Electronic Structure of Chromium Trihalides beyond Density Functional Theory

We explore the electronic band structure of free standing monolayers of chromium trihalides, CrXtextsubscript{3}{, X= Cl, Br, I}, within an advanced emph{ab-initio} theoretical approach based in the use of Greens function functionals. We compare the local density approximation with the quasi-particle self-consistent emph{GW} approximation (QSemph{GW}) and its self-consistent extension (QS$Gwidehat{W}$) by solving the particle-hole ladder Bethe-Salpeter equations to improve the effective interaction emph{W}. We show that at all levels of theory, the valence band consistently changes shape in the sequence Cl{textrightarrow}Br{textrightarrow}I, and the valence band maximum shifts from the M point to the $Gamma$ point. However, the details of the transition, the one-particle bandgap, and the eigenfunctions change considerably going up the ladder to higher levels of theory. The eigenfunctions become more directional, and at the M point there is a strong anisotropy in the effective mass. Also the dynamic and momentum dependent self energy shows that QS$Gwidehat{W}$ adds to the localization of the systems in comparison to the QSemph{GW} thereby leading to a narrower band and reduced amount of halogens in the valence band manifold.