Calculation of the biquadratic spin interactions based on the spin cluster expansion for \textit{ab initio} tight-binding models

We devise a calculation scheme using \textit{ab initio} tight-binding Hamiltonians to evaluate biquadratic magnetic interactions. This approach relies on the spin cluster expansion combined with the disordered local moment (DLM) method, originally developed within the multiple scattering Korringa-Kohn-Rostoker method. Applying it to the two-orbital Hubbard model, we show that the evaluated DLM biquadratic interactions agree well with those obtained from the strongly correlated limit, demonstrating wide applicability {of the method} to various magnetic systems with large local moments. We then apply it to the \textit{ab initio} tight-binding models for bcc Fe and fcc Ni; the resulting magnetic interactions align well with previous studies in the literature. The present approach offers a convenient \textit{ab initio} path for evaluating biquadratic interactions and understanding the electronic mechanisms controlling them.