Alkaline-earth and ytterbium cold atomic gases make it possible to simulate
SU(N)-symmetric fermionic systems in a very controlled fashion. Such a high
symmetry is expected to give rise to a variety of novel phenomena ranging from
molecular Luttinger liquids to (symmetry- protected) topological phases. We
review some of the phases that can be stabilized in a one dimensional lattice.
The physics of this multicomponent Fermi gas turns out to be much richer and
more exotic than in the standard SU(2) case. For N > 2, the phase diagram is
quite rich already in the case of the single-band model, including a molecular
Luttinger liquid (with dominant superfluid instability in the N-particle
channel) for incommensurate fillings, as well as various Mott-insulating phases
occurring at commensurate fillings. Particular attention will be paid to the
cases with additional orbital degree of freedom (which is accessible
experimentally either by taking into account two atomic states or by putting
atoms in the p-band levels). We introduce two microscopic models which are
relevant for these cases and discuss their symmetries and strong coupling
limits. More intriguing phase diagrams are then presented including, for
instance, symmetry protected topological phases characterized by non-trivial
edge states.Comment: 59 pages, 20 figures. Review article published in Annals of Physics.
Some minor errors fixe
