  We theoretically study magnetic properties of two-component cold fermions in
half-filled multilayer Lieb optical lattices, i.e., two, three, and several
layers, using the dynamical mean-field theory. We clarify that the magnetic
properties of this system become quite different depending on whether the
number of layers is odd or even. In odd-number-th layers in an odd-number-layer
system, finite magnetization emerges even with an infinitesimal interaction.
This is a striking feature of the flatband ferromagnetic state in multilayer
systems as a consequence of the Lieb theorem. In contrast, in even-number
layers, magnetization develops from zero on a finite interaction. These
different magnetic behaviours are triggered by the flat bands in the local
density of states and become identical in the limit of the infinite-layer
(i.e., three-dimensional) system. We also address how interlayer hopping
affects the magnetization process. Further, we point out that layer
magnetization, which is a population imbalance between up and down atoms on a
layer, can be employed to detect the emergence of the flat-band ferromagnetic
state without addressing sublattice magnetization.
