Modified interactions in a Floquet topological system on a square lattice and their impact on a bosonic fractional Chern insulator state

We propose a simple scheme for the realization of a topological quasienergy band structure with ultracold atoms in a periodically driven optical square lattice. It is based on a circular lattice shaking in the presence of a superlattice that lowers the energy on every other site. The topological band gap, which separates the two bands with Chern numbers $\pm 1$, is opened in a way characteristic of Floquet topological insulators, namely, by terms of the effective Hamiltonian that appear in subleading order of a high-frequency expansion. These terms correspond to processes where a particle tunnels several times during one driving period. The interplay of such processes with particle interactions also gives rise to new interaction terms of several distinct types. For bosonic atoms with on-site interactions, they include nearest-neighbor density-density interactions introduced at the cost of weakened on-site repulsion as well as density-assisted tunneling. Using exact diagonalization, we investigate the impact of the individual induced interaction terms on the stability of a bosonic fractional Chern insulator state at half filling of the lowest band.

Figure 1

Realizing the chiral $\pi$ model. The original square lattice in (a) is separated into two energy-mismatched sublattices (yellow $\mathcal{A}$ and cyan $\mathcal{B}$) of a larger lattice constant in (b). The vectors ${\mathbf{\delta }}_{\mu }$ with $\mu =\{1,2,3,4\}$ connect the nearest-neighbor sites belonging to distinct sublattices. The vectors ${\mathit{a}}_{1|2}$ are the elementary translation vectors. Solid (dashed) blue lines depict natural (driving-assisted) transitions.

Figure 2

The single-particle band structure of the driven square lattice versus the scaled inverse driving frequency $\beta$ and the scaled driving strength $\alpha$. (a) The width of the topological band gap ${\mathrm{\Delta }}_{\mathrm{sp}}$ (when present) is measured in units of the hopping parameter $J$. (b) The band gap is compared to the width of a single band thus defining the band flatness ratio $\mathcal{F}={\mathrm{\Delta }}_{\mathrm{sp}}/W$.

Figure 3

Many-body topological gap measured in units $J$ for eight bosons moving on a $4\times 4$ lattice of two-site unit cells: (a) micromotion omitted and (b) micromotion included. The points labeled with the letters A, B, C, and D correspond to four typical behavioral patterns further analyzed in Fig. 4.

Figure 4

Many-body gap for eight bosons moving on a $4\times 4$ lattice at four typical parameter sets A, B, C, and D marked in Fig. 3: black lines and open circles, micromotion completely neglected; red lines with solid circles, only reduction of on-site repulsion strength taken into account; blue lines with crosses, on-site repulsion spread onto neighboring sites; and purple lines with diamonds, all third-order corrections included.