We discuss the rich phase diagram of doped AB_2 t -J chains by using data from density matrix renormalization group and exact diagonalization techniques. The J vs δ (hole doping) phase diagram exhibits regions of itinerant ferrimagnetism, incommensurate, resonating valence bond and Nagaoka states, phase separation, and Luttinger liquid (LL) physics. Several features are highlighted, such as the modulated ferrimagnetic structure, the occurrence of Nagaoka spin polarons in the underdoped regime and small values of J = 4t~2/U, where t is the first-neighbor hopping amplitude and U is the on-site repulsive Coulomb interaction, incommensurate structures with nonzero magnetization, and strong-coupling LL physics in the high-doped regime. We also verify that relevant findings are in agreement with the corresponding findings in square and n-leg ladder lattices. In particular, we mention the instability of Nagaoka ferromagnetism against J and δ.
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