Abstract
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. Here, we construct a local Hamiltonian on a fractal lattice whose ground state exhibits topological braiding properties. The fractal lattice is obtained from a second generation Sierpinski carpet with Hausdorff dimension 1.89. We use local potentials to trap and exchange anyons in the model, and the numerically obtained results for the exchange statistics of the anyons are close to the ideal statistics for quasiholes in a bosonic Laughlin state at half filling. For the considered system size, the energy gap is about three times larger for the fractal lattice than for a two-dimensional square lattice, and we find that the braiding results obtained on the fractal lattice are more robust against disorder. We propose a scheme to implement both fractal lattices and our proposed local Hamiltonian with ultracold atoms in optical lattices.
Highlights
For the considered system size, the energy gap is about three times larger for the fractal lattice than for a two-dimensional square lattice, and we find that the braiding results obtained on the fractal lattice are more robust against disorder
Fractional quantum Hall phases exist in systems defined on two-dimensional lattices, where the physical magnetic field is replaced by an artificial magnetic field, which can be much stronger [7–11]
The proposed Hamiltonian is found to lead to a larger energy gap between the topological ground state and the first excited state on the fractal lattice with 64 sites and 4 particles than that on a square lattice of the same size
Summary
Fractional quantum Hall phases exist in systems defined on two-dimensional lattices, where the physical magnetic field is replaced by an artificial magnetic field, which can be much stronger [7–11]. We show that a system with only nearest-neighbor complex hopping and hardcore interactions on a finitegeneration fractal lattice can give rise to anyonic braiding properties, and we propose a scheme to implement the Hamiltonian experimentally with ultracold atoms in optical lattices. We find that the considered system with relatively few sites and particles is already enough to get braiding statistics close to the ideal value for quasiholes in a bosonic Laughlin state at half filling and to produce interesting differences compared to a corresponding model on a two-dimensional lattice with the same number of sites and particles. We observe that the phase acquired by the wave function due to braiding is more robust with respect
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
