We investigate the existence of symmetry-protected topological phases in onedimensional alkaline-earth cold fermionic atoms with general half-integer nuclear spin I at half filling. In this respect, some orbital degrees of freedom are required. They can be introduced by considering either the metastable excited state of alkaline-earth atoms or the p-band of the optical lattice. Using complementary techniques, we show that SU(2) Haldane topological phases are stabilised from these orbital degrees of freedom. On top of these phases, we find the emergence of topological phases with enlarged SU(2I + 1) symmetry which depend only on the nuclear spin degrees of freedom. The main physical properties of the latter phases are further studied using a matrix-product state approach. On the one hand, we find that these phases are symmetryprotected topological phases, with respect to inversion symmetry, when I = 1/2, 5/2, 9/2, . . ., which is directly relevant to ytterbium and strontium cold fermions. On the other hand, for the other values of I(=half-odd integer), these topological phases are stabilised only in the presence of exact SU(2I + 1)-symmetry.

4 pages, 4 figuresInternational audienceA Haldane conjecture is revealed for spin-singlet charge modes in 2N-component fermionic cold atoms loaded into one-dimensional optical lattice. By means of a low-energy approach and DMRG calculations, we show the emergence of gapless and gapped phases depending on the parity of N for attractive interactions at half-filling. In particular, the analogue of the Haldane phase of the spin-1 Heisenberg chain is stabilized for N = 2 with non-local string charge correlation, and pseudo-spin 1/2 edge states. At the heart of this even-odd behavior is the existence of spin-singlet pseudo-spin N/2 operator which governs the low-energy properties of the model for attractive interactions and give rise to the Haldane physics