Interplay between next nearest neighbor hopping and long-range antiferromagnetism in t-J models with doping

We present a numerical study of the competing orders in the 1D t-J model with long-range RKKY-like unfrustrated spin interactions. By circumventing the constraints imposed by Mermin-Wagner's theorem, this Hamiltonian accommodates long-range N'eel order at half-filling. We determine the full phase diagram as a function of the exchange and particle density using the density matrix renormalization group (DMRG) method. We show that in this model, pairing is disfavored and the AFM insulator and metallic phases are separated by a broad regime with phase segregation, before spin-charge separation re-emerges at low densities. Upon doping, interactions induce a confining potential that binds holons and spinons into full fledged fermionic quasi-particles in a range of parameters and densities. In addition, we include next nearest neighbor(NNN) hopping terms t' into the model. By tuning the sign and magnitude of t', the superconducting phase is partially restored for some doping densities. We discuss how this simple toy-model can teach us about the phenomenology of its higher dimensional counterpart.