A Poor Man's Derivation of Quantum Compass-Heisenberg Interaction: Superexchange Interaction in J-J Coupling Scheme

TL;DR

This paper derives the superexchange interaction between 5d electrons in the J-J coupling scheme, revealing how spin-orbit interaction influences the transition from ferromagnetic Ising to antiferromagnetic Heisenberg interactions.

Contribution

It provides a simplified derivation of exchange interactions in the J-J coupling scheme, highlighting the role of hybridization and spin-orbit coupling in determining magnetic interactions.

Findings

Ferromagnetic Ising interaction arises from $ ext{Γ}_7$ and $ ext{Γ}_8$ hybridization.

Antiferromagnetic Heisenberg interaction results from $ ext{Γ}_7$ orbital hybridization.

The ferromagnetic Ising interaction diminishes as spin-orbit interaction increases.

Abstract

The exchange interaction between 5d electrons in t$_{2g}$ orbitals is derived in the J-J coupling scheme which is the appropriate basis in the case of strong spin-orbit interaction. From simple calculations, it is found that ferromagnetic Ising interaction (quantum compass model) occurs due to a selection rule of hybridization between $\Gamma_7$ and $\Gamma_8$ orbitals, while antiferromagnetic Heisenberg interaction appears from hybridization between $\Gamma_7$ orbitals. It is also found that the ferromagnetic Ising interaction becomes small as the spin-orbit interaction increases. Thus, the sign of exchange interaction changes from ferromagnetic Ising to antiferromagnetic Heisenberg interaction as the spin-orbit interaction increases.