Recent experiments with alkaline-earth-like cold atoms, such as strontium or ytterbium, have reached the quantum regime. In these systems, the interactions are almost independent of the nuclear spin state, which enables the realization of large symmetries. This motivates the study of SU(N) fermionic systems in low dimensions which have been widely studied in condensed matter theory .
In this Letter, we have investigated the possible stabilization of topological phases in alkaline-earth cold fermions loaded into an 1D optical lattice. The study of topological phases in general dimensions has become in recent years a very important hot topic in condensed matter. To the best of our knowledge, our work represents the first careful investigation of the nature of 1D fermionic topological phases that can be formed in SU(2I+1) alkaline-earth cold fermions for general half-integer nuclear spins I, which is relevant to the above-cited atoms. To this end, we have used the most powerful methods available in one dimension : density matrix renormalization group (DMRG) computations, low-energy and conformal field theory approaches, matrix-product state technique, and strong-coupling arguments.
In particular, we have emphasized the crucial role played by the introduction of an orbital degree of freedom to the SU(N) Hubbard model. Physically, this two-orbital model corresponds either to (i) the ground state and the metastable excited state of alkaline-earth atoms (ii) or to the second degenerate orbital -band of the optical lattice. Exotic topological phases are absent if one only considers the ground state.
By varying the interaction strengths, our first finding is a "Haldane orbital" phase which is analogous to the Haldane phase of the integer Heisenberg spin chain but in terms of the orbital degrees of freedom (see cartoon below). The existence of this phase is predicted both in weak and strong coupling using analytical tools, and then confirmed numerically with large-scale simulations.
A second different topological phase is found which depends only on the nuclear spin degrees of freedom. It is first identified in the strong-coupling approach and, in order to gain insight on its physical properties, we propose an explicit construction (in the spirit of AKLT, see figure).
It shows that the ground-state of a closely related model is indeed topological, i.e. is non-degenerate and supports non-trivial edge states. Robustness of this state is also analysed in terms of symmetry-protection, which has become a very useful tool recently. We find that the phase is a symmetry-protected topological phase with respect to the lattice inversion symmetry when I=1/2,5/2,9/2,.., which is directly relevant to ytterbium and strontium atoms. We call on state-of-the-art DMRG numerical simulations to confirm that the non-degenerate gapful phase has non-trivial edge states.
Contact : Sylvain Capponi