From the microscopic to the van Hove regime in the XY chain out of equilibrium
FFrom the microscopic to the van Hove regime inthe XY chain out of equilibrium
Walter H. Aschbacher ∗ Aix Marseille Universit ´e, CNRS, CPT, UMR 7332, 13288 Marseille, FranceUniversit ´e de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France
Abstract
Using the framework of rigorous algebraic quantum statistical mechanics, we con-struct the unique nonequilibrium steady state in the isotropic XY chain in which asample of arbitrary finite size is coupled by a bond coupling perturbation of arbitrarystrength to two infinitely extended thermal reservoirs, and we prove that this state isthermodynamically nontrivial. Moreover, extracting the leading second order contri-bution to its microscopic entropy production and deriving its entropy production in thevan Hove weak coupling regime, we prove that, in the mathematically and physicallyimportant XY chain, the van Hove regime reproduces the leading order contribution tothe microscopic regime.
Mathematics Subject Classifications (2010)
Keywords
Open systems; nonequilibrium quantum statistical mechanics; quasifreefermions; Hilbert space scattering theory; nonequilibrium steady state; entropy produc-tion; van Hove weak coupling regime.
In recent years, a broad range of important thermodynamic properties of open quantumsystems have been successfully derived from first principles within the mathematicallyrigorous framework of algebraic quantum statistical mechanics. Not only return to equi-librium type phenomena from states close to equilibrium have been explored but alsofundamental transport processes in systems far from equilibrium have come within reach.In the latter field, an important role is played by the quasifree fermionic systems since, ∗ [email protected] a r X i v : . [ m a t h - ph ] J un W. H. Aschbacher (cid:115) (cid:115) (cid:115) (cid:115) (cid:115) (cid:115) (cid:115) (cid:115) (cid:115) (cid:115) (cid:115)(cid:115) (cid:115) (cid:115) (cid:115) (cid:115) n − n (cid:45)(cid:27) Z S Z R Z L Figure 1: The nonequilibrium setting for the XY chain.on one hand, they allow for a powerful description by means of scattering theory on theone-particle Hilbert space on which the fermionic algebra of observables is built beingthus ideally suited for a rigorous analysis on many levels. On the other hand, they alsoconstitute a class of systems which are indeed realized in nature. One of the most promi-nent representatives of this class is the XY spin chain introduced mathematically in 1961by Lieb et al. [19] who showed that this spin system can be mapped onto a gas of freefermions by using the Jordan-Wigner transformation. Already at the end of the 1960s, thefirst candidates for a possible physical realization have been identified by Culvahouse etal. [12] and, later, by D’Iorio et al. [15] (see also Sologubenko et al. [21] for experimentson more general Heisenberg models). Subsequently, Araki [4] extended the mathemat-ical setup from the finite spin chain to fermions over the two-sided infinite discrete linein the framework of C ∗ -dynamical systems and it is this system whose energy transportproperties we will study in this paper. In order to do so, we fall back upon the paradigmof the theory of open system by coupling a localized sample to two infinitely extendedreservoirs in thermal equilibrium at different temperatures. For this purpose, we cut thetwo bonds between the sites ± n and ± ( n + 1) of the two-sided discrete line meaning thatthe coupling strength in the local Hamiltonian between the corresponding sites, initially atvalue λ = 1 , is set to zero. The piece Z S between these bonds plays the role of the config-uration space of the sample whereas the remaining two half-infinite pieces Z L and Z R toits left and right constitute the configuration spaces of the reservoirs, see Figure 1. Overthese configuration spaces, an initial state is prepared as the product of three thermalequilibrium states. The first central object of interest is then the so-called nonequilibriumsteady state (NESS) defined by Ruelle [20] as the large time limit of the (averaged) tra-jectory of the initial state along the fully coupled time evolution. In this setup, the uniqueNESS has been constructed in Aschbacher and Pillet [6] using Ruelle’s scattering ap-proach (for a special case of the XY model, namely for the case of vanishing anisotropyand external magnetic field parameters given in Remark 2 below, this NESS has alsobeen found by Araki and Ho [5] using a different method). It has been proved in [6] thatthis NESS, henceforth called the XY NESS, is thermodynamically nontrivial in the sensethat its entropy production is strictly positive as soon as the system is truly out of equilib-rium. In the present paper, we generalize the foregoing situation to couplings of arbitrarystrength λ ∈ R . As a first result, we prove that the XY NESS can be embedded into atwo-parameter family of NESS parametrized by λ and n which, for λ (cid:54) = 0 , are all thermo-dynamically nontrivial. This provides us with a physically richer nonequilibrium situation.In particular, it becomes possible to study the van Hove weak coupling regime λ → ofthe entropy production of these NESS and compare it to the leading order contribution ofthe fully microscopic regime. Although one naturally expects that the van Hove regime rom the microscopic to the van Hove regime et al. [7] (which corresponds to n = 0 ) and another one is thespin-fermion system from Jakˇsi´c et al. [18]. Then, as a second result, and this is the mainmotivation of the present paper, we prove that this natural expectation is indeed rigorouslytrue in the mathematically and physically important XY chain out of equilibrium.The present paper is organized as follows. In Section 2, we introduce the nonequi-librium setting, construct a (family of) unique NESS, and derive an explicit expression forits entropy production. In particular, we prove that, in a true nonequilibrium, this NESSis thermodynamically nontrivial for all nonvanishing couplings and all sample sizes. InSection 3, we extract the leading second order contribution to the microscopic entropyproduction. In Section 4, we construct the NESS in the van Hove regime and prove thatthe van Hove entropy production is the leading order contribution to the microscopic en-tropy production. In Appendix A, some spectral properties of the appearing one-particleHamiltonians are summarized. In Appendix B, we construct the wave operator needed inthe derivation of the microscopic NESS and display some results of the lengthy compu-tations involved in its construction. Finally, in Appendix C, we summarize the van Hoveweak coupling theory and derive the necessary decay and positivity properties of thereservoir time correlation function. We begin this section by summarizing the setting for the system out of equilibrium usedin Aschbacher and Pillet [6]. In contradistinction to the presentation there, we skip theformulation of the two-sided XY chain as a spin system and rather focus directly on theunderlying C ∗ -dynamical system structure in terms of Bogoliubov automorphisms on theCAR algebra of observables O over the corresponding one-particle Hilbert space h . Re-call that a C ∗ -dynamical system is a pair ( O , τ ) with R (cid:51) t (cid:55)→ τ t ∈ Aut( O ) , (1)where O is a C ∗ algebra and τ t a strongly continuous group of ∗ -automorphism of O (formore information on the algebraic approach to open quantum systems, see, for example,Aschbacher et al. [7]). Moreover, let us denote the states on O by E ( O ) and recall that astate ω ∈ E ( O ) is called a (gauge invariant) quasifree state with density ρ ∈ L ( h ) satisfying ≤ ρ ≤ if, for all p, q ∈ N and all f i , g j ∈ h with i, j ∈ N , we have ω ( a ∗ ( f p ) . . . a ∗ ( f ) a ( g ) . . . a ( g q )) = δ pq det([( g i , ρf j )] pi,j =1 ) , (2)where a ∗ ( f ) , a ( f ) ∈ L ( F ( h )) with f ∈ h stand for the usual creation and annihilationoperators on the fermionic Fock space F ( h ) over the one-particle Hilbert space h . Here,we used the notation L ( G , H ) for the bounded linear operators from some Hilbert space G into some Hilbert space H (with L ( H ) := L ( H , H ) ), and δ ab denotes the usual Kronecker W. H. Aschbacher symbol. In the following, we will also use the notations N := { } ∪ N and Re( A ) :=( A + A ∗ ) / and Im( A ) := ( A − A ∗ ) / (2ı) for all A ∈ L ( H ) , and L ( H ) and L ( H ) stand forthe finite rank operators and the trace class operators, respectively. Moreover, dΓ( A ) isthe usual second quantization on the fermionic Fock space. Definition 1 (Quasifree setting)
The ingredients for this setting are specified as follows. (a) Observable algebra
Let n ∈ N . The sample and the reservoir configuration spaces are defined by Z S := { x ∈ Z | | x | ≤ n } , (3) Z R := { x ∈ Z | | x | ≥ n + 1 } , (4) whereas the subreservoir spaces are given by Z L := { x ∈ Z | x ≤ − ( n + 1) } , (5) Z R := { x ∈ Z | x ≥ n + 1 } , (6) see Figure 1. The observable algebra is then defined to be the CAR algebra O := A ( h S ⊕ h R ) , (7) where the one-particle Hilbert space consists of the sample and the reservoir space h S := (cid:96) ( Z S ) , (8) h R := (cid:96) ( Z R ) , (9) and the dimension of the sample Hilbert space is denoted by n S := dim( h S ) = 2 n +1 .Moreover, the one-particle Hilbert spaces of the subreservoirs are given by h L := (cid:96) ( Z L ) , (10) h R := (cid:96) ( Z R ) . (11) For a ∈ {S , R , L, R } , using the map i a ∈ L ( h a , h ) defined, for all f ∈ h a , by i a f ( x ) := f ( x ) if x ∈ Z a and i a f ( x ) := 0 if x ∈ Z \ Z a , the total one-particle Hilbert space isnaturally identified with h := (cid:96) ( Z ) (12) through f (cid:55)→ i ∗S f ⊕ i ∗R f for all f ∈ h . Analogously, h R is identified with h L ⊕ h R through f (cid:55)→ i ∗ L i R f ⊕ i ∗ R i R f for all f ∈ h R . (b) Dynamics Let λ ∈ R . The one-particle Hamiltonians h, h λ ∈ L ( h ) are defined by h := Re[ u ] , (13) h λ := h + ( λ − v = h + λv, (14) rom the microscopic to the van Hove regime where the right translation u ∈ L ( h ) is defined by ( uf )( x ) := f ( x − for all f ∈ h and all x ∈ Z . The operator h = h is called the XY Hamiltonian, h the decoupledHamiltonian, and h λ with λ (cid:54) = 0 the coupled Hamiltonian. Moreover, the operator v ∈ L ( h ) , called the bond perturbation, is defined by v := (cid:88) a ∈{ L,R } v a , (15) v a := Re[( δ S ,a , · ) δ R ,a ] , (16) where the coupling functions are specified by δ S ,L := δ − n , δ S ,R := δ n , δ R ,L := δ − ( n +1) , δ R ,R := δ n +1 , (17) and δ x ∈ h with x ∈ Z is given by δ x ( y ) := δ xy for all y ∈ Z . Moreover, for a ∈{S , R , L, R } , the one-particle Hamiltonians h a ∈ L ( h a ) are defined by h a := i ∗ a hi a . (18) The second quantized Hamiltonians on the fermionic Fock space F ( h ) are given by H := dΓ( h ) , (19) V := (cid:88) a ∈{ L,R } V a , (20) V a := dΓ( v a ) , (21) H λ := H + ( λ − V = H + λV, (22) where H is unbounded and V a ∈ O is a local perturbation. Finally, the dynamics τ tλ ∈ Aut( O ) with t, λ ∈ R are defined, for all A ∈ O , by τ tλ ( A ) := e ı tH λ A e − ı tH λ . (23)(c) Initial state The initial state ω ∈ E ( O ) is defined to be quasifree with density ρ ∈ L ( h ) defined by ρ := i S ρ S i ∗S + i R ρ R i ∗R , (24) where ρ S ∈ L ( h S ) and ρ R ∈ L ( h R ) are given by ρ S := (cid:37) β S ( h S ) , (25) ρ R := (cid:88) a ∈{ L,R } i ∗R i a (cid:37) β a ( h a ) i ∗ a i R . (26) Here, for any α ∈ R , the Planck density function (cid:37) α : R → R is defined by (cid:37) α ( e ) := (1 + e αe ) − , (27) W. H. AschbacherPour fixer les id ´ees, we will always assume that the inverse temperatures of thesample and the reservoirs satisfy β S = 0 , < β L ≤ β R < ∞ , (28) and we set β := ( β R + β L ) / and δ := ( β R − β L ) / .Remark 2 As discussed in the introduction, this model has its origin in the XY spin chainwhose formal Hamiltonian reads H XY = − (cid:88) x ∈ Z (cid:110) (1 + γ ) σ ( x )1 σ ( x +1)1 + (1 − γ ) σ ( x )2 σ ( x +1)2 + 2 µ σ ( x )3 (cid:111) , (29)where γ ∈ ( − , denotes the anisotropy, µ ∈ R the external magnetic field, and the Paulibasis of C × is given by σ = (cid:20) (cid:21) , σ = (cid:20) (cid:21) , σ = (cid:20) − ıı 0 (cid:21) , σ = (cid:20) − (cid:21) . (30)Namely, under the Araki-Jordan-Wigner transformation (see, for example, Araki [4]), theHamiltonian from (13) corresponds to the case of the so-called isotropic XY chain (orXX chain) without external magnetic field, i.e. to the case where γ = 0 and µ = 0 . Inorder to treat the anisotropic case γ (cid:54) = 0 , one often uses the so-called selfdual quasifreesetup introduced and developed in Araki [2, 3]. There, one works in the doubled one-particle Hilbert space h ⊕ and the generator of the truly anisotropic XY dynamics hasnontrivial off-diagonal blocks on h ⊕ (which vanish for γ = 0 ). In many respects, the trulyanisotropic XY model is substantially more complicated than the isotropic one (this is true a fortiori if a magnetic field is added whose contribution to the generator acts diagonallyon h ⊕ though). In the following, every once in a while, we will make a remark on thecorresponding issue for the anisotropic case. Remark 3
The Hamiltonian h λ for λ = 0 does not couple the different subsystems to eachother, i.e. we have h = (cid:80) a ∈{S ,L,R } i a h a i ∗ a .In order to construct a NESS, we use the following definition due to Ruelle [20]. If notspecified otherwise, it will always be assumed that n ∈ N and λ ∈ R . Definition 4 (NESS)
A NESS associated with the C ∗ -dynamical system ( O , τ tλ ) and theinitial state ω ∈ E ( O ) is a weak- ∗ limit point for T → ∞ of the net (cid:26) T (cid:90) T d t ω ◦ τ tλ (cid:12)(cid:12)(cid:12) T > (cid:27) . (31) Such a NESS is denoted by ω λ, + ∈ E ( O ) .rom the microscopic to the van Hove regime i.e. if λ = 1 ), the unique quasifreeNESS in the fully anisotropic XY model with magnetic field has been constructed in As-chbacher and Pillet [6]. In order to state the corresponding theorem, we switch to themomentum space representation by using the Fourier transformation f : h → (cid:98) h , where (cid:98) h := L ([ − π, π ]; d k π ) , (32)and f is defined with the sign convention (cid:98) f ( k ) := ( f f )( k ) := (cid:80) x ∈ Z f ( x )e ı kx . Moreover,we also use the notation (cid:98) A := f A f ∗ for all A ∈ L ( h ) . In the following, for any selfadjoint A, B ∈ L ( h ) , we denote by w ( A, B ) ∈ L ( h ) the wave operator w ( A, B ) := s − lim t →∞ e − i tA e i tB ac ( B ) , (33)where ac ( B ) is the spectral projection onto the absolutely continuous subspace of B . Theorem 5 (XY NESS)
There exists a unique quasifree NESS ω , + ∈ E ( O ) associatedwith the C ∗ -dynamical system ( O , τ t ) and the initial state ω ∈ E ( O ) . Moreover, its density ρ , + ∈ L ( h ) has the form ρ , + = w ∗ ( h , h ) ρw ( h , h )= (cid:0) βh − δd (cid:1) − , (34) where, in momentum space, (cid:98) h, (cid:98) d ∈ L ( (cid:98) h ) are the multiplication operators acting, for all ϕ ∈ (cid:98) h and all k ∈ ( − π, π ] , as (cid:98) hϕ ( k ) = (cid:15) ( k ) ϕ ( k ) , (35) (cid:98) dϕ ( k ) = sign( (cid:15) (cid:48) ( k )) (cid:98) hϕ ( k ) , (36) and the dispersion relation (cid:15) : R → R is given by (cid:15) ( k ) := cos( k ) . (37) Proof.
See Aschbacher and Pillet [6]. (cid:3)
In the following, we denote by e ( h λ ) ∈ L ( h ) the usual spectral projection onto theeigenspace of h λ corresponding to the eigenvalue e ∈ spec pp ( h λ ) . Moreover, spec sc ( h λ ) isthe singular continuous spectrum of h λ . We then have the following result. Theorem 6 (Microscopic NESS)
There exists a unique NESS ω λ, + ∈ E ( O ) associatedwith the C ∗ -dynamical system ( O , τ tλ ) and the initial state ω ∈ E ( O ) . Moreover, its density ρ λ, + ∈ L ( h ) has the form ρ λ, + = w ∗ ( h , h λ ) ρw ( h , h λ ) + (cid:88) e ∈ spec pp ( h λ ) e ( h λ ) ρ e ( h λ ) . (38) W. H. AschbacherProof.
Since h λ ∈ L ( h ) , h λ − h ∈ L ( h ) , and spec sc ( h λ ) = ∅ due to Lemma 30 of AppendixA, we can use Aschbacher et al. [8] which yields the assertion. (cid:3) Remark 7
As given in Lemma 30 of Appendix A, the pure point component in (38) isabsent if < | λ | ≤ (see also Jakˇsi´c et al. [17]).We next turn to the energy current observable and its NESS expectation. Definition 8 (Energy current)
The observable Φ λ,a ∈ O with a ∈ { L, R } describing theenergy current flowing from reservoir a into the sample is defined by Φ λ,a := dΓ( ϕ λ,a ) , (39) where the one-particle energy current observable ϕ λ,a ∈ L ( h ) is given by ϕ λ,a := − dd t (cid:12)(cid:12)(cid:12) t =0 e ı th λ i a h a i ∗ a e − ı th λ . (40) Moreover, its NESS expectation value is denoted by J λ,a := ω λ, + (Φ λ,a ) . (41)Let us next turn to the structure of the NESS current. The following proposition shows,on one hand, that the expectation value is independent of the pure point component of theNESS density. On the other hand, it implies that we can later proceed to its computationby exploiting the known density of the XY NESS and the purely absolutely continuousnature of the XY Hamiltonian in the construction of the wave operator. The commutatorof A, B ∈ L ( H ) is denoted by [ A, B ] := AB − BA . Proposition 9 (Energy current structure)
For a ∈ { L, R } , we have J λ,a = tr( w ∗ ( h, h λ ) ρ , + w ( h, h λ ) ϕ λ,a ) . (42) Proof.
Since ϕ λ,a ∈ L ( h ) and using the form (38) of the NESS density, we can write J λ,a = tr( ρ λ, + ϕ λ,a )= tr( w ∗ ( h , h λ ) ρw ( h , h λ ) ϕ λ,a ) + (cid:88) e ∈ spec pp ( h λ ) tr(1 e ( h λ ) ρ e ( h λ ) ϕ λ,a ) . (43)The independence of the current of the pure point component of the NESS density nowfollows as in Aschbacher et al. [8] from the observation that, since the one-particle energycurrent observable from (40) has the form of a commutator, namely ϕ λ,a = − ı[ h λ , i a h a i ∗ a ] ,we have, for all e ∈ spec pp ( h λ ) , that e ( h λ ) ϕ λ,a e ( h λ ) = 0 . (44) rom the microscopic to the van Hove regime w ( h , h λ ) = w ( h , h ) w ( h, h λ ) to the wave operators in (43) (whichis applicable since the perturbations are trace class) and using (34), we get w ∗ ( h , h λ ) ρw ( h , h λ ) = w ∗ ( h, h λ ) ρ , + w ( h, h λ ) . (45)This is the assertion. (cid:3) Remark 10
Note that, due to ϕ λ,a = − ı[ h λ , i a h a i ∗ a ] = − ı λ [ v a , i a h a i ∗ a ] ∈ L ( h ) , we have (cid:80) a ∈{ L,R } ϕ λ,a = ı[ h λ , i S h S i ∗S + λv ] , where i S h S i ∗S + λv ∈ L ( h ) . Hence, dΓ( i S h S i ∗S + λv ) ∈ O ,and (cid:88) a ∈{ L,R } Φ λ,a = dd t (cid:12)(cid:12)(cid:12) t =0 τ tλ (dΓ( i S h S i ∗S + λv )) . (46)Since Definition 4 implies that the NESS is invariant under the corresponding C ∗ dynam-ics, i.e. since ω λ, + ◦ τ tλ = ω λ, + for all λ, t ∈ R , we get the first law of thermodynamics of themicroscopic regime, (cid:88) a ∈{ L,R } J λ,a = 0 . (47)Therefore, we can restrict ourselves to study the objects from Definition 8 for a = L , and,for this case, we drop the index L in the notation.We now arrive at the first of our main theorems. For any coupling strength and anysize of the sample, it yields an explicit expression for the NESS energy current and, thus,for the microscopic entropy production (see, for example, Aschbacher et al. [7]), Ep λ := − (cid:88) a ∈{ L,R } β a J λ,a = 2 δJ λ . (48)Of course, for λ = 0 , we have Ep = 0 . If λ (cid:54) = 0 , we have the following result. Theorem 11 (Microscopic second law of thermodynamics)
For λ (cid:54) = 0 , the microscopicentropy production is given by the absolutely convergent integral Ep λ = δλ (cid:90) π − π d k π S ( | (cid:15) ( k ) | ) Q λ ( | (cid:15) ( k ) | ) , (49) where the functions S : [ − , → R and Q λ : ( − , → R are defined by S ( e ) := e (1 − e ) / [ (cid:37) β L ( e ) − (cid:37) β R ( e )] , (50) Q λ := | (1 − E ) − [(1 − λ E ) − (1 − λ ) E n S +1) ] | , (51) and E ( e ) := e + ı(1 − e ) / . Thus, if the system is truly out of equilibrium, i.e. if δ (cid:54) = 0 ,the microscopic entropy production is strictly positive and the energy current is flowingthrough the sample from the hotter to the colder reservoir. W. H. AschbacherRemark 12
We can rewrite the microscopic entropy production in the form Ep λ = δλ (cid:90) π − π d k π (cid:15) ( k ) | (cid:15) (cid:48) ( k ) | Q λ ( (cid:15) ( k )) sh[ δ(cid:15) ( k )]ch [ β (cid:15) ( k )] + sh [ δ (cid:15) ( k )] , (52)where we used the convenient identity x ) − − (1 + e y ) − ] = sh[( y − x ) / / (ch [( x + y ) /
4] + sh [( x − y ) / for x, y ∈ R . For λ = 1 , we have Q = 1 and, hence, we recover theexpression found in Aschbacher and Pillet [6] (which, in addition, is also independent ofthe sample size). Proof.
In order to analyze the NESS current, we start from (42) and determine its ingre-dients. For convenience, we will work with the objects for a = R in this proof. First, using(40), we can write the one-particle energy current observable as ϕ λ,R = λ δ n , · ) δ n +2 ] . (53)Plugging (53) into (42), we get J λ,R = λ F λ ( n, n + 2)] , (54)where, for all λ ∈ R , the function F λ : Z → C is defined by F λ ( x, y ) := ( w ( h, h λ ) δ x , ρ , + w ( h, h λ ) δ y ) . (55)In order to compute this function and, in particular, the wave operator appearing in it, weswitch to the energy space of the XY Hamiltonian ( i.e. to the space diagonalizing h ) givenin Lemma 27 of Appendix A by (cid:101) h = L ([ − , , C ; d e ) . (56)In this representation, the action of the wave operator on the completely localized wavefunctions δ x ∈ h with x ∈ Z reads (cid:101) w ( h, h λ ) (cid:101) δ x = (cid:101) δ x − λ − (cid:88) i,j =1 Σ − λ,ij ( · − ı0) (cid:37) δ j ,δ x ( · − ı0) (cid:101) δ i , (57)where Σ λ ( e − ı0) ∈ C × and (cid:37) δ j ,δ x ( e − ı0) ∈ C are the boundary values of the interactionmatrix and of the XY resolvent amplitudes, respectively, and δ i , δ i ∈ h are the couplingfunctions, see Proposition 35 of Appendix B. Plugging (57) into (55), we get F λ ( x, y ) = (cid:90) − d e S (0) x,y ( e ) + (cid:88) i =1 (cid:18) − λ (cid:19) i (cid:90) − d e S ( i ) λ,x,y ( e ) , (58) rom the microscopic to the van Hove regime λ ∈ R , x, y ∈ Z , and i = 1 , , the functions S (0) x,y , S ( i ) λ,x,y : ( − , → C read S (0) x,y := (cid:10)(cid:101) δ x , (cid:101) ρ , + (cid:101) δ y (cid:11) , (59) S (1) λ,x,y := (cid:10) ξ x , Σ − λ ( · − ı0) η y (cid:11) + (cid:10) Σ − λ ( · − ı0) η x , ξ y (cid:11) , (60) S (2) λ,x,y := (cid:10) Σ − λ ( · − ı0) η x , ΘΣ − λ ( · − ı0) η y (cid:11) . (61)Here, for all x ∈ Z , the vector-valued functions ξ x , η x : ( − , → C and the matrix-valuedfunction Θ : ( − , → C × are defined, for i, j = 1 , . . . , , by ξ x,i := (cid:10)(cid:101) δ i , (cid:101) ρ , + (cid:101) δ x (cid:11) , (62) η x,i := (cid:37) δ i ,δ x ( · − ı0) , (63) Θ ij := (cid:10)(cid:101) δ i , (cid:101) ρ , + (cid:101) δ j (cid:11) , (64)where (cid:104) · , · (cid:105) d stands for the Euclidean scalar product in C d . We next specialize to the caseat hand, namely to x = n and y = n + 2 . For this case, the ingredients of (59)–(61) arecomputed in Lemma 39 of Appendix B. Plugging these expressions into (58), we get F λ ( n, n + 2) = F ( n, n + 2) + (cid:90) − d e (cid:80) i =0 p i ( e ) λ i Q λ ( e ) , (65)where the function Q λ : ( − , → R , defined by Q λ := | det(Σ λ ( · − ı0) | , has the expansion Q λ = (cid:80) i =0 q i λ i , and the coefficient functions p i , q i : ( − , → C are given in Lemma40 of Appendix B. Subtracting F ( n, n + 2) = 0 from (65) (where the latter follows fromLemma 40, see also (55)), we can write F λ ( n, n + 2) = (cid:90) − d e (cid:80) i =0 p i +1 ( e ) λ i +1 Q λ ( e ) , (66)where we used that q p i − p q i = 0 for i = 1 , . . . , , see Lemma 40 of Appendix B.Transforming the coordinates as e = (cid:15) ( k ) for k ∈ [0 , π ] and using that Im[ p i +1 ( (cid:15) ( k ))] = 0 for i = 0 , , and Im[ p ( (cid:15) ( k ))] = − (cid:15) ( k )[ (cid:37) β L ( (cid:15) ( k )) − (cid:37) β R ( (cid:15) ( k ))] /π from Lemma 40, we get(49). Finally, due to Lemma 41 in Appendix B, the integral in (49) is absolutely convergent,and since the numerator and the denominator are even functions in e , we arrive at theassertion. (cid:3) In this section, we determine the leading order contribution to the microscopic entropyproduction from Theorem 11 for small bond coupling λ . It has the following form.2 W. H. Aschbacher
Figure 2: The function S on [0 , for β L = 1 and β R = 2 . Theorem 13 (Leading order contribution)
For λ → , the microscopic entropy produc-tion has the expansion Ep λ = Ep λ + O ( λ ) , (67) where the second order contribution has the form Ep := 2 δn S + 1 n S (cid:88) i =1 S ( (cid:15) ( k i )) , (68) and the function S : [ − , → R (see Figure 2) and the momenta k i for i = 1 , . . . , n S read S ( e ) := e (1 − e ) / [ (cid:37) β L ( e ) − (cid:37) β R ( e )] , (69) k i := iπn S + 1 . (70) Remark 14
Note that (cid:15) ( k n +1+ i ) = − (cid:15) ( k n +1 − i ) for i = 1 , . . . , n and that S is an even func-tion. Hence, for n > , we can further simplify (68) as Ep = 2 δ/ ( n + 1) (cid:80) ni =1 S ( (cid:15) ( k i )) . Remark 15
It follows from (cid:15) ( k n +1 ) = 0 that, for n = 0 , we have Ep = 0 . Hence, in thiscase, the entropy production from Theorem 11 is carried by higher orders than the secondone. On the other hand, since S ◦ (cid:15) ∈ C ([ − π, π ]; d k π ) is an even function, we get with theidentity from Remark 12 that lim n →∞ Ep = δ (cid:90) π − π d k π (cid:15) ( k ) | (cid:15) (cid:48) ( k ) | sh[ δ(cid:15) ( k )]ch [ β (cid:15) ( k )] + sh [ δ (cid:15) ( k )] . (71)Let us now turn to the proof of Theorem 13. Proof.
In order to extract the second order contribution to the NESS current, we determinethe limit λ → of J λ /λ with the help of a Sokhotski-Plemelj type argument. For this rom the microscopic to the van Hove regime S/Q λ = N/D λ , where N, D λ : R → R read N := 4(1 − (cid:15) ) S ( | (cid:15) | ) , (72) D λ := L λ + λ R λ , (73)and the functions L λ , R λ : R → R are given by L λ := (cid:88) i ∈{ , } d i λ i + d (cid:48) λ , (74) R λ := (cid:88) i ∈{ , , } d i λ − i − d (cid:48) . (75)Moreover, the coefficients d i : R → R for i = 0 , , , (cid:48) , , have the form d = σ n , (76) d = − σ n σ n (cid:15), (77) d = 2 σ n σ n (cid:15) + 4 σ n , (78) d (cid:48) := 4 σ n (cid:15) , (79) d = − σ n σ n (cid:15), (80) d = σ n , (81)where, for all α ∈ R , we used the notation σ α ( k ) := sin( αk ) , σ := σ , and (cid:15) α ( k ) := (cid:15) ( αk ) for all k ∈ R , and we set n := n S + 1 , n := n S , and n := n S − . Moreover, fromnow on, if not stated otherwise, we always assume that | λ | > . In order to apply theSokhotski-Plemelj argument, we analyze the neighborhoods of the roots of d , located at k x := xπn , (82)where x ∈ M := { x ∈ Z | | x | ≤ n S + 1 } . The neighborhoods of these roots are denoted by K x := ( k x − κ x , k x + κ x ) ∩ ( − π, π ) and their size κ x , satisfying < κ x ≤ k / for all x ∈ M ,will be suitably chosen below. Moreover, for all x ∈ M , we define the integrals I λ,x := λ (cid:90) K x d k π N ( k ) D λ ( k ) . (83)Then, we can make the decomposition J λ λ = (cid:88) x ∈ M I λ,x + (cid:88) x ∈ M \ M I λ,x + λ (cid:90) K c d k π N ( k ) D λ ( k ) , (84)where we set M := { , ± n } and K c := ( − π, π ) \ ( ∪ x ∈ M K x ) . In the following, we willsuccessively study all the contributions of the different integration domains in the decom-position (84). The coupling strength will always be assumed sufficiently small without4 W. H. Aschbacher necessarily specifying its size in each estimate. Moreover, if nothing else is indicated, theestimates are supposed to hold for all momenta. Finally, the positive constant C can takedifferent values at each place it appears. Case 1: K x for x ∈ M Let us set κ x := κ := k / for all x ∈ M and let us rewrite (72) by using the identity fromRemark 12. Then, after an eventual shift of (83) to the origin, we have | N ( k x + k ) | ≤ Ck and d ( k x + k ) ≥ Ck for | k | < κ . Since (cid:80) i =1 | d i ( k x + k ) | ≤ Ck , we get D λ ( k x + k ) ≥ Ck for | k | < κ . Hence, for x ∈ M and λ → , we find I λ,x = O ( λ ) . (85) Case 2: K x for x ∈ M \ M In order to determine some size κ x for the neighborhood K x , we estimate (75) from belowas follows. First, we define the function d (cid:48)(cid:48) : R → R by d (cid:48)(cid:48) := 4 σ n σ which allows us towrite (75), shifted to the origin, in the form R λ ( k x + k ) = d (cid:48)(cid:48) ( k x ) (86) + d (cid:48)(cid:48) ( k x + k ) − d (cid:48)(cid:48) ( k x ) (87) + d ( k x + k ) − d (cid:48) ( k x + k ) − d (cid:48)(cid:48) ( k x + k ) (88) + λ d ( k x + k ) (89) + λ d ( k x + k ) . (90)Using | (87) | , | (88) | ≤ C | k | and | (89) | , | (90) | ≤ C , we have R λ ( k x + k ) ≥ σ ( k x ) − C ( | k | + λ ) .Moreover, we note that σ ( k x ) ≥ σ ( k ) =: ϑ > for all x ∈ M \ M . Hence, there exists a κ > which we can choose as κ := ϑ/ (3 n ) ≤ / , s.t., for | k | < κ , we have R λ ( k x + k ) ≥ ϑ. (91)Therefore, we set κ x := κ for all x ∈ M \ M . In order to study (83) in the neighborhood K x = ( k x − κ, k x + κ ) , we make the decomposition I λ,x = A λ,x + N ( k x ) B λ,x /n , where A λ,x := λ (cid:90) K x d k π N ( k ) − N ( k x ) D λ ( k ) , (92) B λ,x := n λ (cid:90) K x d k π D λ ( k ) . (93)Let us first analyze (92). To this end, we further decompose it as A λ,x = A (1) λ,x + A (2) λ,x , where A (1) λ,x := λ (cid:90) κ d k π N + x ( k ) L λ ( k x − k ) + N − x ( k ) L λ ( k x + k ) D λ ( k x + k ) D λ ( k x − k ) , (94) A (2) λ,x := λ (cid:90) κ d k π N + x ( k ) R λ ( k x − k ) + N − x ( k ) R λ ( k x + k ) D λ ( k x + k ) D λ ( k x − k ) , (95) rom the microscopic to the van Hove regime N ± x : R → R are defined by N ± x ( k ) := N ( k x ± k ) − N ( k x ) . Next, let usdecompose (94) as A (1) λ,x = A (1 , λ,x + A (1 , λ,x , where A (1 , λ,x := λ (cid:90) κ d k π N + x ( k ) d ( k x − k ) + N − x ( k ) d ( k x + k ) D λ ( k x + k ) D λ ( k x − k ) , (96) A (1 , λ,x := λ (cid:90) κ d k π N + x ( k )( d + λ d (cid:48) )( k x − k ) + N − x ( k )( d + λ d (cid:48) )( k x + k ) D λ ( k x + k ) D λ ( k x − k ) . (97)In order to bound (96), we use d ( k x ± k ) = d ( k ) for all x ∈ M , expand N ∈ C ∞ ( R \ π Z ) around k x up to second order in ± k , and apply D λ ( k x ± k ) ≥ L λ ( k x ± k ) + ϑλ > for all | k | < κ which follows from (91) and the fact that L λ = (cid:96) λ ≥ , where (cid:96) λ : R → R is (cid:96) λ := σ n − λ σ n (cid:15). (98)Hence, we get | A (1 , λ,x | ≤ Cλ (cid:90) κ d k π k d ( k )([ (cid:96) λ ( k x + k )] + ϑλ )([ (cid:96) λ ( k x − k )] + ϑλ ) . (99)Since d = σ n and (cid:96) λ ( k x ± k ) = σ n ( k ) ± − x +1 λ σ ( n [ k x ± k ]) (cid:15) ( k x ± k ) , we makethe coordinate transformation k = arcsin( λ p ) /n for ≤ p ≤ a λ := σ ( n κ ) /λ , where n κ < π/ . Hence, we get | r.h.s. (99) | ≤ C (cid:90) a λ d p π p [arcsin( λ p )] ([ p + (cid:96) + λ,x ( p )] + ϑ )([ p + (cid:96) − λ,x ( p )] + ϑ ) , (100)where, for Λ λ := ( − /λ , /λ ) , the functions (cid:96) ± λ,x : Λ λ → R are defined by (cid:96) ± λ,x ( p ) := [ (cid:96) − (cid:96) ]( k x ± arcsin( λ p ) /n )= ± − x +1 σ ( n [ k x ± arcsin( λ p ) /n ]) (cid:15) ( k x ± arcsin( λ p ) /n ) . (101)Since | (cid:96) ± λ,x ( p ) | ≤ for all p ∈ Λ λ , we find | r.h.s. (100) | ≤ Cλ (cid:18)(cid:90) d p π p ϑ + (cid:90) a λ d p π p ([ p − + ϑ ) (cid:19) . (102)The integrand of the second integral on the r.h.s. of (102) is bounded and, hence, dueto a λ = σ ( ϑ/ /λ , we find A (1 , λ,x = O ( λ ) . We next turn to the estimate of (97). For thispurpose, we make the decomposition A (1 , λ,x = A (1 , , λ,x + A (1 , , λ,x , where A (1 , , λ,x := λ (cid:90) κ d k π N + x ( k ) d ( k x − k ) + N − x ( k ) d ( k x + k ) D λ ( k x + k ) D λ ( k x − k ) , (103) A (1 , , λ,x := λ (cid:90) κ d k π N + x ( k ) d (cid:48) ( k x − k ) + N − x ( k ) d (cid:48) ( k x + k ) D λ ( k x + k ) D λ ( k x − k ) . (104)6 W. H. Aschbacher
In order to bound (103), we first bound the numerator in (103) by using | N ± x ( k ) | ≤ Ck and | d ( k x ± k ) | ≤ σ ( n k ) for all k ∈ [0 , κ ] , and we treat the denominator in (103) as in (99).Then, proceeding as in (100), we can write | A (1 , , x ( λ ) | ≤ C (cid:90) a λ d p π p arcsin( λ p )([ p + (cid:96) + λ,x ( p )] + ϑ )([ p + (cid:96) − λ,x ( p )] + ϑ ) . (105)Moreover, analogously to (102), we get | r.h.s. (105) | ≤ Cλ (cid:18)(cid:90) d p π p ϑ + (cid:90) a λ d p π p ([ p − + ϑ ) (cid:19) . (106)Extending the integration domain of the second integral on the r.h.s. of (106) to infinity,we get A (1 , , λ,x = O ( λ ) . In order to bound (104), using | d (cid:48) ( k x ± k ) | ≤ C and again | N ± x ( k ) | ≤ Ck for all k ∈ [0 , κ ] , we can proceed as above and get | A (1 , , λ,x | ≤ Cλ (cid:18)(cid:90) d p π pϑ + (cid:90) a λ d p π p ([ p − + ϑ ) (cid:19) , (107)which again implies that A (1 , , λ,x = O ( λ ) . Finally, in order to bound (95), we note that | R λ ( k x ± k ) | ≤ C , and estimating (95) as (104), we get A (2) λ,x = O ( λ ) . Taking all of theforegoing estimates together finally implies that the term (92) does not contribute anythingto the second order of the current, i.e. , for λ → , we have A λ,x = O ( λ ) for all x ∈ M \ M .We next turn to the study of (93). For this purpose, we rewrite (93) using the coordinatetransformation introduced before (100) which leads to B λ,x = (cid:90) a λ − a λ d p π Y λ,x ( p ) , (108)where the function Y λ,x : Λ λ → R is defined by Y λ,x ( p ) := 1(1 − λ p ) / p + (cid:96) λ,x ( p )] + R λ,x ( p ) , (109)and we set (cid:96) λ,x := (cid:96) + λ,x , and the function R λ,x : Λ λ → R is given by R λ,x ( p ) := R λ ( k x +arcsin( λ p ) /n ) . Let us first write B λ,x = B ,x + (cid:2) B λ,x − B ,x (cid:3) , where B ,x := (cid:90) ∞−∞ d p π Y ,x ( p ) , (110)the function Y ,x : R → R is defined by Y ,x ( p ) := 1[ p + (cid:96) ,x ] + R ,x , (111) rom the microscopic to the van Hove regime x ∈ M \ M , by (cid:96) ,x := σ ( k x ) and R ,x := 4 σ ( k x ) > .Let us next decompose the difference as B λ,x − B ,x = B (1) λ,x − B (2) λ,x , where B (1) λ,x := (cid:90) a λ − a λ d p π (cid:2) Y λ,x ( p ) − Y ,x ( p ) (cid:3) , (112) B (2) λ,x := (cid:90) | p |≥ a λ d p π Y ,x ( p ) . (113)Furthermore, we make the decomposition B (1) λ,x = B (1 , λ,x + B (1 , λ,x , where B (1 , λ,x := λ (cid:90) a λ − a λ d p π p (1 − λ p ) / (1 + (1 − λ p ) / ) 1[ p + (cid:96) λ,x ( p )] + R λ,x ( p ) , (114) B (1 , λ,x := (cid:90) a λ − a λ d p π (cid:18) p + (cid:96) λ,x ( p )] + R λ,x ( p ) − p + (cid:96) ,x ] + R ,x (cid:19) . (115)In order to bound (114), we use (1 − λ p ) / ≥ (cid:15) ( ϑ/ for all | p | ≤ a λ , | (cid:96) λ,x ( p ) | ≤ , and(91) which yields | B (1 , λ,x | ≤ Cλ (cid:18)(cid:90) d p π p ϑ + (cid:90) a λ d p π p ( p − + ϑ (cid:19) . (116)Due to the boundedness of the integrand of the second integral, we have B (1 , λ,x = O ( λ ) .In order to estimate (115), we make the decomposition B (1 , λ,x = B (1 , , λ,x + B (1 , , λ,x , where B (1 , , λ,x := (cid:90) a λ − a λ d p π [ p + (cid:96) ,x ] − [ p + (cid:96) λ,x ( p )] ([ p + (cid:96) ,x ] + R ,x )([ p + (cid:96) λ,x ( p )] + R λ,x ( p )) , (117) B (1 , , λ,x := (cid:90) a λ − a λ d p π R ,x − R λ,x ( p )([ p + (cid:96) ,x ] + R ,x )([ p + (cid:96) λ,x ( p )] + R λ,x ( p )) . (118)In order to bound (117), we use | [ p + (cid:96) ,x ] − [ p + (cid:96) λ,x ( p )] | ≤ C (1 + | p | ) | (cid:96) ,x − (cid:96) λ,x ( p ) | and | (cid:96) ,x − (cid:96) λ,x ( p ) | ≤ Cλ | p | for the numerator and, for the denominator, | (cid:96) ,x | , | (cid:96) λ,x ( p ) | ≤ and(91) which implies | B (1 , , λ,x | ≤ Cλ (cid:18)(cid:90) d p π pϑ + (cid:90) a λ d p π p ([ p − + ϑ ) (cid:19) . (119)Extending the second integral to infinity, we get B (1 , , λ,x = O ( λ ) . In order to bound (118),we use | R ,x − R λ,x ( p ) | ≤ C (1 + | p | ) λ and estimate the denominator as above yielding | B (1 , , λ,x | ≤ Cλ (cid:18)(cid:90) d p π ϑ + (cid:90) a λ d p π p ([ p − + ϑ ) (cid:19) . (120)8 W. H. Aschbacher
Extending the second integral to infinity, we again get B (1 , , λ,x = O ( λ ) . We next turn tothe estimate of (113). Again, from | (cid:96) ,x | ≤ and R ,x ≥ ϑ , we have | B (2) λ,x | ≤ C (cid:90) ∞ a λ d p π p − + ϑ , (121)and a coordinate transformation fixing the lower limit of the integral leads to B (2) λ,x = O ( λ ) .Collecting the estimates for (92) and (93), we get that, for x ∈ M \ M and λ → , I λ,x = N ( k x ) n (cid:90) ∞−∞ d p π Y ,x ( p ) + O ( λ ) . (122)It remains to study the last term in (84). Case 3: K c Rewriting the numerator of the integrand with the help of the identity from Remark 12, weimmediately get | N ( k ) | ≤ C . Moreover, the zeroth order contribution of the denominator isbounded from below by d ( k ) ≥ d ( κ ) > for all k ∈ K c . Since we have | D λ ( k ) − d ( k ) | ≤ Cλ , we get D λ ( k ) ≥ d ( κ ) / for all k ∈ K c . Hence, for λ → , we find (cid:90) K c d k π N ( k ) D λ ( k ) = O (1) . (123)We can now extract the nontrivial second order contribution to the NESS current from(84). Using (85), (122), and (123), it follows from (84) that lim λ → J λ λ = 12 (cid:88) x ∈ M \ M N ( k x ) n (cid:90) ∞−∞ d p π Y ,x ( p )= 14 n (cid:88) x ∈ M \ M N ( k x ) R / ,x . (124)This concludes the proof of the theorem. (cid:3) We start this section by introducing what we call the product setting. In this setting, thesample algebra is split off from the total algebra, and we can conveniently focus on thethermodynamics of the sample system. It is defined as follows.
Definition 16 (Product setting)
The ingredients for this setting, partially labelled by atilde, are specified as follows.rom the microscopic to the van Hove regime
The observable algebras of the sample and the reservoir are defined by O S := A ( h S ) , (125) O R := A ( h R ) , (126) and the total observable algebra is defined to be their tensor product, ˜ O := O S ⊗ O R . (127)(b) Dynamics The Hamiltonians of the sample, the reservoir, the decoupled, and the coupled sys-tem are specified by H S := dΓ( h S ) , (128) H R := dΓ( h R ) , (129) ˜ H := H S ⊗ ⊗ H R , (130) ˜ H λ,a := ˜ H + λ ˜ V a , (131) ˜ H λ := ˜ H + λ ˜ V , (132) where the couplings ˜ V a , ˜ V ∈ ˜ O are given by ˜ V a := − ıΓ( − ⊗ a ∗ ( i ∗S δ S ,a ) ⊗ a ( i ∗R δ R ,a )] , (133) ˜ V := (cid:88) a ∈{ L,R } ˜ V a . (134) Correspondingly, the dynamics τ t S ∈ Aut( O S ) , τ t R ∈ Aut( O R ) , and ˜ τ tλ,a , ˜ τ tλ ∈ Aut( ˜ O ) are given, for all A in O S , O R , and ˜ O , respectively, by τ t S ( A ) := e ı tH S A e − ı tH S , (135) τ t R ( A ) := e ı tH R A e − ı tH R , (136) ˜ τ tλ,a ( A ) := e ı t ˜ H λ,a A e − ı t ˜ H λ,a , (137) ˜ τ tλ ( A ) := e ı t ˜ H λ A e − ı t ˜ H λ . (138)(c) Initial state The initial state of the sample ω S ∈ E ( O S ) and of the reservoir ω R ∈ E ( O R ) aredefined to be quasifree with the densities ρ S ∈ L ( h S ) and ρ R ∈ L ( h R ) , respectively.The total initial state ˜ ω ∈ E ( ˜ O ) is defined by ˜ ω := ω S ⊗ ω R . (139)0 W. H. Aschbacher
In order to show the equivalence of this product setting and the quasifree setting fromDefinition 1, we make use of the following lemma. We denote by U ( H ) the unitary opera-tors on the Hilbert space H . Lemma 17 (Exponential law for fermions)
For i = 1 , , let F ( h i ) be the fermionic Fockspaces over the Hilbert spaces h i having vacua Ω i , creation and annihilation operators a ∗ i , a i : h i → L ( F ( h i )) , and second quantizations Γ i : U ( h i ) → L ( F ( h i )) . Then, there existsa unique U ∈ U ( F ( h ⊕ h ) , F ( h ) ⊗ F ( h )) s.t. U Ω = Ω ⊗ Ω , (140) U a ( f ⊕ f ) U ∗ = a ( f ) ⊗ + Γ ( − ) ⊗ a ( f ) , (141) U Γ( U ⊕ U ) U ∗ = Γ ( U ) ⊗ Γ ( U ) , (142) where Ω , a ∗ , a , and Γ are the corresponding objects for F ( h ⊕ h ) .Proof. See, for example, Alicki and Fannes [1]. (cid:3)
The two settings are then equivalent in the following sense.
Lemma 18 (Product setting isomorphism)
Let
Φ : L ( F ( h S ⊕ h R )) → L ( F ( h S ) ⊗ F ( h R )) be defined by Φ( A ) := U AU ∗ , where U ∈ U ( F ( h S ⊕ h R ) , F ( h S ) ⊗ F ( h R )) is the unitary fromLemma 17 corresponding to the decomposition h (cid:39) h S ⊕ h R . Then, Φ is a C ∗ algebra ∗ -isomorphism. Moreover, the following assertions hold.(a) Φ :
O → ˜ O is a C ∗ algebra ∗ -isomorphism.(b) ˜ τ tλ = Φ ◦ τ tλ ◦ Φ − for all λ, t ∈ R (c) ˜ ω = ω ◦ Φ − Proof.
Note that for our sample of finite size, n S < ∞ , we have Γ( −
1) = (cid:89) x ∈ Z S (1 − a ∗ ( i ∗S δ x ) a ( i ∗S δ x )) ∈ O S . (143)Moreover, the couplings are related by Φ( V α ) = ˜ V α . The proof is then analogous to theone of Aschbacher et al. [7], see there for details. (cid:3) We next specify the van Hove weak coupling regime (see also Aschbacher et al. [7]for example). For this purpose, we make use of the weak coupling theory developed byDavies [13, 14] and summarized for our needs in Appendix C.
Definition 19 (Van Hove regime)
Let the operator P S : ˜ O → O S be defined, for all A ∈O S and all B ∈ O R , by P S ( A ⊗ B ) := ω R ( B ) A, (144) rom the microscopic to the van Hove regime and the same notation is used for its extension to ˜ O . Moreover, for a ∈ { L, R } , the two-parameter family of mappings T t S ,λ , T t S ,λ,a : O S → O S with λ, t ∈ R and a ∈ { L, R } aredefined, for all A ∈ O S , by T t S ,λ ( A ) := P S [˜ τ − t ◦ ˜ τ tλ ( A ⊗ , (145) T t S ,λ,a ( A ) := P S [˜ τ − t ◦ ˜ τ tλ,a ( A ⊗ . (146) The van Hove NESS ω S , + ∈ E ( O S ) with density ρ S , + ∈ L ( h S ) and the Davies generator K H,a : O S → O S of subreservoir a ∈ { L, R } are defined, for all A ∈ O S , by ω S , + ( A ) := lim t →∞ lim λ → ω S ( T t/λ S ,λ ( A )) , (147) K H,a ( A ) := dd t (cid:12)(cid:12)(cid:12) t =0 lim λ → T t/λ S ,λ,a ( A ) , (148) if the limits exist. Finally, the van Hove energy current observable Φ S ,a ∈ O S and itsexpectation value J S ,a in the van Hove NESS are given by Φ S ,a := K H,a ( H S ) , (149) J S ,a := ω S , + (Φ S ,a ) . (150) Remark 20
For all A ∈ O S and t > , defining K H ( A ) := dd t (cid:12)(cid:12)(cid:12) t =0 lim λ → T t/λ S ,λ ( A ) , (151)and using K H = (cid:80) a ∈{ L,R } K H,a (see, for example, Spohn and Lebowitz [22]), and theinvariance of the van Hove NESS under the time evolution generated by K H , we get thefirst law of thermodynamics of the van Hove regime, (cid:88) a ∈{ L,R } J S ,a = 0 . (152)Hence, as for the microscopic regime, we set J S := J S ,L .We begin our analysis by constructing the van Hove NESS. In the proof of the followingtheorem (and the subsequent one), we will make use of the reservoir time correlationfunction ψ βa : R → C with a ∈ { L, R } and β ∈ R defined by ψ βa ( t ) := ( δ R ,a , i a (cid:37) β ( h a )e ı th a i ∗ a δ R ,a ) . (153)Moreover, we will use (cid:15) i and π i with i = 1 , . . . , n S which are the simple eigenvalues andthe corresponding eigenprojections of the sample Hamiltonian h S , respectively, given inLemma 31 of Appendix A, and, for a ∈ { L, R } and i = 1 , . . . , n S , we set Ω a,i := (cid:107) π i i ∗S δ S ,a (cid:107) , (154)and the scalar product and the norm in h S are denoted as the ones in h . The NESS canthen be characterized as follows.2 W. H. Aschbacher
Theorem 21 (Van Hove NESS)
There exists a unique quasifree van Hove NESS ω S , + ∈E ( O S ) whose density has the form ρ S , + = 12 (cid:88) a ∈{ L,R } (cid:37) β a ( h S ) . (155) Proof.
Let us introduce the two-parameter family of states ω t S ,λ ∈ E ( O S ) with t, λ ∈ R which, for all A ∈ O S , is defined by ω t S ,λ ( A ) := ω S ( T t S ,λ ( A ))= ˜ ω ◦ ˜ τ tλ ( A ⊗ . (156)Lemma 18, (23), and (24) then imply that their two-point function can be written as ω t S ,λ ( a ∗ ( f ) a ( g )) = (cid:88) a ∈{S , R} F a ( λ, t ) , (157)where, for fixed f, g ∈ h S , the function F a : R → C with a ∈ {S , R} is defined by F a ( λ, t ) := (e ı th λ i S g, i a ρ a i ∗ a e ı th λ i S f ) . (158)In order to study the limit for λ → of F a ( λ, t/λ ) with fixed t > , we apply the weakcoupling theory summarized in Appendix C in a form suitable for the present theorem(and for Theorem 22 below). Its ingredients are specified as follows: H := h , P := i S i ∗S , P := i R i ∗R , U t := e tZ with Z := ı h (satisfying [ U t , P ] = 0 for all t ∈ R ), A := (cid:80) a ∈{ L,R } A a , A a := ı v a , V tλ := e ı th λ , W tλ := i S i ∗S e ı th λ i S i ∗S and R tλ := i R i ∗R e ı th λ i S i ∗S . In order to simplify theverification of the assumptions of the weak coupling theory, we define the operator-valuedfunction A βa,b,c : R → L ( h ) with a, b, c ∈ { L, R } and β ∈ R by A βa,b,c ( r, s, t ) := 2 U r P A a P B βb U s P A c P U t = − δ ab δ ac ψ βa ( s ) (e − ı th δ S ,a , · ) e ı rh δ S ,a , (159)where we set B βb := i b (cid:37) β ( h b ) i ∗ b and the reservoir time correlation function ψ βa is given in(153). Let us begin with the sample contribution. For a = S , we can write (158) as F S ( λ, t ) = ( U − t W tλ P i S g, U − t W tλ P i S f ) . (160)In order to apply assertion (1) of Theorem 42 of Appendix C on each factor of the scalarproduct in (160), we verify the following three assumptions of Theorem 42. Assumption (a) is dim(ran ( P )) = n S < ∞ . Assumption (b) is P AP = 0 and P AP = 0 which followsfrom (15). It remains to verify assumption (c) which reads (cid:82) ∞ d t (cid:107) P AP U t P AP (cid:107) < ∞ .Since P = 2 (cid:80) b ∈{ L,R } B b , we have P AP U t P AP = (cid:88) a,b,c ∈{ L,R } A a,b,c (0 , t, − (cid:88) a ∈{ L,R } ψ a ( t ) ( δ S ,a , · ) δ S ,a , (161) rom the microscopic to the van Hove regime (cid:107) P AP U t P AP (cid:107) ≤ (cid:80) a ∈{ L,R } | ψ a ( t ) | ≤ . In order to analyzethe temporal decay of (153), we proceed to the diagonalization of h a by using Lemma 29of Appendix A. Switching to the energy space (cid:101) h + = L ([ − , e ) of h a , we get, for all β ∈ R and a ∈ { L, R } , that ψ βa ( t ) = 2 π (cid:90) − d e (1 − e ) / (cid:37) β ( e ) e ı te , (162)which, by symmetry, is independent of a . From the asymptotic analysis of Lemma 43 ofAppendix C (or by noting that, for the case β = 0 , we can write ψ a ( t ) = J ( t ) /t , where J is the first order Bessel function), we have, for t → ∞ , that ψ βa ( t ) = O ( t − / ) . (163)Therefore, assumption (c) is also satisfied and we can apply assertion (1) of Theorem 42.This assertion implies that, for any fixed t > , we get lim λ → F S ( λ, t/λ ) = ( g, ρ t SS f ) , (164)where the operator ρ t SS ∈ L ( h S ) is defined, for all t ∈ R , by ρ t SS := 12 i ∗S e t ( K (cid:92) ) ∗ e tK (cid:92) i S . (165)Here, K (cid:92) = (cid:80) a ∈{ L,R } K (cid:92)a ∈ L ( h ) , where, for a ∈ { L, R } , the operator K (cid:92)a ∈ L ( h ) is thespectral average from Theorem 42 of the Davies generator K a ∈ L ( h ) given by K a := (cid:90) ∞ d t U − t P A a P U t P A a P = (cid:88) b ∈{ L,R } (cid:90) ∞ d t A a,b,a ( − t, t, − (cid:90) ∞ d t ψ a ( t )( δ S ,a , · ) e − ı th δ S ,a . (166)For the computation of K (cid:92)a , we make use of the fact that, for any A ∈ L ( h ) , we can write A (cid:92) = (cid:80) n S i =1 i S π i i ∗S Ai S π i i ∗S , where h S = (cid:80) n S i =1 (cid:15) i π i with π i := ( ϕ i , · ) ϕ i is the spectral repre-sentation of the sample Hamiltonian whose simple eigenvalues (cid:15) i and the correspondingorthonormal eigenfunctions ϕ i are given in Lemma 31 of Appendix A. Using this repre-sentation, we find K (cid:92)a = − n S (cid:88) i =1 Ψ a (ı (cid:15) i ) Ω a,i i S π i i ∗S , (167)where Ψ βa denotes the Laplace transform of ψ βa and Ω a,i is given in (154). Hence, for all t ∈ R , we immediately get e tK (cid:92)a = i R i ∗R + n S (cid:88) i =1 e − t Ψ a (ı (cid:15) i )Ω a,i i S π i i ∗S . (168)4 W. H. Aschbacher
Using that [ K (cid:92)a , K (cid:92)b ] = [ K (cid:92)a , ( K (cid:92)b ) ∗ ] = 0 for all a, b ∈ { L, R } and plugging (168) and its adjointinto (165), we find that, for any t > , the sample contribution has the form ρ t SS = 12 i ∗S (cid:89) a ∈{ L,R } e t ( K (cid:92)a ) ∗ (cid:89) b ∈{ L,R } e tK (cid:92)b i S = 12 n S (cid:88) i =1 (cid:89) a ∈{ L,R } e − t Re[Ψ a (ı (cid:15) i )]Ω a,i π i . (169)We next turn to the reservoir contribution. For a = R , we can write (158) as F R ( λ, t ) = (cid:88) a ∈{ L,R } ( R tλ P i S g, B β a a R tλ P i S f ) , (170)and B β a a ≥ and [ B β a a , U t ] = 0 for all t ∈ R and all a ∈ { L, R } . In order to determine thelimit for λ → of F R ( λ, t/λ ) with fixed t > , we apply assertion (2) of Theorem 42. Tothis end, we have to verify that S := (cid:80) a,b,c ∈{ L,R } S a,b,c converges in norm, where S a,b,c := − (cid:90) ∞ d t Re[ P A a P B β b b U t P A c P U − t ]= − (cid:90) ∞ d t Re[ A β b a,b,c (0 , t, − t )]= 12 δ ab δ ac (cid:90) ∞ d t Re[ ψ β a a ( t )(e ı th δ S ,a , · ) δ S ,a ] . (171)Using again Lemma 43, we get (cid:107) Re[ P A a P B β b b U t P A c P U − t ] (cid:107) ≤ | ψ β a a ( t ) | = O ( t − / ) for t → ∞ . Hence, we apply assertion (2) of Theorem 42 which implies, for any fixed t > ,that lim λ → F R ( λ, t/λ ) = ( g, ρ t SR f ) , (172)where ρ t SR ∈ L ( h S ) is defined, for all t ∈ R , by ρ t SR := (cid:90) t d s i ∗S e s ( K (cid:92) ) ∗ S (cid:92) e sK (cid:92) i S . (173)Using the spectral representation of the sample Hamiltonian h S as above, we get S (cid:92)a,b,c = 12 δ ab δ ac n S (cid:88) i =1 Re[Ψ β a a (ı (cid:15) i )] Ω a,i i S π i i ∗S . (174)Plugging (168) and (174) into (173), we find that, for any t > , the reservoir contributesas ρ t SR = 12 (cid:88) a ∈{ L,R } (cid:90) t d s i ∗S (cid:89) b ∈{ L,R } e s ( K (cid:92)b ) ∗ S (cid:92)a,a,a (cid:89) c ∈{ L,R } e sK (cid:92)c i S = 12 n S (cid:88) i =1 (cid:80) a ∈{ L,R } Re[Ψ β a a (ı (cid:15) i )]Ω a,i (cid:80) b ∈{ L,R } Re[Ψ b (ı (cid:15) i )]Ω b,i − (cid:89) c ∈{ L,R } e − t Re[Ψ c (ı (cid:15) i )]Ω c,i π i . (175) rom the microscopic to the van Hove regime i = 1 , . . . , n S , we have Re[Ψ βa (ı (cid:15) i )] = 2(1 − (cid:15) i ) / (cid:37) β ( (cid:15) i ) , (176) Ω a,i = 2 n S + 1 (1 − (cid:15) i ) , (177)where (cid:15) i = (cid:15) ( k i ) and k i = iπ/ ( n S +1) , and both expressions are independent of a ∈ { L, R } .Using (169) and (175), we then find the density (155) since ω S , + ( a ∗ ( f ) a ( g )) = lim t →∞ lim λ → ω t S ,λ ( a ∗ ( f ) a ( g ))= lim t →∞ ( g, [ ρ t SS + ρ t SR ] f )= ( g, ρ S , + f ) . (178)Moreover, it follows from the quasifreeness of the initial state and Lemma 18 that the vanHove NESS is again quasifree. This is the assertion. (cid:3) Now we are able to determine the energy current expectation in the van Hove NESSor the van Hove entropy production given by (see, for example, Aschbacher et al. [7]) Ep S := − (cid:88) a ∈{ L,R } β a J S ,a = 2 δJ S . (179)In the following, tr denotes the trace over h S . Theorem 22 (Van Hove second law of thermodynamics)
The van Hove entropy pro-duction has the form Ep S = 2 δn S + 1 tr[ S ( h S )] , (180) where S is given in Theorem 13. Hence, if the system is truly out of equilibrium and n > , the van Hove entropy production is strictly positive.Proof. The van Hove energy current observable is given in Definition 19 by Φ S ,a = K H,a ( H S )= dd t (cid:12)(cid:12)(cid:12) t =0 lim λ → T t/λ S ,λ,a ( H S ) . (181)Moreover, for n ∈ N , we know from (128) that the sample Hamiltonian has the form H S = 12 (cid:88) x ∈ Z (cid:48)S [ a ∗ ( i ∗S δ x ) a ( i ∗S δ x +1 ) + a ∗ ( i ∗S δ x +1 ) a ( i ∗S δ x )] , (182)6 W. H. Aschbacher where we set Z (cid:48)S := Z S \ { n } . For n = 0 , we have H S = 0 since h S = 0 . Let us firstconsider (146) on the observable A = a ∗ ( f ) a ( g ) for any f, g ∈ h S . We then get T t S ,λ,a ( a ∗ ( f ) a ( g )) = (cid:88) b ∈{S , R} G b,a ( λ, t ) , (183)where, for fixed f, g ∈ h S , the map G b,a : R → O S with b ∈ {S , R} and a ∈ { L, R } isdefined by G S ,a ( λ, t ) := a ∗ ( f S ,a ( λ, t )) a ( g S ,a ( λ, t )) , (184) G R ,a ( λ, t ) := ω R ( a ∗ ( f R ,a ( λ, t )) a ( g R ,a ( λ, t ))) 1 S , (185)and, for any f ∈ h , the function f b,a : R → h b with b ∈ {S , R} and a ∈ { L, R } is given by f b,a ( λ, t ) := i ∗ b e − ı th e ı th λ,a i S f. (186)In order to study the limit λ → of G b,a ( λ, t/λ ) for fixed t > , we again apply theweak coupling theory from Appendix C with similar ingredients as in the proof of Theorem21, namely, H := h , P := i S i ∗S , U t := e tZ with Z := ı h , A a := ı v a , V tλ,a := e ı th λ,a , W tλ,a := i S i ∗S e ı th λ,a i S i ∗S , and R tλ,a := i R i ∗R e ı th λ,a i S i ∗S . Let us start with the sample contribution(184). Since (cid:107) a ( f ) (cid:107) = (cid:107) f (cid:107) for all f ∈ h S , it is enough to study the weak coupling limit of f S ,a ( λ, t ) = i ∗S U − t W tλ,a P i S f. (187)The assumptions (a) , (b) , and (c) of Theorem 42 are again verified as in the proof ofTheorem 21 with, in particular, (cid:107) P A a P U t P A a P (cid:107) ≤ | ψ a ( t ) | = O ( t − / ) for t → ∞ ,where here and in the following, we use the same notations as in the proof of Theorem21. It then follows from assertion (1) of Theorem 42 and (168) that, for t > , we have lim λ → f S ,a ( λ, t/λ ) = i ∗S e tK (cid:92)a P i S f = n S (cid:88) i =1 e − t Ψ a (ı (cid:15) i ) Ω a,i π i f. (188)We next turn to the reservoir contribution (185). In this case, we have to study the weakcoupling limit of G R ,a ( λ, t ) = (cid:88) b ∈{ L,R } ( R tλ,a P i S g, B β b b R tλ,a P i S f )1 S . (189)Since the additional assumption in assertion (2) of Theorem 42 about the norm conver-gence of (304) is satisfied due to (171) and the line following it, we get from (168) and(174) that, for t > , lim λ → G R ,a ( λ, t/λ ) = (cid:88) b ∈{ L,R } (cid:90) t d s (e sK (cid:92)a P i S g, S (cid:92)a,b,a e sK (cid:92)a P i S f ) 1 S = 12 n S (cid:88) i =1 Re[Ψ β a a (ı (cid:15) i )]Re[Ψ a (ı (cid:15) i )] (cid:104) − e − t Re[Ψ a (ı (cid:15) i )]Ω a,i (cid:105) ( g, π i f ) 1 S . (190) rom the microscopic to the van Hove regime K H,a ( a ∗ ( f ) a ( g )) = (cid:88) b ∈{S , R} dd t (cid:12)(cid:12)(cid:12) t =0 lim λ → G b,a ( λ, t/λ )= − n S (cid:88) i,j =1 (cid:2) Ψ a (ı (cid:15) i )Ω a,i + ¯Ψ a (ı (cid:15) j )Ω a,j (cid:3) a ∗ ( π i f ) a ( π j g )+ 12 n S (cid:88) i =1 Re (cid:2) Ψ β a a (ı (cid:15) i ) (cid:3) Ω a,i ( g, π i f ) 1 S . (191)Applying (191) to (182), plugging the resulting expression into the van Hove NESS fromTheorem 21, and using (176), (177), and (cid:80) x ∈ Z (cid:48)S Re[( i ∗S δ x , π i i ∗S δ x +1 )] = (cid:15) i , we arrive at theassertion. (cid:3) We finally get the following result.
Theorem 23 (Van Hove is second order)
The van Hove entropy production is the lead-ing second order contribution to the small coupling expansion of the microscopic entropyproduction, Ep S = Ep . (192) Proof.
Due to Lemma 31 in Appendix A which states that the eigenvalues of the sampleHamiltonian have the form (cid:15) i = (cid:15) ( k i ) for all i = 1 , . . . , n S , we immediately get the assertionby comparing (68) in Theorem 13 and (180) in Theorem 22. (cid:3) Remark 24
In Aschbacher et al. [7], an assertion like Theorem 23 has been derived forthe simple electronic black box model (SEBB) with one-dimensional sample system. Theassumptions made there on the SEBB model compare to Lemma 30 and 31 of AppendixA, Definition 1, and Lemma 43 of Appendix C.
Remark 25
In Aschbacher and Spohn [9], a simple sufficient condition has been estab-lished which ensures the strict positivity of the entropy production as soon as the mi-croscopic regime is related to the van Hove regime as in Theorem 23. In order to beable to apply this criterion, one assumption on the so-called effective coupling and an-other one on the triviality of some commutants have to be satisfied. Whereas it has beenshown in [9] that the entropy production can still be strictly positive if the latter conditionis violated, the present case is an example showing that the criterion is not necessarydue to violation of the former condition. In order to formulate this condition precisely, werewrite the couplings (133) as ˜ V a = (cid:80) i =1 V (1) S ,a,i ⊗ V (1) R ,a,i , where V (1) S ,a, = Re[ a ( i ∗S δ S ,a )Γ( − ,8 W. H. Aschbacher V (1) S ,a, = − Im[ a ( i ∗S δ S ,a )Γ( − , and V (1) R ,a, = Re[ a ( i ∗R δ R ,a )] , (193) V (1) R ,a, = Im[ a ( i ∗R δ R ,a )] . (194)Moreover, we define the matrix-valued reservoir correlation function R a : R → C × by R a,ij ( t ) := ω R ( τ t R ( V (1) R ,a,i ) V (1) R ,a,j ) . (195)The effective coupling conditions from [9] then requires that, for all a ∈ { L, R } and forall energies (cid:15) ∈ spec( H S ) − spec( H S ) , the temporal Fourier transform of the reservoircorrelation matrix should be positive definite, ˆ R a ( (cid:15) ) > . (196)Now, due to (128), and (207) from Appendix A, we have on F ( h S ) = C ⊕ ( ⊕ n S α =1 h ∧ α S ) that spec( H S ) = { } ∪ (cid:32) n S (cid:91) α =1 { (cid:15) i + . . . + (cid:15) i α } ≤ i <... . Since ˆ ψ β a a ( (cid:15) − (cid:15) ) = 0 due to Lemma 43 in Appendix C, the effective coupling condition (196) is not satisfied forall energy differences. Remark 26
As we indicated repeatedly in the appendix, the derivation of Theorem 23for the full anisotropic XY model with an additional external magnetic field is much morecomplicated. This is also true for the derivation of a theorem like Theorem 23 for theisotropic case and general observables. We will study these question for more generalquasifree systems elsewhere.
A Spectral properties
In this section, we display some spectral properties of the different Hamiltonians appear-ing in the model. In the first lemma, we introduce what we will call the energy space ofthe XY Hamiltonian h being the direct integral decomposition of the absolutely continuoussubspace w.r.t. which h is diagonal, namely (cid:101) h := L ([ − , , C ; d e ) . (199) rom the microscopic to the van Hove regime (cid:101) f ∈ L ( (cid:98) h , (cid:101) h ) is defined, for all ϕ ∈ (cid:98) h , by (cid:101) f ϕ ( e ) := (2 π ) − / (1 − e ) − / [ ϕ (arccos( e )) , ϕ ( − arccos( e ))] , (200)and the momentum space (cid:98) h = L ([ − π, π ]; d k π ) has been introduced in (32). We will usethe notation (cid:101) f := (cid:101) ff f for all f ∈ h , and (cid:101) A := (cid:101) ff A f ∗ (cid:101) f ∗ for all A ∈ L ( h ) , where the Fouriertransform f : h → (cid:98) h is also given after (32). Moreover, the Euclidean scalar product in thefiber C is denoted by (cid:104) · , ·(cid:105) . Lemma 27 (XY Hamiltonian)
The XY Hamiltonian h ∈ L ( h ) has purely absolutely con-tinuous spectrum with spec( h ) = [ − , , and it is diagonal in (cid:101) h .Proof. In momentum space (cid:98) h , the Hamiltonian (cid:98) h acts as the multiplication by the dis-persion relation (cid:15) ( k ) from Theorem 5. Moreover, a simple computation shows that (cid:101) f is asurjective isometry with (cid:101) f − = (cid:101) f ∗ : (cid:101) h → (cid:98) h acting on all η =: [ η , η ] ∈ (cid:101) h as (cid:101) f ∗ η ( k ) = (2 π ) / (1 − (cid:15) ( k )) / [ χ [0 ,π ] ( k ) η ( (cid:15) ( k )) + χ [ − π, ( k ) η ( (cid:15) ( k ))] . (201)This implies the assertion. (cid:3) Remark 28
For γ (cid:54) = 0 , the energy space for the XY Hamiltonian, now acting on h ⊕ (seeRemark 2), takes the form L ([ − , −| γ | ] , C ; d e ) ⊕ L ([ | γ | , , C ; d e ) (and additional C -valued factors if µ (cid:54) = 0 ). Moreover, the nondiagonal matrix-multiplication operator by whichits Fourier transform acts in momentum space L ([ − π, π ]; d k π ) ⊕ has to be diagonalized.The subreservoir Hamiltonians have similar properties. Let us introduce the spaces h + := (cid:96) ( N ) and (cid:98) h + := L ([0 , π ]; π d k ) , and (cid:101) h + := L ([ − , e ) . (202) Lemma 29 (Subreservoir Hamiltonians)
The subreservoir Hamiltonians h a ∈ L ( h a ) with a ∈ { L, R } have purely absolutely continuous spectrum with spec( h a ) = [ − , , and theyare diagonal in (cid:101) h + .Proof. We use the unitary mappings h a t a → h + s → (cid:98) h + (cid:101) s → (cid:101) h + , where the ingredients aregiven by t L f ( x ) := f ( − ( x + n )) and t R f ( x ) := f ( x + n ) for all f ∈ h a , by the Fourier-sinetransform s ( f )( k ) := (cid:80) ∞ x =1 f ( x ) sin( xk ) for all f ∈ h + , and by the energy transformationwhich, for all ϕ ∈ (cid:98) h + , has the form (cid:101) s ϕ ( e ) := 2 / π − / (1 − e ) − / ϕ (arccos( e )) . (203)0 W. H. Aschbacher In (cid:98) h + , the subreservoir Hamiltonians act by multiplication with (cid:15) ( k ) , and applying the en-ergy transformation, we get the assertion. (cid:3) Next, we turn to the coupled and decoupled Hamiltonians. We denote by spec sc ( A ) , spec ac ( A ) , and spec pp ( A ) the singular continuous, the absolutely continuous, and the purepoint spectrum of the operator A , respectively. Lemma 30 ([De]coupled Hamiltonian)
For all λ ∈ R , it holds that spec sc ( h λ ) = ∅ and spec ac ( h λ ) = [ − , . Moreover, the coupled Hamiltonian has the properties spec pp ( h λ ) = ∅ for all < | λ | ≤ and card(spec pp ( h λ )) ≤ for all | λ | > . The decoupled Hamiltoniansatisfies card(spec pp ( h )) = n S .Proof. For the first two assertions and the fact that card(spec pp ( h λ )) < ∞ for all λ ∈ R ,see, for example, Hume and Robinson [16] (in fact, this is all what is used in Theorem 6).Next, let λ ∈ R \ { } , and let us assume that there exist (cid:54) = f ∈ h and e ∈ [ − , s.t. h λ f = ef . Written out and evaluated at any x ∈ Z , this equation reads f ( x + 1) + f ( x −
1) + ( λ − (cid:88) i =1 ( δ i , f ) δ i ( x ) = 2 ef ( x ) , (204)where δ , δ ∈ h are given in Lemma 32 of Appendix B. It follows from [16] that eigenfunc-tions corresponding to eigenvalues in [ − , satisfy f ( x ) = 0 for all | x | ≥ n + 1 . Hence,plugging x = ± ( n + 1) into (204), we find that f ( ± n ) = 0 . The eigenvalue equation thenbecomes h λ f = hf = ef which leads to spec pp ( h λ ) ∩ [ − ,
1] = ∅ for all | λ | > . Let usnext consider the eigenvalue equation h λ f = ef for (cid:54) = f ∈ h and e ∈ R with | e | > (see also Lemma 32 of Appendix B). Plugging x = ± n, ± ( n + 1) into (204) and setting f n := [ f ( − n ) , f ( − n − , f ( n + 1) , f ( n )] ∈ C , we get Σ λ ( e ) f n = 0 , (205)where the matrix Σ λ ( e ) ∈ C × is given in Lemma 32 and Lemma 36 of Appendix B. If det(Σ λ ( e )) (cid:54) = 0 , we again get h λ f = hf = ef . Hence, the eigenvalues are the solutions of det(Σ λ ( e )) = 0 , where, analogously to Proposition 38 of Appendix B, we have det(Σ λ ) = (cid:89) σ = ± [(1 − λ ) E n +2 + σ ( λ E − , (206)and E ( e ) = e − sign( e )( e − / stems from (233) of Appendix B. Using that < E ( e ) < for all e ∈ R with | e | > , none of the two factors in (206) vanishes if < | λ | ≤ . On theother hand, if | λ | > , the factor with σ = 1 has at most one root (depending on the sizeof | λ | , it may have no root for small n but for sufficiently large n , it has one root), whereasthe factor with σ = − has exactly one root for all n ∈ N . Finally, for λ = 0 , we know fromRemark 3, that h does not couple the subsystems to each other. Using Lemma 29 andLemma 31, we then arrive at the assertion. (cid:3) rom the microscopic to the van Hove regime Lemma 31 (Sample Hamiltonian)
The spectrum of the sample Hamiltonian h S ∈ L ( h S ) consists of n S nondegenerate eigenvalues which, for i = 1 , . . . , n S , have the form (cid:15) i = (cid:15) ( k i ) , k i := iπn S + 1 . (207) The corresponding orthonormal eigenfunctions ϕ i ∈ h S are given, for all x ∈ Z S , by ϕ i ( x ) = (cid:18) n S + 1 (cid:19) / sin (cid:18)(cid:20) x + n S + 12 (cid:21) k i (cid:19) . (208) Proof.
Note that the sample Hamiltonian h S = i ∗S hi S ∈ L ( h S (cid:39) C n S ) is the usual discreteLaplacian acting by application of the matrix [ h S ] ij = ( δ ij +1 + δ ij − ) for i, j = 1 , . . . , n S (see, for example, B ¨ottcher and Grudsky [10]). (cid:3) B Wave operator
In this section, we use the stationary approach to scattering theory in order to computethe wave operators appearing in the NESS expectation value of the energy current ob-servable. To this end, we first express the resolvent of the coupled Hamiltonian by theresolvent of the XY Hamiltonian. For any operator A ∈ L ( H ) , we denote by res( A ) theresolvent set of A and by r z ( A ) := ( A − z ) − ∈ L ( H ) the resolvent of A at the point z ∈ res( A ) . Lemma 32 (Coupled resolvent)
For z ∈ res( h ) ∩ res( h λ ) , we have r z ( h λ ) = r z ( h ) − λ − (cid:88) i,j =1 Σ − λ,ij ( z ) ( r ¯ z ( h ) δ j , · ) r z ( h ) δ i , (209) where v = (cid:80) i =1 ( δ i , · ) δ i and δ := [ δ i ] i =1 , δ := [ δ i ] i =1 ∈ h are given by δ := [ δ S ,L , δ R ,L , δ R ,R , δ S ,R ] , (210) δ := [ δ R ,L , δ S ,L , δ S ,R , δ R ,R ] . (211) Moreover, the interaction matrix Σ λ ( z ) ∈ C × is defined, for i, j = 1 , . . . , , by Σ λ,ij ( z ) := δ ij + λ −
12 ( δ i , r z ( h ) δ j ) . (212)2 W. H. AschbacherRemark 33
For γ (cid:54) = 0 , the wave operator in the selfdual setting acts on h ⊕ and hasnonvanishing off-diagonal components (whereas for γ = 0 it is block-diagonal). The inter-action matrix then lies in C × . Proof.
In order to simplify the notation, we drop the indices of the resolvents. Using theresolvent identity r ( h λ ) = r ( h ) − ( λ − r ( h ) vr ( h λ ) , we have, for all f ∈ h , r ( h λ ) f + λ − (cid:88) j =1 ( δ j , r ( h λ ) f ) r ( h ) δ j = r ( h ) f. (213)Taking the scalar product of (213) with δ i for all i = 1 , . . . , , we get (cid:0) λ − A (cid:1) ξ = η, (214)where the components of ξ, η ∈ C and A ∈ C × are defined, for i, j = 1 , . . . , , by ξ i := ( δ i , r ( h λ ) f ) , (215) η i := ( δ i , r ( h ) f ) , (216) A ij := ( δ i , r ( h ) δ j ) . (217)Moreover, defining B ∈ C × by B ij := ( δ i , r ( h λ ) δ j ) for i, j = 1 , . . . , , the resolvent identityimplies that, for any λ ∈ R , we have (1 + λ − A )(1 − λ − B ) = 1 , so λ − A is invertible.We now solve (214) for ξ and plug the resulting expression into (213). This yields theassertion. (cid:3) We next introduce the following abbreviations.
Definition 34 (Boundary values)
Let z ∈ res( h ) , e ∈ R , ε > , and f, g ∈ h . We define (cid:37) f,g ( z ) := ( f, r z ( h ) g ) , (218) γ f,g ( e, ε ) := 12 π ı ( (cid:37) f,g ( e + ı ε ) − (cid:37) f,g ( e − ı ε )) , (219) and, if the limits exist, we write (cid:37) f,g ( e ± ı0) := lim ε → + (cid:37) f,g ( e ± ı ε ) , (220) γ f,g ( e ) := lim ε → + γ f,g ( e, ε ) . (221)Let us recall from Lemma 27 of Appendix A that (cid:101) h = L ([ − , , C ; d e ) is the energyspace of the XY Hamiltonian h . The wave operator then looks as follows. Proposition 35 (Wave operator)
In the energy space (cid:101) h of the XY Hamiltonian, the actionof the wave operator is given, for all f ∈ h , by (cid:101) w ( h, h λ ) (cid:101) f = (cid:101) f − λ − (cid:88) i,j =1 Σ − λ,ij ( · − ı0) (cid:37) δ j ,f ( · − ı0) (cid:101) δ i , (222) rom the microscopic to the van Hove regime where, for all e ∈ ( − , , the boundary interaction matrix Σ λ ( e − ı0) ∈ C × is defined by Σ λ,ij ( e − ı0) := δ ij + λ − (cid:37) δ i ,δ j ( e − ı0) . (223) Proof.
In order to compute the wave operator with the help of stationary scattering theory,we rewrite it in its weak abelian form (see, for example, Yafaev [23]), w ( h, h λ ) = w − lim ε → + ε (cid:90) ∞ d t e − εt ac ( h )e − i th e i th λ ac ( h λ ) . (224)Applying Parseval’s identity to (224), and using that r e − ı ε ( h ) = − ı (cid:82) ∞ d t e ı t ( h − ( e − ı ε )) , weget, for all f, g ∈ h , that ( f, w ( h, h λ ) g ) = lim ε → + επ (cid:90) ∞−∞ d e ( r e − ı ε ( h )1 ac ( h ) f, r e − ı ε ( h λ )1 ac ( h λ ) g ) . (225)Moreover, if the limit ε → + of ε ( r e − ı ε ( h ) f, r e − ı ε ( h λ ) g ) exists for all f, g ∈ h and almostall e ∈ R (the set of full measure depending on f and g ) and using that ac ( h ) = 1 and spec( h ) = [ − , , we can write ( f, w ( h, h λ ) g ) = (cid:90) − d e lim ε → + επ ( r e − ı ε ( h ) f, r e − ı ε ( h λ ) g ) . (226)In order to compute the limit in (226), we express the resolvent r e − ı ε ( h λ ) of the coupledHamiltonian in terms of the resolvent r e − ı ε ( h ) of the XY Hamiltonian. Plugging (209) intothe scalar product on the r.h.s. of (226), we have επ ( r e − ı ε ( h ) f, r e − ı ε ( h λ ) g ) = γ f,g ( e, ε ) − λ − (cid:88) i,j =1 γ f,δ i ( e, ε ) Σ − λ,ij ( e − ı ε ) (cid:37) δ j ,g ( e − ı ε ) , (227)where we used that επ ( r e − ı ε ( h ) f, r e − ı ε ( h ) g ) = γ f,g ( e, ε ) (which follows from the resolventidentity). Now, we know that, for all f, g ∈ h and almost all e ∈ [ − , , the limit (cid:37) f,g ( e ± ı0) = ± π ı d( f, ζ ( e ) g )d e + p . v . (cid:90) − d e (cid:48) e (cid:48) − e d( f, ζ ( e (cid:48) ) g )d e (cid:48) (228)exists, where the p . v . -integral is Cauchy’s principle value, the mapping ζ : B ( R ) → L ( h ) denotes the projection-valued spectral measure of the XY Hamiltonian h with B ( R ) theBorel sets on R , and we used that d( f, ζ ( e ) g ) = χ [ − , ( e ) d( f, ζ ( e ) g )d e d e. (229)Moreover, it follows from (219) and (228) that γ f,g ( e ) = d( f, ζ ( e ) g )d e . (230)4 W. H. Aschbacher
Hence, we find that ( f, w ( h, h λ ) g ) = ( f, g ) − λ − (cid:88) i,j =1 (cid:90) − d e γ f,δ i ( e ) Σ − λ,ij ( e − ı0) (cid:37) δ j ,g ( e − ı0) , (231)where the invertibility of the interaction matrix is assured as in the proof of Lemma 32and we used (cid:82) − d e γ f,g ( e ) = ( f, ac ( h ) g ) = ( f, g ) in the first term on the r.h.s. of (231).In order to write the derivatives in (230) entering (231) more explicitly, we switch to theenergy space representation using (cid:101) h of Lemma 27. This lemma implies that d( f, ρ ( e ) g )d e = (cid:104) (cid:101) f ( e ) , (cid:101) g ( e ) (cid:105) , (232)where we recall that (cid:104)· , ·(cid:105) denotes the Euclidean scalar product in the fiber C of the directintegral (cid:101) h , and (cid:101) f = (cid:101) ff f for all f ∈ h . Hence, plugging (230) and (232) into (231), we arriveat the assertion. (cid:3) In order to completely determine the wave operator, we have to compute the bound-ary values and the inverse of the interaction matrix. To this end, we define the function E : R → C by E ( e ) := (cid:40) e + ı(1 − e ) / , if | e | ≤ ,e − sign( e )( e − / , if | e | > . (233)Let us start with the computation of some XY resolvent amplitudes for completely localizedwave functions (which is also used in Appendix A). Lemma 36 (Resolvent amplitudes)
For x ∈ Z , we have − E | x | +1 ( e )1 − E ( e ) = (cid:40) (cid:37) δ ,δ x ( e − ı0) , if | e | < ,(cid:37) δ ,δ x ( e ) , if | e | > . (234) Proof.
Let x ∈ Z with x ≥ , e ∈ ( − , , and ε > sufficiently small. We first rewritethe momentum space representation of the resolvent amplitude in the form of a contourintegral over the positively oriented unit circle T as (cid:37) δ ,δ x ( e − ı ε ) = 1ı π (cid:73) T d z z x z − e − ı ε ) z + 1 . (235)Then, using Cauchy’s residue theorem and taking the limit (cid:15) → + , we get the first ex-pression in (234) for x ≥ . Moreover, due to the parity invariance of the XY Hamiltonian, [ h, θ ] = 0 , where θ : h → h is defined, for all f ∈ h , by ( θf )( x ) := f ( − x ) , we also have (cid:37) δ ,δ − x ( e − ı (cid:15) ) = (cid:37) δ ,δ x ( e − ı (cid:15) ) . The second assertion is derived similarly. (cid:3) rom the microscopic to the van Hove regime Remark 37
For γ (cid:54) = 0 and µ = 0 , one gets an analogous expression for (235) but, in thiscase, there is a nontrivial numerator, and the polynomial in the denominator becomesbiquadratic, z + az + 1 , where a depends on γ , e , and ε . Moreover, if both γ (cid:54) = 0 and µ (cid:54) = 0 , this polynomial changes to z + az + bz + az + 1 , where a depends on µ and γ ,and b on µ , γ , e , and ε . Hence, the computation of the roots becomes increasingly andsubstantially more complicated (see also, for example, Carey and Hume [11]).We next turn to the computation of the inverse of the boundary value interaction ma-trix from Proposition 35. For the convenience of the reader who wants to work with thisnonequilibrium model and for reasons of a possible future extension, we display the de-tailed results of the computations. Proposition 38 (Inverse boundary interaction matrix)
For all e ∈ ( − , , we have Σ − λ ( e − ı0) = 1∆ λ ( e ) (cid:20) M λ ( e ) N λ ( e ) σ N λ ( e ) σ σ M λ ( e ) σ (cid:21) . (236) Here, the determinant ∆ λ := det[Σ λ ( · − ı0)] : ( − , → C reads ∆ λ = 1 − E n +4 (1 − E ) − E (1 − E n +2 )(1 − E ) λ + E (1 − E n )(1 − E ) λ , (237) the matrix-valued functions M λ , N λ : ( − , → C × have the structure M λ := (cid:20) a λ b λ c λ a λ (cid:21) , N λ := (cid:20) d λ e λ e λ f λ (cid:21) , (238) and the functions a λ , . . . , f λ : ( − , → C are defined by a λ := 1 − E n +4 (1 − E ) − E (1 − E n +2 )(1 − E ) ( λ + λ ) + E (1 − E n )(1 − E ) λ , (239) b λ := − E (1 − E n +2 )(1 − E ) (1 − λ ) + E (1 − E n )(1 − E ) ( λ − λ ) , (240) c λ := − E (1 − E n +4 )(1 − E ) (1 − λ ) + E (1 − E n +2 )(1 − E ) ( λ − λ ) , (241) d λ := − E n +1 − E (1 − λ ) , (242) e λ := − E n +2 − E ( λ − λ ) , (243) f λ := − E n +3 − E ( λ − λ ) . (244) Proof.
For all e ∈ ( − , , the matrix Σ λ ( e − ı0) ∈ C × has the structure Σ λ ( e − ı0) = (cid:20) A λ ( e ) B λ ( e ) C λ ( e ) D λ ( e ) (cid:21) , (245)6 W. H. Aschbacher where the matrix-valued functions A λ , . . . , D λ : ( − , → C × are defined by A λ := 11 − E (cid:20) − λE (1 − λ ) E (1 − λ ) E − λE (cid:21) , (246) B λ := (1 − λ ) E n +1 − E (cid:20) EE E (cid:21) (247) C λ := σ B λ σ , (248) D λ := A λ . (249)A lengthy calculation then leads to the assertion. (cid:3) In the following lemmas, we display some ingredients used in the proof of Theorem11. Recall from there that, for all x ∈ Z and i, j = 1 , . . . , , the vector-valued functions ξ x , η x : ( − , → C and the matrix-valued function Θ : ( − , → C × are given by ξ x,i = (cid:10)(cid:101) δ i , (cid:101) ρ , + (cid:101) δ x (cid:11) , (250) η x,i = (cid:37) δ i ,δ x ( · − ı0) , (251) Θ ij = (cid:10)(cid:101) δ i , (cid:101) ρ , + (cid:101) δ j (cid:11) . (252)The first lemma displays the explicit form of these functions. Lemma 39 (Ingredients, 1)
For x ∈ Z , the functions ξ x , η x : ( − , → C and Θ :( − , → C × read ξ x = Eπ ı(1 − E ) (cid:88) σ = ± ρ σ (cid:2) E σ ( x + n +1) , E σ ( x + n ) , E σ ( x − n ) , E σ ( x − n − (cid:3) , (253) η x = − E − E (cid:2) E | x + n | , E | x + n +1 | , E | x − n − | , E | x − n | (cid:3) , (254) Θ = Eπ ı(1 − E ) (cid:88) σ = ± ρ σ E σ E σ (2 n +1) E σ (2 n +2) E − σ E σ n E σ (2 n +1) E − σ (2 n +1) E − σ n E σ E − σ (2 n +2) E − σ (2 n +1) E − σ , (255) where, for σ = ± , the function (cid:37) σ : R → R is defined by (cid:37) σ ( e ) := (cid:0) ( β + σδ ) e (cid:1) − . (256) Moreover, for all e ∈ ( − , , we have Σ − λ ( e − ı0) η x = 1∆ λ ( e ) (cid:2) a λ,x ( e ) , b λ,x ( e ) , c λ,x ( e ) , d λ,x ( e ) (cid:3) , (257) rom the microscopic to the van Hove regime where the component functions a λ,x , . . . , d λ,x : ( − , → C are given, for x = n , by a λ,n := − E n +1 − E , (258) b λ,n := − E n +2 − E λ, (259) c λ,n := − E (1 − E n +2 )(1 − E ) λ + 2 E (1 − E n )(1 − E ) λ , (260) d λ,n := − E (1 − E n +2 )(1 − E ) + 2 E (1 − E n )(1 − E ) λ , (261) and, for x = n + 2 , by a λ,n +2 := − E n +3 − E λ, (262) b λ,n +2 := − E n +4 − E λ , (263) c λ,n +2 := − E (1 − E n +4 )(1 − E ) + 2 E (1 − E n +2 )(1 − E ) λ , (264) d λ,n +2 := − E (1 − E n +2 )(1 − E ) λ + 2 E (1 − E n )(1 − E ) λ . (265) Proof.
Note that, for all x ∈ Z , we have (cid:101) δ x = (cid:18) Eπ ı(1 − E ) (cid:19) / [ E x , E − x ] , (266)and that the density ρ , + ∈ L ( h ) of the XY NESS given in Theorem 5 acts, for all η ∈ (cid:101) h , asthe matrix multiplication operator (cid:101) ρ , + η = diag( (cid:37) + , (cid:37) − ) η. (267)Hence, we get (253) and (255), and the expressions in (254) are given in Lemma 36.Moreover, (257) directly follows from Proposition 38. (cid:3) We next display some further ingredients used in the proof of Theorem 11. Recallfrom there that, for x = n and y = n + 2 , the function F λ ( x, y ) = ( w ( h, h λ ) δ x , ρ , + w ( h, h λ ) δ y ) reads F λ ( n, n + 2) = F ( n, n + 2) + (cid:90) − d e (cid:80) i =0 p i ( e ) λ i (cid:80) i =0 q i ( e ) λ i , (268)where the coefficient functions p i , q i : ( − , → C with i = 1 , . . . , are displayed in thefollowing lemma. Due to their complicated form, we do not display the formulas for general x ∈ Z . W. H. Aschbacher
Lemma 40 (Ingredients, 2)
The numerator functions p i : ( − , → C with i = 1 , . . . , have the form p = (1 − E n +4 ) π ı E n +1 (1 − E ) ( (cid:37) + E + (cid:37) − ) , (269) p = (1 + E )(1 − E n +2 )(1 − E n +4 ) π ı E n +1 (1 − E ) (cid:37) − , (270) p = − E )(1 − E n +2 )(1 − E n +4 ) π ı E n +1 (1 − E ) ( (cid:37) + E + (cid:37) − ) , (271) p = − (1 + E n +4 )(1 + 3 E + E ) − E n (1 + E + 5 E + 3 E ) π ı E n +1 (1 − E ) (cid:37) − + E π ı(1 − E ) (cid:37) + , (272) p = (1 + E n +4 )(1 + 4 E + E ) − E n (1 + 10 E + E ) π ı E n +1 (1 − E ) ( (cid:37) + E + (cid:37) − ) , (273) p = 2 E (1 + E )(1 − E n )(1 − E n +2 ) π ı E n (1 − E ) (cid:37) − , (274) p = − E (1 + E )(1 − E n )(1 − E n +2 ) π ı E n (1 − E ) ( (cid:37) + E + (cid:37) − ) , (275) p = − E (1 − E n ) π ı E n (1 − E ) (cid:37) − , (276) p = E (1 − E n ) π ı E n (1 − E ) ( (cid:37) + E + (cid:37) − ) , (277) and the denominator functions q i : ( − , → C with i = 1 , . . . , read q = q = q = q = 0 , q = − (1 − E n +4 ) E n (1 − E ) , (278) q = 2(1 + E )(1 − E n +2 )(1 − E n +4 ) E n (1 − E ) , (279) q = − (1 + E )(1 − E n )(1 − E n +4 ) + 4 E (1 − E n +2 ) E n (1 − E ) , (280) q = 2 E (1 + E )(1 − E n )(1 − E n +2 ) E n (1 − E ) , (281) q = − E (1 − E n ) E n (1 − E ) . (282) Moreover, on the energies e = (cid:15) ( k ) with k ∈ [0 , π ] , we have Im[ p i +1 ( (cid:15) ( k ))] = 0 for i = 0 , , and, for i = 1 , we find Im[ p ( (cid:15) ( k ))] = − (cid:15) ( k ) π [ (cid:37) β L ( (cid:15) ( k )) − (cid:37) β R ( (cid:15) ( k ))] . (283) rom the microscopic to the van Hove regime Moreover, on these energies, the denominator functions have the form q ( (cid:15) ( k )) = sin (2( n + 1) k )4 sin ( k ) , (284) q ( (cid:15) ( k )) = − sin((2 n + 1) k ) sin(2( n + 1) k ) cos( k )sin ( k ) , (285) q ( (cid:15) ( k )) = sin(2 nk ) sin(2( n + 1) k ) cos(2 k ) + 2 sin ((2 n + 1) k )2 sin ( k ) , (286) q ( (cid:15) ( k )) = − sin((2 n + 1) k ) sin(2 nk ) cos( k )sin ( k ) , (287) q ( (cid:15) ( k )) = sin (2 nk )4 sin ( k ) . (288) Proof.
This follows from Lemma 39 and a lengthy calculation. (cid:3)
Finally, in the following lemma, we discuss the ingredients needed in order to derivethe absolute convergence of the NESS current integral in the proof of Theorem 11. Recallfrom there that the function Q λ : ( − , → R is given by Q λ = | ∆ λ | = | det(Σ λ ( · − ı0) | = (cid:88) i =0 q i λ i . (289)This function has the following property. Lemma 41 (Boundedness)
For λ ∈ R \ { } , the inverse of Q λ is bounded. In particular,for λ = ± , we have Q ± = 1 .Proof. For λ = 1 , we have Σ ( e − ı0) = 1 ∈ C × for all e ∈ [ − , from (223). Since thedeterminant (237) contains even powers of λ only, we thus have Q ± = 1 on [ − , . Wenext observe that Q λ = − E n (1 − E ) (cid:81) i =1 P λ,i , (290)where the polynomials P λ,i : ( − , → C with i = 1 , . . . , are defined by P λ, := 1 + E n +2 − (1 + E n ) λ , (291) P λ, := − E n +2 + (1 − E n ) λ , (292) P λ, := 1 − E n +2 − E (1 − E n ) λ , (293) P λ, := − − E n +2 + E (1 + E n ) λ . (294)0 W. H. Aschbacher
From now on, let λ ∈ R \ { , ± } . Then, it easily follows from (291)–(294) that, for all n ∈ N , there are no unimodular roots of P λ, and P λ, , whereas the only unimodularroots of P λ, and P λ, are E = ± . We next study the order of these roots. Specializing(291)–(294) for n = 0 , we see that, in this case, the roots are simple and we get Q λ = − (1 − E ) ( E + 1 − λ )(1 + (1 − λ ) E ) , (295)which implies the assertion for n = 0 . Next, let n > and let us first consider P λ, and E = 1 . From the factorization P λ, = ( E − R λ, , where we set R λ, := E n (1 + E ) + (1 − λ ) n − (cid:88) i =0 E i , (296)we get R λ, (1) = 2( n + 1 − nλ ) . Hence, if λ (cid:54) = ( n + 1) /n , the polynomial P λ, has a simpleroot at E = 1 . On the other hand, if λ = ( n + 1) /n , we can write P λ, = ( E − S , where S := E n + 1 n n − (cid:88) i =0 (2 n − i ) E n − − i , (297)and now we have S (1) = 2( n + 1) (cid:54) = 0 . Hence, in this case, P λ, has a double rootat E = 1 . Since P λ, ( − E ) = P λ, ( E ) , the same conclusions hold for the root E = − .Moreover, P λ, can be treated similarly (we again have the two cases λ different or equalto ( n +1) /n ) and the conclusions remain unchanged. Hence, the order of the roots E = ± in the denominator of (290) is not exceeding and, since the numerator cancels thesesingularities of /Q λ , we arrive at the assertion. (cid:3) C Van Hove weak coupling theory
The material of the following theorem is taken from Davies [13, 14]. We do not displayhis assertions in full generality but rather adapt them to our special case at hand. Theingredients are as follows. Let H be a Hilbert space, P ∈ L ( H ) a projection, and P :=1 − P ∈ L ( H ) . Let U t ∈ L ( H ) with t ∈ R be a strongly continuous one-parameter groupof isometries s.t., for all t ∈ R , we have [ U t , P ] = 0 . (298)The generator of U t is denoted by Z . Moreover, let A ∈ L ( H ) with A ∗ = − A , and let V tλ ∈ L ( H ) with λ, t ∈ R be the one-parameter group generated by Z + λA . Besides, theoperators W tλ , R tλ ∈ L ( H ) with λ, t ∈ R are defined by W tλ := P V tλ P , (299) R tλ := P V tλ P . (300) rom the microscopic to the van Hove regime dim(ran ( P )) < ∞ , we define the spectral average X (cid:92) ∈ L ( H ) of an operator X ∈ L ( H ) by X (cid:92) := lim T →∞ T (cid:90) T − T d t U t P XP U − t . (301)We then have the following result. Theorem 42 (Van Hove weak coupling limit)
Let us assume the validity of the condi-tions(a) dim(ran ( P )) < ∞ ,(b) P AP = 0 and P AP = 0 ,(c) (cid:82) ∞ d t (cid:107) P AP U t P AP (cid:107) < ∞ .Then, the following assertions hold.(1) For t > and all ψ ∈ H , we have lim λ → sup t ∈ [0 ,t ] (cid:13)(cid:13) U − t/λ W t/λ λ P ψ − e tK (cid:92) P ψ (cid:13)(cid:13) = 0 , (302) where K (cid:92) ∈ L ( H ) is the spectral average of K ∈ L ( H ) given by K := (cid:90) ∞ d t U − t P AP U t P AP . (303) (2) Let B ∈ L ( H ) satisfy B ≥ and [ B, U t ] = 0 for all t ∈ R , and let the operator S B ∈ L ( H ) be defined by S B := − (cid:90) ∞ d t Re[ P AP BU t P AP U − t ] , (304) where the integral is assumed to converge in norm. For t > and all ϕ, ψ ∈ H , wehave lim λ → sup t ∈ [0 ,t ] (cid:12)(cid:12)(cid:12)(cid:12) ( R t/λ λ P ϕ, BR t/λ λ P ψ ) − (cid:90) t d s (e sK (cid:92) P ϕ, S (cid:92)B e sK (cid:92) P ψ ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (305) Proof.
See Davies [13, 14]. (cid:3)
Recall from (153) that, for a ∈ { L, R } and β ∈ R , the function ψ βa : R → C is given by ψ βa ( t ) = ( δ R ,a , i a (cid:37) β ( h a )e ı th a i ∗ a δ R ,a ) . (306)We then have the following lemma which is used in the proof of Theorem 21. As usual,in the proofs below, the constant C can take different values at each place it appears.Moreover, the Laplace transform at the points ı (cid:15) with (cid:15) ∈ R is denoted by Ψ βa (ı (cid:15) ) := (cid:82) ∞ d t ψ βa ( t ) e − ı (cid:15)t and the temporal Fourier transform by ˆ ψ βa ( (cid:15) ) := (cid:82) ∞−∞ d t ψ βa ( t ) e − ı (cid:15)t .2 W. H. Aschbacher
Lemma 43 (Reservoir time correlation)
For a ∈ { L, R } and β ∈ R , there exists a con-stant C > s.t., for all t ∈ R , we have (cid:12)(cid:12) ψ βa ( t ) (cid:12)(cid:12) ≤ C (1 + | t | ) − / . (307) Moreover, for all (cid:15) ∈ R , it holds that Re[Ψ βa (ı (cid:15) )] = 12 ˆ ψ βa ( (cid:15) ) = (cid:40) − (cid:15) ) / (cid:37) β ( (cid:15) ) , | (cid:15) | < , | (cid:15) | ≥ . (308) Proof.
In order to prove the first assertion, we switch to the energy space representationof the subreservoir Hamiltonians h a from Lemma 29 already used in the derivation of (162)in order to make the decomposition ψ βa = π ( ϕ β + ¯ ϕ − β ) , where the function ϕ β : R → C ,independent of a ∈ { L, R } , is defined by ϕ β ( t ) := (cid:90) d e (1 − e ) / (cid:37) β ( e ) e ı te . (309)After the coordinate transformation e (cid:55)→ − e and one partial integration, we get ϕ β ( t ) = ı2 t + e ı t ı t (cid:88) i =1 ϕ β,i ( t ) , (310)where the functions ϕ β,i : R → C have the form ϕ β,i ( t ) := (cid:82) d e φ β,i ( e ) e − ı te , the integrands φ β,i : (0 , → [0 , ∞ ) are defined by φ β, ( e ) := 12 e − / (2 − e ) / (cid:37) β (1 − e ) , (311) φ β, ( e ) := 12 e / (2 − e ) − / (cid:37) β (1 − e ) , (312) φ β, ( e ) := β e / (2 − e ) / (cid:37) β (1 − e ) (cid:37) − β (1 − e ) , (313)and we used the fact that (cid:37) (cid:48) β ( e ) = − β(cid:37) β ( e ) (cid:37) − β ( e ) (the prime denotes the derivative w.r.t. e ). First, using | φ β, ( e ) | ≤ Ce − / for all e ∈ (0 , , we immediately get | ϕ β, ( t ) | ≤ C for all ≤ | t | ≤ . On the other hand, if | t | > , we write ϕ β, ( t ) = (cid:90) / | t | d e φ β, ( e ) e − ı te + (cid:90) / | t | d e φ β, ( e ) e − ı te . (314)Using directly | φ β, ( e ) | ≤ Ce − / for all e ∈ (0 , for the first integral and one partialintegration and | φ (cid:48) β, ( e ) | ≤ Ce − / for all e ∈ (0 , for the second integral, we can bound themodulus of both integrals by C | t | − / for | t | > . Moreover, for i = 2 , , since | φ β,i ( e ) | ≤ C for all e ∈ (0 , , we have | ϕ β,i ( t ) | ≤ C for all ≤ | t | ≤ . If | t | > , using | φ (cid:48) β,i ( e ) | ≤ C e − / for all e ∈ (0 , and one partial integration, we get | ϕ β,i ( t ) | ≤ C | t | − . These facts imply rom the microscopic to the van Hove regime Re[Ψ βa (ı (cid:15) )] = 12 (cid:90) ∞−∞ d t ψ βa ( t ) e − ı t(cid:15) (315) = lim δ → + π (cid:90) ∞−∞ d t e − δ | t | e − ı t(cid:15) (cid:90) − d e ξ β ( e ) e ı te , (316)where, in the first equality, we used that ψ βa ( − t ) = ¯ ψ βa ( t ) for all t ∈ R (leading to a Fouriertransform), and, in the second, we set ξ β ( e ) := (1 − e ) / (cid:37) β ( e ) and we used (162) andLebesgue’s theorem of dominated convergence. Since (cid:82) ∞−∞ d t e − δ | t | e ı t ( e − (cid:15) ) = 2 δ/ ( δ + ( e − (cid:15) ) ) , using Fubini’s theorem, and transforming the coordinates as e (cid:55)→ ( e − (cid:15) ) /δ , we get Re[Ψ βa (ı (cid:15) )] = lim δ → + π (cid:90) (1 − (cid:15) ) /δ − (1+ (cid:15) ) /δ d e ξ β ( (cid:15) + δe )1 + e . (317)If | (cid:15) | > , using ≤ ξ β ( e ) ≤ for all e ∈ [ − , , we directly get Re[Ψ βa (ı (cid:15) )] = 0 . For | (cid:15) | ≤ , we decompose the numerator as ξ β ( (cid:15) + δe ) = ξ β ( (cid:15) ) + [ ξ β ( (cid:15) + δe ) − ξ β ( (cid:15) )] . Due to | ξ β ( (cid:15) + δe ) − ξ β ( (cid:15) ) | ≤ Cδ | e | for all e in the integration interval, the difference term vanishesand we get Re[Ψ βa (ı (cid:15) )] = 2 ξ β ( (cid:15) ) . Since ξ β ( ±
1) = 0 , we arrive at (308). (cid:3)
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