  We study the formation of particle-imbalanced quantum droplets in a
one-dimensional optical lattice containing a binary bosonic mixture at zero
temperature. To understand the effects of the imbalance from both the few- and
many-body perspectives, we employ density matrix renormalization group (DMRG)
simulations and perform the extrapolation to the thermodynamic limit. In
contrast to the particle-balanced case, not all bosons are paired, resulting in
an interplay between bound states and individual atoms that leads to intriguing
phenomena. Quantum droplets manage to sustain a small particle imbalance,
resulting in an effective magnetization. However, as the imbalance is further
increased, a critical point is eventually crossed, and the droplets start to
expel the excess particles while the magnetization in the bulk remains
constant. Remarkably, the unpaired particles on top of the quantum droplet
effectively form a super Tonks-Girardeau (hard-rod) gas. The expulsion point
coincides with the critical density at which the size of the super
Tonks-Girardeau gas matches the size of the droplet.
