  Conformal field theory has recently been applied to derive few-body
Hamiltonians whose ground states are lattice versions of fractional quantum
Hall states. The exact lattice models involve interactions over long distances,
which is difficult to realize in experiments. It seems, however, that such
long-range interactions should not be necessary, as the correlations decay
exponentially in the bulk. This poses the question, whether the Hamiltonians
can be truncated to contain only local interactions without changing the
physics of the ground state. Previous studies have in a couple of cases with
particularly much symmetry obtained such local Hamiltonians by keeping only a
few local terms and numerically optimizing the coefficients. Here, we
investigate a different strategy to construct truncated Hamiltonians, which
does not rely on optimization, and which can be applied independent of the
choice of lattice. We test the approach on two models with bosonic
Laughlin-like ground states with filling factor $1/2$ and $1/4$, respectively.
We first investigate how the coupling strengths in the exact Hamiltonians
depend on distance, and then we study the truncated models. For the case of
$1/2$ filling, we find that the truncated model with truncation radius
$\sqrt{2}$ lattice constants on the square lattice and $1$ lattice constant on
the triangular lattice has an approximate twofold ground state degeneracy on
the torus, and the overlap per site between these states and the states
constructed from conformal field theory is higher than $0.99$ for the lattices
considered. For the model at $1/4$ filling, our results give some hints that a
truncation radius of $\sqrt{5}$ on the square lattice and $\sqrt{7}$ on the
triangular lattice might be enough, but the finite size effects are too large
to judge whether the topology is, indeed, present in the thermodynamic limit.
