Diffuse and Propagating Modes in the Heisenberg Paramagnet

Abstract

The spectral weight function of the frequency $\omega$ and wave number $q$ for the spin pair correlation function in a Heisenberg system is studied in the paramagnetic region. It is suggested that this function will exhibit propagating modes at short wavelengths and at temperatures $T$ not much greater than the transition temperature $T_c$. This suggestion follows from the approximation used to evaluate the moments of the spectral weight function and from the assumption that the generalized diffusivity, of which the pair correlation function is a functional, contains no $\delta$-function terms $\delta \left(\omega \right)$ and is a smooth monotonic function. The region for which the dispersion equation has real solutions (propagating modes) $\omega ={\omega }_B\left(\mathrm{q}\right)$ is obtained by estimating the moment-fluctuation ratio $R\left(\mathrm{q}\right)=\frac{{{〈{\omega }^2〉}_{\mathrm{q}}}^2}{{〈{({\omega }^2-{〈{\omega }^2〉}_{\mathrm{q}})}^2〉}_{\mathrm{q}}}$ for the pair correlation function. When $T>\mathrm{}{T}_c$ and when $q$ is greater than a critical wave number $q_c$, the estimate gives $R\left(\mathrm{q}\right)>\mathrm{}1\mathrm{}$ and thereby predicts propagating modes. An approximate nonlinear integral equation for the susceptibility is used to estimate $R\left(\mathrm{q}\right)$. It is shown that the critical wave number $q_c$ is proportional to the inverse square root of the static susceptibility, $q_c\sim {\chi }^{-\frac{1}{2}}\mathrm{}\left(T\right)\mathrm{}$. This approximation yields an expression for $R\left(\mathrm{q}\right)$ at high temperature which is in substantial agreement with the exact high-temperature evaluation of $R\left(\mathrm{q}\right)$. The exact and approximate evaluations of $R\left(\mathrm{q}\right)$ for high temperature predict that $R\left(\mathrm{q}\right)<\mathrm{}1\mathrm{}$ for all values of wave vector q in the first Brillouin zone, and consequently suggest that there are no high-frequency propagating modes at high temperatures.