Estimation of biquadratic and bicubic Heisenberg effective couplings from multiorbital Hubbard models

We studied a multi-orbital Hubbard model at half-filling for two and three orbitals per site on a two-site cluster via full exact diagonalization, in a wide range for the onsite repulsion U, from weak to strong coupling, and multiple ratios of the Hund coupling J H to U. The hopping matrix elements among the orbitals were also varied extensively. At intermediate and large U, we mapped the results into a Heisenberg model. For two orbitals per site, the mapping is into a S = 1 Heisenberg model where by symmetry both nearest-neighbor (S i · S j ) and (Si·Sj)2 are allowed, with respective couplings J 1 and J 2. For the case of three orbitals per site, the mapping is into a S = 3/2 Heisenberg model with (S i · S j ), (Si·Sj)2, and (Si·Sj)3 terms, and respective couplings J 1, J 2, and J 3. The strength of these coupling constants in the Heisenberg models depend on the U, J H, and hopping amplitudes of the underlying Hubbard model. Our study provides a first crude estimate to establish bounds on how large the ratios J 2/J 1 and J 3/J 1 can be. We show that those ratios appear rather limited and, as a qualitative guidance, we conclude that J 2/J 1 is less than 0.4 and J 3/J 1 is less than 0.2, establishing bounds on effective models for strongly correlated Hubbard systems. Moreover, the intermediate Hubbard U regime was found to be the most promising to enhance J 2/J 1 and J 3/J 1.