Many-body Green's function theory of ferromagnetic Heisenberg systems with single-ion anisotropies in more than one direction

TL;DR

This paper develops a comprehensive many-body Green's function approach to analyze ferromagnetic Heisenberg systems with multiple directional single-ion anisotropies, providing analytical expressions for magnetization components and expectation values.

Contribution

It generalizes previous uniaxial models to include multiple anisotropy directions and derives analytical formulas for all magnetization components without favoring any spatial axis.

Findings

Analytical expressions for magnetization components in systems with multiple anisotropies.

Numerical results for 2D and 3D systems showing anisotropy effects.

Inclusion of dipole-dipole interactions in 2D case.

Abstract

The behaviour of ferromagnetic systems with single-ion anisotropies in more than one direction is investigated with many-body Green's function theory generalizing earlier work with uniaxial anisotropies only. It turns out to be of advantage to construct Green's functions in terms of the spin operators S^x, S^y and S^z, instead of the commonly used S^+,S^- and S^z operators. The exchange energy terms are decoupled by RPA and the single-ion anisotropy terms by a generalization of the Anderson-Callen decoupling. We stress that in the derivation of the formalism none of the three spatial axes is special, so that one is always able to select a reference direction along which a magnetization component is not zero. Analytical expressions are obtained for all three components of the magnetization and the expectation values <(S^x)^2>, <(S^y)^2> and <(S^z)^2> for any spin quantum number S. The formalism considers both in-plane and out-of-plane anisotropies. Numerical calculations illustrate the behaviour of the magnetization for 3-dimensional and 2-dimensional systems for various parameters. In the 2-dimensional case, the magnetic dipole-dipole coupling is included, and a comparison is made between in-plane and out-of-plane anisotropies.