Fractional topological phase in one-dimensional flat bands with nontrivial topology

We consider a topologically nontrivial flat-band structure in one spatial dimension in the presence of nearest- and next-nearest-neighbor Hubbard interaction. The noninteracting 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 quasiparticles emerges. At filling $\frac{1}{3}$, 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 flat bands” 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.

Figure 1

Trivial unpolarized model with $v^x=v^z=0\ne v^y$ (top) with orbitals localized on the sites. Nontrivial SSH model with polarized orbitals localized on the bonds between two sites (bottom).

Figure 2

Gapped ground state at filling $\nu =\frac{1}{3}$ and $V_1>V_2$. Every third bond is filled with a particle (green dot).

Figure 3

The low-lying part of the spectrum of the interacting Hamiltonian $H\left(\lambda \right)$, with interaction parameters $V_1=\frac{1}{10}$, $V_2=\frac{1}{20}$ for system size $L=15$ and periodic boundary conditions. We note that the ground state is threefold degenerate for each value of $\lambda$.

Figure 4

The low-lying part of the energy spectrum of the 1DFTP with $L=15$ sites and $F=5$ fermions. The interaction parameters are $V_1=\frac{1}{10}$ and $V_2=0$ (left panel), and $V_1=\frac{1}{10}$ and $V_2=\frac{1}{20}$ (right panel).

Figure 5

The numerically calculated Berry phase for the interacting Hamiltonian $H\left(\lambda \right)$ with $V_1=\frac{1}{10}$, $V_2=\frac{1}{20}$ as a function of the interpolation parameter $\lambda$ for $L=3$ (blue circles), $L=6$ (red squares), and $L=9$ (green diamonds). The black line is the analytic result for the noninteracting case, valid for $L=3$.

Figure 6

The map $k\mapsto (v^x\left(k\right),v^y\left(k\right))$ of Eq. (14), for various values of $\lambda$: $\lambda =0$ (black line), $\lambda =\frac{1}{5}$ (red dashed), $\lambda =\frac{1}{2}$ (green dotted), $\lambda =\frac{4}{5}$ (blue dash-dotted), $\lambda =1$ (black dot). The black cross marks the origin, corresponding to $E=0$, which lies on the curve with $\lambda =\frac{1}{2}$.