Anyons are mainly studied and considered in two spatial dimensions. For fractals, the scaling dimension that characterizes the system can be non integer and can take values between that of a standard one-dimensional or two-dimensional system. Generating Hamiltonians that meet locality conditions and support anyons is not a simple task. Here, we construct a local Hamiltonian on a fractal lattice which realizes physics similar to the fractional quantum Hall effect. The fractal lattice is obtained from a second generation Sierpinski carpet, which has 64 sites, and is characterized by a Hausdorff dimension of 1.89. We demonstrate that the proposed local Hamiltonian acting on the fractal geometry has Laughlin-type topological order by creating anyons and then studying their charge and braiding statistics. We also find that the energy gap between the ground state and the first excited state is approximately three times larger for the fractal lattice than for a standard square lattice with 64 sites, and the model on the fractal lattice is significantly more robust against disorder. We propose a scheme to implement fractal lattices and our proposed local Hamiltonian for ultracold atoms in optical lattices. The discussed scheme could also be utilized to study integer quantum Hall phases and the physics of other quantum systems on fractal lattices.
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