  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.
