Institut Fourier - UFR de Math´ ematiques (Grenoble UI) Open Quantum Systems Maison Jean Kuntzmann - 29 novembre au 02 d´ ecembre 2010 Lieb-Robinson Bounds and Construction of Dynamics Valentin A. ZAGREBNOV Universit´ e de la M´ editerran´ ee Centre de Physique Th´ eorique - Luminy - UMR 6207 • Motivation: Two Ways to Infinite W ∗ -Dynamical Systems • From Lieb-Robinson Bounds to Infinite Dynamics • Dynamics of a Harmonic Lattice • On-Site and Multiple-Site Anharmonisities Based on the paper in RMP 22(2010)207-331 by B.Nachtergaele, B.Schlein, R.Sims,Sh.Starr and VZ =[0] 0
Grenoble - Open Quantum Systems 1.Motivation: Two Ways to Infinite W ∗ -Dynamical Systems • The traditional way is to first define the dynamics of anhar- monic quantum lattice in finite volume (which can be done by standard means), and then studying the limit in which the vol- ume tends to infinity [Amour,Levy-Bruhl,Nourrigat (2010)] =[1]. • We follow a different approach. The main difference is that we study the thermodynamic limit of anharmonic perturbations of an infinite harmonic lattice system described by an explicit W ∗ -dynamical system. (a) It appears that controlling the continuity of the limiting dy- namics is more straightforward in our approach and we are able to show that the resulting dynamics for the class of anharmonic lattices that we study is indeed weakly continuous and we obtain a W ∗ -dynamical system for the infinite system. 1
Grenoble - Open Quantum Systems (b) Common to both approaches, ours [0] and [1], is the crucial role of an estimate the speed of propagation of perturbations in the system, commonly referred to as Lieb-Robinson bounds (1972). • Recall: Let A and B be two observables of a spatially extended system, localized in regions X and Y , respectively, and τ t denotes the time evolution of the system then, a Lieb-Robinson bound is an estimate of the form: � [ τ t ( A ) , B ] � ≤ Ce − a ( d ( X,Y ) − v | t | ) , where C, a , and v are positive constants and d ( X, Y ) denotes the distance between X and Y . (c) The Lieb-Robinson bounds for anharmonic lattice systems were recently proved in by [Nachtergaele,Raz,Schlein,Sims (2009)] =[2]. 2
Grenoble - Open Quantum Systems 2.From Lieb-Robinson Bounds to Infinite Dynamics • To each x in lattice Γ, we associate a Hilbert space H x . In many relevant systems, one considers H x = L 2 ( R , dq x ). Then H Λ = � x ∈ Λ H x for finite subset Λ ⊂ Γ and the local algebra of observables over Λ is A Λ = � x ∈ Λ B ( H x ) where B ( H x ) denotes the algebra of bounded linear operators on H x . • Let local Hamiltonians H loc := { H x } x ∈ Γ , here H x are on-site self-adjoint operators in H x , and interactions Φ( X ) ∈ A X . We consider self-adjoint Hamiltonians: H Λ = H loc + H Φ � � Λ = H x + Φ( X ) , Λ x ∈ Λ X ⊂ Λ They generate a dynamics { τ Λ with domain � x ∈ Λ D ( H x ). t } , which is the one parameter group of automorphisms defined by τ Λ t ( A ) = e itH Λ A e − itH Λ for any A ∈ A Λ . 3
Grenoble - Open Quantum Systems • Consider the unitary propagator U Λ ( t, s ) = e itH loc e − i ( t − s ) H Λ e − isH loc Λ Λ and its associated interaction-picture evolution defined by τ Λ t, int ( A ) = U Λ (0 , t ) A U Λ ( t, 0) for all A ∈ A Γ . • Then for n ≤ m with X ⊂ Λ n ⊂ Λ m one gets � t d τ Λ m t, int ( A ) − τ Λ n � � t, int ( A ) = U Λ m (0 , s ) U Λ n ( s, t ) A U Λ n ( t, s ) U Λ m ( s, 0) ds . ds 0 • Let A ∈ A X . Then with help of the operators: A ( t ) = e − itH loc Λ n A e itH loc Λ n = e − itH loc X A e itH loc ˜ X B ( s ) = e − isH loc e isH loc H int Λ m ( s ) − H int � � ˜ Λ n ( s ) Λ n Λ n 4
Grenoble - Open Quantum Systems • One obtains the estimate: � t � τ Λ m t, int ( A ) − τ Λ n τ Λ n � � � �� � � � ˜ , ˜ t, int ( A ) � ≤ A ( t ) B ( s ) � ds . � � � � s − t � 0 • Application of the Lieb-Robinson bound [2] implies that the sequence { τ Λ n t, int ( A ) } is Cauchy in norm, uniformly for t ∈ [ − T, T ]: � � τ Λ m t, int ( A ) − τ Λ n � sup t, int ( A ) � → 0 as n, m → ∞ . � � t ∈ [ − T,T ] e it � x ∈ X H x A e − it � � x ∈ X H x � • Since is localised in X and � � e itH loc A e − itH loc e it � x ∈ X H x A e − it � τ Λ t ( A ) = τ Λ = τ Λ � x ∈ X H x � , Λ Λ t, int t, int an analogous statement then follows for { τ Λ n ( A ) } . t 5
Grenoble - Open Quantum Systems • If all local Hamiltonians H x are bounded , { τ t } is strongly con- tinuous. If the H x are allowed to be densely defined unbounded self-adjoint operators, we only have weak continuity and the dy- namics is more naturally defined on a von Neumann algebra. • Theorem. Under the conditions stated above, for all t ∈ R , A ∈ A Γ , the norm limit Λ → Γ τ Λ lim t ( A ) = τ t ( A ) exists in the sense of non-decreasing exhaustive sequences of finite volumes Λ and defines a group of ∗− automorphisms τ t on the completion of A Γ . The convergence is uniform for t in a compact set. 6
Grenoble - Open Quantum Systems 3.Dynamics of a Harmonic Lattice • Consider a system of harmonic oscillators restricted to cubic subsets Λ L = ( − L, L ] d ⊂ Z d , with harmonic n.n. couplings: d x + ω 2 q 2 H h p 2 λ j ( q x − q x + e j ) 2 � � L = x + x ∈ Λ L j =1 x ∈ Λ L L 2 ( R , dq x ). Here p x , q x are sin- in the Hilbert space H Λ L = � gle site momentum and position operators satisfying the CCR: [ p x , p y ] = [ q x , q y ] = 0 and [ q x , p y ] = iδ x,y , valid for all x, y ∈ Λ L , the numbers λ j ≥ 0 and ω ≥ 0 are the pa- rameters of the system, and the Hamiltonian is assumed to have periodic boundary conditions . It is well-known that Hamiltonians of this form can be diagonalized in the Fourier space . 7
Grenoble - Open Quantum Systems • Using this diagonalization, one can determine the action of the dynamics corresponding to H h L on the Weyl algebra W ( D = ℓ 2 (Λ L )). In fact, by setting , � W ( f ) = exp i Re[ f ( x )] q x + Im[ f ( x )] p x x ∈ Λ L for each f ∈ ℓ 2 (Λ L ), and symplectic form σ ( f, g ) = Im[ � f, g � ]. • Limiting harmonic dynamics is quasi-free on W ( D ): it is a one- parameter group of *-automorphisms τ t (Bogoliubov transfor- mations) τ t ( W ( f )) = W ( T t f ) , f ∈ D where T t : D → D is a group of real-linear, symplectic transformations, σ ( T t f, T t g ) = σ ( f, g ). • As � W ( f ) − W ( g ) � = 2 for all f � = g ∈ D , one should not expect τ t to be strongly continuous. 8
Grenoble - Open Quantum Systems • In the present context, it suffices to regard a W ∗ -dynamical sys- tem as a pair {M , α t } where M is a von Neumann algebra and α t is a weakly continuous, one parameter group of ∗ -automorphisms of M . • For the harmonic systems a specific W ∗ -dynamical system arises as follows. Let ρ be a state on W and denote by ( H ρ , π ρ , Ω ρ ) the corresponding GNS representation. Assume that ρ is both regular and τ t -invariant. For the algebra M , take the weak- closure of π ρ ( W ) in L ( H ρ ) and let α t be the weakly continuous, one parameter group of ∗ -automorphisms of M obtained by lift- ing τ t to M . • Lieb-Robinson bounds for harmonic lattice and f, g ∈ ℓ 2 (Γ): � [ τ t ( W ( f )) , W ( g )] � ≤ c a e v a | t | � | f ( x ) | | g ( y ) | F a ( d ( x, y )) . x,y 9
Grenoble - Open Quantum Systems 4.On-Site and Multiple-Site Anharmonisities • Our first Lien-Robinson estimate involves perturbations defined as finite sums of on-site terms . To each site x ∈ Γ, we will For Λ ⊂ Γ we set P Λ = associate an element P x ∈ W ( D ). x ∈ Λ P x , and note that ( P Λ ) ∗ = P Λ ∈ W ( D ). We will denote by � τ (Λ) the dynamics that results from applying Dyson expansion t to the W ∗ -dynamical system {M , τ 0 t } and P Λ . • Theorem: There exist positive numbers c a and v a , for which the estimate � �� � τ (Λ) � ≤ c a e ( v a + c a κC a ) | t | � � � ( W ( f )) , W ( g ) | f ( x ) | | g ( y ) | F a ( d ( x, y )) t � � � x,y holds for all t ∈ R and for any functions f, g ∈ D . 10
Grenoble - Open Quantum Systems • Multiple-site anharmonisity: For any finite subset Λ ⊂ Γ, we will set P Λ = � X ⊂ Λ P X where Here we will again let τ (Λ) the sum is over all subsets of Λ. t denote the dynamics resulting from Dyson expansion applied to the W ∗ -dynamical system {M , τ 0 t } and the perturbation P Λ . • Theorem: There exist positive numbers c a and v a for which one has second Lien-Robinson estimate � � �� � ≤ c a e ( v a + c a κ a C 2 τ (Λ) a ) | t | � � � ( W ( f )) , W ( g ) | f ( x ) | | g ( y ) | F a ( d ( x, y )) , � t � � x,y for all t ∈ R and for any functions f, g ∈ D . 11
Recommend
More recommend
