Exact Parent Hamiltonian for the Quantum Hall States in a Lattice

We study lattice models of charged particles in uniform magnetic fields. We show how longer range hopping can be engineered to produce a massively degenerate manifold of single-particle ground states with wave functions identical to those making up the lowest Landau level of continuum electrons in a magnetic field. We find that in the presence of local interactions, and at the appropriate filling factors, Laughlin’s fractional quantum Hall wave function is an exact many-body ground state of our lattice model. The hopping matrix elements in our model fall off as a Gaussian, and when the flux per plaquette is small compared to the fundamental flux quantum one only needs to include nearest and next-nearest neighbor hoppings. We suggest how to realize this model using atoms in optical lattices, and describe observable consequences of the resulting fractional quantum Hall physics.

Figure 1

All single-particle eigenvalues for the hopping Hamiltonian in Eq. (1) with $\varphi =1/3$ on a $12\times 12$ lattice with periodic boundary conditions. The index $n$ labels eigenvalues from smallest to largest. The white disks use hopping matrix elements given by Eq. (2), the black disks are the same model with only nearest and next-nearest neighbor hopping, and the grey boxes have only nearest neighbor hopping (the Hofstadter Hamiltonian). Energies are all measured in units of $t$. The energies of the white and black disks are nearly indistinguishable, and the white disks obscure some of the black ones. The lowest $1/3$ of the white disks are all degenerate.

Figure 2

First 100 eigenvalues for 4 particles on a $4\times 4$ lattice with periodic boundary conditions, $\varphi =1/2$, and hard-core repulsion. The two states at $ϵ=-4$ are Laughlin states (11); the degeneracy stems from the toroidal geometry. There is a distinct energy gap of $0.566t_{nn}$ to the lowest excited states, where $t_{nn}$ is the nearest neighbor hopping amplitude.

Figure 3

Schematic plot of $⟨n⟩$ vs $\mu$, for lattice bosons described by Eqs. (1, 2), with hard-core interactions added. The steps correspond to incompressible particle or hole fractional quantum Hall states at $\nu =1/2$. This structure will be visible in the density profile of a trapped gas. Similar structure will be seen with fermions, but with plateaus at fillings with odd denominators.