An exact perturbation expansion, to the order 1/S2, is derived for the Heisenberg ferromagnet. The equivalence of the Dyson-Maleev (DM), Holstein-Primakoff (HP) and matching-of-matrix-element (MME) transformations is proven. They give identical T5/2 and T4 coefficients. For the planar ferromagnet, an exact expansion in anisotropy against exchange field proves the equivalence of the HP and MME transformations. Several explicit results and numerical estimates are given.
Rastelli, E., & Lindgård, P-A. (1979). Exact Results for Spin-Wave Renormalisation in Heisenberg and Planar Ferromagnets. JOURNAL OF PHYSICS C-SOLID STATE PHYSICS, 12(10), 1899-1916. https://doi.org/10.1088/0022-3719/12/10/021