Temperature-dependent renormalization of magnetic interactions by thermal, magnetic, and lattice disorder from first principles

We put forth an ab initio framework to calculate local moment magnetic interaction parameters, renormalized to treat both the lattice and magnetic systems as a function of temperature $T$. For bcc Fe, magnetic and lattice thermal disorders act in opposition, the former strengthening the Heisenberg-like interactions, while the latter decreasing them. Below ${T}_{C}$, $J$ stays nearly independent of $T$, while around and above ${T}_{C}$, it exhibits a sharp decrease. This remarkable behavior reflects an intricate spin-lattice coupling and its evolution with $T$, in which magnetic interactions and interatomic bonds are each renormalized by the other. This finding is consistent with magnetization data and with the observed softening of magnon and phonon modes at high temperatures. Magnetization as well as magnon and phonon mode softening are discussed.