  Laughlin states have recently been constructed on fractal lattices and have
been shown to be topological in such systems. Some of their properties are,
however, quite different from the two-dimensional case. On the Sierpinski
triangle, for instance, the entanglement entropy shows oscillations as a
function of particle number and does not obey the area law despite being
topologically ordered, and the particle density is non-uniform in the bulk.
Here, we investigate these deviant properties in greater detail on the
Sierpinski triangle, and we also study the properties on the Sierpinski carpet
and the T-fractal. We find that the density variations across the fractal are
present for all the considered fractal lattices and for most choices of the
number of particles. The size of anyons inserted into the lattice Laughlin
state also varies with position on the fractal lattice. We observe that
quasiholes and quasiparticles have similar sizes and that the size of the
anyons typically increases with decreasing Hausdorff dimension. As opposed to
periodic lattices in two dimensions, the Sierpinski triangle and carpet have
inner edges. We construct trial states for both inner and outer edge states. We
find that oscillations of the entropy as a function of particle number are
present for the T-fractal, but not for the Sierpinski carpet. Finally, we
observe deviations from the area law for several different bipartitions on the
Sierpinski triangle.
