  Chromium triiodide is an intrinsically magnetic van der Waals material down
to the single-layer limit. Here, we provide a first-principles description of
finite-temperature magnetic and spectral properties of monolayer (ML) CrI$_3$
based on fully charge self-consistent density functional theory (DFT) combined
with dynamical mean-field theory, revealing a formation of local moments on Cr
from strong local Coulomb interactions. We show that the presence of local
dynamical correlations leads to a modification of the electronic structure of
ferromagnetically ordered CrI$_3$. In contrast to conventional DFT+$U$
calculations, we find that the top of the valence band in ML CrI$_3$
demonstrates essentially different orbital character for minority and majority
spin states, which is closer to the standard DFT results. This leads to a
strong spin polarization of the optical conductivity upon hole doping, which
could be verified experimentally.
