  By considering a quantum critical Lipkin-Meshkov-Glick model we analyze a new
type of Landau-Zener transitions where the population transfer is mediated by
interaction rather than from a direct diabatic coupling. For this scenario, at
a mean-field level the dynamics is greatly influenced by quantum interferences.
In particular, regardless of how slow the Landau-Zener sweep is, for certain
parameters almost no population transfer occurs, which is in stark contrast to
the regular Landau-Zener model. For moderate system sizes, this
counterintuitive mean-field behaviour is not duplicated in the quantum case.
This can be attributed quantum fluctuations and the fact that multi-level
Landau-Zener-St\"uckelberg interferences have a `dephasing' effect on the above
mentioned phenomenon. We also find a discrepancy between the quantum and
mean-field models in terms of how the transfer probabilities scale with the
sweep velocity.
