  The quantum XY model shows a Berezinskii-Kosterlitz-Thouless (BKT) transition
between a phase with quasi long-range order and a disordered one, like the
corresponding classical model. The effect of the quantum fluctuations is to
weaken the transition and eventually to destroy it. However, in this respect
the mechanism of disappearance of the transition is not yet clear. In this work
we address the problem of the quenching of the BKT in the quantum XY model in
the region of small temperature and high quantum coupling. In particular, we
study the phase diagram of a 2D Josephson junction array, that is one of the
best experimental realizations of a quantum XY model. A genuine BKT transition
is found up to a threshold value $g^\star$ of the quantum coupling, beyond
which no phase coherence is established. Slightly below $g^\star$ the phase
stiffness shows a reentrant behavior at lowest temperatures, driven by strong
nonlinear quantum fluctuations. Such a reentrance is removed if the dissipation
effect of shunt resistors is included.
