Longitudinal Susceptibility of a Planar Ferromagnet

The longitudinal susceptibility ${\chi }_{\mathrm{zz}}(\overrightarrow{\mathrm{q}},\omega )$ of a spin-½ planar ferromagnet is investigated. The random-phase approximation is used and particular emphasis is placed on the long-wave-length behavior. The static susceptibility for temperatures in the vicinity of the Curie point is calculated. Correlation lengths above and below $T_C$ are obtained. In the region below $T_C$ the expression for ${\chi }_{\mathrm{zz}}(\overrightarrow{\mathrm{q}},\omega )$ is compared with the susceptibility of an ideal magnon gas. Departures from ideal-gas behavior near the Curie point are studied. The relevance of the findings to recent hydrodynamic- and dynamic-scaling-law theories is discussed.