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Physical review. B, Condensed matter

Static and dynamic properties of the classical XY chain

R. W. Gerling, David Paul Landau

Published Feb 1, 1983• 11 citations
Publisher

Abstract

We present results for the static and dynamic behavior of a one-dimensional classical $\mathrm{XY}$ 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.

DOI
10.1103/physrevb.27.1719

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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,\ensuremathω)S(q,\ensuremath{\omega})S(q,\ensuremathω). 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 q\ensuremathκ\frac{q}{\ensuremath{\kappa}}\ensuremathκq​. The energy-energy correlation function shows diffusive behavior for high temperatures and non-diffusive behavior for low temperatures.
Results are presented for the static and dynamic behavior of a classical XY-chain in a magnetic field. Equilibrium configurations were 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. The energy-energy correlation function shows a t−1 behavior for times t≳J−1. The correlation function <Sxi(0)Sxi+r(t)≳ decays to the square of the magnetization for large times, and the correlation function <Syi(0)Sy1+r(t)≳ shows oscillatory behavior.
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.
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.
We have investigated the time-dependent behavior of the classical XY chain in a symmetry-breaking anisotropy field using an ultrafast spin-dynamics method in which we integrate the equation of motion with equilibrium spin configurations determined by Monte Carlo simulations. All three polarizations of the scattering function S(q,\ensuremath{\omega}) were determined for temperature kB{\mathit{k}}_{\mathit{B}}kB​T=0.2J and fields varying from 0.1J to 0.6J. Substantial differences are found as compared with the isotropic ferromagnetic chain in a transverse magnetic field. The soliton contribution, single-spin-wave peaks, and both the sum and difference two-spin-wave contributions are separated. Half-widths and integrated intensities are extracted and compared with theoretical predictions based on mapping onto the sine-Gordon model.
Using spin-dynamics techniques we have performed large-scale computer simulations of the dynamic behavior of the classical three component XY model (i.e., the anisotropic limit of an easy-plane Heisenberg ferromagnet), on square lattices of size up to 1922{192}^{2}1922, for several temperatures below, at, and above TKT{\mathit{T}}_{\mathrm{KT}}TKT​. The temporal evolution of spin configurations was determined numerically from coupled equations of motion for individual spins by a fourth-order predictor-corrector method, with initial spin configurations generated by a hybrid Monte Carlo algorithm. The neutron-scattering function S(q,\ensuremath{\omega}) was calculated from the resultant space-time displaced spin-spin correlation function. Pronounced spin-wave peaks were found both in the in-plane and the out-of-plane scattering function over a wide range of temperatures. The in-plane scattering function Sxx{\mathit{S}}^{\mathit{xx}}Sxx also has a large number of clear but weak additional peaks, which we interpret to come from two-spin-wave scattering. In addition, we observed a small central peak in Sxx{\mathit{S}}^{\mathit{xx}}Sxx, even at temperatures well below the phase transition. We used dynamic finite-size scaling theory to extract the dynamic critical exponent z. We find z=1.00(4) for all T\ensuremath{\le}TKT{\mathit{T}}_{\mathrm{KT}}TKT​, in excellent agreement with theoretical predictions, although the shape of S(q,\ensuremath{\omega}) is not well described by current theory. \textcopyright{} 1996 The American Physical Society.