Space-Time Correlations in Exchange-Coupled Paramagnets at Elevated Temperatures

Abstract

The frequency-dependent correlations ${\Phi }_n\left(\omega \right)\equiv {〈{S_0}^z\mathrm{}\left(t\right)\mathrm{}{S_n}^z\left(t^{\prime }\right)〉}_{\left(\omega \right)}$, $n=0,1,2,3, \mathrm{and} 4$, and their inverse-lattice Fourier transforms $\Phi (\mathbf{K},\omega )$, have recently been exactly calculated by Carboni and Richards for finite linear chains consisting of $N$ ($N=6,7,8,9, \mathrm{and} 10$) spins ($S=\frac{1}{2}$) in contact with a heat bath at temperature $T\to \infty$ and interacting via a nearest-neighbor isotropic Heisenberg exchange interaction. Carboni and Richards have used plausible extrapolation procedures to predict the corresponding correlations in the thermodynamic limit $N\to \infty$. As the extension of these exact calculations to two and three dimensions is of prohibitive difficulty, we have constructed an alternative theory, based upon a simple two-parameter Gaussian representation of the generalized diffusivity, for calculating these correlations for general spin and the dimensionality. To test the accuracy of this phenomenological theory (which is, however, free of any arbitrariness in the sense that the diffusivity is exactly specified by the second and the fourth frequency moments of the Fourier transform of the frequency-dependent correlation function, which are known at infinite temperature), we first compare our results for ${\Phi }_n\left(\omega \right)$ with those given by Carboni and Richards for the one-dimensional spin-½ system and find the agreement to be excellent for $n=0\mathrm{}$, good for $n=1,2\mathrm{}$ and adequate for $n=3,4\mathrm{}$. The comparison of the corresponding results for $\Phi (\mathbf{K},\omega )$ reveals the agreement to be quite satisfactory for $\left|\mathbf{K}\right|\lesssim \frac{1}{2}\pi$ but only adequate for the higher-K range, i.e., $\frac{1}{2}\pi \lesssim \mathbf{K}\lesssim \pi$. Further support in favor of our phenomenological theory is obtained from a comparison with another set of available exact "computer experiment" results, whereby ${\Phi }_0\left(\omega \right)$, ${\Phi }_0\mathrm{}\left(t\right)\mathrm{}$, and ${\Phi }_1\mathrm{}\left(t\right)\mathrm{}$ are accurately known for a three-dimensional (simple-cubic) lattice of infinite spins, i.e., $S\to \infty$. (This is the limit in which the classical spin system studied by Windsor corresponds to the quantum spin system of interest to us.) The agreement of our result with Windsor's is excellent. The salient features of our results are as follows: (i) In two dimensions, the divergence of ${\Phi }_0\left(\omega \right)$ for $\omega \to 0$ becomes less sharp and disappears completely in three dimensions. (ii) The cutoff frequency, beyond which ${\Phi }_0\left(\omega \right)$ is effectively zero, increases with the dimensionality, being in three dimensions about twice what it is in one dimension. (iii) There exists a system of reduced units, i.e., ${{\Phi }_n}^{\prime }\left(\omega \right)\to {\Phi }_n\left(\omega \right){\mathrm{}\left[S\right(S+1\mathrm{}\left)\right]}^{-1\mathrm{}}$, $I^{\prime }\to {\mathrm{}\left[S\right(S+1\mathrm{}\left)\right]}^{\frac{1}{2}}I$, and ${\omega }^{\prime }\to \left(\frac{\omega }{I^{\prime }}\right)$, in which the function $I^{\prime }{{\Phi }_n}^{\prime }\left({\omega }^{\prime }\right)$ is approximately the same for all spins, and the accuracy of this law of corresponding states seems to increase with the increase in the dimensionality.