Magnetic anisotropy from interligand hopping in strongly correlated insulators: application to the magnon spectrum of CrI$_3$

TL;DR

This paper investigates how ligand spin-orbit coupling influences magnetic anisotropy and magnon spectra in strongly correlated insulators, applying the findings to explain experimental results in CrI$_3$.

Contribution

It introduces a novel method to calculate exchange interactions at arbitrary distances considering ligand SOC effects, enhancing understanding of magnetic anisotropy in materials like CrI$_3$.

Findings

Anisotropic exchange interactions are derived for CrI$_3$.

Long-range hopping induces single-ion anisotropy.

Calculated magnon spectrum aligns with experimental data.

Abstract

Spin-orbit coupling (SOC) gives rise to complex magnetic states such as spin liquids, skyrmion crystals, and topological spin-wave excitations. We consider exchange interactions in multi-orbital Mott insulators where SOC is strong on ligand ions. SOC on the ligands enables electron hopping accompanied by spin flips and fluctuations in the orbital state of the ligand hole. These processes generate anisotropic exchange interactions and greatly increase the number of possible exchange paths. The number grows further with the inclusion of hopping between ligands, which mediates interactions between more distant spins. We propose an effective method to calculate exchange interactions at arbitrary separations between spins. Applying it to monolayer CrI$_3$, we obtain anisotropic interactions between nearest-neighbor and next-nearest-neighbor Cr spins, as well as single-ion anisotropy induced by long-range hopping. In this material, magnetic anisotropy stabilizes long-range ferromagnetic order and opens a magnon gap at the Dirac points, which defines a nontrivial magnon band topology. Using Hubbard model parameters from first-principles calculations, the resulting spectrum agrees well with the spin-wave dispersion observed experimentally in bulk CrI$_3$, except that the calculated Dirac gap is much smaller.