  By using a combination of several non-perturbative techniques -- a
one-dimensional field theoretical approach together with numerical simulations
using density matrix renormalization group -- we present an extensive study of
the phase diagram of the generalized Hund model at half-filling. This model
encloses the physics of various strongly correlated one-dimensional systems,
such as two-leg electronic ladders, ultracold degenerate fermionic gases
carrying a large hyperfine spin 3/2, other cold gases like Ytterbium 171 or
alkaline-earth condensates. A particular emphasis is laid on the possibility to
enumerate and exhaust the eight possible Mott insulating phases by means of a
duality approach. We exhibit a one-to-one correspondence between these phases
and those of the two-leg Hubbard ladder with interchain hopping. Our results
obtained from a weak coupling analysis are in remarkable quantitative agreement
with our numerical results carried out at moderate coupling.
