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 JH 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 (S i.S j)2 are allowed, with respective couplings J1 and J2 . For the case of three orbitals per site, the mapping is into a S=3/2 Heisenberg model with (S i.S j), (S i.S j)2, and (S i.S j)3 terms, and respective couplings J1 , J2 , and J3 . The strength of these coupling constants in the Heisenberg models depend on the U, JH , and hopping amplitudes of the underlying Hubbard model. Our study provides a first crude estimate to establish bounds on how large the ratios J2/J1 and J3/J1 can be. We show that those ratios appear rather limited and, as a crude guidance, we conclude that J2/J1 is less than 0.4 and J3/J1 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 J2/J1 and J3/J1 .

A deep investigation of NiO and MnO through the first principle calculations and Monte Carlo simulations

“…where Ŝi indicates the magnetic moment direction of the ith site of transition metal atoms. The J i,j , and B indicate the strength of Heisenberg exchange and bi-quadratic exchange interaction [25][26][27] between the ith and jth sites, respectively. The summation notation < i, j > for bi-quadratic interactions indicates that sites i and j are nearest neighbors, and each pair is included once.…”

Entanglement Negativity and Concurrence in Some Low-Dimensional Spin Systems

“…The AKLT model extended the notion of valence bond states to spins higher than . The AKLT model for has and as both positive, and the ground state is exactly solvable, which is defined as The higher-order Heisenberg model for any spin- S can be written generically as Thus, for the bilinear–biquadratic–bicubic model with , the higher-order Heisenberg Hamiltonian reads: For , the diagonalization of this Hamiltonian for a two-site system gives four energy levels [ 70 ] Considering an offset energy, as in the case of an integer spin, where , , is an offset energy [ 70 ], and using Equation ( 19 ), the analytical expressions for rations of exchange constants and in terms of for …”