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Physical review. B, Condensed matter

Time-dependent behavior of classical spin chains at infinite temperature

R. W. Gerling, David Paul Landau

Published Nov 1, 1990• 44 citations
Publisher

Abstract

An ultrafast, vectorized spin-dynamics method is used to study the time-dependent properties of classical XY and Heisenberg spin chains at infinite temperature. The decay of the energy-energy and spin-spin correlation functions is oscillatory for short times and at long times is consistent with classical diffusion, although the approach to the asymptotic behavior is extremely slow. We have also calculated S(q,\ensuremath{\omega}) and find clear indication of spin-wave peaks in both models.

DOI
10.1103/physrevb.42.8214

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We present results for the static and dynamic behavior of a one-dimensional classical XY\mathrm{XY}XY chain. Equilibrium configurations are produced by an importance-sampling Monte Carlo method. Static properties are evaluated from these data and a spin-dynamics method is then used to evaluate time-displaced correlation functions and to calculate the scattering function $S(q,\ensuremath{\omega})$. The transverse spin-spin correlation functions decay exponentially for large times. The longitudinal spin-spin correlation functions show oscillatory behavior, especially at low temperatures. The longitudinal-scattering function displays spin-wave peaks, and the transverse-scattering function shows central peaks (relaxational behavior) which change to fluctuation-broadened spin-wave peaks for large values of $\frac{q}{\ensuremath{\kappa}}$. The energy-energy correlation function shows diffusive behavior for high temperatures and non-diffusive behavior for low temperatures.
A coupled-mode theory for the spin response function is used to calculate time-dependent properties of the classical spin chain at infinite temperature. The behaviour of the spin autocorrelation function is consistent with computer simulation data, in that the exponent of the time tail decreases on moving from intermediate to asymptotic time regions. Results for the spin response function and memory function in the two time regions are obtained by a combination of asymptotic analysis and numerical studies. For a large wavevector, near the zone boundary, the spin response function displays a very pronounced oscillatory time dependence. However, there is next to no structure in the corresponding spectral function, in accord with simulation data. The asymptotic analysis shows that, in the infinite-temperature limit, coupled-mode theory does not comply with usual hydrodynamic assumptions, a view that has also emerged from analysis of some of the computer simulation data.
The time-dependent behavior of the classical, ferromagnetic xy chain in a symmetry-breaking magnetic field was studied using a very fast, vectorized spin-dynamics method. The equations of motion were integrated using starting configurations determined by Monte Carlo simulations. By calculating spin-spin correlation functions and taking space-time Fourier transforms, we determined S(q,\ensuremath{\omega}) for a wide range of fields and temperatures. We identify and measure the characteristics of single spin-wave peaks and two spin-wave sum peaks, as well as central peaks due to both soliton and two spin-wave difference processes. These results are discussed in light of existing theory and experiments.
The energy and spin diffusion behaviors in the one-dimensional classical Heisenberg spin chain have been systematically investigated using the equilibrium diffusion method. The spatiotemporal autocorrelation functions for energy and spin are calculated at finite and infinite temperatures. As conserved quantities, the spreading of excess energy and spin can be used to determine their actual diffusion behaviors. At low temperatures, the energy diffusion shows almost ballistic behavior, and spin shows superdiffusion behavior for finite chain size. For energy diffusion, normal diffusion behavior can be obtained when the temperature is higher than 0.75. For spin diffusion, normal diffusion behavior is observed at infinite temperature.
Results are presented for the static and time-dependent behavior of the classical one-dimensional XY-model at finite temperatures. Equilibrium properties were determined using an importance sampling Monte Carlo method. A spin-dynamics computer simulation method was then used to evaluate spin-spin and energy-energy time-displaced correlation functions out to time t=10J−1 using initial states chosen from the equilibrium distribution. The transverse spin-spin auto correlation function decreases smoothly as a function of time. The longitudinal spin-spin correlation functions, however, show temperature dependent oscillatory behavior. At moderately high temperature the energy-energy correlations show diffusive t−1/2 behavior at long times with only a slow temperature dependence of the diffusion constant.