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Physical review. B, Condensed matter

Time-dependent correlations for spin Van der Waals systems

Raf Dekeyser, M. Howard Lee

Published Jan 1, 1979• 54 citations
Publisher

Abstract

The time-dependent autocorrelation function has been derived for the $x$ component of the total spin for the $S=\frac{1}{2}$ constant-coupling anisotropic Heisenberg model, i.e., the Van der Waals system. For $T>{T}_{c}$, the time correlation is shown to be Gaussian for both $\mathrm{XY}$-like and Ising-like regimes of the model. For $T<{T}_{c}$, the correlation function is still Gaussian if the model is $\mathrm{XY}$-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.

DOI
10.1103/physrevb.19.265

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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}${\mathit{t}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\kappa}}}$, as t\ensuremath{\rightarrow}\ensuremath{\infty}, is found to depend discontinuously on the spin-spin interaction strength R=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.
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.
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.
A phenomenological dynamic scaling method is proposed and utilized to interpret finite-size effects in the time-dependent correlation function for the classical spin van der Waals model. We claim that the equations of motion for the classical spin van der Waals model are well defined only within a certain characteristic time. The order of magnitude for the size-dependent maximum time interval of the time-dependent correlation function, analytically obtained in the large N limit via the method of Laplace by Dekeyser and Lee, is found to be related to our characteristic time. As a by-product of this work, time scaling variables are found with which we can completely collapse the time-dependent correlation functions of the classical spin van der Waals model into one curve. \textcopyright{} 1996 The American Physical Society.