Randomized Low-rank Construction and Representation of the RPA Screening

The random phase approximation (RPA) plays a central role in many-body physics and first-principles electronic structure theory. However, conventional implementations are computationally demanding: evaluating the non-interacting polarizability via the sum-over-states approach scales as O(N⁴) with system size. We introduce a randomized low-rank representation of the polarizability based on randomized singular value decomposition, which efficiently captures its dominant subspace. Combined with our recently developed stochastic Chebyshev-Jackson pseudoband approximation, the overall scaling of the RPA correlation energy and dielectric matrix evaluation is reduced to approximately O(N²). Critically, our approach displays a high degree of parallelism on modern GPU architectures, avoiding explicitly storing large intermediate objects and heavy communications. We analyze the physical significance of the effective rank across different materials and physical regimes, and demonstrate that the algorithm achieves good time-to-solution for various system sizes with a small and controllable error.