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

Abstract

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 ${({\mathbf{S}}_{i}\cdot {\mathbf{S}}_{j})}^{2}$ 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 ), ${({\mathbf{S}}_{i}\cdot {\mathbf{S}}_{j})}^{2}$ , and ${({\mathbf{S}}_{i}\cdot {\mathbf{S}}_{j})}^{3}$ 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 .

Keywords

• Heisenberg model
• multiorbital Hubbard model
• strongly correlated systems
• exact diagonalization