Quantum and classical spin clusters: disappearance of quantum numbers and Hamiltonian chaos

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.

Quantum integrability and action operators in spin dynamics

“…However, a general method exists for the evaluation of that invariant on a dense set of phase points with full measure. This method was proposed earlier, [12,13] and its usefulness for practical applications was demonstrated by numerical implementations.…”

“…, S N ) which satisfy dI/dt = 0, but for which {H, I} cannot be evaluated due to lack of smoothness. [12,13] The phase flow generated by a given Hamiltonian must belong to one of two distinct types. (i) Regular flow: The entire 2N -dimensional phase space is foliated into N -dimensional tori; individual phase points wind around these tori periodically or quasi-periodically.…”

“…[13] Determine, via time average (9), two analytic invariants I F = I F (J 1 , J 2 ) and I G = I G (J 1 , J 2 ) which are functionally independent of H = H(J 1 , J 2 ) and functionally independent of each other. The dependence of these invariants on the actions J 1 , J 2 is, in general, not known explicitly, but for individual tori their numerical values can be determined everywhere in phase space.…”

“…We describe and employ a computational procedure for the construction of classical invariants, whose properties depend sensitively on whether the underlying Hamiltonian is integrable or not. [12,13] In Sec. III we propose the existence of a unitary transformation which converts any quantum one-spin system into a function of a single quantum action -an operator with specific spectral properties.…”

“…Such an expression defines quantum integrability and implies that the time evolution of any non-stationary operator can be determined explicitly. We employ quantum invariants, constructed by a procedure analogous to the one used previously for classical invariants, [13] to produce analytical and numerical evidence in support of the proposition that all one-spin systems are quantum integrable. In Sec.…”