Existing first-order Green's-function theories of the Heisenberg ferromagnet all lead to magnon energies for which the temperature renormalization is wave vector independent. Such theories can describe phase transitions only at a temperature $T_C$ for which all spin-wave excitations have vanishingly small energy. It has become increasingly evident, particularly for systems of low dimensionality, that such an approximation is quite unphysical, paramagnetic magnons often being physically well defined over much of the Brillouin zone to quite elevated temperatures. This paper describes a rather general method for introducing wave-vector-dependent magnon renormalization into the Green's-function formalism, enabling approximations of obvious physical significance to be made directly in terms of the magnon dispersion relation. The theory is developed in detail for the simplest nontrivial approximation and applied to the problem of the two-dimensional Heisenberg ferromagnet. A phase transition is found to a state of zero magnetization and infinite susceptibility. We also discuss the weakly anisotropic two-dimensional ferromagnet, which supports long-range order at low temperatures, and study the approach to the isotropic limit.

• Received 18 May 1970

DOI:https://doi.org/10.1103/PhysRevB.3.1749