  Quantum Monte Carlo (QMC) is applied to obtain the fundamental
(quasiparticle) electronic band gap, $\Delta_f$, of a semiconducting
two-dimensional (2D) phosphorene whose optical and electronic properties fill
the void between graphene and 2D transition metal dichalcogenides. Similarly to
other 2D materials, the electronic structure of phosphorene is strongly
influenced by reduced screening, making it challenging to obtain reliable
predictions by single-particle density functional methods. Advanced GW
techniques, which include many-body effects as perturbative corrections, are
hardly consistent with each other, predicting the band gap of phosphorene with
a spread of almost 1 eV, from 1.6 to 2.4 eV. Our QMC results, from infinite
periodic superlattices as well as from finite clusters, predict $\Delta_f$ to
be about 2.4 eV, indicating that available GW results are systematically
underestimating the gap. Using the recently uncovered universal scaling between
the exciton binding energy and $\Delta_f$, we predict the optical gap of 1.75
eV that can be directly related to measurements even on encapsulated samples
due to its robustness against dielectric environment. The QMC gaps are indeed
consistent with recent experiments based on optical absorption and
photoluminescence excitation spectroscopy. We also predict the cohesion of
phosphorene to be only slightly smaller than that of the bulk crystal. Our
investigations not only benchmark GW methods and experiments, but also open the
field of 2D electronic structure to computationally intensive but highly
predictive QMC methods which include many-body effects such as electronic
correlations and van der Waals interactions explicitly.
