The spin dynamics of two dimensional XY ferromagnet are reexamined at low temperatures in the framework of the Mori continued fraction formalism using the Gaussian approximation. In this formalism, the terms on denominators for the Laplace transform of the relaxation function R(t) are related to the frequency moments of the relaxation shape function R(q,w). In the Gaussian approximation scheme, we truncate the continued fraction for R(t) on the second stage. Adopting this approximation, we calculated up to the sixth moment. The moments are writing in terms the static spin correlation functions. In the estimation of the fourth and sixth moment at finite temperature, the four and six-spin correlation functions may be approximated by a sum of products of appropriate pair correlation functions(mode-mode decoupling). In this work we calculated the static spin correlation functions for the expressions to the fourth and sixth moments, needed in the study of the dynamics.
Numerical calculations of quantum-mechanical time correlation functions are reported for finite Heisenberg linear chains containing up to 10 spin-\textonehalf{} particles. The results, obtained by manipulation of eigenvalues and eigenfunctions, give Fourier transforms in histogram form. Two-spin correlation functions show a sharp rise near zero frequency and highly non-Gaussian behavior at infinite temperature. The short-time behavior is estimated and compared with classical calculations and other theories. Four-spin functions are calculated and compared with results of a simple decoupling approximation. Spatial Fourier transforms of two-spin functions as needed for neutron scattering cross sections are computed. These are limited to fairly short wavelengths by the finiteness of the system. The temperature dependence of two-spin functions shows a decrease in the near-zero frequency component with decreasing temperature. Computed correlation functions are used to predict the frequency dependence of the ESR linewidth in the linear-chain salt
Cu(NH3)4S
O4\ifmmode\cdot\else\textperiodcentered\fi{}
H2O, and comparison is made with experiment. Very good agreement is found using the same exchange constant
J as inferred from specific-heat and magnetic-susceptibility data. The Gaussian approximation, on the other hand, is in extremely poor agreement.
Time-dependent statistics of the Ising model proposed by Glauber for the one-dimensional chain are extended to the example of a two-dimensional square lattice. Each spin is assumed to change its state through the interaction with a heat bath. The equations of motion for both the single spin and the spin correlation functions are solved approximately by using a decoupling procedure where the many-body correlation functions are taken as sums of products of pair correlation functions. As a special case, our theory allows the approximate calculation of the equilibrium properties of the system and it turns out that, in this case, our result is an improvement over the Bethe approximation. Both the frequency-dependent magnetic susceptibility and the decay of the magnetic moment to the equilibrium state are calculated above and below the Curie temperature. The fluctuation–dissipation theorem developed by Glauber for the linear chain is shown to hold in the two-dimensional case also.