  The microscopic mechanism of itinerant ferromagnetism is a long-standing
problem due to the lack of non-perturbative methods to handle strong magnetic
fluctuations of itinerant electrons. We have non-pertubatively studied
thermodynamic properties and magnetic phase transitions of a two-dimensional
multi-orbital Hubbard model exhibiting ferromagnetic ground states. Quantum
Monte-Carlo simulations are employed, which are proved in a wide density region
free of the sign problem usually suffered by simulations for fermions. Both
Hund's coupling and electron itinerancy are essential for establishing the
ferromagnetic coherence. No local magnetic moments exist in the system as a
priori, nevertheless, the spin channel remains incoherent showing the
Curie-Weiss type spin magnetic susceptibility down to very low temperatures at
which the charge channel is already coherent exhibiting a weakly
temperature-dependent compressibility. For the SU(2) invariant systems, the
spin susceptibility further grows exponentially as approaching zero temperature
in two dimensions. In the paramagnetic phase close to the Curie temperature,
the momentum space Fermi distributions exhibit strong resemblance to those in
the fully polarized state. The long-range ferromagnetic ordering appears when
the symmetry is reduced to the Ising class, and the Curie temperature is
accurately determined. These simulations provide helpful guidance to searching
for novel ferromagnetic materials in both strongly correlated $d$-orbital
transition metal oxide layers and the $p$-orbital ultra-cold atom optical
lattice systems.
