Breakdown of the spin-wave approximation for a Heisenberg ferromagnet.

Abstract

By comparing the exact free energy with the free energy of the random-phase approximation (RPA), we find that the spin-wave description of a spin-s Heisenberg ferromagnet breaks down near the temperature T\ifmmode\bar\else\textasciimacron\fi{}=0.18zJs, where J is the ferromagnetic coupling constant and z is the number of nearest neighbors. The scaling of the crossover temperature with zJs agrees with the early results of Vaks, Larkin, and Pikin. We calculate this crossover temperature by expanding the RPA free energy in powers of 1/z on a d-dimensional hypercubic lattice. While the zeroth- and first-order terms in the RPA expansion agree with the terms in the exact expansion, the second-order term disagrees with the exact 1/${\mathit{z}}^{2}$ free energy. Below T\ifmmode\bar\else\textasciimacron\fi{}, the difference between the RPA and exact free energies is negligible. So for TT\ifmmode\bar\else\textasciimacron\fi{}, the RPA summation is justified and the spin-wave description is appropriate. Above T\ifmmode\bar\else\textasciimacron\fi{}, however, the spin-wave interactions become highly nonlinear and the RPA free energy deviates from the exact result. Because all momentum states contribute to the energy above T\ifmmode\bar\else\textasciimacron\fi{}, the transverse free energy enters an equipartition regime and the transverse specific heat tends to zero. As a result, the crossover is marked by a peak in the fluctuation specific heat. The crossover temperature is unchanged if a more sophisticated spin-wave theory is used in place of the RPA. The predicted crossover has been observed as a shoulder in measurements of the total specific heat.