Statistical Transmutation in Floquet Driven Optical Lattices

We show that interacting bosons in a periodically driven two dimensional (2D) optical lattice may effectively exhibit fermionic statistics. The phenomenon is similar to the celebrated Tonks-Girardeau regime in 1D. The Floquet band of a driven lattice develops the moat shape, i.e., a minimum along a closed contour in the Brillouin zone. Such degeneracy of the kinetic energy favors fermionic quasiparticles. The statistical transmutation is achieved by the Chern-Simons flux attachment similar to the fractional quantum Hall case. We show that the velocity distribution of the released bosons is a sensitive probe of the fermionic nature of their stationary Floquet state.

Figure 1

(a) Lowest Floquet band exhibiting approximately flat moat. (b) Fragment of a square optical lattice with two sites per unit cell [29] that gives rise to an approximately flat moat in the lowest Floquet band upon resonant driving. (c) Left panel shows the band structure of the undriven optical lattice. The driving frequency $\omega$ is of the order of the gap causing resonant coupling between the lowest and the first excited bands at $\mathbf{k}=0$. Middle panel: enlargement of the lowest and shifted first excited bands. Right panel: Blow up of the driven lowest Floquet band exhibiting a double-well feature in $k_x+k_y$ direction indicating the appearance of the moat in the Brillouin zone.

Figure 2

Panel (a): characteristic radius $k_0$ plotted as a function of dimensionless $U_0/E_R$ at two values of $k_L\mathrm{\Delta }/\pi =0.01$ (circles) and 0.02 (squares). Panel (b): the flatness measure $F$ of the moat versus $U_0/E_R$ is shown. For definition, see the main text.

Figure 3

Excitation spectrum in units of $E_R$ is plotted versus $n/n_0$. The per particle energy, $E_{gs}/E_R$, represented by the fully filled lowest Landau level, is shown by the thick grey (red) curve. The thin black lines represent excited energies. The dashed line represents the chemical potential of a condensate ${\mu }_{\mathrm{con}1}/E_R$ corresponding to the interaction parameter $g\sim 1$. The comparison between chemical potentials of these candidate states results in a phase diagram presented in the inset. For such strong interactions the transition from composite fermion state to zero temperature BEC takes place at densities $n/n_0\sim 1$. Upon lowering $g$, the transition shifts towards smaller $n/n_0$, as schematically shown in the inset.

Figure 4

Dimensionless velocity distribution $\rho (v)·v_l$, where $m{v}_l=\pi n_l/k_0$, of an expanding gas of composite fermions plotted vs dimensionless velocity $v/v_l$ at fixed values of density $n=n_l$, $l=0,3,5$ and zero temperature. Inset: Dimensionless velocity distribution of condensed bosons with density $n=n_0$. The temperatures are marked. At low temperatures, the distribution shows a sharp peak at zero velocity, which is absent in the distribution of the composite fermions.