We present a framework for analyzing Mott insulators using a material-based tight-binding model. We start with a realistic multiorbital Hubbard model and derive an effective model for the localized electrons through the second-order perturbation theory with respect to intersite hopping. This effective model, known as the Kugel-Khomskii model, is described by $\mathrm{SU}\left(N\right)$ generators, where $N$ is the number of localized states. We solve this model by the mean-field theory that takes local correlations into account and reveal spin-orbital ordered states. To include spatial correlations, we apply the classical Monte Carlo based on the path-integral approach with $\mathrm{SU}(N$) coherent states, and also derive the equation of motion for spin-orbital degrees of freedom. Our approach enables quantitative analysis of Mott insulator materials with a small intersite quantum correlation. The $5d$-pyrochlore oxide is used here as a demonstration.

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• Magnetic order
• Orbital order
• Order parameters
• Spin-orbit coupling
• Mott insulators
• Strongly correlated systems
• Extended Hubbard model
• Hubbard model
• Mean field theory
• Monte Carlo methods