  The ground state (GS) properties of the quasi-one-dimensional $AB_2$ Hubbard
model are investigated taking the effects of charge and spin quantum
fluctuations on equal footing. In the strong-coupling regime, we derive a
low-energy Lagrangian suitable to describe the ferrimagnetic phase at half
filling and the phases in the hole-doped regime. At half filling, a
perturbative spin-wave analysis allows us to find the GS energy, sublattice
magnetizations, and Lieb total spin per unit cell of the effective quantum
Heisenberg model, in very good agreement with previous results. In the
challenging hole doping regime away from half filling, we derive the
corresponding $t\textrm{-}J$ Hamiltonian. Under the assumption that charge and
spin quantum correlations are decoupled, the evolution of the second-order
spin-wave modes in the doped regime unveils the occurrence of spatially
modulated spin structures and the emergence of phase separation in the presence
of resonating-valence-bond states. We also calculate the doping-dependent GS
energy and total spin per unit cell, in which case it is shown that the spiral
ferrimagnetic order collapses at a critical hole concentration. Notably, our
analytical results in the doped regime are in very good agreement with density
matrix renormalization group studies, where our assumption of spin-charge
decoupling is numerically supported by the formation of charge-density waves in
anti-phase with the modulation of the magnetic structure.
