Cubic-Scaling Canonical GW Formulation for Solids based on Tensor Hyper-Contraction

We present a cubic-scaling canonical GW formulation for solids based on tensor hyper-contraction (THC). The THC factorization results in a separable form of electron repulsion integrals (ERIs) in a generic discretized single-particle basis, and therefore allows a reformulation of the canonical GW (THC-GW) that shares the same complexity as in the space-time formalism. The THC-GW formalism scales linearly with respect to the number of k-points, and cubicly with respect to the size of the unit cell. Due to the compactness of discretized single-particle bases compared to the real-space grid, THC-GW has a much smaller prefactor in its complexity than the space-time formalism. The low scaling of THC-GW does not rely on a single-particle picture or sparsity of matrix elements. Therefore, it is directly applicable to different variants of GW, including the fully self-consistent GW, and the accuracy is solely controlled by the error of the THC factorization.