Displacive Quantum Critical Point in Superconducting Hydrides: The Case of H_{3}S

H_{3}S sulfur hydride has been widely investigated for its high superconducting critical temperature T_{c} of 203 K at about p_{c}=155  GPa. Despite being the precursor of superconducting hydrides, a detailed picture of its structural phase diagram in an extended temperature and pressure range is still missing. To determine it with inclusion of both thermal and quantum effects, we carry out path integral molecular dynamics combined to a MACE neural network potential trained on BLYP density functional theory configurations. The resulting H_{3}S phase diagram is characterized by the displacive transition between the centrosymmetric Im3[over ¯]m and polar R3m phases, which originates from a quantum critical point (QCP) located at p_{QCP}≈134  GPa. We show that the experimental T_{c} peak falls into a centrosymmetric region of large nuclear quantum fluctuations above the displacive QCP, as measured by local phonon Green's functions resolved in imaginary time, where fluctuating moments are at play. We study the critical behavior of the system in the proximity of the QCP by a finite-size scaling analysis, showing that it belongs to the 4D Ising universality class. We finally discuss its implications for the superconducting state.