Evidence for Quantum Chaos in a One-Dimensional Classically Integrable System

Mario Castagnino, Olímpia Lombardi

Ciann‐Dong Yang, Chia-Hung Wei

Javad Vahedi, A. Ashouri, S. Mahdavifar

Using one-dimensional spin-1/2 systems as prototypes of quantum many-body systems, we study the emergence of quantum chaos. The main purpose of this work is to answer the following question: how the spin-orbit interaction, as a pure quantum interaction, may lead to the onset of quantum chaos? We consider the three integrable spin-1/2 systems: the Ising, the XX, and the XXZ limits and analyze whether quantum chaos develops or not after the addition of the Dzyaloshinskii-Moriya interaction. We find that depending on the strength of the anisotropy parameter, the answer is positive for the XXZ and Ising models, whereas no such evidence is observed for the XX model. We also discuss the relationship between quantum chaos and thermalization.

Steven Tomsovic, Denis Ullmo

We discuss and give evidence for the existence of a new mechanism for quantum dynamical tunneling. It may occur when a quantum system's underlying classical dynamics are far from integrability. In these circumstances, we show that the dominant tunneling contributions arise through chaos-assisted processes. This leads to behavior that is drastically different from that found in integrable and quasi-integrable systems. In particular, one can observe a marked crossing mechanism when a chaotic level passes nearby the tunneling ones, and the distributions of splitting (due to tunneling) can be modeled using properly designed ensembles of random matrices. Such tunneling should be amenable to experimental detection.

Sven Gnutzmann∥, Fritz Haake, Marek Kuś

Hamiltonians built of SU 3 generators are presented which allow for qualitatively different classical limits. The generators of the group SU 3 give eight independent observables. In the classical limit two invariant functions of these observables, the so-called Casimir functions c 2 and c 3 , take fixed values in the ranges ½ c 2 (2/3) and -(2/9) c 3 (2/9) (an SU 2 system, in contrast, has just one such Casimir function, the squared total angular momentum). Generically, this leads to a six-dimensional phase space. However, for two special (`degenerate') pairs of values c 2 = (2/3), c 3 = -(2/9) and c 2 = (2/3), c 3 = (2/9) the classical dynamics is only four-dimensional. One and the same Hamilton function of the generators may be integrable in the four-dimensional phase space but generate chaos in the six-dimensional case. We give two examples of Hamiltonians for which that alternative does arise; one of these is closely related to the Lipkin Hamiltonian familiar from nuclear shell models. The transition from integrable to chaotic classical behaviour is accompanied by the usual quantum transition from level clustering to level repulsion.

A. G. Lazerson, Alexey Alexeevich Boikov