  Recent years have seen a tremendous rise of two-dimensional (2D) magnetic
materials, several of which verified experimentally. However, most of the
theoretical predictions to date rely on ab-initio methods, at zero temperature
and fluctuations-free, while one certainly expects detrimental quantum
fluctuations at finite temperatures. Here we present the solution of the
quantum Heisenberg model for honeycomb/hexagonal lattices with anisotropic
exchange interaction up to third nearest neighbors and in an applied field in
arbitrary direction, that answers the question whether long-range magnetization
can indeed survive in the ultrathin limit of materials, up to which
temperature, and what the characteristic excitation (magnon) frequencies are,
all essential to envisaged applications of magnetic 2D materials. We find that
long-range magnetic order persists at finite temperature for materials with
overall easy-axis anisotropy. We validate the calculations on the examples of
monolayer CrI3, CrBr3 and MnSe2. Moreover, we provide an easy-to-use tool to
calculate Curie temperatures of new 2D computational materials.

Statistical Mechanics;Mesoscale and Nanoscale Physics;Strongly Correlated Electrons