Stochastic Formalism for Fast Spin-Resolved <i>GW</i>

We extend the stochastic <i>GW</i> (s<i>GW</i>) formalism to fully spin-polarized systems, encompassing both collinear and noncollinear spin configurations. For noncollinear systems, where Kohn-Sham states are complex two-component spinors, we develop a complex-valued stochastic basis that preserves the real-valued external stochastic charge applied at time zero. This basis enables an unbiased evaluation of the random-phase approximation (RPA) screened interaction for spinors. Through error analysis and tests on real materials, we show that the performance of collinear s<i>GW</i> retains the same time complexity as the spin-unpolarized s<i>GW</i>. The noncollinear s<i>GW</i> incurs a computational cost two to three times higher than the spin-unpolarized version, while preserving linear scaling with low multiplicity. By unifying collinear and noncollinear treatments within a single scalable framework, our work paves the way for routine many-body predictions in large scale magnetic and spin-orbit-coupled material systems.