Deterministic and stochastic spin diffusion in classical Heisenberg magnets

Abstract

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 <Sj(t) *Sj), from anomalous spin diffusion (at. > l/2) to normal spin diffusion (a1 = l/2) at some characteristic time lag that depends on the precision of the numerical integration.

Breakdown of hydrodynamics in the classical 1D Heisenberg model

“…The picture that emerges is that although the energy diffusion shows a classical diffusive behavior, surprisingly, the spin diffusion shows a nonclassical behavior that is manifested in all measured quantities. In particular the asymptotic behavior of the autocorrelation is of the form Co(t) ~~(t\nt)~l /2n (we show later in the paper that this functional form explains the results of [11,16] and those of [13]). There is thus a breakdown of the usual hydro-dynamic assumptions as originally suggested by Bloembergen [17] and van Hove [18].…”

“…Theoretically, both analytical tools [3,4] and computer simulations [5][6][7][8][9][10][11][12][13][14][15][16] have been widely used. In particular, Muller [11] analyzed in some detail the time dependence of the spin autocorrelation function of the classical Heisenberg model for dimensionalities rf-»l, 2, and 3 at infinite temperatures and observed a power-law long-time tail whose exponent deviated from the classical diffusion theory prediction: ad*=d/2.…”

“…They conclude there is no anomalous diffusion in d=\ much less in higher dimensions and that the asymptotic behavior for the autocorrelation function is only reached at very long times. Subsequently Jian-Min Liu et al [16] suggested that the computational error in the numerical integration of the equations of motion affects the long-time decay of the autocorrelation causing a crossover from anomalous spin diffusion (a> y) to classical spin diffusion at some characteristic time that depends on the accuracy of the numerical integration. In the present work we give a much more detailed analysis of this problem by simulating the (/-dependent energy and spin correlations as well as the respective current-current correlations.…”

Spin diffusion in the one-dimensional classical Heisenberg model

“…Although the phenomenology of spin diffusion is an old concept [3,4], its validity in classical Heisenberg model has been vigourously debated in recent times. Although much effort [13][14][15][16][17][18][19][20] has been devoted to understand whether spin diffusion in this system is normal or anomalous, a convincing conclusion is yet to be reached. Settling this question is not only conceptually important e.g., in understanding transport properties of spin systems, but also has direct implications in routinely performed experiments e.g., NMR and ESR in magnetic compounds [5][6][7][8][9].…”

“…Nevertheless, they suggested that the problem is computationally difficult since the non-asymptotic behaviour of the spin ACF is quite pronounced. In yet another work [15], it was concluded that in numerical simulation it is not possible to observe the t −1/2 behavior, even if it exists, due to the fact that the numerical scheme introduces computational errors and this violated the conservation of total spin S. The error propagation affects the decay of the ACF and makes it anomalous. It was suggested that the correlation function may show a crossover from non-diffusive to diffusive behavior and the characteristic crossover time will de-pend on the precision of the numerical scheme employed.…”