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Journal of Applied Physics

Computer simulations of the classical one-dimensional XY-model

David Paul Landau, John Thomchick

Published Mar 1, 1979• 9 citations
Publisher

Abstract

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.

DOI
10.1063/1.327182

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Physical review. B, Solid state•Oct 1, 1974•9 citations
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.
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.
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.
We have studied the dynamics of the classical Heisenberg chain at finite and infinite temperatures by computer simulation of an array of 4000 spins. The evaluation of the spin-spin and energy-energy wave-vector- and time-dependent correlation functions out to time $10{J}^{\ensuremath{-}1}$ made it possible to determine the spin and energy diffusion constants. At infinite temperature we find ${D}_{S}=(1.33\ifmmode\pm\else\textpm\fi{}0.10)JS{a}^{2}$ and ${D}_{E}=(3\ifmmode\pm\else\textpm\fi{}1)JS{a}^{2}$. Values of DS{D}_{S}DS​ have also been determined, with less precision, for ferromagnetic and antiferromagnetic interactions at temperatures kBT∣J∣=1.0and0.5\frac{{k}_{B}T}{|J|}=1.0 \mathrm{and} 0.5∣J∣kB​T​=1.0and0.5. Our results are compared with various theoretical estimates. In addition, we calculate the time-dependent spin and energy self-correlation functions.
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.
Computer calculations of the time-dependent spin and energy correlation functions of the classical one-dimensional XY\mathrm{XY}XY model at infinite temperature are reported. Plots of 〈SziSzi(t)〉〈{S}_{z}^{i}{S}_{z}^{i}(t)〉〈Szi​Szi​(t)〉, 〈SziSzi+1(t)〉〈{S}_{z}^{i}{S}_{z}^{i+1}(t)〉〈Szi​Szi+1​(t)〉, $〈{\ensuremath{\epsilon}}^{i}{\ensuremath{\epsilon}}^{i}(t)〉$, and (12)[〈SxiSxi(t)〉+〈SyiSyi(t)〉](\frac{1}{2})[〈{S}_{x}^{i}{S}_{x}^{i}(t)〉+〈{S}_{y}^{i}{S}_{y}^{i}(t)〉](21​)[〈Sxi​Sxi​(t)〉+〈Syi​Syi​(t)〉] out to times Jt=9Jt=9Jt=9 are given (${\ensuremath{\epsilon}}^{i}$ is the energy-density operator and $\ensuremath{\hbar}J$ is the exchange integral). Comparison is made with the exact results for the spin-1/2 XY\mathrm{XY}XY model. After an appropriate scaling of the exchange integral and normalization to the classical value at t=0t=0t=0 the values of the spin-1/2 functions are close to the values of their classical counterparts for times up to $\mathrm{Jt}\ensuremath{\approx}4$.