  The synthesis of the high temperature superconductor LaH$_{10}$ requires
pressures in excess of 100 GPa, wherein it adopts a face-centered cubic
structure. Upon decompression, this structure undergoes a distortion which
still supports superconductivity, but with a much lower critical temperature.
Previous calculations have shown that quantum and anharmonic effects are
necessary to stabilize the cubic structure, but have not resolved the low
pressure distortion. Using large scale path-integral molecular dynamics enabled
by a machine learned potential, we show that a rhombohedral distortion appears
at sufficiently low pressures, even with quantum and anharmonic effects. We
also highlight the importance of quantum zero point motion in stabilizing the
cubic structure.
