[PPT] - Open Quantum Systems Maison Jean Kuntzmann - 29 novembre au 02 d PowerPoint Presentation

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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

Based on the paper in RMP 22(2010)207-331 by B.Nachtergaele, B.Schlein, R.Sims,Sh.Starr and VZ =[0]

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1.Motivation: Two Ways to Infinite W ∗-Dynamical Systems

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 study the thermodynamic limit of anharmonic perturbations

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.

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(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).

system, localized in regions X and Y , respectively, and τt denotes the time evolution of the system then, a Lieb-Robinson bound is an estimate of the form: [τt(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].

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2.From Lieb-Robinson Bounds to Infinite Dynamics

In many relevant systems, one considers Hx = L2(R, dqx). Then HΛ =

the algebra of bounded linear operators on Hx.

self-adjoint operators in Hx, and interactions Φ(X) ∈ AX. We consider self-adjoint Hamiltonians: HΛ = Hloc

+ HΦ

Hx +

Φ(X), with domain

They generate a dynamics {τΛ

which is the one parameter group of automorphisms defined by τΛ

for any A ∈ AΛ.

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UΛ(t, s) = eitHloc

e−i(t−s)HΛ e−isHloc

and its associated interaction-picture evolution defined by τΛ

for all A ∈ AΓ .

τΛm

d ds

˜ A(t) = e−itHloc

˜ B(s) = e−isHloc

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A(t)

B(s)

sequence {τΛn

sup

as n, m → ∞.

is localised in X and τΛ

A e−itHloc

, an analogous statement then follows for {τΛn

(A)}.

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self-adjoint operators, we only have weak continuity and the dy- namics is more naturally defined on a von Neumann algebra.

A ∈ AΓ, the norm limit lim

exists in the sense of non-decreasing exhaustive sequences of finite volumes Λ and defines a group of ∗−automorphisms τt on the completion of AΓ. The convergence is uniform for t in a compact set.

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3.Dynamics of a Harmonic Lattice

subsets ΛL = (−L, L]d ⊂ Zd, with harmonic n.n. couplings: Hh

p2

λj (qx − qx+ej)2 in the Hilbert space HΛL =

gle site momentum and position operators satisfying the CCR: [px, py] = [qx, qy] = 0 and [qx, py] = iδx,y, valid for all x, y ∈ ΛL, the numbers λj ≥ 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

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the dynamics corresponding to Hh

ℓ2(ΛL)). In fact, by setting W(f) = exp

Re[f(x)]qx + Im[f(x)]px

for each f ∈ ℓ2(ΛL), and symplectic form σ(f, g) = Im[f, g].

parameter group of *-automorphisms τt (Bogoliubov transfor- mations) τt(W(f)) = W(Ttf), f ∈ D where Tt : D → D is a group

τt to be strongly continuous.

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tem as a pair {M, αt} where M is a von Neumann algebra and αt is a weakly continuous, one parameter group of ∗-automorphisms

arises as follows. Let ρ be a state on W and denote by (Hρ, πρ, Ωρ) the corresponding GNS representation. Assume that ρ is both regular and τt-invariant. For the algebra M, take the weak- closure of πρ(W) in L(Hρ) and let αt be the weakly continuous,

ing τt to M.

[τt(W(f)), W(g)] ≤ caeva|t|

|f(x)| |g(y)| Fa (d(x, y)) .

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4.On-Site and Multiple-Site Anharmonisities

as finite sums of on-site terms. To each site x ∈ Γ, we will associate an element Px ∈ W(D). For Λ ⊂ Γ we set P Λ =

τ(Λ)

the dynamics that results from applying Dyson expansion to the W ∗-dynamical system {M, τ0

the estimate

(W(f)) , W(g)

|f(x)| |g(y)|Fa (d(x, y)) holds for all t ∈ R and for any functions f, g ∈ D.

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For any finite subset Λ ⊂ Γ, we will set P Λ =

the sum is over all subsets of Λ. Here we will again let τ(Λ)

denote the dynamics resulting from Dyson expansion applied to the W ∗-dynamical system {M, τ0

(W(f)) , W(g)

|f(x)| |g(y)|Fa (d(x, y)) , for all t ∈ R and for any functions f, g ∈ D.

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system and let δ denote the infinitesimal generator of αt. Given any P = P ∗ ∈ M, set δP to be the bounded derivation with domain D(δP) = M satisfying δP(A) = i[P, A] for all A ∈ M. It follows that δ + δP generates a one-parameter group of ∗- automorphisms αP of M which is the unique solution of the integral equation αP

In addition, the estimate

for all t ∈ R and A ∈ M.

that the perturbed dynamics is also weakly continuous. Hence, for P = P ∗ ∈ M the pair {M, αP

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Let τ0

denote a non-decreasing, exhaustive sequence of finite subsets

Consider a family of perturbations P Λn. Then, for each f ∈ ℓ1(Γ) and t ∈ R fixed, the limit lim

(W(f)) exists in norm. The limiting dynamics, which we denote by τt, is weakly continuous.

that it holds for the finite volume dynamics. This completes the proof of Theorem.

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