  Experiments and theory have shown that cell monolayers and epithelial tissues
exhibit solid-liquid and glass-liquid transitions. These transitions are
biologically relevant to our understanding of embryonic development, wound
healing, and cancer. Current models typically consider purely two-dimensional
monolayers with no overlaps between neighboring cells. In reality, overlaps are
important, and they may be precursors of cell extrusion -- a key biophysical
process to maintain homeostasis in epithelial tissues. Here, we use a
multi-phase field model to study the solid-liquid transition in a confluent
monolayer of deformable cells which can overlap. When cells overlap rather than
deform, we find that the melting transition changes from continuous to
discontinuous, and that there is an intermittent regime close to the
transition, where solid and liquid states alternate over time. By studying the
dynamics of $5$- and $7$-fold disclinations in the hexagonal lattice formed by
the cell centers, we observe that these correlate with spatial fluctuations in
the cellular overlap, and that cell extrusion tends to initiate near $5$-fold
disclinations.
