Abstract

We present a method to construct an efficient approximation to the bare exchange and screened direct interaction kernels of the Bethe–Salpeter Hamiltonian for periodic solid state systems via the interpolative separable density fitting technique. We show that the cost of constructing the approximate Bethe–Salpeter Hamiltonian can be reduced to nearly optimal as $\mathcal{𝒪}\left(N_k\right)$ with respect to the number of samples in the Brillouin zone $N_k$ for the first time. In addition, we show that the cost for applying the Bethe–Salpeter Hamiltonian to a vector scales as $\mathcal{𝒪}\left(N_{k}log{N}_k\right)$. Therefore, the optical absorption spectrum, as well as selected excitation energies, can be efficiently computed via iterative methods such as the Lanczos method. This is a significant reduction from the $\mathcal{𝒪}\left(N_k^2\right)$ and $\mathcal{𝒪}\left(N_k^3\right)$ scaling associated with a brute force approach for constructing the Hamiltonian and diagonalizing the Hamiltonian, respectively. We demonstrate the efficiency and accuracy of this approach with both one-dimensional model problems and three-dimensional real materials (graphene and diamond). For the diamond system with $N_k=2197$, it takes $6$ hours to assemble the Bethe–Salpeter Hamiltonian and $4$ hours to fully diagonalize the Hamiltonian using $169$ cores when the brute force approach is used. The new method takes less than $3$ minutes to set up the Hamiltonian and $24$ minutes to compute the absorption spectrum on a single core.

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