A Lieb-Robinson Bound for the Toda System

Transcription

1 1 FRG Workshop May 18, 2011 A Lieb-Robinson Bound for the Toda System Umar Islambekov The University of Arizona joint work with Robert Sims

2 2 Outline: 1. Motivation 2. Toda Lattice 3. Lieb-Robinson bounds

3 3 The Harmonic System Let Λ be a finite subset of Z d. The harmonic Hamiltonian H Λ h : X Λ R is given by H Λ h (x) = x Λ p 2 x + ω 2 q 2 x + where x = (q x, p x ) x Λ and ω, λ j 0. d λ j (q x q x+ej ) 2, For any integer L 1 and each subset Λ L = ( L, L] d Z d, the flow Φ h,l t : X ΛL X ΛL corresponding to H Λ L h may be explicitly computed. j=1

4 4 A Lieb-Robinson Bound for the Harmonic System The following is a Lieb-Robinson bound for the harmonic system (H. Raz, R. Sims [3]). Theorem Let X, Y be finite subsets of Z d and take L 0 to be the minimal integer such that X, Y Λ L0. For any L L 0, denote by αt h,l the dynamics corresponding to H Λ L h. For any µ > 0 and any observables A, B A (1) Λ L0 with supports in X and Y respectively, there exists positive numbers C and v h, both independent of L, such that the bound {αt h,l (A), B} C A 1, B 1, min( X, Y )e µ(d(x,y ) v h t ) holds for all t R.

5 5 General Set-Up We will consider the Toda system in Z. To each integer n Z, we associate an oscillator with position q n R and momentum p n R. The state of the system is described by a sequence x = {(q n, p n )} n Z, and the phase space is denoted by X. The (infinite volume) Hamiltonian H T : X R { } for the Toda lattice is given by H T (x) = n Z p 2 n 2 + V (q n+1 q n ) where V (r) = e r + r 1 and x = {(q n, p n )} n Z.

6 6 Hamilton s equations for this system are easy to write down: for each n Z, q n (t) = H T p n (t) = p n (t), ṗ n (t) = H T q n (t) = V (q n+1 q n ) V (q n q n 1 ) = e (qn(t) q n 1(t)) e (q n+1(t) q n(t)).

7 7 Change of Variables A convenient change of variables (commonly referred to as Flaschka variables [1], [2]) is: for each n Z and t R, set a n (t) = 1 2 e (q n+1(t) q n(t))/2 and b n (t) = 1 2 p n(t). The corresponding system of equations of motion are ȧ n (t) = a n (t) (b n+1 (t) b n (t)) ḃ n (t) = 2 ( an(t) 2 an 1(t) 2 ). (1)

8 8 We will consider the Toda Hamiltonian restricted to the Banach space M = l (Z) l (Z). Each x M will be written as x = {(a n, b n )} n Z. The norm on M is given by x M = max(sup n a n, sup b n ). n For the Toda system one can prove existence and uniqueness of the global solution on M. This is done in two stages. First one proves local existence of a solution and then extends it globally.

9 9 Local Existence Theorem If x 0 = (a 0, b 0 ) M then there exist δ > 0 and a unique solution (a(t), b(t)) = {(a n (t), b n (t))} n Z in C (I, M), where I = ( δ, δ), of the Toda equations (1) such that (a(0), b(0)) = (a 0, b 0 ).

10 10 Global Existence Corresponding to each x 0 M, define the following operators H(t), P(t) : l 2 (Z) l 2 (Z), t I, by setting [H(t)f ] n = a n (t)f n+1 + a n 1 (t)f n 1 + b n (t)f n, [P(t)f ] n = a n (t)f n+1 a n 1 (t)f n 1. A short calculation shows that P(t) and H(t) are a Lax-Pair associated to (1), i.e., d H(t) = [P(t), H(t)]. dt

11 11 Since P(t) is skew-symmetric, it generates a two-parameter family of unitary propagators U(t, s) [4]. Moreover, the Lax equation implies that H(t) = U(t, s)h(s)u(t, s) (t, s) I. Hence H(t) 2 = H(0) 2 and therefore max ( a(t), b(t) ) H(t) 2 = H(0) 2, implying that the solution can be globally extended.

12 12 Class of observables we consider We will denote by A (1) the set of all observables A for which ( ) A A 1, = sup max n Z a n, A b n is finite. An observable A is said to be supported in X Z if the observables A a n and A b n are identically zero for all n Z \ X. The support of an observable A is the minimal set on which A is supported.

13 13 We will denote by α t the Toda dynamics, i.e., α t : A A defined by setting α t (A) = A Φ(t), where Φ(t) is the corresponding Toda flow. If A A and B A are functions of q n s and p n s, n Z, then the Poisson bracket between them is defined as {A, B}(x) = ( A B A ) B, q n p n p n q n n Z where x = {(q n, p n )} n Z. If A A and B A are functions of a n s and b n s, n Z, then the modified Poisson bracket is {A, B}(x) = 1 ( A B a n A ) B, 4 a n c n c n a n n Z where x = {(a n, b n )} n Z and c n = b n+1 b n.

14 14 Main Result The following is a Lieb-Robinson bound for the Toda System. Theorem Let x 0 = {(a n, b n )} n Z M. Then, for every µ > 0 there exists a number v = v(µ, x 0 ) for which given any observables A, B A (1) with finite supports X and Y respectively, the estimate {α t (A), B}(x 0 ) A 1, B 1, sup n holds for all t R. Here v(µ, x 0 ) = 18c a n n X,m Y ( e µ ), µ e µ( n m v t ) where c = c(x 0 ) = H(0) 2.

15 15 Sketch of the Proof A short calculation shows that {α t (A), B}(x 0 ) A 1, B 1, sup a n n ( ) 1 a n (t) 2 a m + b n (t) a m n X,m Y + 1 ( ) a n (t) 4 b m+1 + a n (t) b m + b n (t) b m+1 + b n (t) b m. Let Φ n (t) denote the nth component of the flow Φ(t, x 0 ), i.e., ( ) a n (t) Φ n (t) =. b n (t) Clearly, Φ n (t) = Φ n (0) + t 0 ( a n (s) (b n+1 (s) b n (s)) 2 ( a 2 n(s) a 2 n 1 (s)) Let Φ n(t) = Φn(t) z, where z {a m, b m, b m+1 }. WLOG we take z = a m. Then differentiating the above equality with respect to z ) ds.

16 16 where D n+e (s) = Φ n(t) = δ m (n) ( ( 1 0 ) + e 1 t 0 D n+e (s)φ n+e(s)ds, (b n+1 (s) b n (s)) δ 0 (e) a n (s)( δ 0 (e) + δ 1 (e)) 4(a n (s)δ 0 (e) a n 1 (s)δ 1 (e)) 0 For any v = (x, y) R 2 take v = max( x, y ). Then by taking the norm of both sides we get Φ n (t) δm (n) + c 1 where c 1 = 6 H(0) 2. e 1 t 0 Φ n+e (s) ds, ).

17 17 By iterating the above inequality we obtain Φ n (t) which implies that for any µ > 0 k= n m (3c 1 t ) k, k! Φ n(t) e µ( n m v t ). where v = v(µ, x 0 ) = 18 H(0) 2 (e µ µ ). Remark Similar Lieb-Robinson bounds can be proven in general for the Toda Heirarchy.

18 18 Similar Result for Another Class of Observables Similar Lieb-Robinson bound is valid for another class of observables A (2), consisting of all observables A and B for which A 2, = max A 2 a n n, A 2 b n n <.

19 19 Some Numerics In this example we consider the Toda system in the finite volume and assume periodic boundary conditions. The number of particles N = 100. a n (0) = 1/2 n and b n (0) = 0 n 50, b 50 (0) = 1. The values of a n (t), 1 n 100, over the time interval [0, 10] are plotted below.

20 20 Perturbations of the Toda system Let W : R [0, ) such that W C 2 (R) and W, W L (R). We define a Hamiltonian H W : X R { } by setting H W T (x) = H T (x) + n Z W (a n ), where x = {(a n, b n )} n Z. Hence the corresponding system of equations of motion are ȧ n (t) = a n (t) (b n+1 (t) b n (t)), ḃ n (t) = 2 ( an(t) 2 an 1(t) 2 ) + R n (t), where R n (t) = 1 4 (W (a n (t))a n (t) W (a n 1 (t))a n 1 (t)).

21 21 Local Existence It is easy to see that there exists a unique local C 2 solution (a(t), b(t)) on I (x 0 ) = ( δ, δ) of the above Toda equations if the initial condition x 0 = (a 0, b 0 ) M. Let αt W denote the perturbed Toda dynamics, i.e., αt W : A A defined by setting α W t (A) = A Φ W t, where Φ W t is the corresponding perturbed Toda flow. Define the operators P(t) and H(t), t I (x 0 ) as before. Let U(t, s) be the family of unitary propagators for P(t). A simple calculation shows that d H(t) = [P(t), H(t)] + R(t), dt where R(t) is a bounded linear operator in l 2 (Z) given by [R(t)f ] n = R n (t)f n.

22 22 Let H(t) = U(t, s) H(t)U(t, s). Then H(t) 2 = H(t) 2 and by the similar earlier calculations we get d dt H(t) = U(t, s) R(t)U(t, s). Hence t H(t) = H(0) + U(τ, s) R(τ)U(τ, s)dτ. 0 By taking the norm we get H(t) 2 H(0) By Gronwall s lemma he have W t H(t) 2 c 1 e c 2 t, 0 H(τ) 2 dτ. where c 1 = H(0) 2 and c 2 = 1 2 W, implying the global existence of the solution.

23 23 Another Main Result Theorem Let x 0 = {(a n, b n )} n Z M. Then, for every µ > 0 and T > 0 there exists a number v = v(µ, x 0, W, T ) for which given any observables A, B A (1) with finite supports X and Y respectively, the estimate {α W t (A), B}(x 0 ) A 1, B 1, sup n holds for all t ( T, T ). Here v(µ, x 0, W, T ) = c a n n X,m Y ( e µ ), µ e µ( n m v t ) where c = 2 ( 3 4 W + 18 ) H(0) 2 W ( ) e T 2 W W T 2.

24 24 Sketch of the Proof We follow the argument as in the proof of the previous theorem. In fact, keeping similar notation, one gets Φ n (t) δm (n) + t f (s) Φ n+e (s) ds, e 1 where f (s) = c 3 H(s) c 2, with c 3 = 1 4 W + 6. Iteration implies that Φ n(t) k= n m k= n m 0 ( 1 t ) k 3 f (τ)dτ k! 0 g(t) k [ ] g(t) n m e n m e g(t), (2) k! n m where g(t) = 3c 1c 3 c 2 (e c2 t 1) c 2 t. Given any µ > 0. Then as we argued in the previous theorem one can show that Φ n(t) e µ( n m c(eµ µ ) t ), where c = 3c 1c 3 (e δc 2 1) δc c 2.

25 25 Some Numerics In this example we consider a perturbed Toda system in the finite volume and assume periodic boundary conditions. We take N = 100 and W (x) = 1 2 cos(2πx). a n(0) = 1/2 n and b n (0) = 0 n 50, b 50 (0) = 1. The values of a n (t), 1 n 100, over the time interval [0, 10] are plotted below.

26 26 Bibliography H. Flaschka. The Toda Lattice. I Phys. Rev. B9, (1974). H. Flaschka. The Toda Lattice. II Progr. Theoret. Phys. 51, (1974). H. Raz, R. Sims. Estimating the Lieb-Robinson Velocity for Classical Anharmonic Lattice Systems J Stat Phys, 137: (2009). G. Teschl. Jacobi Operators and Completely Integrable Nonlinear Lattices Mathematical Surveys and Monographs 72 AMS R. Abraham, J.E. Marsden, and T. Ratiu. Manifolds, Tensor Analysis, and Applications 2nd edition, Springer, New York, 1983.