Hubbard Trimer with Spin-Orbit Coupling: Hartree-Fock Solutions, (Non)Collinearity, and Anisotropic Spin Hamiltonian

We present unrestricted and generalized Hartree-Fock solutions (UHF and GHF, respectively) for the single-band Hubbard model of an equilateral triangle. Spin-orbit coupling (SOC) is treated self-consistently, and HF stability and properties of different spin structures are studied in detail. The GHF solution switches from noncollinear to collinear when crossing a high-symmetry point in parameter space (spanned by the amplitudes of spin-conserving and spin-dependent hopping, i.e., kinetic energy and SOC, respectively). The collinear GHF solution represents a simple example to disprove the notion that a collinear vector spin density in a Slater determinant necessarily entails a defined spin projection. Spin Hamiltonian parameters for the anisotropic interaction between three spin-1/2 centers are extracted from HF energies and subsequently compared to exact results from effective Hamiltonian theory. This provides an unambiguous benchmark for interpreting broken-symmetry mean-field solutions in terms of spin configurations and puts this semiclassical approach (frequently applied in broken-symmetry density functional theory) on a firmer basis.