Exact Parent Hamiltonian for the Quantum Hall States in a Optical Lattice

TL;DR

This paper constructs a lattice model with engineered hopping to realize fractional quantum Hall states, specifically Laughlin's wavefunction, as exact ground states, and discusses experimental realization with optical lattices.

Contribution

It introduces an exact parent Hamiltonian for quantum Hall states in optical lattices, bridging continuum and lattice models with practical implementation strategies.

Findings

Lattice models with Gaussian-decaying hopping produce Landau level-like degeneracy.

Laughlin's fractional quantum Hall wavefunction is an exact ground state at specific fillings.

Proposes feasible experimental realization using atoms in optical lattices.

Abstract

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 wavefunctions 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 wavefunction 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.