)ϕ(vj)) = 1 2 " (ui, vj) -(ui, T vj) + (ui, T vj)It immediately follows that det(B) = 2 -k ωFock(a(Qu1) • • • a(Qu k )a * (Qv k ) • • • a * (Qv1)),where ωFock denotes the Fock-vacuum state on CAR(h ⊕ h). The fact thata(Qu) = a * (Qu) = Qu = u , for any u ∈ h yields the result.2 For u, v ∈ V such that u < v set ∆uv ≡ 2 |ω(ϕ(u)ϕ(v))| u v = 2 |Ωuv| u v ,and for any graph p on V set ∆(p) ≡ Y uv∈p u<v ∆uv. Note that ∆uv and hence ∆(p) values in the interval [0, 1]. The following lemma, which controls the contribution of the exit graph g to the sum (4.61) is immediate. Lemma 4.6 For any p ∈ P(W ) |Ω(p)| ≤ 2 -|W |/2 ∆(p) Y w∈W w ! Applying this bound to g ∈ Ex(Π) and using Lemma 4.4 we finally get from Formula (4.61): Lemma 4.7 Under the hypotheses of Lemma 4.2 the following estimate holds |ωT (A1, . . . , An)| ≤ 2 -|V |/2 Y

Lemma 4. 8

 8 For any ordered partition Π of V one has |Ex(Π)| ≤ 4 |V | |Π|! and for any g ∈ Ex(Π) |Θ(g)| ≤ 4 |V | . Proof. We set |V | = 2N , Π = (V1, . . . , Vn) and |Vi| = ki. To construct an exit graph we must first select n exit points xi ∈ Vi. Thus, there are k1k2 • • • kn exit points configurations. Each exit point xi has now to be paired with a different vertex yi ∈ V , subject to some constraints. Releasing these constraints we obtain the upper bound 2N (2N -1) • • • (2Nn + 1) on the number of such pairings. Thus, |Ex(Π)| ≤ 2N (2N -1) • • • (2Nn + 1)k1 • • • kn = 2N n ! k1 • • • knn!.

[ 0

 0 ,t] n |ωT (A1, . . . , An)| dt1 • • • dtn ≤ 2 -N Y t] n ∆(g) dt1 • • • dtn.Lemma 4.9 Let g a graph with vertex set V . Denote by Nc(g) number of connected components of its skeleton g/Π. Then one has Z[0,t] n ∆(g) dt1 • • • dtn ≤ C n-Nc(g) t Nc(g) , fij)τ t (ϕ(f kl )))| dt « .

A

  fja j )τ -s j (ϕ(f l(j)b j )))| for j < l(j),|ω(ϕ(f l(j)b j )τ s j (ϕ(fja j )))| for j > l(j), , in ∆(g). It follows that ∆(g) ≤ Y j∈I\{r} ∆e j (sj). dsr ≤ C n-1 t.In the general case, g/Π is the disjoint union of Nc(g) connected subgraphs. Applying the above estimate to each of them yields the result.2Inserting the estimate (4.67) into Equ. (4.66) and using Lemma 4.8 we finally obtain, taking into account the fact that the skeleton of an exit graph can have at most n/2 connected componentsZ [0,t] n |ωT (A1, . . . , An)| dt1 • • • dtn ≤ n t n/2 n!,which concludes the proof of Part 1.

t -n/ 2 Z

 2 [0,t] n ωT (A1, . . . , An) dt1 • • • dtn = X p∈P(Π) ε(p) t -n/2 Z [0,t] n Ω(p) dt1 • • • dtn. (4.68)By Lemmata 4.6 and 4.9 one has, as t → ∞,t -n/2 Z [0,t] n Ω(p) dt1 • • • dtn = O(t Nc(p)-n/2 ).

π∈P n/ 2 X

 2 p j ∈P 0 (V π(2j-1) ,V π(2j) ) ε(p1 ∨ • • • ∨ p n/2 ) Ω(p1 ∨ • • • ∨ p n/2 ).By Lemma 4.1 we haveε(p1 ∨ • • • ∨ p n/2 ) = ε(V π(1) , . . . , V π(n) )ε(p1) • • • ε(p n/2 ), Ω(p1 ∨ • • • ∨ p n/2 ) = Ω(p1) • • • Ω(p n/2 ),

  1.13)where K and n k 's are finite and f kj , g kj ∈ v. We denote nA ≡ max k n k and F (A) ≡ {f kj , g kj } (to indicate the dependence of K on A we will also denote it by KA). O(v) is a * -subalgebra of O, and if v is dense in h, then O(v) is norm dense in O.

  and we shall indicate explicitly the dependence of ω +

λ on β and µ by ω + λ, β, µ . Similarly, we shall indicate explicitly the dependence of L(A, B) on λ, β, µ by L λ, β, µ . Since ω + λ, βeq, µeq

  For |λ| ≤ λV the QCLT holds for the linear span of {Φj, Jj | 1 ≤ j ≤ M } w.r.t. (O, τ λ , ω λ, βeq, µeq ). The associated fluctuation algebra W is commutative.

	1.22)
	(3)

Remark 1. Parts (1) and (2) of Theorem 1.7 are proven in

[START_REF] Jakšić | The Green-Kubo formula for locally interacting fermionic open systems[END_REF]

. Part (3) is a special case of Theorem 1.6. Parts (1) and (3) relate linear response to thermodynamical forces to fluctuations in thermal equilibrium and constitute the Fluctuation-Dissipation Theorem for our model. The physical aspects of linear response theory and Fluctuation-Dissipation Theorem are discussed in classical references

[DGM, KTH]

. Remark 2. The arguments in

[START_REF] Jakšić | The Green-Kubo formula for locally interacting fermionic open systems[END_REF]

do not establish that the functions

t → ω + λ, βeq, µeq `Aτ t λ (B) ´,

(1.23

) are absolutely integrable for A, B ∈ {Φj, Jj | 1 ≤ j ≤ M } and in Part (2) L λ, βeq, µeq (A, B) is defined by L λ, βeq, µeq (A, B) = lim t→∞ Z t -t ω + λ, βeq, µeq (Aτ s λ (B)) ds. The absolute integrability of the correlation functions (1.23) is a delicate dynamical problem resolved in Part (3) for |λ| ≤ λV . Remark 3. Remarks 4 and 6 after Theorem 1.5 in [JOP4] apply without changes to Theorem 1.7. Remark 7 is also applicable and allows to extend the Fluctuation-Dissipation Theorem to a large class of so called centered observables. Remark 4. Although the time-reversal Assumption (D) plays no role in Part (3) of Theorem 1.7, it is a crucial ingredient in proofs of Parts (1) and (2) (see [JOP4, AJPP3] for a discussion). The Fluctuation-Dissipation Theorem fails for locally interacting open fermionic systems which are not time-reversal invariant. A class of concrete models for which (A)-(B)-(D) hold is easily constructed following the examples discussed at the end of Subsection 1.2. Let G1, . . . , GM be admissible graphs. Then (A)-(D) hold if hj = ℓ 2 (Gj) (or ℓ 2 (Gj) ⊗ C L ), hj = -∆G j , and d is the subspace of finitely supported elements of h. A physically important class of allowed interactions is