Unraveling of the fractional topological phase in one-dimensional flatbands with nontrivial topology

We consider a topologically non-trivial flat band structure in one spatial dimension in the presence of nearest and next nearest neighbor Hubbard interaction. The non-interacting band structure is characterized by a symmetry protected topologically quantized Berry phase. At certain fractional fillings, a gapped phase with a filling-dependent ground state degeneracy, and fractionally charged quasi-particles emerges. At filling 1/3, the ground states carry a fractional Berry phase in the momentum basis. These features at first glance suggest a certain analogy to the fractional quantum Hall scenario in two dimensions. We solve the interacting model analytically in the physically relevant limit of a large band gap in the underlying band structure, the analog of a lowest Landau level projection. Our solution affords a simple physical understanding of the properties of the gapped interacting phase. We pinpoint crucial differences to the fractional quantum Hall case by studying the Berry phase and the entanglement entropy associated with the degenerate ground states. In particular, we conclude that the `fractional topological phase in one-dimensional flatbands' is not a one-dimensional analog of the two-dimensional fractional quantum Hall states, but rather a charge density wave with a nontrivial Berry phase. Finally, the symmetry protected nature of the Berry phase of the interacting phase is demonstrated by explicitly constructing a gapped interpolation to a state with a trivial Berry phase.