Fast optical absorption spectra calculations for periodic solid state systems

Abstract

We present a method to construct an efficient approximation to the bare\nexchange and screened direct interaction kernels of the Bethe-Salpeter\nHamiltonian for periodic solid state systems via the interpolative separable\ndensity fitting technique. We show that the cost of constructing the\napproximate Bethe-Salpeter Hamiltonian scales nearly optimally as\n$\\mathcal{O}(N_k)$ with respect to the number of samples in the Brillouin zone\n$N_k$. In addition, we show that the cost for applying the Bethe-Salpeter\nHamiltonian to a vector scales as $\\mathcal{O}(N_k \\log N_k)$. Therefore the\noptical absorption spectrum, as well as selected excitation energies can be\nefficiently computed via iterative methods such as the Lanczos method. This is\na significant reduction from the $\\mathcal{O}(N_k^2)$ and $\\mathcal{O}(N_k^3)$\nscaling associated with a brute force approach for constructing the Hamiltonian\nand diagonalizing the Hamiltonian respectively. We demonstrate the efficiency\nand accuracy of this approach with both one-dimensional model problems and\nthree-dimensional real materials (graphene and diamond). For the diamond system\nwith $N_k=2197$, it takes $6$ hours to assemble the Bethe-Salpeter Hamiltonian\nand $4$ hours to fully diagonalize the Hamiltonian using $169$ cores when the\nbrute force approach is used. The new method takes less than $3$ minutes to set\nup the Hamiltonian and $24$ minutes to compute the absorption spectrum on a\nsingle core.\n