  There is a growing interest in searching for topology in fractal dimensions
with the aim of finding different properties and advantages compared to the
integer dimensional case. It has previously been shown that the Laughlin state
can be adapted to fractal lattices. A key element in doing so is to replace the
uniform background charge by a background charge that resides only on the
lattice sites. This motivates the study of Hofstadter type models on fractal
lattices, in which the magnetic field is present only at the lattice sites.
Here, we study such models for hardcore bosons on finite lattices derived from
the Sierpinski carpet and on square lattices with open boundary conditions. We
find that the system sizes that we can investigate with exact diagonalization
are generally too small to judge whether these local models are topological or
not. Studying the particle densities on the lattices derived from the
Sierpinski carpet, we find that the densities tend to accumulate in the regions
that are locally similar to a square lattice. Such accumulation seems to be
incompatible with the uniform densities in fractional quantum Hall systems,
which might suggest that the models are not topological. Our computations
provide guidance for future searches for topology in finite systems. We also
propose a scheme to implement both fractal lattices and our proposed local
Hamiltonian with ultracold atoms in optical lattices, which could allow for
quantum simulators to go beyond the numerical results presented here.
