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Quantum critical (QC) phenomena can be accessed by studying quantum magnetsunder an applied magnetic field ($B$). The QC points are located at theendpoints of magnetization plateaus and separate gapped and gapless phases. Inone dimension, the low-energy excitations of the gapless phase form a Luttingerliquid (LL), and crossover lines bound insulating (plateau) and LL regimes, aswell as the QC regime. Alternating ferrimagnetic chains have a spontaneousmagnetization at $T=0$ and gapped excitations at zero field. Besides theplateau at the fully polarized (FP) magnetization; due to the gap, there isanother magnetization plateau at the ferrimagnetic (FRI) magnetization. Wedevelop spin-wave theories to study the thermal properties of these chainsunder an applied magnetic field: one from the FRI classical state, and otherfrom the FP state, comparing their results with quantum Monte Carlo data. Wedeepen the theory from the FP state, obtaining the crossover lines in the $T$vs. $B$ low-$T$ phase diagram. In particular, from local extreme points in thesusceptibility and magnetization curves, we identify the crossover between anLL regime formed by excitations from the FRI state to another built fromexcitations of the FP state. These two LL regimes are bounded by an asymmetricdome-like crossover line, as observed in the phase diagram of other quantummagnets under an applied magnetic field.

Magnetic field - temperature phase diagram of ferrimagnetic alternating chains: spin-wave theory from a fully polarized vacuum