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Physical review. B, Condensed matter

Nonequilibrium statistical mechanics of the spin-1/2 van der Waals model. II. Autocorrelation function of a single spin and long-time tails

Raf Dekeyser, M. Howard Lee

Published Apr 1, 1991• 15 citations
Publisher

Abstract

The autocorrelation function of a single spin 〈s(t)\ensuremath{\cdot}s(0)〉 has been obtained from the time-evolution solutions given in the preceding paper by carrying out the ensemble averages explicitly. Slow decay is found in the transverse component and only at high temperatures (T>${\mathit{T}}_{\mathit{c}}$). The exponent \ensuremath{\kappa}, where 〈${\mathit{s}}^{\mathit{x}}$(t)${\mathit{s}}^{\mathit{x}}$(0)〉\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\kappa}}}$, as t\ensuremath{\rightarrow}\ensuremath{\infty}, is found to depend discontinuously on the spin-spin interaction strength R=${\mathit{J}}_{\mathit{z}}$/J: \ensuremath{\kappa}=2 if R=0, \ensuremath{\kappa}=3 if 0R\ensuremath{\le}2 (R\ensuremath{\ne}1), \ensuremath{\kappa}=\ensuremath{\infty} if R=1, and \ensuremath{\kappa}=\ensuremath{\infty} if R>2. Slow decay in this model is attributed to nondimensional effects, e.g., cooperativity. Physical and mathematical mechanisms of the slow decay are described.

DOI
10.1103/physrevb.43.8131

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The autocorrelation function of a single spin 〈s(t)\ensuremath{\cdot}s(0)〉 has been obtained from the time-evolution solutions given in the preceding paper by carrying out the ensemble averages explicitly. Slow decay is found in the transverse component and only at high temperatures (T>Tc{\mathit{T}}_{\mathit{c}}Tc​). The exponent \ensuremath{\kappa}, where 〈sx{\mathit{s}}^{\mathit{x}}sx(t)sx{\mathit{s}}^{\mathit{x}}sx(0)〉\ensuremath{\sim}t\ensuremath−\ensuremathκ{\mathit{t}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\kappa}}}t\ensuremath−\ensuremathκ, as t\ensuremath{\rightarrow}\ensuremath{\infty}, is found to depend discontinuously on the spin-spin interaction strength R=Jz{\mathit{J}}_{\mathit{z}}Jz​/J: \ensuremath{\kappa}=2 if R=0, \ensuremath{\kappa}=3 if 0R\ensuremath{\le}2 (R\ensuremath{\ne}1), \ensuremath{\kappa}=\ensuremath{\infty} if R=1, and \ensuremath{\kappa}=\ensuremath{\infty} if R>2. Slow decay in this model is attributed to nondimensional effects, e.g., cooperativity. Physical and mathematical mechanisms of the slow decay are described.
The spin van der Waals model has received some attention in recent years as a soluble model, useful for demonstrating collective behavior analytically. By solving the Heisenberg equation of motion, we obtain the time evolution of a single spin in this model when N\ensuremath{\rightarrow}\ensuremath{\infty}, where N is the number of spins in the model. Our solution is characterized by rotations in spin space having the appearance of a Larmor precession. In the accompanying paper, we use this time-evolution solution to obtain the spin autocorrelation function and determine its long-time behavior.
The time-dependent autocorrelation function has been derived for the xxx component of the total spin for the S=12S=\frac{1}{2}S=21​ constant-coupling anisotropic Heisenberg model, i.e., the Van der Waals system. For T>TcT>{T}_{c}T>Tc​, the time correlation is shown to be Gaussian for both XY\mathrm{XY}XY-like and Ising-like regimes of the model. For T<TcT<{T}_{c}T<Tc​, the correlation function is still Gaussian if the model is XY\mathrm{XY}XY-like; but it is oscillatory if the model is Ising-like. Critical slow down appears only with the Ising-like system for this time-correlation function.
A statistical theory of the localized model of ferromagnetism is developed by considering the effective density matrices for a single spin, and two nearest-neighbor spins, respectively. The Ising Model and the Heisenberg model, and even the case of the anisotropic exchange interaction, are treated within a unified mathematical scheme. The result is reduced to that of the Bethe approximation in the case of the Ising model, and the constant coupling approximation in the case of the Heisenberg model. The method is generalized to calculate the static spin-correlation function. It is obtained in Ornstein-Zernike type. In the nonequilibrium state, the expectation value of the spin decays by the homogeneous damping as well as the spin-diffusion process. This relaxation equation is different from that by Van Hove because of the existence of the homogeneous damping. The diffusion constant does not vanish even at the Curie point TC, contrary to Van Hove's result, although the damping constant vanishes at TC. The latter fact causes the critical slowing down of the regression of fluctuation. The solution of this relaxation equation gives the time-dependent spin-correlation function. The method can be applied to the antiferromagnetic case, too.
Extending the Monte Carlo method to dynamic critical phenomena we investigated the time-dependent correlation functions in the two-dimensional one-spin-flip Ising model and the critical behavior of the associated relaxation times. These relaxation times are the following: ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}$, characterizing the approach of the order parameter to equilibrium after a change of temperature $\ensuremath{\Delta}T$ of the system; ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}$ characterizing the slowing down of the order-parameter correlation and autocorrelation functions, respectively; ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}$, characterizing the slowing down of the energy correlation and autocorrelation functions; and finally ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}$, characterizing the cross-correlation function. We give estimates for the associated exponents ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}1.90\ifmmode\pm\else\textpm\fi{}0.10$, and ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}\ensuremath{\approx}1.60\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0.95\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0$, which are consistent with the dynamic scaling hypothesis and with exact inequalities. A detailed comparison with recent high-temperature-expansion estimates is performed, and the reliability of the Monte Carlo results is carefully analyzed.
A systematic approximation scheme is proposed that permits solution of the kinetic equations, recently derived by the authors, describing the high-temperature behavior of the spin autocorrelation function ${\ensuremath{\Gamma}}_{\mathrm{ab}}(t)$. A numerical solution and approximate analytical expressions are given in the first approximation and the importance of the neglected terms is discussed. It is shown that the direct autocorrelation function has an approximate Gaussian behavior for short times and decays then to zero through damped oscillations. A diffusion equation is derived for the short-wave-number Fourier components of ${\ensuremath{\Gamma}}_{\mathrm{ab}}(t)$. We also compare this theory with previous work on the subject.