Abstract
It is generally accepted that the effective magnetic field acting on a magnetic moment is given by the gradient of the energy with respect to the magnetization. However, in ab initio spin dynamics within the adiabatic approximation, the effective field is also known to be exactly the negative of the constraining field, which acts as a Lagrange multiplier to stabilize an out-of-equilibrium, non-collinear magnetic configuration. We show that for Hamiltonians without mean-field parameters both of these fields are exactly equivalent, while there can be a finite difference for mean-field Hamiltonians. For density-functional theory (DFT) calculations the constraining field obtained from the auxiliary Kohn-Sham Hamiltonian is not exactly equivalent to the DFT energy gradient. This inequality is highly relevant for both ab initio spin dynamics and the ab initio calculation of exchange constants and effective magnetic Hamiltonians. We argue that the effective magnetic field and exchange constants have the highest accuracy in DFT when calculated from the energy gradient and not from the constraining field.
Highlights
The magnetization dynamics of both insulating and metallic materials can, in many cases, be described within the framework of atomistic spin dynamics
In ab initio spin dynamics within the adiabatic approximation, the effective field is known to be exactly the negative of the constraining field, which acts as a Lagrange multiplier to stabilize an out-of-equilibrium, noncollinear magnetic configuration
We argue that the effective magnetic field and exchange constants have the highest accuracy in density functional theory (DFT) when calculated from the energy gradient and not from the constraining field
Summary
The magnetization dynamics of both insulating and metallic materials can, in many cases, be described within the framework of atomistic spin dynamics (see Refs. [1,2] for an overview). [1,2] for an overview) This approach is valid when the electronic Hamiltonian can be mapped onto an effective model of localized spins with constant magnetic moment lengths and interaction parameters that are independent of the spin configuration. While this is generally fulfilled by magnetic insulators, these assumptions may not be valid for magnetic metals, for magnets with noncollinear states [3,4], or for systems far away from equilibrium. This implies that exchange constants and effective magnetic Hamiltonians should be derived from the energy gradient and not from the constraining field
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