Phase diagram of the one dimensional t₁-t₂-J model: Ferromagnetism, triplet pairing, charge and pair density waves

We present a density matrix renormalization group (DMRG) study of an extended $t-J$ model with hopping to the first and second neighbors -- the one dimensional t₁-t₂-J model. The full phase diagram as a function of the density n and exchange strength J, for both positive and negative values of t₂, is obtained. For t₂=-0.5 we observe that, in the strongly interacting region, Nagaoka Ferromagnetism (FM) is accompanied by a triplet pair density wave (PDW) upon doping. As the spin exchange J increases, a charge density wave (CDW) phase emerges and then gives way to singlet superconductivity (SC). This phase behaves as a singlet PDW with vanishing spacial average of the order parameter and a spin gap. When t₂=0.5, the physics is basically reminiscent of the conventional $t-J$ model, undergoing a transition from a metallic to a SC phase as a function of J.