Anyon-Hubbard model in optical lattices (2017)
Abstract. Anyons can be considered to be a third class of particles with nontrivial exchange statistics that interpolate between fermions and bosons (they do not obey Bose-Einstein or Fermi-Dirac statistics). For two anyons under particle exchange, the wave function acquires a fractional phase e^(iθ) , giving rise to fractional statistics with 0 θ π. We study the properties of a collective of anyons loaded in an one-dimensional optical lattice at a zero temperature. We study a Hubbard model of anyons that takes into account the hopping of the particles along the lattice and the local two-body interaction between them. With the aim to proposing a realistic setup, Keilmann et al. introduces an exact mapping between anyons and bosons in one-dimension (the fractional version of the Jordan-Wigner transformation) [1]. We used this exact mapping and we studied the anyon-Hubbard Hamitonian in terms of bosonic operators. Thus, the model is a modified Bose-Hubbard model where the tunneling depends on the local density and the interchange angle (t → te^{iθn_j} ). The study was performed by means of the density matrix renormalization group (DMRG), which has allowed us to obtain the phase diagram for different values of the statistical angle θ and densities ρ = N/L. We observe the gapped (Mott insulator) and gapless (superfluid) phases that characterized the phase diagram and we calculated these phase diagram for higher densities. The phase transition was studied using the block von Neumman entropy, and we were able to observe the superfluid to Mott insulator transition. In particular, we use the estimator proposed by Lauchli and Kollath to determine the critical points, which has enabled us to present the evolution of the critical point with the global density and the statistical angle. On the other hand, when we change the local interaction in the system, anyons interacting via repulsive local three-body interactions, the quantum phase transition is driven by the statistics and the appearence of Mott insulator states, for the system with ρ = 1, depends on the anyonic angle. We showed the phases diagram and it was possible to study the influence of the many-body interactions on critical point position.