High-Field Fractional Quantum Hall Effect in Optical Lattices

We consider interacting bosonic atoms in an optical lattice subject to a large simulated magnetic field. We develop a model similar to a bilayer fractional quantum Hall system valid near simple rational numbers of magnetic flux quanta per lattice cell. Then we calculate its ground state, magnetic lengths, fractional fillings, and find unexpected sign changes in the Hall current. Finally we study methods for detecting these novel features via shot noise and Hall current measurements.

Figure 1

Wave function for (a) $\alpha =1/2$ and (b) $\alpha =1/3$, in linear geometry with trap $V(y)=m{\omega }^2{y}^2/2$ and $\omega ={10}^{-3}\pi /m{d}^2$. The dots show the numerical wave function and the solid curves the analytic approximations for $y/d\equiv 0$ mod $n$ (upper curve) and $y/d\ne 0$ mod $n$ (lower curve).

Figure 2

Areas of validity of different fractions ${\alpha }_c$, in a one-dimensional harmonic trap. The solid (dotted) curves show regions with overlap between exact numerical wave functions and their analytical approximations larger than 99% (99.9%) and resulting energy difference smaller than $0.01/m{d}^2$ ($0.001/m{d}^2$). The dashed lines are areas of validity obtained from Eq. (5) taking $\gg$ to mean a ratio larger than 50.

Figure 3

Two-point functions: $\nu =1/2$ Laughlin state (solid line); $\tilde{\nu }=2/3$ 221 state $g_{11}=g_{22}$ (dashed line) and $g_{12}=g_{21}$ (dotted line).

Figure 4

Dimensionless Hall current $I_d=I/(2^{3/2}ma\sqrt{u\Omega }/\omega )$ against dimensionless number of atoms per unit length $N_d=N/(2^{3/2}L{\Omega }^{3/2}m\sqrt{u}/\omega )$ for $V_1(y)=m{\omega }^2{y}^2/2$, $V_2(y)=V_1(y)-may$ at small $\alpha$ and $a$, for no disorder (dotted straight line), maximal disorder (constant density of states) (black curve), and Lorentzian disorder of width $um\Omega /10\pi$ ( gray curve). Numbers under the curve are filling factors at trap center. The inset shows the $\nu =1/2$ to $\nu =1$ corner.