aa r X i v : . [ qu a n t - ph ] F e b On the Dynamics of XY Spin Chains with Impurities
Giuseppe Genovese ∗ Insitut für Mathematik, Universität ZürichWinterthurerstrasse 190, CH-8057 Zürich, CH
July 30, 2018
Abstract
We provide a theoretical set up for studying the dynamics in quantum spin chain models with inhomogeneoustwo-body interaction. We frame in our formalism models that can be mapped into fermion systems withquadratic Hamiltonian, namely XY chains with transverse field. Local and global existence results of thedynamics are discussed.
In this paper we will treat time dependent Hamiltonians in many body quantum mechanics. Since thesubject is intrinsically problematic, we will analyse one of the simplest significant class of models available,namely quasi-free Fermi particle systems in one dimension.Nowadays quantum mechanics is a common background to diverse branches of science, and time dependentformalism finds different attitudes in the community. However it arises quite naturally in statistical mechanicsin the context of non equilibrium dynamics.The philosophy of non autonomous systems is in few words that the action of the environment on the systemis described in an effective theory by putting a time dependence in the Hamiltonian. This is fairly a coarseapproximation, since it does not take into account the structure of the environment, nor the effect thatthe system has on it. Nonetheless this picture is very helpful for many practical purposes, and it deservescertainly to be thoroughly studied to improve our comprehension of non equilibrium phenomena.The XY spin chains constitute a very appropriate model to study the basic features of many body quantumsystems, and indeed there is a sizeable literature on the topic (see for instance [1], [3], [4], [5], [7], [8], [14],[16], [17], [20], [23], [24], [28] and reference therein, albeit this list is far to be exhaustive). In particular thedynamical properties of such models are non trivial. For example the global transverse magnetisation of theXY model has been observed (see [5] or [4]) to approach a non equilibrium limit when a global transversefield is abruptly switched on. This kind of features has been the subject of an intensive study by the ’70: werefer for reviews to [19][4] and for more recent developments to [17][24] (and references therein).More recently spin chains have gained increasing interest from different perspectives: e.g. in the statisticalmechanics of non equilibrium phenomena they provide a simple quantum model for studying the effectof reservoirs on an extensive system (in the set-up presented for instance in [16] or [10]), along with theproperties of the quantum phase transition associated to these models (as for instance in [8]); from thequantum information side one dimensional spin systems are appealing test models for information transferprotocols, also because nowadays we have experimental realisations (with few atoms) of such systems [28].On the other hand impurities change drastically the scenario, both in equilibrium and in dynamics. Theydestroy many of the symmetries of the model, so rendering thermodynamics very difficult to analyse. At ∗ email: [email protected] Our starting point is a system of N quantum particles on the grid. We do not still specify the statistics ofthe particles: we describe them by creation and annihilation operators c, c † , satisfying the canonical anti-2ommutation rules ( + sign) or the canonical commutation rules ( − sign), respectively if we are dealing witha system of particle obeying to Fermi-Dirac or Bose-Einstein statistics: [ c j , c k ] ± = 0 , h c † j , c k i ± = δ jk . (1)We attach a quantum particle to each site, mathematically a Hilbert space H i , and the states space ofthe system is the Fock space F ≡ L N N Ni =1 H i . We are interested in general quadratic time dependentHamiltonians: H N ( t ) = X j,k ∈ Z J jk ( t ) c † j c k + 12 K ∗ jk ( t ) c † j c † k + 12 K jk ( t ) c j c k . (2) Hypotheses on the interaction.
We require:1. stability: J jk ( t ) ≥ J > −∞ ;2. ℓ finiteness: sup k P j | J jk ( t ) | , sup j P k | J jk ( t ) | < ∞ and sup k P j | K jk ( t ) | , sup j P k | K jk ( t ) | < ∞ ,uniformly in t ;3. boundedness and piecewise continuity in t ∈ [ t , T ) . The last hypothesis will allow us to use Fourier inverse transform also in t (in the meaning of distributions).Of course J ( t ) is a Hermitian matrix and K ( t ) is symmetric or anti-symmetric according to the statistics weare considering.These requirements make it possible to pass to Fourier transform: the Hamiltonian becomes H N ( t ) = 1 N X q,p ∈ Z ( N ) α qp ( t ) a † q a p + 12 β qp ( t ) a † q a † p + 12 β ∗ qp ( t ) a q a p , where Z ( N ) denotes the set { (2 πk ) /N, k = 1 , . . . , N } and ( c j = N P q ∈ Z ( N ) e − iqj a q c † j = N P q ∈ Z ( N ) e iqj a † q ; ( a q = P Nj =1 e iqj c j a † q = P j e − iqj c † j (3)and α qp ( t ) ≡ X j,k J jk e i ( qj − pk ) ; β qp ( t ) ≡ X j,k K jk e i ( qj + pk ) . (4)The fundamental feature of this Hamiltonian is that for t ′ = t ′′ in general [ H N ( t ′ ) , H N ( t ′′ )] = 0 . As it is wellknown, this condition implies a non trivial dynamics. We can regard the time evolution of our system as adynamical system indexed by q ∈ Z ( N ) with the Heisenberg equations of motion d t a q ( t ) = [ a q ( t ) , − iH ( t )] , q ∈ Z ( N ) . These are the equations for the evolution semigroup. We will show that it is actually given by a one-parametersemigroup of Bogoliubov-Valatin transformations [6], namely a q ( t ) = 1 N X p A qp ( t ) a p + B qp ( t ) a † p , (5)where A qp ( t ) , B qp ( t ) become L function on [0 , π ] × [0 , π ] for N → ∞ uniformly in t and A qp ( t ) = δ qp and B qp ( t ) = 0 . We seek a solution satisfying id t a q ( t ) = 1 N X p i ˙ A qp ( t ) a p + i ˙ B qp ( t ) a † p . (6)3e want to evaluate the commutator in the Heisenberg equations and compare it with the last formula.This is easily done: we notice h a p , a † q ′ a p ′ i ∓ = δ q ′ p a ′ p h a p , a † q ′ a † p ′ i ∓ = δ q ′ p a † p ′ ± δ pp ′ a † q ′ [ a p , a q ′ a p ′ ] ∓ = 0 h a † p , a † q ′ a p ′ i ∓ = ± δ pp ′ a † q ′ h a † p , a † q ′ a † p ′ i ∓ = 0 (cid:2) a † p , a q ′ a p ′ (cid:3) ∓ = δ pp ′ a q ′ ± δ pq ′ a p ′ , and hence id t a q ( t ) = 1 N X q ′ p ′ a p ′ (cid:18) A qq ′ ( t ) α q ′ p ′ ( t ) + 12 B qq ′ ( t )( β ∗ q ′ p ′ ( t ) ± β ∗ p ′ q ′ ( t )) (cid:19) + 1 N X q ′ p ′ a † p ′ (cid:18) ± B qq ′ ( t ) α q ′ p ′ ( t ) + 12 A qq ′ ( t )( β q ′ p ′ ( t ) ± β p ′ q ′ ( t )) (cid:19) . This leads us to the following linear system of first order ordinary differential equations ( i ˙ A qp ( t ) = N P q ′ A qq ′ ( t ) α q ′ p ( t ) + B qq ′ ( t ) β ∗ q ′ p ( t ) i ˙ B qp ( t ) = ± N P q ′ B qq ′ ( t ) α q ′ p ( t ) + A qq ′ ( t ) β q ′ p ( t ) . (7)So the system (7) represents the evolution equations for the one-parameter semigroup of Bogoliubov-Valatintransformations that gives the quantum dynamics. It is perhaps remarkable that this structure, quite naturalfor quadratic Hamiltonians on the grid (see also [21] and [8]), emerges also in other scenarios, as shown forinstance in the more involved problem of effective dynamics for mean field interacting particles [26].As long as N remains finite, classical theorems on linear systems of first order ordinary differential equationsassure existence and uniqueness of a global solution in the interval of definition [ t , T ] . This solves completelythe problem of existence of dynamics at finite size. On the other hand, to find an explicit form for the timeevolution is a more challenging task. A general approach in literature to cope with this kind of problems isby series expansions of the solution [11].Of course the thermodynamic limit N → ∞ requires much more attention. In the limit we are led to considerthe integro-differential system i ˙ A ( q, p ; t ) = R π dq ′ ( A ( q, q ′ ; t ) α ( q ′ , p ; t ) + B ( q, q ′ ; t ) β ∗ ( q ′ , p ; t )) ± i ˙ B ( q, p ; t ) = R π dq ′ ( B ( q, q ′ ; t ) α ( q ′ , p ; t ) + A ( q, q ′ ; t ) β ( q ′ , p ; t )) A ( q, p ; t ) = δ ( q − p ) B ( q, p ; t ) = 0 , (8)where α ( q ′ , p ; t ) , β ( q ′ , p ; t ) are C q,p C t ([0 , π ] × × [ t , T ]) functions, obtained as natural limit of α q ′ p ( t ) , β q ′ p ( t ) (that is straightforward by our hypotheses). It is suggestive here to represent the above system as ddt A = Γ( t ) A ( t ) , (9)that stands for ddt (cid:18) A ( q, p ; t ) B ( q, p ; t ) (cid:19) = Z π dq ′ (cid:18) − iα ( q ′ , p ; t ) − iβ ∗ ( q ′ , p ; t ) ∓ iβ ( q ′ , p ; t ) ∓ iα ( q ′ , p ; t ) (cid:19) (cid:18) A ( q, q ′ ; t ) B ( q, q ′ ; t ) (cid:19) . (10)4et us analyse at first the simplest case in which Γ( t ) = Γ( t ) is constant in time. It is evidently a compactlinear map from the space L ([0 , π ] ) of square integrable functions on [0 , π ] × [0 , π ] into itself. Thedynamics exists globally, and the semigroup is given by exponentiation. This is the starting point of theanalysis performed in [21].In dealing with time dependent generators of the dynamics, the easiest occurrence is that as long as t runsin its interval of definition, Γ( t ) remains bounded and the solution keeps lying into its domain. By ourhypotheses this is straightforward to verify in every interval [ t ′ , t ′′ ] ⊂ [ t , T ) such that the couplings arecontinuos. In this way Γ( t ) is a strongly continuous operator from L C t ([0 , π ] × [ t ′ , t ′′ ]) into itself and onecan derive local existence of the dynamics from general theorems (for detailed discussion on the topic see forinstance [22][27]): Theorem (Local Existence) . Consider a quadratic Hamiltonian as in (2), with summable and stable cou-plings and piece-wise continuous dependance on time in t ∈ [ t , T ) , according to Hypotheses 1, 2, 3. Then ineach interval [ t ′ , t ′′ ] ⊂ [ t , T ) in which the Hamiltonian is continuous in t , the quantum dynamics is definedas one-parameter continuously t -differentiable semigroup of Bogoliubov–Valatin transformations. What is more challenging of course is to extend the local existence result to a global one. As a preliminaryobservation, we see that Γ( t ) represents a family of compact operators in L ([0 , π ] × ) for t ∈ [ t , T ) ,uniformly bounded in [ t , T ] . Therefore the spectrum of Γ( t ) is uniformly bounded in t : the Hille-Yoshidatheorem allows us to conclude that Γ( t ) is a family of generators of strongly continuous semigroups ofcontractions and there is a number w such that the resolvent set of Γ( t ) is contained in [ w, + ∞ ) uniformly in t .Another crucial property we have for free is that the domain of the generators is the entire space L ([0 , π ] × ) independently of t . Unfortunately, since the family Γ( t ) is not by assumptions strongly continuous, this isnot enough to ensure existence of the dynamics for all times. Anyway we can recover a similar property byrequiring that in each discontinuity point of Γ( t ) in [ t , T ) the left and right limit exists. In this way, wecan establish a local existence theorem locally around each discontinuity, because each interval containinga discontinuity splits into two intervals in which the previous local theorem holds, and then we can pastetogether the contributions. We sketch an argument somewhat more explicit: let t ∗ be a discontinuity point,we focus on the interval [ t ∗ − δ, t ∗ + δ ] for an arbitrary δ > , such that the unique discontinuity falls at time t ∗ . We write for every t ∈ [ t ∗ − δ, t ∗ + δ ] A ( t ) = A ( t ∗ − δ ) + Z tt ∗ − δ dt ′ Γ( t ′ ) A ( t ′ )= A ( t ∗ − δ ) + Z Tt ∗ − δ dt ′ χ ([ t ∗ − δ, t ])Γ( t ′ ) A ( t ′ ) . Now we can make discrete approximations of Γ( t ) in [ t ∗ − δ, t ∗ + δ ] (that is possible in virtue to the requiredregularity): for a given even integer N > we divide [ t ∗ − δ, t ∗ + δ ] in N intervals of length τ = δN and wehave a discretised version of the last formula: A h − τ N X k =1 Γ kh A k = A ( t ∗ − δ ) , with t = hτ , A k = A ( kτ ) and Γ kh = I ( k ≤ h )Γ( kτ ) . This is a linear system of equations, where Γ is adiagonal matrix with only the first h entries non zero; it has a unique solution provided that det( I − τ Γ) = 0 .In our setting, due to the freedom in the choice of δ , it is not hard to verify this condition uniformly in N .Then one can take the limit N → ∞ to have the existence of the solution for the flux.Thus we have local existence also in presence of (bounded) discontinuities and we can extend the previoustheorem to all intervals [ t ′ , t ′′ ] ⊂ [ t , T ) . In order to recover the whole [ t , T ] we need a compactificationcondition, namely that the limit lim t → T − Γ( t ) exists (it is finite by hypothesis). In this way one obtainsexistence of the flux for all t ∈ [ t , T ] by standard arguments.We summarise our results in the following 5 heorem (Global Existence) . Consider a quadratic Hamiltonian as in (2), with summable and stable cou-plings and piece-wise continuous dependance on time in t ∈ [ t , T ) , according to Hypotheses 1, 2, 3, witha countable number of discontinuity points t ∗ i . Additionally we assume that the left and right limits of theHamiltonian in each discontinuity point t ∗ i exist, and furthermore the existence of the left limit t → T − as well. Then the dynamics is globally defined in [ t , T ] as one-parameter semigroup of Bogoliubov-Valatintransformations. The main advantage in this context is that one never deals with unbounded generators. This permits us toprofit from piece-wise compactness, and so we need to require just suitable conditions to glue the solutions.The very question of existence of the dynamics therefore relies in our opinion only in the asymptotic behaviourfor t → T . We will discuss this point later on with the aid of a specific example. The derivation presented above is very general in its simplicity, not depending on the particular quantumstatistics of the particles. However, being interested in quantum spin, hereafter we will deal exclusively withfermions.Of course, all the information for the temporal dependance of observables, and thus, the state of the system,is encoded into the equations (7) at finite size and (8) in the thermodynamic limit.In order to clarify this point let us introduce some standard notations. We consider our system at initialtime t in equilibrium at inverse temperature β . The partition function reads Z N ( β ) ≡ Tr h e − βH N ( t ) i , and the thermal average of an operator A is defined by h A i β,t ≡ Z N ( β ) Tr h Ae − βH N ( t ) i . In particular, due to the fermion statistics, we have (cid:10) a † q a p (cid:11) β,t = δ qp e βω p (cid:10) a † q a † p (cid:11) β,t = 0 h a q a p i β,t = 0 , where ω p are the initial eigen-frequencies of the system. Since the Hamiltonian is quadratic we know thatat least in principle it can be set in a diagonal form by a unitary transformation.Following the classical reference [25], we introduce a generic time dependent observable as a function f N from a subset of the grid S ⊆ { , .., N } over the algebra of creation and annihilation operator. We allow thisfunction to have also a possible explicit dependence on time, even if this will play no role in our exposition.Thus we write f N ( S ; t ) = X V ⊆ S f ( V ; t ) 1 N | V | X q j ,p j ,j ∈ V : Y j ∈ V η q j ξ q j : e P j i ( p j − q j ) j + h.c. and : : denotes as usual the Wick product for the operators η q = a † q + a − q ξ q = a †− q − a q . For sake of brevity we will skip hereafter the locution + h.c. to indicate that every operator must be selfadjoint. The crucial point here is that the Hamiltonian is quadratic in creation and annihilation operators.6hus the thermal average is evaluated by using the Wick theorem: h f ( S ; t ) i β,t = X V ⊆ S f ( V ; t ) 1 N | V | X q j ,p j ,j ∈ V * : Y j ∈ V η q j ξ q j : + β,t e P j i ( p j − q j ) j + h.c. = X V ⊆ S f ( V ; t ) 1 N | V | X q j ,p j ,j ∈ V e P j i ( p j − q j ) j X all pairings Π V ( − π ′ Y ( h,k ) ∈ Π V h : η q h ξ q k : i β,t where π ′ gives the parity of a given pairing Π V of the set { q j , p j : j ∈ V } . However we can take advantageby switching to the Heisenberg picture: we transfer the time dependence from the thermal average to theoperators, and compute the thermal average at initial time t . Since time evolution is given by a Bogoliubov-Valatin semigroup, we have η q ( t ) = X p (cid:0) A ∗ qp ( t ) + B − q,p ( t ) (cid:1) a † p + (cid:0) B ∗ qp ( t ) + A − q,p ( t ) (cid:1) a p ξ q ( t ) = X p (cid:0) A ∗− q,p ( t ) − B q,p ( t ) (cid:1) a † p + (cid:0) B ∗− q,p ( t ) − A q,p ( t ) (cid:1) a p and so h f ( S ; t ) i t = X V ⊆ S f ( V ; t ) 1 N | V | X q j ,p j ,j ∈ V e P j i ( p j − q j ) j X all pairings Π V ( − π ′ Y ( h,k ) ∈ Π V h : η q h ( t ) ξ q k ( t ) : i β,t For a given pair ¯ q, ¯ p the only contributing term is σ ¯ q ¯ p ( β, t ) = 1 N X q ′ A ∗ ¯ qq ′ ( t ) B ∗ q ′ − ¯ p ( t ) − B − ¯ qq ′ ( t ) A ¯ q ′ p ( t ) + B ∗− ¯ qq ′ ( t ) B − ¯ q ′ p ( t ) − A ∗ ¯ qq ′ ( t ) A ¯ q ′ p ( t )1 + e βω q ′ . (11)Therefore the time dependence of the state of the system is all in that formula. We have obtained h f ( S ; t ) i t = X V ∈ S f ( V ; t )Σ( V, β, t ) , (12)with Σ( V, β, t ) = 1 N | V | X q j ,p j ,j ∈ V e P j i ( p j − q j ) j X all pairings Π V ( − π ′ Y ( h,k ) ∈ Π V σ ¯ q h ¯ p k ( β, t ) . (13)This is the time dependent state of the system. We notice that as long as N remains finite, the dynamics iswell defined and our derivation of the state is exact. The problem arises as N → ∞ : in this case, as we haveseen in the previous Section, we need a further assumption on the interaction in order to have existence anduniqueness of the flow.To provide a simple example, we see how the number operator N = N P j c † j c j reads: hN i t = hN ( t ) i = 1 N X j X q,p e ij ( q − p ) (cid:10) a † q ( t ) a p ( t ) (cid:11) = Z π dq e βω q Z π dp (cid:0) | A ( q, p, t ) | − | B ( q, p, t ) | (cid:1) . (14)As a final remark we observe that in the hypotheses of the global existence theorem, the state asymptoticallyapproaches a limit, that is the stationary state of the system. But it is the nature of the time dependentHamiltonian that determines whether it is an equilibrium state, viz. an ergodic theorem holds, or it is a nonequilibrium stationary state. An example of this phenomenon is global versus local transverse perturbationsin the XY chains, as it will be further discussed below.7 Preliminaries on Spin Chains
Now we turn our attention to quantum spin systems, ruled by nearest neighbour two body ferromagneticinteraction on Z . We will be interested specifically in those models admitting a mapping into quasi freefermions analysed before. The most general Hamiltonian for describing such a class of systems is H N = − N ∗ X j =1 ( J xj ( t ) S xj S xj +1 + J yj ( t ) S yj S yj +1 ) − N X j =1 h zj ( t ) S zj . (15)The space this operator acts on is a tensor product of / -spin vector spaces H j = C , each spanned by thevectors spin up and spin down : H N ≡ N Nj =1 H j = C ⊗ N ; the corresponding matrix algebra of × matrices GL ( C ) is spanned by the Pauli matrices σ x , σ y , σ z plus the identity I . As usual the thermodynamic limitfor the system is performed in the Fock space, defined as F ≡ L N H N . The spin operators attached toeach site j , S xj , S yj , and S zj , are defined in terms of Pauli matrices as S ij = σ i , i = x, y, z . The boundaryconditions are defined by the value of N ∗ : e.g. if N ∗ = N − we are dealing with open (or free) boundaryconditions at extrema, while if N ∗ = N we mean that N + 1 = N , so periodic boundary conditions areassumed. The following notations is usually adopted: A j ≡ I ⊗ I ⊗ ... ⊗ A ⊗ I ⊗ ... ⊗ I , i.e. the operator A j acts as A on the Hilbert space of the j -th spin and as the identity on the others. So each observable, forfinite N , belongs to the tensor product algebra GL ( C ) ⊗ N . Finally we will assume the same space and timeregularity for the couplings as in Section 2.As mentioned above, these spin chains correspond to particular quadratic forms in the Fermi operators, themapping being given by the Jordan-Wigner transformation, introduced in [18]. We put ( c j = ( σ xj − iσ yj ) N j − k =1 ( − σ zk ) c † j = ( σ xj + iσ yj ) N j − k =1 ( − σ zk ) , and their inverses σ xj = ( c † j + c j ) N j − k =1 ( − σ zk ) σ yj = − i ( c † j − c j ) N j − k =1 ( − σ zk ) σ zj = 2 c † j c j − I . It is easily seen that the c operators satisfy fermionic anti-commutation relations: [ c j , c k ] + = 0 , [ c † j , c k ] + = δ jk . (16)The morphism of algebras naturally induces a morphism of spaces: each c j ( c † j ) acts on a two level vectorspace H ′ j as an annihilation (creation) operator, spanned by the vectors | i j (hole) and | i j (particle), asusual in the theory of Fermi systems. It is important to notice that this map does not preserve locality.Therefore, up to a constant term, the Hamiltonian (15) is transformed in the following quadratic Hamiltonianfor a one dimensional Fermi gas: H N = − N ∗ X j =1 ( g + g xyj ( t ))( c † j c j +1 − c j c † j +1 ) − N ∗ X j =1 γ xyj ( t )( c † j c † j +1 − c j c j +1 ) − N X i =1 ( h + h zj ( t )) c † j c j . (17)Here we will make a particular choice: we will look at impurities as (time dependent) perturbations of theisotropic XY Hamiltonian in a transverse field. The reason why we opt for perturbing around the isotropicinstead of the (somehow more general) anisotropic model is merely technical and it will be clear in thefollowing. We put g + g j = J xj + J yj , γ j = J xj − J yj , (18)8nd P j g j = P j γ j = P j h j = 0 . In other words, the impurities are thought as fluctuations around the XXHamiltonian. However in our framework it is more convenient to work in the Fourier space. By setting ˜ g q − p = N ∗ X k =1 g k,k +1 e ik ( q − p ) ; ˜ γ q + p = N ∗ X k =1 γ k,k +1 e ik ( q + p ) ; ˜ h q − p = N ∗ X k =1 h k e ik ( q − p ) , the Hamiltonian (17) can be written as H N = 1 N X q,p ∈ Z ( N ) h ( g cos q + h ) δ qp + h q − p ( t ) + e iq g q − p ( t )(1 + e − i ( q − p ) ) i a † q a p + (cid:20) γ q + p ( t ) (cid:18) e iq − e − ip (cid:19)(cid:21) a † q a † p + h.c. For spin chains we can identify α qp = ( g cos q + h ) δ qp + h q − p ( t ) + e iq g q − p ( t )(1 + e − i ( q − p ) ) , (19) β ∗ qp = γ q + p ( t ) (cid:18) e iq − e − ip (cid:19) . (20)We would like to apply the theory developed in Section 2 to the last class of Hamiltonians. Rather then thedynamical variables A qp ( t ) , B qp ( t ) , it turns out more convenient to deal with the site variables X k,q ( t ) = 1 N X q ′ A qq ′ ( t ) e − ik ( q − q ′ ) , V k,q ( t ) = 1 N X q ′ B qq ′ ( t ) e − ik ( q + q ′ ) . (21)These are in fact conceived to take into account the perturbation in the dynamics due to the impurities.The systems of equations (7) or (8) can be written in terms of these new variables as i ˙ A = Hom( A, B ) + Inhom(
X, V ) i ˙ B = Hom( A, B ) + Inhom(
X, V ) where with Hom (respectively
Inhom ) we have denoted the part of (7), (8) depending on homogeneous(inhomogeneous) couplings. Here it is important that X k ( t ) , V k ( t ) enter only in the inhomogeneous part ofthe last expressions. By Duhamel formula we obtain A qp ( t ) and B qp ( t ) in an integral form A qp ( t ) = e − iω p ( t − t ) δ qp − iN X k e − i ( q − p ) k Z tt dt ′ e − iω p ( t − t ′ ) h h k ( t ′ ) X ′ k,q ( t ′ )+ g k ( t ′ )( e − iq X ′ k +1 q ( t ′ ) + e ip X ′ k,q ( t ′ )) + γ k ( t ′ )( e − iq V ′ k +1 q ( t ′ ) − e ip V ′ kq ( t ′ )) i B qp ( t ) = − ie − iω p ( t − t ) δ q, − p + iN X k e − i ( q + p ) k Z tt dt ′ e − iω p ( t − t ′ ) h h k ( t ′ ) V ′ kq ( t ′ )+ g k ( t ′ )( e − iq V ′ k +1 q ( t ′ ) + e ip V ′ kq ( t ′ )) + γ k ( t ′ )( e − iq X ′ k +1 q ( t ′ ) − e ip X ′ k,q ( t ′ )) i . Here one can see the structure of the interaction. The quantities g k ( t ) , γ k ( t ) are linked respectively to thevariable X, V of the sites k, k + 1 , which reflects the nearest neighbour two body inhomogeneous interaction.On the other hand the transverse field is associated to the each single site. The eigen-frequencies of therelated homogeneous system ω p depend only on g, h . Of course we could allow g, h to depend on time aswell, simply by substituting every ω p ( t − t ) with R tt dt ′ ω p ( t ′ ) . Nevertheless this would just make the theory9ore complicated, without adding any substantial new feature. Therefore we will suppose that g, h arefixed. With a further step we can obtain a closed system of integral equations for the site variables, simplyby using the definition of X k ( t ) and V k ( t ) . We will skip all the calculations and give only the result in thelimit N → ∞ : X k ( q ; t ) = e − iω q ( t − t ) − i X j Z tt dt ′ e iq ( k − j ) − ih ( t − t ′ ) h e iφ k − j − J k − j − ( t − t ′ ) (cid:0) g j ( t ′ ) X j ( q ; t ′ ) − γ j ( t ′ ) V j ( q ; t ′ ) (cid:1) + e iφ k − j J k − j ( t − t ′ ) (cid:0) h j ( t ′ ) X j ( q ; t ′ ) + e − iq g j ( t ′ ) X j +1 ( q ; t ′ ) + e − iq γ j V j +1 ( q ; t ′ ) (cid:1) i (22) V k ( q ; t ) = − ie − iω q ( t − t ) + i X j Z tt dt ′ e − iq ( k − j ) − ih ( t − t ′ ) h e − iφ k − j − J ∗ k − j − ( t − t ′ ) (cid:0) g j ( t ′ ) V j ( q ; t ′ ) − γ j ( t ′ ) X j ( q ; t ′ ) (cid:1) + e − iφ k − j J ∗ j − k ( t − t ′ ) (cid:0) h j ( t ′ ) V j ( q ; t ′ ) + e − iq g j ( t ′ ) V j +1 ( q ; t ′ ) + e − iq γ j X j +1 ( q ; t ′ ) (cid:1) i . (23)Here we have used (see for instance [13]) π Z π dpe − ipn e − i ( g cos p + h )( t − t ′ ) = e − ih ( t − t ′ )+ iφ n J n ( g ( t − t ′ )) , where J n ( t ) are the Bessel functions of first kind and order n (see [2] for definitions and properties), whichappear quite naturally in this context, and φ n = nπ/ . It is remarkable that we can profit from this exactrepresentation only by adopting the assumption γ = 0 : in this way the unperturbed spectrum is proportionalto cos q ; otherwise it assumes a more involved form and we can no longer identify a known special function.This essentially concludes our analysis. In the rest of this Section we will point out some further mathematicaldetail in the theory.The variables X, V are the marginal Fourier transform of the former
A, B with respect to one argument. Inthe thermodynamic limit they are ℓ ,k L ,q C t ( Z × [0 , π ] × [ t , T ]) functions: this is the natural space whereseeking a solution. In addition it is worthwhile to notice that one can recover the A, B variables via a simpleFourier transform in order to obtain the formula for the state of the system as in (11).The main feature of our characterisation of the dynamics is to capture the properties of the chain right viathe presence of the Bessel functions. They witness indeed the internal structure of the chain that matcheswith the external perturbation given by the impurities. So we have a very interesting effect, that can bedescribed as follows: ergodicity is easily broken by the dynamics as one can see by the non thermalisation ofobservables, as the local transverse magnetisation (see for instance [1] or the Appendix B); and indeed thereis a strong memory effect given by the time-convolution integral in (22) and (23) with the Bessel functions J n ( t ) , which naturally gives the time irreversibility of the dynamics.Finally it is useful to recall the following formula (from formulas 6.671 in [13]): ˆ J n (Ω) ≡ Z + ∞ dτ J n ( gτ ) e i Ω τ = χ ( | Ω | ≤ g ) e in arcsin(Ω /g ) + ig n e iφ n χ ( | Ω | ≥ g ) / ( p Ω − g + Ω) n q | Ω − (cid:0) g (cid:1) | , (24)that permits us to represent J ( t ) = Z + ∞ d Ω e − i Ω t ˆ J n (Ω) . Thus we can decompose the perturbations in frequencies (via Fourier transform) and study the problem ofresonances between the three quantities in the play: the proper frequencies of the perturbation, the strengthof the perturbation, the proper frequencies of the chain. Such a scheme is particularly evident in the instanceof perturbations periodic or quasi periodic in t . This is just a conceptual picture and to implement it in aconcrete example is quite a hard task, even in the simplest cases. We will give a hint of that in the sequel.Nevertheless we believe that it is rather explicative: in general of course the inverse of a continuous functionvia Fourier transform is not integrable; in our framework this case may arise only because of resonances,manifesting themselves by means of non integrable singularities. Existence of the dynamics will correspondto a proper renormalisation of these resonances. 10 Inhomogeneous Transverse Field in the XY Chain
In this last Section we will focus on the explicit form the system of equations (22), (23) takes when theimpurities act only at the level of the transverse field. Our treatment will be twofold: at first we will dealwith the XY chain, giving a more exhaustive account of the choice γ = 0 ; then we will turn to the dynamicsof the XX chain, that has a considerably simpler form, nonetheless capturing all the main features (encodedin the Bessel functions). We will show explicitly the mechanism behind existence of local and global solutionsby briefly discussing the case of oscillation of a single impurity.For the XY model with impurities in the transverse field we have α qp = g (cos q + h ) δ qp + h q − p ( t ) , β qp = − iγ sin qδ p, − q . (25)Therefore the evolution equation for the dynamics reads i ˙ A qp ( t ) = A qp ( t )( g cos p + h ) + iγ sin pB q, − p ( t ) + 1 N X q ′ A qq ′ ( t ) h q ′ − p ( t ) (26) i ˙ B qp ( t ) = − B qp ( t )( g cos p + h ) − iγ sin pA q, − p ( t ) − N X q ′ B qq ′ ( t ) h q ′ − p ( t ) . (27)It is convenient to analyse in primis the unperturbed system ( h = 0 ) and reduce the system in diagonalform. That is done via the same transformation one introduces in order to diagonalise the Hamiltonian (thisprocedure appeared for the first time in the paper of Lieb, Mattis and Schultz [20] and it is reviewed inAppendix A). It is more practical to arrange (26) and (27) in the vectorial form i ˙ A qp ( t ) = Γ p A qp ( t ) , with A qp = A q,p B q,p A q, − p B q, − p and Γ p = ( g cos p + h ) 0 0 − iγ sin p − ( g cos p + h ) − iγ sin p iγ sin p ( g cos p + h ) 0 iγ sin p − ( g cos p + h ) . This is in fact the matrix to be diagonalised in order to find the spectrum of the model. This is done bymeans of the unitary matrix (47), and the eigenvalues for the energy levels are given by (46). Thus thesystem reads i ˙ A ′ qp ( t ) = E p − E p E p
00 0 0 − E p A ′ qp ( t ) solved by A ′ qp = e − iE p ( t − t ) A ′ qp ( t ) , A ′ qp ( t ) = δ qp cos φ q ,B ′ qp = e iE p ( t − t ) B ′ qp ( t ) , B ′ qp ( t ) = − iδ q, − p sin φ q . In the original variables the solution is instead given by A qp ( t ) = δ qp (cid:18) cos( E p ( t − t )) − i g cos p + hE p sin( E p ( t − t )) (cid:19) B qp ( t ) = − γ sin pE p δ q, − p sin( E p ( t − t )) .
11e note that, by plugging the last two expression in (14), we easily recover the non-ergodicity of thetransverse magnetisation m z firstly achieved in [5] (see also [19][17]). We have Z dp | A qp ( t ) | = cos ( E q ( t − t )) + ( g cos q + h ) E q sin ( E p ( t − t )) + Z dp | B qp ( t ) | = (cid:18) γ sin qE q sin( E q ( t − t )) (cid:19) , and so h m z i = Z dq e βE q cos ( E q ( t − t )) (cid:20) (cid:18) ( g cos q + h ) E q − γ sin qE q (cid:19) tan ( E q ( t − t )) (cid:21) (28)We notice two important features of this formula: because of the − sign in (14), the limit h → does notgive the equilibrium value m z ; in the isotropic limit γ → one recovers the thermalisation of m z , whichmirrors [ H XX , m z ] = 0 , the conservation of the magnetic momentum in the XX chain. As for the longitudinalmagnetisation the situation is more involved, due to its more complicate representation in terms of Fermioperators. However it is remarkable that this observable approaches its equilibrium value for t → ∞ [3][17].When a perturbative transverse field is added, we describe the system in terms of X, V variables. Since therelations linking
A, B with
X, V are linear, we have X ′ k,q ( t ) = X q ′ A ′ qq ′ ( t ) e − ik ( q ′ − q ) , V ′ k,q ( t ) = X q ′ B ′ qq ′ ( t ) e − ik ( q ′ + q ) , (29)and also the following equations for the eigenvariables i ˙ A ′ qp ( t ) = E p A ′ qp ( t ) + 1 N X k h k ( t ) X ′ k,q ( t ) e − ik ( q − p ) (30) i ˙ B qp ( t ) = − E p B ′ qp ( t ) − N X k h k ( t ) V ′ k,q ( t ) e − ik ( q − p ) . (31)Now these equations are decoupled and they can be solved separately by Duhamel principle: A ′ qp ( t ) = e − iE p ( t − t ) δ qp cos φ q − iN X k e − i ( q − p ) k Z tt dt ′ e − iE p ( t − t ′ ) h k ( t ′ ) X ′ k,q ( t ′ ) (32) B ′ qp ( t ) = − ie − iE p ( t − t ) δ q, − p sin φ q + iN X k e − i ( q + p ) k Z tt dt ′ e − iE p ( t − t ′ ) h k ( t ′ ) V ′ kq ( t ′ ) . (33)As we have already seen (here ω p = E p ), there are closed equations for X ′ k,q , V ′ kq : X ′ k,q ( t ) = e − iω q ( t − t ) cos φ q − iN X j Z tt dt ′ X p e i ( q − p )( k − j ) e − iω p ( t − t ′ ) ! h j ( t ′ ) X ′ jq ( t ′ ) (34)and V ′ kq ( t ) = − ie − iω q ( t − t ) sin φ q + iN X j Z tt dt ′ X p e i ( q + p )( k − j ) e − iω p ( t − t ′ ) ! h j ( t ′ ) V ′ jq ( t ′ ) . (35)These equations, although formally identical to the ones we got in the previous Section, are technically moreinvolved, essentially because lim N N X p e i ( q ± p )( k − j ) e − iE p ( t − t ′ ) = 12 π Z π − π dpe i ( q ± p )( k − j ) e − iE p ( t − t ′ ) , E p on cos p , is not a known function. Thus for the XY chain ( i.e. for γ = 0 ) we still have a clear theoretical picture for the dynamics, but in general we do not have an exactanalytical formulation of it.In the XX Hamiltonian there is a further simplification: if we set ω q = ( g cos q + h ) , we have α qp ( t ) = ω q δ qp + h q − p ( t ) , β qp = 0 . (36)Since the term β qp vanishes, equations (7), (8) decouple (they are in fact the same equation), and thedynamics of the system is described by A qp ( t ) . Hence the equation we are interested in is i ˙ A qp ( t ) = A qp ω q + 1 N X q ′ A qq ′ ( t ) h q ′ − p ( t ) . (37)In terms of the site variables previously introduced, this translates into a system of equations involving onlythe X k,q ( t ) associated to the k -th impurity. At finite N we get X k,q ( t ) = e − iω q ( t − t ) − iN X j Z tt dt ′ X p e i ( q − p )( k − j ) e − i ( g cos p + h )( t − t ′ ) ! h j ( t ′ ) X jq ( t ′ ) . (38)Passing to the limit N → ∞ equations (22), (23) simplify a lot X k ( q, t ) = e − iω q ( t − t ) − i X j e iq ( k − j ) Z tt dt ′ e − ih ( t − t ′ ) J j − k ( g ( t − t ′ )) h j ( t ′ ) X j ( q, t ′ ) . (39)The simplest, but already quite rich, case to study in this context is the one of a single impurity localised inthe site ˆ k , h j ( t ) = δ j ˆ k h ( t ) , for which X k ( q ; t ) = e − iω q ( t − t ) − ie iq ( k − ˆ k ) Z tt dt ′ e − ih ( t − t ′ ) J k − ˆ k ( g ( t − t ′ )) h ( t ′ ) X ˆ k ( q ; t ′ ) . (40)It is immediate to see that the state of the system is completely determined by X ˆ k ( q ; t ) ≡ X ( q ; t ) . This isdefined by X ( q ; t ) = e − iω q ( t − t ) − i Z tt dt ′ J ( g ( t − t ′ )) h ( t ′ ) X ( q ; t ′ ) . (41)Here we have incorporated in the field its zero mode, and so our unperturbed system has now eigen frequencies ω q = g cos q , a choice somehow natural in this simplified situation.We can now differentiate this integral equation in order to obtain the linear Schrödinger equation i∂ t X ( q ; t ) = ( ω q + h ( t )) X ( q ; t ) − g Z tt dτ ( gJ ( g ( t − τ )) − iω q J ( g ( t − τ ))) h ( τ ) X ( q ; τ ) , (42)which reads more familiarly in the configuration space of the chain Z i∂ t X k ( t ) = ( − g ∆ + h ( t )) X k ( t ) − W [ X, ∆ X ; t ] . (43)Here − ∆ is the Laplace operator on Z , with eigenvalues cos q , and W [ X, ∆ X ; t ] = g Z tt dτ ( gJ ( g ( t − τ )) − iJ ( g ( t − τ )) ∆) h ( τ ) X k ( τ ) . The equation (43) describes the evolution of a single spin coupled to an external field h ( t ) and to the restof the chain via the parameter g . It is a Schrödinger equation with Floquet Hamiltonian, but the time13ependance exhibits explicitly a gentle memory effect via the convolution operator. This eventually makesit easier to study with respect to the case of the purely multiplicative potential. In its integral form it canbe symbolically rewritten as X = X + KX , where K is an integral operator acting on continuous functions(in t ). Equations of this form are usually approached by seeking a solution like X = (1 − K ) − X by meansof Fredohlm theory (for an exhaustive treatment we refer to the original paper by I. Fredholm and themonograph by I. Gohberg, S. Goldberg and M. A. Kaashoek [12]). Now we switch on the field, for examplecontinuously, at time t , and let the system evolve up to time T > t . The integral operator has a continuouskernel in each closed interval [ t , T ] : this automatically implies existence of the solution and we also havethe explicit form of it by Fredholm expansion. This is morally the local existence theorem of Section 2. Butas we send T → ∞ the situation becomes different. Continuity and boundedness of the perturbation are nolonger sufficient and necessary conditions in order to ensure existence, and one needs for example a suitablerelaxation property at infinity. A possibility is, as already discussed, that the limit lim t →∞ h ( t ) = ¯ h exists.In this case the results reported in [1] (see also [4]) show that the existence of the solution is not affected bythe specific time dependence of the perturbation: the system thermalise to an equilibrium state given by thelimiting Hamiltonian. It is remarkable that the situation is very different for a perturbation of the same timedependance, but spatially homogenous on the whole chain. In this occurrence the system thermalise as well,but the final state is not an equilibrium state, meaning that the transverse magnetisation does not go to zeroin the limit h → , once the asymptotic limit T → ∞ has been taken. As mentioned in the Introduction,this aspect is exhaustively reviewed in [4][19] it is discussed in relation to more general dynamical propertiesof simple classical and quantum systems.More challenging and still unsolved is instead the case of transverse field with an asymptotic oscillatingbehaviour. Even the occurrence (somehow basic in this context) of time periodic perturbation is hard tomanage, although partial results have been achieved: [1], [9]. These situations indeed do not fall in thehypotheses required by the global existence theorem stated in Section 2. Therefore a proof of the existenceof the dynamics for all time will need some new ideas and additional tools.However this problem is inscribed in a well known and vastly ranging topic in theoretical physics, namelysemiclassical interaction between light and matter. In our framework we figure out the non autonomous spinsystems as feeling the presence of an oscillating classical external field. As we have discussed, a rigorousmathematical approach to this question is hard to pursue and we can point out at least two (probably notindependent) hard aspects of the problem: the global existence of the time evolution is ultimately relatedto the preservation of the infinite dimensional analogous of the KAM tori in presence of a perturbation; theanalysis of the final state has possibly to take into account an infinite number of crossing of the critical pointof the model. In this paper we have made an attempt to describe with a unified formalism the quantum motion of thosespin chain models which can be mapped into free fermions, when the Hamiltonian explicitly depends ontime. We have discussed on a general level the problem of existence of dynamics locally and globally intime. Then we have deepened the subject by studying the time evolution for spin chains with a more specificsystem of integral equations.The central feature emerging by our analysis is that we can a priori distinguish two cases: on one hand wehave time dependent perturbations approaching a determined limit in the extrema of the definition intervalin time; in this scenario one has global existence of time evolution and the system approaches a final state.Therefore the main investigations are focused on the properties of such a state, which are far to be trivial ingeneral (they are eventually connected to the quantum phase transition of the model). On the other handwe have the case of temporal dependance with asymptotic oscillations. We can single out this last situationas the more challenging one to deal with, because the possible global existence of the dynamics requires amore delicate study.Of course in the context of spin chains the next natural step will be to obtain a similar picture in the case ofthe XYZ model, which is mapped into fermions with a quartic interaction. This class of models is certainly14ore appropriate to describe accurately the properties of low dimensional quantum systems and a fortiorithe effect of the defects. One of the main obstacles to extend our analysis to these models is that it is notpossible to recover the state so easily as we have done in Section 3, once one has the dynamical variables, bymeans of the Wick theorem. Therefore one should recur to a perturbative expansion which will render muchmore involved the whole analysis. However, in principle it is possible to suitably modify our approach toobtain (at least in a perturbative scheme) the equations of the dynamics in the case of fermions with quarticinteraction. Then one could study the effects given by the presence of impurities, at least in some simplifiedcase.
Acknowledgements
I would like to thank H. Narnhofer, for a useful correspondence about the papers [21]. Moreover I am gratefulto A. Giuliani, F. Illuminati and especially to B. Schlein for many valuable comments and suggestions.This work has been supported by the ERC Grant MAQD 240518.
A Diagonalisation of the XY Chain
This appendix is devoted to review the procedure of diagonalisation developed in [20] (see also [4]). After aFourier transform of the Fermi operators, the Hamiltonian of the XY model becomes (up to the irrelevantaddendum P q e iq = 1 ) H N ( γ, h ) = − X q (2 g cos q + 2 h ) a † q a q + e − iq a † q a †− q − e iq a q a − q = − X q h ( g cos q + h )( a † q a q + a †− q a − q ) − iγ sin q ( a † q a †− q + a q a − q ) i = − X q H q ( g, γ, h ) , (44)bearing in mind that X q g ( q ) α q α − q = 12 X q ( g ( q ) − g ( − q )) α q α − q X q g ( q ) α q β q = 12 X q g ( q ) α q β q + g ( − q ) α − q β − q α q β q = 12 [ α q , β q ] + δ qp δ αβ † , for every couple of Fermi operators such that [ α q , β p ] + = δ qp . We notice that all the H q acts in independentsubspaces, that is H q = I ⊗ ... ⊗ h ( g cos q + h )( a † q a q + a †− q a − q ) − iγ sin q ( a † q a †− q + a q a − q ) i ⊗ ... ⊗ I and so we always have [ H p , H q ] = 0 ∀ p, q ∈ Z ( N ) . This simply means that we can diagonalise all the modesindependently, thus we can operate by fixing q . Each H q is a quadratic form in creation and annihilationoperators. In order to render it more symmetric, we will write (up to a constant term cos q + h ) H q = 12 (cid:16) ( g cos q + h )([ a † q , a q ] + [ a †− q , a − q ]) − iγ sin q ([ a † q , a †− q ] + [ a q , a − q ]) (cid:17) ,
15r in matrix form H q = (cid:16) a † q a q a †− q a − q (cid:17) Γ − i Γ − Γ − i Γ i Γ Γ i Γ − Γ a q a † q a − q a †− q , (45)with Γ = g cos q + h and Γ = γ sin q . This matrix can be easily diagonalised, thereby obtaining theeigenvalues for the energy of the system: E q = ( g cos q + h ) + γ sin q. (46)The Bogoliubov-Valatin transformation W that diagonalises the matrix can be directly verified to be thetensor product of (SU(2) × SU(2)) rotations around the y -axis: W = π O q = − π W q ,W q = (cid:18) I cos φ q − σ y sin φ q σ y sin φ q I cos φ q (cid:19) , (47)that is the Hamiltonian is diagonal in the new operators b q = a q cos φ q + ia †− q sin φ q b † q = a † q cos φ q − ia − q sin φ q , (48)with cos 2 φ q = ( g cos q + h ) /E q , sin φ q = − γ sin q/E q . Sometimes it can be useful to get rid of imaginaryunit and make everything real. Let us introduce the unitary matrix T = (cid:18) e i π e − i π (cid:19) , with σ y T = T † σ y , and T † σ y T = ( T † ) σ y = − iσ z σ y = − σ x , and define ˆ W q = (cid:18) I cos φ q − T † σ y T sin φ q T † σ y T sin φ q I cos φ q (cid:19) , as the transformation that diagonalises our bilinear form with phase adjustment. Thus for the new operatorobtained in this way we have ˆ b q = a q cos φ q + a †− q sin φ q ˆ b † q = a † q cos φ q + a − q sin φ q , (49)The Hamiltonian in the new variables becomes − X q H q = − X q (cid:16) E q [ b † q , b q ] + E q [ b †− q , b − q ] (cid:17) = − X q (cid:16) E q b † q b q + E q b †− q b − q (cid:17) = − X q E q b † q b q . (50)Analogously we find − X q H q = − X q E q ˆ b † q ˆ b q .
16t turns useful to specify the transformation that diagonalises our initial Fermi Hamiltonian in c, c † : denotingby b ≡ ( b q , b † q ) and c ≡ ( c j , c † j ) one has b = U † cU where U = W ◦ F T is the composition of a Fourier transform and our Bogoliubov-Valatin transformation.Thus the relations defining the matrix elements U qj are b q = √ N P Nj =1 e iqj (cid:16) cos φ q c j + i sin φ q c † j (cid:17) b † q = √ N P Nj =1 e − iqj (cid:16) cos φ q c † j − i sin φ q c j (cid:17) , (51)with the obvious (more useful) inverses: ( c j = √ N P q e − iqj cos φ q b q + ie iqj sin φ q b † q ,c † j = √ N P q e iqj cos φ q b † q − ie − iqj sin φ q b q . (52) B Quench of an Impurity in the XX Chain
In this appendix we provide a very simple example which witnesses the break of ergodicity of the dynamicsof the XX chain. We will consider an abrupt change of transverse magnetic field in t = t acting only on one k -th spin of the chain: h ( t ) = (cid:26) , t ≤ t ; h, t > t . (53)We are going to study the time behaviour of the local transverse magnetisation. Since we deal with the XXmodel, the dynamics is described by the sole A variables (see formulas (4), (7) and (8)). Following [1] wehave for the local magnetisation along z in the k -th site h m zk ( t ) i = 1 N X q,p e ik ( q − p ) (cid:10) a † q ( t ) a p ( t ) (cid:11) = 1 N X q,p e ik ( q − p ) X q ′ ,p ′ A ∗ qq ′ ( t ) A pp ′ ( t ) D a † q ′ a p ′ E = 1 N X q ′ X q,p e ik ( q − p ) A ∗ qq ′ ( t ) A pq ′ ( t )1 + e βgω q ′ , (54)where we recall ω q = g cos q . Noticing that | X k,q | = 1 N X qp e ik ( q − p ) A ∗ qq ′ ( t ) A pq ′ ( t ) , we immediately get as N → ∞ h m zk ( t ) i = 2 π Z π − π dp e βω p | X k,p | . (55)The variable X ( q ; t ) is the unknown of the equation (43). It is convenient to study it into its integral form: X ( q ; t ) = e − iω q ( t − t ) − ih Z tt dt ′ J ( g ( t − t ′ )) X ( q ; t ′ ) . (56)Let us perform the change of variables Y ( p ; t ) = X ( p, t ) e ω p ( t − t ) . Of course the magnetisation is unaffectedby it, since it rests proportional to | Y | . The integral equation becomes Y t ( p ; t ) = 1 − ih Z t − t dt ′ J ( t ′ ) e iω p t ′ Y t ( p ; t − t ′ ) . (57)17ow we would like to send t → −∞ , that corresponds to study the asymptotic dynamics. The limitingequation reads Y ( p ; t ) = 1 − ih Z + ∞ dt ′ J ( t ′ ) e iω p t ′ Y ( p ; t − t ′ ) . (58)Now we pass to Fourier transform in time, and we get (1 + ih ˆ J ( ω p + ξ )) ˆ Y ( ξ, t ) = δ ( ξ ) , where (see (24) for n = 0 ) ˆ J ( ω p + ξ ) = χ ( g < ω p + ξ ) + iχ ( g > ω p + ξ ) p | g − ( ω p + ξ ) | . Then we obtain Y ( p, t ) = Z dξe iξt δ ( ξ ) 1 − ih ˆ J ( ω p + ξ )1 + h | J ( ω p + ξ ) | = 1 − ih ˆ J ( ω p )1 + h | J ( ω p ) | (59)Therefore the asymptotic value of Y is constant in time and | Y ( p ; t ) | = | Y ( p ; ∞ ) | = (1 + h | J ( ω p ) | ) − = sin p sin p + h /g . (60)Hence h m zk ( ∞ ) i = 2 π Z π − π dp e βω p ( h ) sin p sin p + h /g = 1 + h m zk i − h g π Z π − π dp e βω p ( h ) p + h /g , (61)where we denoted by h m zk i = 0 the initial equilibrium value of the magnetisation. Therefore we have a niceexplicit formula h m zk ( ∞ ) i = − h π Z π − π dp e βω p ( h ) g sin p + h . (62)We can see immediately that as h = 0 the magnetisation does not approach its initial value. An analysis ofthe opposite situation is also interesting. Consider the transverse field to be abruptly switched off form h to . By the aid of (62) we can only argue that the magnetisation in this case should thermalise. In fact thecalculations are slightly more involved, because the equilibrium system is given by the XX chain with oneimpurity, which is fairly non trivial. It has to be diagonalised in order to recover the eigen-frequencies ω p and the initial state. The spectrum of this system presents a band ω p = g cos q and an isolated eigenvalue ω p = p g + h . These results are in [1] (see also [9]), where it has been proven the ergodic behaviour thetransverse magnetisation. References [1] D. B. Abraham, E. Barouch, G. Gallavotti, A. Martin-Löf.
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