Swift GW beyond 10,000 electrons using fractured stochastic orbitals

We introduce the concept of fractured stochastic orbitals (FSOs), short vectors that sample a small number of space points and enable an efficient stochastic sampling of any general function. As a first demonstration, FSOs are applied in conjunction with simple direct-projection to accelerate our recent stochastic $GW$ technique; the new developments enable accurate prediction of $G_{0}W_{0}$ quasiparticle energies and gaps for systems with up to $N_{e}>10,000$ electrons, with small statistical errors of $\pm0.05\,{\rm eV}$ and using less than 2000 core CPU hours. Overall, stochastic $GW$ scales now linearly (and often sub-linearly) with $N_{e}.$