Decomposition and embedding in the stochastic $GW$ self-energy

We present two new developments for computing excited state energies within the $GW$ approximation. First, calculations of the Green's function and the screened Coulomb interaction are decomposed into two parts: one is deterministic while the other relies on stochastic sampling. Second, this separation allows constructing a subspace self-energy, which contains dynamic correlation from only a particular (spatial or energetic) region of interest. The methodology is exemplified on large-scale simulations of nitrogen-vacancy states in a periodic hBN monolayer and hBN-graphene heterostructure. We demonstrate that the deterministic embedding of strongly localized states significantly reduces statistical errors, and the computational cost decreases by more than an order of magnitude. The computed subspace self-energy unveils how interfacial couplings affect electronic correlations and identifies contributions to excited-state lifetimes. While the embedding is necessary for the proper treatment of impurity states, the decomposition yields new physical insight into quantum phenomena in heterogeneous systems.