Topological edge states in the one-dimensional super-lattice Bose-Hubbard model

We analyze interacting ultra-cold bosonic atoms in a one-dimensional (1D) super-lattice potential with alternating tunneling rates t_1 and t_2 and inversion symmetry, which is the bosonic analogue of the Su-Schrieffer-Heeger (SSH) model. A Z2 topological order parameter is introduced which is quantized for the Mott insulating (MI) phases. Depending on the ratio t_1/t_2 the n=1/2 MI phase is topologically non-trivial, which results in many-body edge states at open boundaries. In contrast to the SSH model the bosonic counterpart lacks chiral symmetry and the edge states are no longer mid-gap. This leads to a generalization of the bulk-edge correspondence, which we discuss in detail. The edge states can be observed in cold atom experiments by creating a step in the effective confining potential, e.g. by a second heavy atom species, which leads to an interface between two MI regions with filling n=1 and n=1/2. Shape and energy of the edge states as well as conditions for their occupation are determined analytically in the strong coupling limit and in general by density-matrix renormalization group (DMRG) simulations.