Classical and quantum Chaos plus RMT and some applications Arul - PDF document

1 Classical and quantum Chaos plus RMT and some applications Arul Lakshminarayan Department of Physics IIT Madras, Chennai. arul@iitm.ac.in Notes for the Bangalore Summer School on Statistical Physics, 9-13, July, 2018. ICTS, Bangalore.

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Contents 1 Hamiltonian Classical Chaos 5 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Welcome to 1.5 degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 Stroboscopic Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Kicked Hamiltonian Systems, Justforkix. . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.1 Important Area-Preserving Maps in 2D . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.2 Poincar´ e Surface of Section: Hamiltonian Flows to Maps . . . . . . . . . . . . . . 20 1.4 Poincar´ e Recurrence Theorem, Ergodicity, Mixing . . . . . . . . . . . . . . . . . . . . . 21 1.4.1 Recurrence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4.2 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.4.3 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5 Fully chaotic, exactly solvable, model systems . . . . . . . . . . . . . . . . . . . . . . . . 26 1.5.1 The Baker Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.5.2 The cat map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2 Hamiltonian Chaos: Quantum Mechanics 37 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1.1 But is there quantum chaos? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.1.2 Spectrum of Chaotic systems: Eigenfunctions . . . . . . . . . . . . . . . . . . . . 40 2.1.3 Billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.1.4 Scars of the classical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.1.5 Nonlinear oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.2 Quantum Maps: Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.2.2 Quantum maps on the plane and cylinder . . . . . . . . . . . . . . . . . . . . . . 50 2.2.3 Standard map on the cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.2.4 Quantum maps on the torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.2.5 General toral quantum maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.2.6 Quantum standard map on the torus . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.2.7 Quantum bakers map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.3 Eigenfunction of quantum maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.4 Eigenvalues and quantum chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.4.1 The straircase function, density of states, and spectral fluctuations . . . . . . . . 66 2.4.2 Semiclassics of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3

4 CONTENTS 3 Random Matrix Theory and applications to quantum chaos 75 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.2 Ensembles of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.2.1 The Gaussian Orthogonal Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2.2 Gaussian Unitary ensemble: GUE . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.3 More on the nearest neighbor spacing and ratio of spacings . . . . . . . . . . . . . . . . 82 3.4 Eigenfunctions of random matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.4.1 GOE eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.4.2 GUE eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.5 Trace constrained Wishart matrices: Entanglement . . . . . . . . . . . . . . . . . . . . . 87 3.5.1 The Marchenko-Pastur law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.6 Applications to Quantum chaos and thermalization . . . . . . . . . . . . . . . . . . . . . 90

Chapter 1 Hamiltonian Classical Chaos 1.1 Preliminaries Hamiltonian systems are a very special class of dynamical systems that are of central concern to physics. They encompass the classical mechanics of Newton and are the starting point for quantum and statistical physics. The dynamics completely specified by specifying the values of a minimum number of coordinates of the particles involved and their conjugate momenta. This minimum number, the number of degrees of freedom , or DoF, is a crucial number and is determined by the number of constraints on the system. Thus a simple pendulum is a 1 DoF system although it moves in a plane. But the Kepler problem is really 6 degrees of freedom (for the two masses). The fact that it finally reduces to the motion of one particle on a plane and in fact even further, so that an effective one-dimensional potential in radial coordinates is emerges is not due to mechanical constraints but to a large number of constants of motion. Let the number of DoF be d . Let us state at the very outset that this is a very critical number. All that we understand reasonably well today, in detail are utmost 2 DoF systems! Detailed structures and behaviour of say even 3 DoF systems is not yet achieved. One of the key reasons for this surprising in- ability is “chaos”, which is potentially present for any system whose d > 1; however even the complexity arises from the intricate mixtures of chaotic and regular orbits in large parts of phase space. Let the coordinates that make up the DoF be labelled ( q 1 , q 2 , . . . , q d ). Then the phase-space is the space that also includes the conjugate momenta ( p 1 , p 2 , . . . , p d ). Thus the phase-space has 2 d dimensions. This is the proper space in which Hamiltonian mechanics is to be studied. Every physical system is assumed to be described by an Hamiltonian H which is a scalar function on the phase-space and completely determines the future and past of the 2 d variables. The central equations are the Hamilton equations of motion: dq i dt = ∂H ∂p i (1.1) dp i dt = − ∂H , 1 ≤ i ≤ d. ∂q i These 2 d first order ordinary differential equations are determined by the single phase-space function H . Note the − sign in the momentum equation. Given any initial point ( q (0) , p (0)) it evolves to ( q ( t ) , p ( t )) after a time t under the Hamiltonian equations of motion. Here q (0) is written for the d position variable collectively, and similarly for momentum. Thus the Hamitonian generates a flow in the phase-space and classical mechanics is all about studying this flow. The Hamiltonian H could in general depend on all the 2 d phase space variables as well as time t . If it depends on time t , the system is said to be 5