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Interaction driven polarization shift in the $t-V-V'$ lattice fermion model at half filling: emergent Haldane phase

We study the $t-V-V'$ model in one dimension at half-filling. It is known that for large enough $V$ fixed, as $V'$ is varied, the system goes from a charge-density wave into a Luttinger liquid, then a bond-order, and then a second charge density wave phase. We find that the Luttinger liquid state is further split into two, separating parts with distinct values of the many-body polarization Berry phase. Inside this phase, the variance of the polarization is infinite in the thermodynamic limit, meaning that even if the polarization differs, it would not be measurable. However, in the gapped phases on each side of the Luttinger liquid, the polarization takes a different measurable value, implying topologically distinction. The key difference is that the large-$V'$ phases are link-inversion symmetric, while the small-$V'$ one is site-inversion symmetric. We show that large-$V'$ phase can be related to an $S=1$ spin chain, and exhibits many features of the Haldane phase. The lowest lying states of the entanglement spectrum display different degeneracies in the two cases, and we also find string order in the large-$V'$ phase. We also study the system under open boundary conditions, and suggest that the number of defects is related to the topology.