  We propose to realize one-dimensional topological phases protected by SU($N$)
symmetry using alkali or alkaline-earth atoms loaded into a bichromatic optical
lattice. We derive a realistic model for this system and investigate it
theoretically. Depending on the parity of $N$, two different classes of
symmetry-protected topological (SPT) phases are stabilized at half-filling for
physical parameters of the model. For even $N$, the celebrated spin-1 Haldane
phase and its generalization to SU($N$) are obtained with no local symmetry
breaking. In stark contrast, at least for $N=3$, a new class of SPT phases,
dubbed chiral Haldane phases, that spontaneously break inversion symmetry,
emerge with a two-fold ground-state degeneracy. The latter ground states with
open-boundary conditions are characterized by different left and right boundary
spins which are related by conjugation. Our results show that topological
phases are within close reach of the latest experiments on cold fermions in
optical lattices.
