Lieb-Robinson Bounds for the Toda Lattice Item Type text; Electronic Dissertation Authors Islambekov, Umar Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 01/10/2021 18:00:29 Link to Item http://hdl.handle.net/10150/294026 Lieb-Robinson Bounds for the Toda Lattice by Umar Islambekov A Dissertation Submitted to the Faculty of the Department of Mathematics In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy In the Graduate College The University of Arizona 2 0 1 3 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Dissertation Committee, we certify that we have read the disser- tation prepared by Umar Islambekov entitled Lieb-Robinson Bounds for the Toda Lattice and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy. Date: May 9, 2013 Jan Wehr Date: May 9, 2013 Kenneth McLaughlin Date: May 9, 2013 Robert Sims Date: May 9, 2013 Thomas Kennedy Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dis- sertation requirement. Date: May 9, 2013 Robert Sims 3 Statement by Author This dissertation has been submitted in partial fulfillment of re- quirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to bor- rowers under rules of the Library. Brief quotations from this dissertation are allowable without spe- cial permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or repro- duction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. Signed: Umar Islambekov 4 Acknowledgments I would like to show my sincere gratitude to my advisor Professor Robert Sims for supervising my PhD research. His continuous patience, support and motivation will never be forgotten. Also, I extend my thanks to our co-author Professor Gerald Teschl for his participation and contributions in our joint research. Special gratitude also goes to my dissertation committee members Professors Thomas Kennedy, Kenneth McLaughlin and Jan Wehr for their help, productive discussions and useful comments. 5 Table of Contents Abstract ..................................... 6 Chapter 1. Introduction .......................... 7 Chapter 2. Present Study ......................... 12 References .................................... 15 Appendix A { Lieb-Robinson Bounds for the Toda Lattice ..... 16 Appendix B { An Estimate for Infinite Range Perturbations ... 57 Appendix C { Supplemental material to Section 2.4 ......... 60 6 Abstract We study locality properties of the Toda lattice in terms of Lieb-Robinson bounds. The estimates we prove produce a finite Lieb-Robinson velocity depending on the initial condition. Then we establish analogous results for certain perturbations of the Toda system. Finally, we obtain generalizations of our main results in the setting of the Toda hierarchy. 7 Chapter 1 Introduction A concept of locality is inherent in many physical systems. In relativistic systems, the support of a time-evolved local observable stays confined to a light cone, i.e. locality is preserved under the time evolution. In non-relativistic systems, the effect of a small perturbation at some site is immediately felt at all sites of the system. In other words, strictly speaking locality is not preserved by the dynamics for such systems. In 1972, E. Lieb and D. Robinson showed that for non-relativistic quantum spin systems with finite range interactions there exists nevertheless an approximate form of locality, called quasi-locality [6]. The estimates they proved, commonly referred to as Lieb-Robinson bounds, are quantitative locality bounds which show that except for exponentially small corrections, the support of a local observable evolving under the dynamics remains essentially contained in a ball with radius growing linearly in time. In later years, works on proving analogues of Lieb-Robinson bounds for certain classical oscillator systems appeared in the literature [2], [7], [8]. We will now give a brief summary of these results because of their close relevance to the main topic of the present dissertation: to investigate quasi-locality in terms of Lieb-Robinson bounds for the Toda system and its perturbations. We begin by setting the general framework following [5] and describe how quasi-locality is studied in the context of classical oscillator systems. Consider a system of particles situated at the sites of a finite lattice Λ ⊂ Zd. Suppose the position and the momentum of a particle at site x 2 Λ are equal to real numbers qx and px respectively. The set of all sequences x = f(qx; px)gx2Λ, which describe all the states of the system, is called phase space and denoted by XΛ. Time 8 evolution of the system is governed by Hamilton's equations of motion: for each x 2 Λ, @H @H q_x(t) = andp _x(t) = − ; (1.0.1) @px @qx where H is the corresponding Hamiltonian for the system. In cases when this system can be solved, the flow generated by H and associated with an initial condition x is defined as Φt(x) = f(qx(t); px(t))gx2Λ. To study the properties of the solutions, it is convenient to introduce observables (complex-valued functions defined on phase space). For example, observables Qx and Px defined by Qx(x) = qx and Px(x) = px, respectively give the position and the momentum of a particle located at site x 2 Λ. The set of all observables over XΛ is denoted by AΛ. In such systems, quasi-locality is studied in terms of the Poisson bracket { the generator of the dynamics, i.e. d α (A) = α (fA; Hg) = fα (A);Hg ; (1.0.2) dt t t t where the Poisson bracket of observables A and B is defined as X @A @B @A @B fA; Bg = · − · : (1.0.3) @qx @px @px @qx x2Λ A Lieb-Robinson bound is described as follows in terms of the Poisson bracket. Fix two disjoint finite sets X and Y of Zd. Let Λ ⊂ Zd be a set containing the union of X and Y . Consider observables A and B supported in X and Y respectively, i.e., e.g. A does not depend on qx and px whenever x is outside X. Since X and Y are disjoint, the Poisson bracket of A and B is zero. A Lieb-Robinson bound is said to hold for a system with Hamiltonian H, if for some initial condition x there exist constants µ, C, and v such that an estimate −µ(d(X;Y )−vjtj) jfαt(A);Bg (x)j ≤ Ce ; (1.0.4) 9 holds for all t sufficiently small. Here αt is the dynamics (the map A 7! A ◦ Φt) corresponding to H and d(X; Y ) is the distance between the supports of A and B. This bound shows that the supports of B and the time-evolved A remain essentially apart from each other for times jtj ≤ d(X; Y )=v up to exponentially small errors in d(X; Y ), making Lieb-Robinson bounds useful only for short times, in contrast with more common long-time decay estimates. The number v is called a Lieb-Robinson velocity and it provides an upper bound on how fast the disturbances propagate through the system. One is also interested in what the constants C and v depend on. Most importantly, these constants must be independent of the size of the lattice whenever it is finite. Shortly after Lieb and Robinson published their above mentioned result, Mar- chioro et al. explored the problem of estimating the velocity of disturbances in clas- sical oscillator systems [7]. In their paper, first the authors show that for a system of harmonic oscillators there is a cone in spacetime with slope v such that if an observ- able A is supported in a ball of radius less than vt and an observable B is supported at p the origin, then the Poisson bracket fαt(A);Bg decays as 1= t and is exponentially small in time if the support of A has radius greater than vt (Proposition 3.1). The number v is the system's Lieb-Robinson velocity and it is the slope of the cone which separates these two asymptotic behaviors. Then the authors investigate two different anharmonic models with the aim of finding a dependence of the distance between the supports of A and B on time for which fαt(A);Bg is exponentially small. The first model they consider is a plane rotator model with compact configuration space. Adapting the approach employed for quantum lattice models by Lieb and Robinson in [6], they establish the existence of a finite Lieb-Robinson velocity for such a system (Theorem 4.1). However, this result is not accompanied by any explicit estimates on the Poisson bracket. Next, they deal with a model having a quartic on-site potential and prove a nontrivial quasi-locality result (Theorem 5.2). They show that whenever the distance between the supports of the local observables grows faster than t4=3, 10 the relevant Poisson bracket is exponentially small in time for almost every initial condition with respect to a suitably chosen measure.