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Lattice effects on Laughlin wave functions and parent Hamiltonians

We investigate lattice effects on wave functions that are lattice analogues of bosonic and fermionic Laughlin wave functions with number of particles per flux $ν=1/q$ in the Landau levels. These wave functions are defined analytically on lattices with $μ$ particles per lattice site, where $μ$ may be different than $ν$. We give numerical evidence that these states have the same topological properties as the corresponding continuum Laughlin states for different values of $q$ and for different fillings $μ$. These states define, in particular, particle-hole symmetric lattice Fractional Quantum Hall states when the lattice is half-filled. On the square lattice it is observed that for $q\leq 4$ this particle-hole symmetric state displays the topological properties of the continuum Laughlin state at filling fraction $ν=1/q$, while for larger $q$ there is a transition towards long-range ordered anti-ferromagnets. This effect does not persist if the lattice is deformed from a square to a triangular lattice, or on the Kagome lattice, in which case the topological properties of the state are recovered. We then show that changing the number of particles while keeping the expression of these wave functions identical gives rise to edge states that have the same correlations in the bulk as the reference lattice Laughlin states but a different density at the edge. We derive an exact parent Hamiltonian for which all these edge states are ground states with different number of particles. In addition this Hamiltonian admits the reference lattice Laughlin state as its unique ground state of filling factor $1/q$. Parent Hamiltonians are also derived for the lattice Laughlin states at other fillings of the lattice, when $μ\leq 1/q$ or $μ\geq 1-1/q$ and when $q=4$ also at half-filling.