Role of real-space micromotion for bosonic and fermionic Floquet fractional Chern insulators

Fractional Chern insulators are the proposed phases of matter mimicking the physics of fractional quantum Hall states on a lattice without an overall magnetic field. The notion of Floquet fractional Chern insulators refers to the potential possibilities to generate the underlying topological band structure by means of Floquet engineering. In these schemes, a highly controllable and strongly interacting system is periodically driven by an external force at a frequency such that double tunneling events during one forcing period become important and contribute to shaping the required effective energy bands. We show that in the described circumstances it is necessary to take into account also third order processes combining two tunneling events with interactions. Referring to the obtained contributions as micromotion-induced interactions, we find that those interactions tend to have a negative impact on the stability of fractional Chern insulating phases and discuss implications for future experiments.

Figure 1

Honeycomb lattice as a triangular Bravais lattice with a two-site basis. The inequivalent sites A and B are marked with filled and open dots, respectively. Nearest neighbors are connected by the vectors ${\mathbf{\delta }}_{\mu }$, while the set of next-nearest neighbor vectors ${\mathit{a}}_j$ also defines the elementary translations.

Figure 2

Comparison of the importance of various Fourier harmonics. The shaded background shows the range of the dimensionless shaking amplitudes $\alpha$ considered in the calculations. The red lines depict the behavior of ${\mathcal{J}}_0\left(\alpha \right)$ and ${\mathcal{J}}_1^2\left(\alpha \right)$, setting the scale of, respectively, the static components of the Hamiltonian and the first harmonic. The dashed blue lines show the behavior of the (omitted) subleading contributions ${\mathcal{J}}_{2,3}^2\left(\alpha \right)$.

Figure 3

Typical energy spectra of the driven honeycomb lattice with micromotion-induced interactions ignored [panel (a)], and taken into account [panel (b)] plotted versus the auxiliary flux ${\gamma }_1$ inserted along the first (commensurate) lattice direction. Here, the energies of many body states in the eight-fermion system numerically calculated at $\beta =0.3, v=5$, and ${\gamma }_2=0$ are shown. Colored (black) lines denote the states in the ground-state manifold (other) total quasimomentum sectors, and two lowest-energy states of each quasimomentum sector are included. The dark red arrow illustrates the definition of the local many body gap referred to in the text.

Figure 4

Phase diagram showing the many body gap $\mathrm{\Delta }$ as a function of the inverse shaking frequency $\beta$ and the dimensionless interaction strength $u$ for eight-boson system. Micromotion-induced interactions are omitted in panel (a) and taken into account to the third order in panel (b). The full black lines delimit the regions of positive many body gaps (shown in red shades) and differ considerably between the panels. The white dashed line indicates the parameter regime where the interaction strength $U$ is equal to the single-particle band gap. Well below this line mixing between single-particle bands becomes small.

Figure 5

Same as Fig. 4 but for eight-fermion system with dimensionless NN repulsion energy $v$. Note that the scale of the color bar and correspondingly the typical many body gaps are significantly smaller than for bosons.

Figure 6

Quasihole excitation spectra for eight-boson system calculated at the inverse dimensionless driving strength $\beta =0.1$. Formation of an isolated manifold of $4\times 16$ states is visible for strong interactions with $u=5$, and gradually closes when interaction becomes weaker.

Figure 7

Dispersion of the upper band of the Haldane model drawn along the diagonal of the rhombic BZ.