  In order to study conditions for the appearance of ferromagnetism in a wide
filling region, we investigate numerically three types of one-dimensional
Lieb-ferrimagnetic Hubbard models: a periodic diamond (PD) chain, a periodic
alternately-attached leg (PAAL) chain and an open diamond (OD) chain. All of
these models have a flat band (or equivalently, degenerate single-electron
eigenvalues). The PD and OD chains commonly have a local-loop structure.
Nagaoka's theorem holds only in the PD chain. At half-filling, it have been
rigorously proven that all of these models are ferrimagnet. Away from
half-filling, however, quite different magnetic properties are found. In the
fillings 1/3< rho_e <1/2, the ground state of the PD chain for a
infinitely-large U is the extended ferromagnetic state, that is, the saturated
ferromagnetic state or the spiral state for odd or even number of electrons,
respectively. In the PAAL chain, on the other hand, there is no magnetic order.
Thus, the flat band is found to be not a sufficient condition of the extended
ferromagnetic state. We find, moreover, that the saturated ferromagnetism
appears in the OD chain, although the Nagaoka theorem does not hold on this
chain. This indicates that the local-loop structure plays an important role on
the appearance of the extended ferromagnetic state.
