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We present low-scaling algorithms for $GW$ and constrained random phaseapproximation based on a symmetry-adapted interpolative separable densityfitting (ISDF) procedure that incorporates the space-group symmetries ofcrystalline systems. The resulting formulations scale cubically with respect tosystem sizes and linearly with the number of $\mathbf{k}$-points, regardless ofthe choice of single-particle basis and whether a quasiparticle approximationis employed. We validate these methods through comparisons with publishedliterature and demonstrate their efficiency in treating large-scale systemsthrough the construction of downfolded many-body Hamiltonians for carbon dimerdefects embedded in hexagonal boron nitride supercells. Our work highlights theefficiency and general applicability of ISDF in the context of large-scalemany-body calculations with $\mathbf{k}$-point sampling beyond densityfunctional theory.

Low-Scaling algorithms for $GW$ and constrained random phase approximation using symmetry-adapted interpolative separable density fitting