  We present detailed spectral calculations for small Lieb lattices having up
to $N=4$ number of cells, in the regime of half-filling, an instance of
particular relevance for the nano-magnetism of discrete systems such as quantum
dot arrays, due to the degenerate levels at mid-spectrum. While for the Hubbard
interaction model -and even number of sites- the ground state spin is given by
the Lieb theorem, the inclusion of long range interaction -or odd number of
sites- make the spin state not a priori known, which justifies our approach. We
calculate also the excitation energies, which are of experimental importance,
and find significant variation induced by the interaction potential. One
obtains insights on the mechanisms involved that impose as ground state the
Lieb state with lower spin rather than the Hund one with maximum spin for the
degenerate levels, showing this in the first and second order of the
interaction potential for the smaller lattices. The analytical results concorde
with the numerical ones, which are performed by exact diagonalization
calculations or by a combined mean-field and configuration interaction method.
While the Lieb state is always lower in energy than the Hund state, for strong
long-range interaction, when possible, another minimal spin state is imposed as
ground state.
