Anab initioeffective Hamiltonian for magnetism including longitudinal spin fluctuations

We discuss the use of the magnetic force theorem using different reference states upon which the perturbative approach is based. Using a fixed spin disordered local moment state one finds good Curie (or Ne\'el) temperatures, and good energetics for planar spin spirals in the $3d$ magnets Fe, fcc Co, Ni, Mn, and Cr, though worse agreement for small θ spin spirals. On the other hand, the ferromagnetic reference state provides excellent energetics for small θ spin spirals in Fe, fcc Co, and Ni, and by extension magnon energies under the assumption of adiabacity. However, planar spin spiral energetics and transition temperatures show worse agreement. The reasons for this, and for the case of fcc Co where both approaches work very well, are discussed. We further provide an extension of the mapping of the quantum problem to include longitudinal fluctuations within force theorem based approaches, and discuss the role they will play in magnetic phase transitions. This construction is tested using planar spin spirals where q is fixed but the moment is allowed to relax. It is demonstrated that results from this approach and directly calculated ab initio values agree very well.