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Physical review. B, Solid state

Time-dependent correlation functions of the classical one-dimensional XY model at infinite temperature

David L. Huber

Published Oct 1, 1974• 9 citations
Publisher

Abstract

Computer calculations of the time-dependent spin and energy correlation functions of the classical one-dimensional $\mathrm{XY}$ model at infinite temperature are reported. Plots of $〈{S}_{z}^{i}{S}_{z}^{i}(t)〉$, $〈{S}_{z}^{i}{S}_{z}^{i+1}(t)〉$, $〈{\ensuremath{\epsilon}}^{i}{\ensuremath{\epsilon}}^{i}(t)〉$, and $(\frac{1}{2})[〈{S}_{x}^{i}{S}_{x}^{i}(t)〉+〈{S}_{y}^{i}{S}_{y}^{i}(t)〉]$ out to times $Jt=9$ 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 $\mathrm{XY}$ model. After an appropriate scaling of the exchange integral and normalization to the classical value at $t=0$ the values of the spin-1/2 functions are close to the values of their classical counterparts for times up to $\mathrm{Jt}\ensuremath{\approx}4$.

DOI
10.1103/physrevb.10.2955

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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)〉, 〈\ensuremathϵi\ensuremathϵi(t)〉〈{\ensuremath{\epsilon}}^{i}{\ensuremath{\epsilon}}^{i}(t)〉〈\ensuremathϵi\ensuremathϵi(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 (\ensuremathϵi{\ensuremath{\epsilon}}^{i}\ensuremathϵi is the energy-density operator and \ensuremathℏJ\ensuremath{\hbar}J\ensuremathℏJ 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 Jt\ensuremath≈4\mathrm{Jt}\ensuremath{\approx}4Jt\ensuremath≈4.
The infinite-temperature space- and time-dependent spin-correlation functions grx(t){g}_{r}^{x}(t)grx​(t) are studied for the one-dimensional XY\mathrm{XY}XY model. Numerical calculations are performed to obtain the exact autocorrelation function g0x(t){g}_{0}^{x}(t)g0x​(t) for chains containing 5, 7, and 9 spins (S=12S=\frac{1}{2}S=21​). This yields exact results for the first 16 moments of the frequency autocorrelation function of the infinite chain, and estimates for a few of the higher moments as well. The analysis suggests that g0x(t){g}_{0}^{x}(t)g0x​(t) for the infinite chain is identical to $\frac{\mathrm{exp}(\ensuremath{-}{J}^{2}{t}^{2})}{4}$. We show that grx(t){g}_{r}^{x}(t)grx​(t) for $r\ensuremath{\ne}0$ vanishes identically for all values of time, implying a wave-vector-independent relaxation shape function. Our result for g0x(t){g}_{0}^{x}(t)g0x​(t) is compared with that obtained by Huber for the classical ($S=\ensuremath{\infty}$) chain.
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.
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.
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