Chaos in spin clusters: Correlation functions and spectral properties

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.

Regular and Chaotic Time Evolution in Spin Clusters

“…Considering the effect of a single boundary torus, they concluded that decay ∼ t −2.05 would occur; considering a hierarchy of tori, Meiss and Ott [10] find decay like ∼ t −0.96 . While these theoretical predictions seem to accommodate our previous results (more rapid decay, ∼ t −1.5 ) [8] fairly well, the examples presented here involve considerably more slowly decaying time correlation functions, implying the existence of an even stronger trapping mechanism, which is yet to be understood.…”

“…If that is the case, then the intensity spectrum, which is (as a consequence of the Wiener-Khinchin theorem) equal to the Fourier transform of the autocorrelation function, has a singularity ∼ ω −β as ω → 0. In previous work [8] we have already found power-law decay with β < 0 of autocorrelation functions for particular chaotic trajectories of the 2-spin model (1); in that case the intensity spectrum was non-divergent in the low-frequency limit. Here we report calculations for different trajectories of the same general model with very slow autocorrelation decay and consequent low-frequency divergence in the intensity spectrum.…”

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.…”

“…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].…”

Time correlation functions of three classical Heisenberg spins on an isosceles triangle and on a chain

“…[15,16] For the equilateral triangle, such comparisons were only made for T → ∞. [17] In addition, exact and numerical results were presented for the classical dimer [18,19,20], the fourspin ring [21], and the N -spin equivalent neighbor model, which includes the equilateral triangle. [22] For three spins on an equilateral triangle, with equal Heisenberg exchange constants J 1 , the spin sites on each triangle are translationally invariant.…”