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The ground state (GS) properties of the quasi-one-dimensional $AB_2$ Hubbardmodel are investigated taking the effects of charge and spin quantumfluctuations on equal footing. In the strong-coupling regime, we derive alow-energy Lagrangian suitable to describe the ferrimagnetic phase at halffilling and the phases in the hole-doped regime. At half filling, aperturbative spin-wave analysis allows us to find the GS energy, sublatticemagnetizations, and Lieb total spin per unit cell of the effective quantumHeisenberg model, in very good agreement with previous results. In thechallenging hole doping regime away from half filling, we derive thecorresponding $t\textrm{-}J$ Hamiltonian. Under the assumption that charge andspin quantum correlations are decoupled, the evolution of the second-orderspin-wave modes in the doped regime unveils the occurrence of spatiallymodulated spin structures and the emergence of phase separation in the presenceof resonating-valence-bond states. We also calculate the doping-dependent GSenergy and total spin per unit cell, in which case it is shown that the spiralferrimagnetic order collapses at a critical hole concentration. Notably, ouranalytical results in the doped regime are in very good agreement with densitymatrix renormalization group studies, where our assumption of spin-chargedecoupling is numerically supported by the formation of charge-density waves inanti-phase with the modulation of the magnetic structure.

Lieb and hole-doped ferrimagnetism, spiral, resonating valence-bond states, and phase separation in large-U $AB_{2}$ Hubbard chains