Figure 1: Model and implementation.
From: Local models of fractional quantum Hall states in lattices and physical implementation
(a) The considered spin lattice Hamiltonian (3) is a sum of local two- and three-body interactions. (b) For suitable parameters, it is effectively equivalent to the Fermi–Hubbard-like Hamiltonian in equation (5), of which we here show the kinetic energy parts Hkin,σ, σε{↑, ↓}. Specifically, each arrow/line/wiggle from position n to m on the lattice represents the contribution 
to Hkin,σ with 
and φ given in the figure. (c) N fermions trapped in the optical lattice potential we propose to use for implementing the Fermi–Hubbard-like Hamiltonian. (d) We encode the spin-up and -down states in four internal hyperfine levels of the fermions. The blue/red states feel the blue/red potential in (c) and are hence trapped at the blue/red lattice sites. In this setting, we can implement the nearest-neighbour hopping terms through Raman transitions as indicated with the dashed blue and red lines in (d). For this we need the three standing wave laser fields shown in (b). Er1 and Er2 are directed along the x- and y axis, respectively, and are illustrated with the red dashed lines. Eb3 is directed along the z axis and is illustrated with the blue round shadow (explicit expressions for the fields are given in Table 3). The next-nearest-neighbour hopping terms are implemented as a combination of hops induced by the fields listed above and tunnelling between nearest-neighbour sites in the blue/red lattice.
