Effective magnetic interactions in spin-orbit coupled $d^4$ Mott insulators

TL;DR

This paper investigates how spin-orbit coupling influences magnetic interactions in $d^4$ Mott insulators, revealing phase transitions and potential spin-orbital liquid states driven by the interplay of superexchange, spin-orbit coupling, and Hund's coupling.

Contribution

It introduces a comprehensive analysis of the effective magnetic Hamiltonian in $d^4$ systems, highlighting the role of anisotropic orbital interactions and phase transitions induced by tuning interaction ratios.

Findings

Identification of a phase transition from nonmagnetic to magnetic phases.

Discovery of spin-orbital entangled states and frustration effects.

Role of spin-orbit coupling in generating a triplon condensate.

Abstract

Transition metal compounds with the $(t_{2g})^4$ electronic configuration are expected to be nonmagnetic atomic singlets both in the weakly interacting regime due to spin-orbit coupling, as well as in the Coulomb dominated regime with oppositely aligned $L=1$ and $S=1$ angular momenta. However, starting with the full multi-orbital electronic Hamiltonian, we show the low energy effective magnetic Hamiltonian contains isotropic superexchange spin interactions but anisotropic orbital interactions. By tuning the ratio of superexchange to spin-orbit coupling $J_\mathrm{SE}/\lambda$, we obtain a phase transition from nonmagnetic atomic singlets to novel magnetic phases depending on the strength of Hund's coupling, the crystal structure and the number of active orbitals. Spin-orbit coupling plays a non-trivial role in generating a triplon condensate of weakly interacting excitations at antiferromagnetic ordering vector $\vec k=\vec \pi$, regardless of whether the local spin interactions are ferromagnetic or antiferromagnetic. In the large $J_\mathrm{SE} / \lambda$ regime, the localized spin and orbital moments produce anisotropic orbital interactions that are frustrated or constrained even in the absence of geometric frustration. Orbital frustration leads to frustration in the spin channel opening up the possibility of spin-orbital liquids with both spin and orbital entanglement.