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One-dimensional superlattices with periodic spatial modulations of onsitepotentials or tunneling coefficients can exhibit a variety of propertiesassociated with topology or symmetry. Recent developments of ring-shapedoptical lattices allow a systematic study of those properties in superlatticeswith or without boundaries. While superlattices with additional modulatingparameters are shown to have quantized topological invariants in the augmentedparameter space, we also found localized or zero-energy states associated withsymmetries of the Hamiltonians. Probing those states in ultracold-atoms ispossible by utilizing recently proposed methods analyzing particle depletion orthe local density of states. Moreover, we summarize feasible realizations ofconfigurable optical superlattices using currently available techniques.

physical review. a/physical review, a

Topology, edge states, and zero-energy states of ultracold atoms in 1D  optical superlattices with alternating onsite potentials or hopping  coefficients