Floquet Fractional Chern Insulators

We show theoretically that periodically driven systems with short range Hubbard interactions offer a feasible platform to experimentally realize fractional Chern insulator states. We exemplify the procedure for both the driven honeycomb and the square lattice, where we derive the effective steady state band structure of the driven system by using the Floquet theory and subsequently study the interacting system with exact numerical diagonalization. The fractional Chern insulator state equivalent to the $1/3$ Laughlin state appears at $7/12$ total filling ($1/6$ filling of the upper band). The state also features spontaneous ferromagnetism and is thus an example of the spontaneous breaking of a continuous symmetry along with a topological phase transition. We discuss light-driven graphene and shaken optical lattices as possible experimental systems that can realize such a state.

Figure 1

A graphene flake is irradiated with light of frequency $\omega$, while a gate voltage is applied via a back gate (yellow) to change the band filling. In the high-frequency regime, the incident light changes the single-particle band structure into an effective Floquet band structure that acquired a gap at the Dirac points (shown are the Floquet bands for the electric field configuration $A_x=A_y=1.7$, $\varphi =\pi /2$, and $\omega =10t_1$).

Figure 2

Topological phase diagram of periodically driven graphene at a high-frequency regime ($\omega =10t_1$) for circularly polarized light. The phase difference $\varphi$ and the field amplitude $A_x=A_y$ allow us to tune the Chern number of the lower spin-degenerate bands of the effective Floquet-Bloch Hamiltonian (3) between $c_1=-1$ and $c_1=+1$. The black dot indicates the parameter values for which exact diagonalization calculations are presented in Figs. 3 and 4.

Figure 3

Low-lying portion of the energy spectrum of Hamiltonian (5) on a $L_x\times L_y=4\times 6$ with $N=8$ particles. Encircled is the ground state manifold with threefold topological (quasi)degeneracy of the fractional Chern insulator and ($N+1=9$)-fold degeneracy as a precursor of spontaneous symmetry breaking toward a ferromagnetic phase in the thermodynamic limit. The good quantum numbers total momentum $Q=k_x+L_x{k}_y$, total spin $S$, and total spin in $z$ direction $S_z$ are indicated.

Figure 4

(a) Spectral evolution of the energy spectrum of Fig. 3 in the sector with $S_z=1$ upon inserting a flux ${\gamma }_y$ into the system, which is synonymous to twisting the boundary conditions in the $y$ direction with a complex phase $e^{\mathrm{i}{\gamma }_y}$. The three fractional Chern insulator ground states evolve independently of the rest of the spectrum and trade places, which signals their topological degeneracy and the charge fractionalization. (b) The energy spectrum of Fig. 3 plotted against the total spin $S(S+1)$ reveals the tower of states which evidence the ferromagnetic nature of the ground state.