Nonperturbative calculation of exchange coupling parameters

Exchange coupling parameters $J_{ij}$ within the Heisenberg model and its extensions are crucial for understanding magnetic behavior at the atomic level. To calculate $J_{ij}$ from first principles, perturbative methods based on the magnetic force theorem (MFT), commonly referred to as Liechtenstein's method, have been widely employed. However, the quantitative accuracy of $J_{ij}$ obtained through this perturbative technique remains uncertain. In this paper, we nonperturbatively calculate $J_{ij}$ for several systems of both fundamental and applied significance, including perovskite SrMnO$_3$, neodymium-magnet compounds, and elemental 3$d$ transition metals, and compare these results with those obtained using the conventional MFT-based method. The nonperturbative approach provides consistent results for magnetic configurations exhibiting a certain degree of magnetic disorder. In contrast, due to its inherent theoretical limitations, the MFT-based method struggles to accurately evaluate the magnetic energy in such configurations. As discussed later in this paper, these discrepancies arise from neglected contributions within MFT-based methods. Our nonperturbative scheme is highly versatile and holds strong prospects as a new platform for analyzing and designing spintronic materials, where developing accurate spin models is essential.