Accurate and efficient predictions of the quasiparticle properties of complex materials remain a major challenge due to the convergence issue and the unfavorable scaling of the computational cost with respect to the system size. Quasiparticle $GW$ calculations for two dimensional (2D) materials are especially difficult. The unusual analytical behaviors of the dielectric screening and the electron self-energy of 2D materials make the conventional Brillouin zone (BZ) integration approach rather inefficient and require an extremely dense $k$-grid to properly converge the calculated quasiparticle energies. In this work, we present a combined non-uniform sub-sampling and analytical integration method that can drastically improve the efficiency of the BZ integration in 2D $GW$ calculations. Our work is distinguished from previous work in that, instead of focusing on the intricate dielectric matrix or the screened Coulomb interaction matrix, we exploit the analytical behavior of various terms of the convolved self-energy $Sigma(mathbf{q})$ in the small $mathbf{q}$ limit. This method, when combined with another accelerated $GW$ method that we developed recently, can drastically speed-up (by over three orders of magnitude) $GW$ calculations for 2D materials. Our method allows fully converged $GW$ calculations for complex 2D systems at a fraction of computational cost, facilitating future high throughput screening of the quasiparticle properties of 2D semiconductors for various applications. To demonstrate the capability and performance of our new method, we have carried out fully converged $GW$ calculations for monolayer C$_2$N, a recently discovered 2D material with a large unit cell, and investigate its quasiparticle band structure in detail.
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