Interplay of Coulomb interaction and spin-orbit coupling

TL;DR

This paper uses the Gutzwiller variational approach to study how Coulomb interaction and spin-orbit coupling influence a three-orbital Hubbard model, revealing renormalization effects, Hund's rules signatures, and magnetic anisotropy.

Contribution

It introduces a detailed analysis of the interplay between Coulomb interaction and spin-orbit coupling in a multi-orbital Hubbard model, highlighting renormalization and magnetic effects.

Findings

Renormalization of spin-orbit coupling in the paramagnetic phase

Signatures of Hund's rules near half band-filling with strong Coulomb interaction

Spin-orbit coupling reduces magnetic moment and induces anisotropy

Abstract

We employ the Gutzwiller variational approach to investigate the interplay of Coulomb interaction and spin-orbit coupling in a three-orbital Hubbard model. Already in the paramagnetic phase we find a substantial renormalization of the spin-orbit coupling that enters the effective single-particle Hamiltonian for the quasi-particles. Only close to half band-filling and for sizable Coulomb interaction we observe clear signatures of Hund's atomic rules for spin, orbital, and total angular momentum. For a finite local Hund's-rule exchange interaction we find a ferromagnetically ordered state. The spin-orbit coupling considerably reduces the size of the ordered moment, it generates a small ordered orbital moment, and it induces a magnetic anisotropy. To investigate the magnetic anisotropy energy, we use an external magnetic field that tilts the magnetic moment away from the easy axis $(1,1,1)$.