  Laughlin states have recently been constructed on fractal lattices, and the
charge and braiding statistics of the quasiholes were used to confirm that
these states have Laughlin type topology. Here, we investigate density,
correlation, and entanglement properties of the states on a fractal lattice
derived from a Sierpinski triangle with the purpose of identifying similarities
and differences compared to two-dimensional systems and with the purpose of
investigating whether various probes of topology work for fractal lattices.
Similarly to two-dimensional systems, we find that the connected
particle-particle correlation function decays roughly exponentially with the
distance between the lattice sites measured in the two-dimensional plane, but
the values also depend on the local environment. Contrary to two-dimensional
systems, we find that the entanglement entropy does not follow the area law if
one defines the area to be the number of nearest neighbor bonds that cross the
edge of the selected subsystem. Considering bipartitions with two bonds
crossing the edge, we find a close to logarithmic scaling of the entanglement
entropy with the number of sites in the subsystem. This also means that the
topological entanglement entropy cannot be extracted using the Kitaev-Preskill
or the Levin-Wen methods. Studying the entanglement spectrum for different
bipartitions, we find that the number of states below the entanglement gap is
robust and the same as for Laughlin states on two-dimensional lattices.

Strongly Correlated Electrons;Quantum Physics