Spin–orbit coupling in quantum materials: emergent phenomena, their modelling and examples from two-dimensional magnets

Acknowledging spin–orbit coupling (SOC) as a pivotal microscopic ingredient in quantum materials, this paper provides a comprehensive overview of spin–orbit-driven phenomena and effects in magnetic, non-magnetic and multiferroic materials. As for magnets, the review discusses SOC-induced magnetic anisotropy, exotic spin–spin interactions, and their implications on the emergence of complex spin textures. It also explores band-splitting effects in non-magnetic solids, with a focus on Rashba effects and spin-valley coupling. Additionally, the emergence of ferroelectric polarization from non-collinear spin textures in multiferroics is examined. The paper analyzes methods for estimating SOC-related quantities from first principles within the density functional theory, particularly exchange coupling tensors and magnetoelectric coupling tensors. Finally, it presents case studies on two-dimensional magnetic materials, including the characterization of peculiar Kitaev-like exchange coupling and the investigation of multiferroicity in NiI2 monolayers. Overall, the paper delves into the microscopic mechanisms related to SOC, by offering insights into the diverse effects and manifestations of SOC in the quantum materials world.