Interaction-induced phases in the half-filled Bernevig-Hughes-Zhang model in one dimension

We explore the ground-state properties of a one-dimensional model with two orbitals per site, where, in addition to atomic energies $\ifmmode\pm\else\textpm\fi{}M$ and intra- and interorbital hoppings, the intraorbital Hubbard $(U)$ and nearest-neighbor density-density $(V)$ repulsions are included. Our results are primarily based on a Jastrow-Slater wave function and variational Monte Carlo methods but also corroborated by density-matrix renormalization group calculations. In the noninteracting limit, when varying $M>0$, a gapless point separates a trivial phase from a topological one. The inclusion of a finite Hubbard-$U$ repulsion does not give rise to any phase transition within the topological region, inducing a smooth crossover into a Haldane (spin gapped) insulator; notably, the string-order parameter, which characterizes the latter phase, is already finite in the noninteracting limit. Most importantly, at finite values of $U$, the transition between the trivial and topological states is not direct, since an emergent insulator, which shows evidence of sustaining gapless spin excitations, intrudes between them. A small-$V$ interaction further stabilizes the intermediate insulator, while a sufficiently large value of this nearest-neighbor repulsion gives rise to two different charge-density wave insulators, one fully gapped and another still supporting gapless spin excitations. Our results demonstrate the richness of multiorbital Hubbard models, in the presence of a topologically nontrivial band structure, and serve as a basis for future investigations on similar two-dimensional models.