  Recent discovery of correlated electronic phases in twisted heterostructures
raised a surge of interests in studying models and materials with flat bands
where the electronic excitations are nearly dispersionless in momentum space.
As such, the kinetic energy is quenched and the correlations are enhanced,
giving rise to a plethora of unusual magnetic, superconducting and transport
behaviors. Finding materials whose energy bands are completely flat is rather
challenging, yet those whose dispersion is flat only in a portion of the
momentum space might be more accessible in material search. In this work, we
propose a partially flat-band system on a square lattice. Using the Hubbard
model, it is demonstrated that the suppression of the electronic kinetic energy
in the flat portion of the band dispersion drives the system to Stoner
ferromagnetism even at very weak interactions, i.e., much smaller than the
bandwidth, with significantly enhanced Curie temperature. While the low-energy
magnon modes are well defined collective excitations, flat magnon bands can be
observed at high energies. We show that the strong interaction leads to
reduction of the flat portion of the magnon band. However, tuning the chemical
potential at a strong interaction regime may lead to spin density wave at
finite wave vectors. Then, focusing on the non-magnetic correlated phase and
using dynamical mean-field theory, we demonstrate the appearance of a flat-band
induced sharp peak in the density of states in addition to the
correlation-induced Mott bands. Furthermore, the large seebeck coefficient and
the figure of merit of the proposed partially flat-band model, compared to
symmetric regular band models, put them in the category of efficient
thermoelectric materials.
