Integrable and nonintegrable classical spin clusters

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

“…If it weren't for the slow convergence of the time averages along chaotic trajectories due to low-flux cantori [18][19][20], the entire chaotic region would be represented by a single isolated point in the constant-energy section of the invariant-surface. In the full ( M 2 x , M 2 z , E)-space, the points associated with chaotic regions form string-like objects.…”

“…In a diagram I λ versus E λ , the array of (2σ + 1) 2 points form a regular pattern, which, however, is likely to be visually distorted due to the generally complicated nonlinear dependence of E and I on J 1 , J 2 . But we do not, in fact, need to know the 2 nd integral of the motionÎ explicitly to carry out this scheme; we can, following an idea of Peres [24], always construct it via time averages similar to the procedure previously employed for the classical model [18]: Take any operator with [Â,Ĥ] = 0, which represents a dynamical variable independent ofĤ, and consider the matrix elementsÂ(t) in the energy representation,…”

“…In Ref. 18 we have numerically verified for specific choices of A and an integrable 2-spin model that the function I A (S 1 , S 2 ) is a piecewise smooth function of the phasespace variables with the singularities attributable to separatrices in the phase flow [22].…”

“…18 we described a method for the numerical construction of invariants, a prescription applicable to quite general situations: Pick any dynamical variable A which is independent of H and determine its time average over the phase-space trajectory specified by initial condition (S 1 , S 2 ):…”

“…The present paper is, in fact, the fourth part of our study of the dynamics of integrable and nonintegrable spin clusters. In the first three parts, the focus was, respectively, on the following properties of classical 2-spin clusters: integrability criteria and analytic structure of invariants [17]; geometric structure of analytic and nonanalytic invariants [18]; time-dependent correlation functions and spectral properties [19,20].…”