We investigate the correlation effects on spin–orbit coupling (SOC) in a two-orbital Hubbard model on a square lattice by applying the variational Monte Carlo method. We consider an effective SOC constant λeff in the one-body part of the variational wave function and mainly discuss the cases of the electron number per site n = 1, that is, quarter filling. We find that λeff is proportional to the bare value λ and depends on the electron–electron interactions through (U′ − J′) in a relatively wide parameter range in the paramagnetic (PM) phase, where U′ is the interorbital Coulomb interaction and J′ is the pair hopping interaction. Increasing the electron–electron interactions in the PM phase leads to a transition to an effective one-band state, in which the upper band becomes empty due to the enhanced λeff. We also construct phase diagrams considering magnetic order. The carrier doping effect on λeff is also investigated. We find that λeff enhances in a strongly correlated region around the Mott transition point and it is necessary to include the correlation effects beyond the Hartree–Fock approximation to describe the enhanced SOC properly. 

Enhanced Spin–Orbit Coupling in a Correlated Metal

Enhancement of an Effective Spin–Orbit Coupling in a Correlated Metal

In atomic physics, the Hund's rule states that the largest spin and orbital state is realized due to the interplay of spin-orbit coupling (SOC) and Coulomb interactions. Here, we show that in ferromagnetic solids the effective SOC and the orbital magnetic moment can be dramatically enhanced by a factor of $1/[1\ensuremath{-}(2{U}^{\ensuremath{'}}\ensuremath{-}U\ensuremath{-}{J}_{H}){\ensuremath{\rho}}_{0}]$, where $U$ and ${U}^{\ensuremath{'}}$ are the on-site Coulomb interaction within the same orbitals and between different orbitals, respectively, ${J}_{H}$ is the Hund's coupling, and ${\ensuremath{\rho}}_{0}$ is the average density of states. This factor is obtained by using the two-orbital as well as five-orbital Hubbard models with SOC. We also find that the spin polarization is more favorable than the orbital polarization, being consistent with experimental observations. The theory is also extended to study the spin fluctuations and long-range Coulomb interactions, and can be applied to understand the enhanced orbital magnetic moment and giant Faraday effect in ferromagnetic nanogranules in recent experiments. This present paper provides a fundamental basis for understanding the enhancements of SOC and orbital moment by Coulomb interactions in ferromagnets, which would have wide applications in spintronics. 

Enhanced spin-orbit coupling and orbital moment in ferromagnets by electron correlations

We investigate the interplay of electronic correlations and spin-orbit coupling (SOC) for a one-band and two-band honeycomb lattice models. The main difference between the two models concerning SOC is that in the one-band case the SOC is a purely nonlocal term in the basis of the ${p}_{z}$ orbitals, whereas in the two-band case with ${p}_{x}$ and ${p}_{y}$ as basis functions, it is purely local. In order to grasp the correlation effects on nonlocal spin-orbit coupling, we apply the TRILEX approach that allows to calculate nonlocal contributions to the self-energy approximately. For the two-band case, we apply dynamical mean-field theory. In agreement with previous studies, we find that for all parameter values in our study, the effect of correlations on the spin-orbit coupling strength is that the bare effective SOC parameter is increased. However, this increase is much weaker in the nonlocal than in the local SOC case. Concerning the TRILEX method, we introduce the necessary formulas for calculations with broken SU(2) symmetry.

Comparing the effective enhancement of local and nonlocal spin-orbit couplings on honeycomb lattices due to strong electronic correlations

We investigate the interplay of electronic correlations and spin-orbit coupling (SOC) for a one-band and a two-band honeycomb lattice model. The main difference between the two models concerning SOC is that in the one-band case the SOC is a purely non-local term in the basis of the $p_z$ orbitals, whereas in the two-band case with $p_x$ and $p_y$ as basis functions it is purely local. In order to grasp the correlation effects on non-local spin-orbit coupling, we apply the TRILEX approach that allows to calculate non-local contributions to the self-energy approximatively. For the two-band case we apply dynamical mean-field theory. In agreement with previous studies, we find that for all parameter values in our study, the effect of correlations on the spin-orbit coupling strength is that the bare effective SOC parameter is increased. However, this increase is much weaker in the non-local than in the local SOC case. Concerning the TRILEX method, we introduce the necessary formulas for calculations with broken SU(2) symmetry.

Markus Richter

Papers

Comparing the effective enhancement of local and nonlocal spin-orbit couplings on honeycomb lattices due to strong electronic correlations

Comparing the effective enhancement of local and non-local spin-orbit couplings on honeycomb lattices due to strong electronic correlations