Range of biquadratic and triquadratic Heisenberg effective couplings deduced 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 $({\mathbf{S}}_i\cdot {\mathbf{S}}_j)$ 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 mappping is into a $S=3/2$ Heisenberg model with $({\mathbf{S}}_i\cdot {\mathbf{S}}_j)$, $({\mathbf{S}}_i\cdot {\mathbf{S}}_j{)}^2$, 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 allows to establish bounds on how large the ratios $J_2/J_1$ and $J_3/J_1$ can be. We show that those ratios are severely limited and, as a crude 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.

8 pages, 7 figures

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