Spin-orbit coupling in periodic systems with broken time-reversal symmetry: Formal and computational aspects

We discuss the treatment of spin-orbit coupling (SOC) in time-reversal-symmetry-broken periodic systems for relativistic electronic structure calculations of materials within the generalized noncollinear Kohn-Sham density functional theory (GKSDFT). We treat SOC self-consistently and express the GKS orbitals in a two-component spinor basis. Crucially, we present a methodology (and its corresponding implementation) for the simultaneous self-consistent treatment of SOC and exact nonlocal Fock exchange operators. The many advantages of the inclusion of nonlocal Fock exchange in the self-consistent treatment of SOC, as practically done in hybrid exchange-correlation functionals, are both formally derived and illustrated through numerical examples: (i) it imparts a local magnetic torque (i.e., the ability of the two-electron potential to locally rotate the magnetization with respect to a starting guess configuration) that is key to converge to the right solution in noncollinear DFT regardless of the initial guess for the magnetization; (ii) because of the local magnetic torque, it improves the rotational invariance of noncollinear formulations of the DFT; (iii) it introduces the dependence on specific pieces of the spinors (i.e., those mapped onto otherwise missing spin blocks of the complex density matrix) into the two-electron potential, which are key to the correct description of the orbital- and spin-current densities and their coupling with the magnetization; and (iv) when space-inversion symmetry is broken, it allows for the full breaking of time-reversal symmetry in momentum space, which would otherwise be constrained by a sum rule linking the electronic band structure at opposite points in the Brillouin zone (k̲ and -k̲). The presented methodology is implemented in a developmental version of the public crystal code. Numerical tests are performed on the model system of an infinite radical chain of Ge2H with both space-inversion and time-reversal symmetries broken, which allows to highlight all the above-mentioned effects.