  The discovery of atomically thin two-dimensional (2D) magnetic semiconductors
has triggered enormous research interest recently. In this work, we use
first-principles many-body perturbation theory to study a prototypical 2D
ferromagnetic semiconductor, monolayer chromium tribromide (CrBr$_3$). With
broken time-reversal symmetry, spin-orbit coupling, and excitonic effects
included through the full-spinor $GW$ and $GW$ plus Bethe-Salpeter equation
($GW$-BSE) methods, we compute the frequency-dependent layer polarizability
tensor and dielectric function tensor that govern the optical and
magneto-optical properties. In addition, we provide a detailed theoretical
formalism for simulating magnetic circular dichroism, magneto-optical Kerr
effect, and Faraday effect, demonstrating the approach with monolayer CrBr$_3$.
Due to reduced dielectric screening in 2D and the localized nature of the Cr
$d$ orbitals, we find strong self-energy effects on the quasiparticle band
structure of monolayer CrBr$_3$ that give a 3.8 eV indirect band gap. Also,
excitonic effects dominate the low-energy optical and magneto-optical responses
in monolayer CrBr$_3$ where a large exciton binding energy of 2.3 eV is found
for the lowest bright exciton state with excitation energy at 1.5 eV. We
further find that the magneto-optical signals demonstrate strong dependence on
the excitation frequency and substrate refractive index. Our theoretical
framework for modeling optical and magneto-optical effects could serve as a
powerful theoretical tool for future study of optoelectronic and spintronics
devices consisting of van der Waals 2D magnets.
