No abstract

We present a direct link between manifestations of classical Hamiltonian chaos and quantum nonintegrability effects as they occur in quantum invariants. In integrable classical Hamiltonian systems, analytic invariants (integrals of the motion) can be constructed numerically by means of time averages of dynamical variables over phase-space trajectories, whereas in near-integrable models such time averages yield nonanalytic invariants with qualitatively different properties. Translated into quantum mechanics, the invariants obtained from time averages of dynamical variables in energy eigenstates provide a topographical map of the plane of quantized actions (quantum numbers) with properties which again depend sensitively on whether or not the classical integrability condition is satisfied. The most conspicuous indicator of quantum chaos is the disappearance of quantum numbers, a phenomenon directly related to the breakdown of invariant tori in the classical phase flow. All results are for a system consisting of two exchange-coupled spins with biaxial exchange and single-site anisotropy, a system with a nontrivial integrability condition.

We have carried out an extensive simulation study for the spin autocorrelation function at T=∞ of the one-dimensional classical Heisenberg model with four different types of isotropic bilinear nearest-neighbor coupling: uniform exchange, alternating exchange, and two kinds of random exchange. For the long-time tails of all but one case, the simulation data seem incompatible with the simple ∼t−1/2 leading term predicted by spin diffusion phenomenology.

This computer simulation study rjrovides further evidence that spin diffusion in the onedimensional classical Heisenberg 'model at T = o is anomalous: (Si(t) * Si)-t-"1 with at > l/2. However, the exponential instability of the numerically integrated phase-space trajectories transforms the deterministic transport of spin fluctuations into a computationally generated stochastic process in which the global conservation laws are still satisfied to high precision. This may cause a crossover in l/2) to normal spin diffusion (a1 = l/2) at some characteristic time lag that depends on the precision of the numerical integration.

No abstract

We investigate dynamic correlation functions for a pair of exchange-coupled classical spins with biaxial exchange and/or single-site anisotropy. This represents a Hamiltonian system with two degrees of freedom for which we have previously established the integrability criteria. We discuss the impact of (non-)integrability on the autocorrelation functions and their spectral properties. We point out the role of long-time anomalies caused by low-flux cantori, which dominate the convergence properties of time averages and determine the long-time asymptotic behavior of autocorrelation functions in nonintegrable cases.

. Chaos in spin clusters: quantum invariants and level statistics.

Cartesian coordinates, generalized coordinates, canonical coordinates, and, if you can solve the problem, action-angle coordinates. That is not a sentence, but it is classical mechanics in a nutshell. You did mechanics in Cartesian coordinates in introductory physics, probably learned generalized coordinates in your junior year, went on to graduate school to hear about canonical coordinates, and were shown how to solve a Hamiltonian problem by finding the action-angle coordinates. Perhaps you saw the action-angle coordinates exhibited for the harmonic oscillator, and were left with the impression that you (or somebody) could find them for any problem. Well, you now do not have to feel badly if you cannot find them. They probably do not exist!