  We conducted ab initio molecular dynamics simulations of the delta phase of
the hydrous mineral AlOOH at ambient temperature and high pressure. Nuclear
quantum effects were included through a Langevin dynamics in a bath of quantum
harmonic oscillators.
  We confirm that under increasing pressure delta-AlOOH undergoes a phase
transition from a $P2_1nm$ structure with asymmetric and disordered O-H bonds
to a stiffer $Pnnm$ phase with symmetric hydrogen bonds, which should be stable
within the pressure and temperature ranges typical for the Earth's mantle.
  The transition is initially triggered by proton tunneling, which makes the
mean proton position to coincide with the midpoint of the O-O distance, at
pressures as low as 10 GPa. However, only at much larger pressures, around 30
GPa as previously found by other calculations, the Pnnm phase with symmetric
hydrogen bonds is stable from the classical point of view.
  The transition is also characterized through the analysis of the H-O
stretching modes, which soften considerably and fade out around 10 GPa in the
P2_1nm structure, when thermal and nuclear quantum effects are taken into
account in the simulations. At variance, the harmonic picture is not adequate
to describe the highly anharmonic effective potential that is seen by the
protons at the transition.
  Finally, we propose that the picture of a dynamical transition to the
high-symmetry and proton-centered Pnnm phase, which is brought about by the
onset of proton tunneling, could be confirmed by quasi-elastic neutron
scattering and vibrational spectroscopy under pressure.
