Figure 3: Topological properties of the Hamiltonian.
From: Local models of fractional quantum Hall states in lattices and physical implementation
(a) Energy spectrum in the 
=0 subspace on the torus for different lattice sizes (in green, N=LxLy) obtained by exact diagonalization. The number to the right of each energy level is the degeneracy and only the energies of the five lowest states (n=0, 1 , 2 , 3, 4) are displayed. Note that the spectrum in the complete Hilbert space is the same except that each state is replaced by 2S+1 degenerate states, where S is the total spin quantum number of the state. As the two lowest states have S=0, the results suggest that there are two degenerate ground states and a gap to the first excited state in the thermodynamic limit like for the ν=1/2 Laughlin state in the continuum59,60. (b) Energies of the five lowest states in the 
=0 subspace on the torus for twisted boundary conditions in the x direction (θx is the twist angle) and a lattice size of 6 × 5. Twisting the boundary conditions corresponds to gradually inserting a flux line through the hole of the torus (inset), and we observe that the two ground states flow into each other under this operation. φ1=0.07 × 2π and φ2=0.03 × 2π in both panels.
