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We propose to realize one-dimensional topological phases protected by SU($N$)symmetry using alkali or alkaline-earth atoms loaded into a bichromatic opticallattice. We derive a realistic model for this system and investigate ittheoretically. Depending on the parity of $N$, two different classes ofsymmetry-protected topological (SPT) phases are stabilized at half-filling forphysical parameters of the model. For even $N$, the celebrated spin-1 Haldanephase and its generalization to SU($N$) are obtained with no local symmetrybreaking. 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 withopen-boundary conditions are characterized by different left and right boundaryspins which are related by conjugation. Our results show that topologicalphases are within close reach of the latest experiments on cold fermions inoptical lattices.

"Haldane" phases with ultracold fermionic atoms in double-well optical lattices