Fast optical absorption spectra calculations for periodic solid state systems

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 scales nearly optimally as $\mathcal{O}(N_k)$ with respect to the number of samples in the Brillouin zone $N_k$. In addition, we show that the cost for applying the Bethe-Salpeter Hamiltonian to a vector scales as $\mathcal{O}(N_k \log N_k)$. 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 $\mathcal{O}(N_k^2)$ and $\mathcal{O}(N_k^3)$ 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.