We investigate Luttinger liquid superlattices, periodic structures composed of two kinds of one-dimensional systems of interacting electrons. We calculate several properties of the low-energy sector: the effective charge and spin velocities, the compressibility, various correlation functions, the Landauer conductance, and the Drude weight. The low-energy properties are subsumed into effective parameters, much like homogeneous onedimensional systems. A generic result is the weighted average nature of these parameters, in proportion to the spatial extent of the underlying subunits, pointing to the possibility of ''engineered'' structures. As a specific realization, we consider a one-dimensional Hubbard superlattice, which consists of a periodic arrangement of two long Hubbard chains with different coupling constants and different hopping amplitudes. This system exhibits a rich phase diagram with several phases, both metallic and insulating. We have found that gapless insulating phases are present over a wide range of parameters.

Anyons are particles with fractional statistics that exhibit a nontrivial change in the wavefunction under an exchange of particles. Anyons can be considered to be a general category of particles that interpolate between fermions and bosons. We determined the position of the critical points of the one-dimensional anyon-Hubbard model, which was mapped to a modified Bose-Hubbard model where the tunneling depends on the local density and the interchange angle. We studied the latter model by using the density matrix renormalization group method and observed that gapped (Mott insulator) and gapless (superfluid) phases characterized the phase diagram, regardless of the value of the statistical angle. The phase diagram for higher densities was calculated and showed that the Mott lobes increase (decrease) as a function of the statistical angle (global density). The position of the critical point separating the gapped and gapless phases was found using quantum information tools, namely the block von Neumann entropy. We also studied the evolution of the critical point with the global density and the statistical angle and showed that the anyon-Hubbard model with a statistical angle $\theta =\pi/4$ is in the same universality class as the Bose-Hubbard model with two body interactions.Comment: 9 pages, 8 figures. Comments are welcom

Using the density matrix renormalization group method, we determine the phase diagram of the Bose-Hubbard model with local two-and three-body interactions, describing polar molecules in one-dimensional optical lattices. The difference in the block von Neumann entropy with different system sizes was used to establish the critical points. We found that the quantum critical point position increases with the three-body interaction. We show that the model studied is in the same universality class as the model with pure two-body interactions.

We calculate the correlation functions and the DC conductivity of Luttinger liquid superlattices, modeled by a repeated pattern of interacting and free Luttinger liquids. In a specific realization, where the interacting subsystem is a Hubbard chain, the system exhibits a rich phase diagram with four different phases: two metals and two compressible insulators. In general, we find that the effective low energy description amalgamates features of both types of liquids in proportion to their spatial extent, suggesting the interesting possibility of 'engineered' Luttinger liquids.