  Chromium triiodide is a magnetic van-der-Waals material with weak inter-layer
interactions. It is one of the first materials for which intrinsic magnetism
was observed down to the single-layer limit. This remarkable discovery fostered
a whole new field of 2D magnetism and magnetic layered heterostructure research
holding high promisses for spintronic applications. First-principles electronic
structure calculations have an outstanding role in this field not only to
describe the properties of existing 2D magnets, but also to predict new
materials, and thus to guide the experimental progress. So far the most 2D
magnet studies are based on standard density functional theory (DFT), which
poorly addresses the effects of strong electron correlations. Here, we provide
a first-principles description of finite-temperature magnetic and spectral
properties of monolayer CrI$_3$ based on fully charge self-consistent DFT
combined with dynamical mean field theory (DFT+DMFT), revealing a formation of
local moments on Cr from strong local Coulomb interactions. We show that local
dynamical correlations play an important role in the electronic structure of
CrI$_3$. In contrast to conventional DFT+$U$ calculations, we find that the top
of the valence band in monolayer CrI$_3$ demonstrates essentially different
orbital character for minority and majority spin states. This results in a
strong spin-polarization of the optical conductivity upon hole doping, which
could be verified experimentally.
