Nonlinear dynamics of a Heisenberg ferromagnet.

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Using an expansion technique we study the dynamics of a spin-s Heisenberg ferromagnet with nearest-neighbor exchange coupling J. The transverse-mode frequencies are calculated by expanding a self-energy function in powers of 1/z, where z is the number of nearest neighbors on the lattice. To zeroth-order in 1/z, the mode frequencies agree with the random-phase approximation. To the next order in 1/z, the coupling between the transverse and longitudinal fluctuations is responsible for a shift in the spin-wave (SW) frequencies ${\mathrm{\ensuremath{\omega}}}_{\mathbf{k}}$ and for the appearance of a second pole in the correlation function at an energy close to zJs. This second mode is excited by longitudinal fluctuations, which force the local spin to precess about the mean field with frequency zJs rather than with the spin-wave frequency ${\mathrm{\ensuremath{\omega}}}_{\mathbf{k}}$. Because of its interactions with the surrounding spins, this precession can propagate through the lattice. When ${\mathrm{\ensuremath{\omega}}}_{\mathbf{k}}$ is close to zJs, the precessional mode may be observable as a splitting of the transverse resonance into two peaks. Since the coupling between the longitudinal and transverse fluctuations becomes exponentially small at low temperatures, the precessional mode only appears above the crossover temperature T\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\approxeq}0.2zJs. Because the SW approximation mishandles the subtle interplay between the longitudinal and transverse fluctuations, it misses the second pole in the correlation function. In agreement with earlier work we find that the SW approximation breaks down above the temperature T\ifmmode\bar\else\textasciimacron\fi{}, when the exponential coupling terms in the correlation function become significant. Although the precessional mode is induced by longitudinal fluctuations, it is fundamentally a transverse excitation and must be distinguished from the longitudinal mode, which has zero frequency.