Microscopic origin of temperature-dependent magnetism in spin-orbit-coupled transition metal compounds

A few $4d$ and $5d$ transition metal compounds with various electron fillings were recently found to exhibit magnetic susceptibilities $\chi$ and magnetic moments that deviate from the well-established Kotani model. This model has been considered for decades to be the canonical expression to describe the temperature dependence of magnetism in systems with non-negligible spin-orbit coupling effects. In this work, we uncover the origin of such discrepancies and determine the applicability and limitations of the Kotani model by calculating the temperature dependence of the magnetic moments of a series of $4d$ (Ru-based) and $5d$ (W-based) systems at different electron fillings. For this purpose, we perform exact diagonalization of $ab~initio$-derived relativistic multiorbital Hubbard models on finite clusters and compute their magnetic susceptibilities. Comparison with experimentally measured magnetic properties indicates that contributions such as a temperature independent $\chi_0$ background, crystal field effects, Coulomb and Hund's couplings, and intersite interactions - not included in the Kotani model - are specially crucial to correctly describe the temperature dependence of $\chi$ and magnetic moments at various electron fillings in these systems. Based on our results, we propose a generalized approach to describe their magnetism.