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

A few <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mn>4</a:mn><a:mi>d</a:mi></a:mrow></a:math> and <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mn>5</b:mn><b:mi>d</b:mi></b:mrow></b:math> transition metal compounds with various electron fillings were recently found to exhibit magnetic susceptibilities <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>χ</c:mi></c:math> and magnetic moments that deviate from the well-established Kotani model. This model has been considered for decades to be the canonical expression for descriing the temperature dependence of magnetism in systems with nonnegligible spin-orbit coupling effects. In this paper, 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 <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mn>4</d:mn><d:mi>d</d:mi></d:mrow></d:math> (Ru-based) and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mn>5</e:mn><e:mi>d</e:mi></e:mrow></e:math> (W-based) systems at different electron fillings. For this purpose, we perform exact diagonalization of -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 <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:msub><f:mi>χ</f:mi><f:mn>0</f:mn></f:msub></f:math> background, crystal field effects, Coulomb and Hund's couplings, and intersite interactions—not included in the Kotani model—are especially crucial for correctly describing the temperature dependence of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>χ</g:mi></g:math> and magnetic moments at various electron fillings in these systems. Based on our results, we propose a generalized approach beyond the Kotani model to accurately describe their magnetism.