  Recent experiments with Yb-173 and Sr-87 isotopes provide new possibilities
to study high spin two-orbital systems. Within these experiments part of the
atoms are excited to a higher energy metastable electronic state mimicking an
additional internal (orbital) degree of freedom. The interaction between the
atoms depends on the orbital states, therefore four different scattering
channels can be identified in the system characterized by four independent
couplings. When the system is confined into a one-dimensional chain the
scattering lengths can be tuned by changing the transverse confinement, and
driven through four resonances. Using the new available experimental data of
the scattering lengths we analyze the phase diagram of the one-dimensional
system as the couplings are tuned via transverse confinement, and the
populations of the two orbital states are changed. We found that three orders
compete showing power law decay: a state with dominant density wave
fluctuations, another one with spin density fluctuations, and a third one
characterized by exotic Fulde-Ferrell-Larkin-Ovchinnikov-like pairs consisting
one atom in the electronic ground state and one in the excited state. We also
show that sufficiently close to the resonances the compressibility of the
system starts to diverge indicating that the emerging order is unstable and
collapses to a phase separated state with a first order phase transition.
