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H$_3$S is believed to the most possible high-temperature superconducting($T_{\text{c}}$) phase of hydrogen sulfide at $\sim$200 GPa. It's isotopesubstitution of hydrogen (H) by deuterium (D), however, shows an anomalous$T_{\text{c}}$ decrease of $\sim$100 K at 140 to 160 GPa, much larger than theBardeen-Cooper-Schrieffer theory prediction. Using ab initio path-integralmolecular dynamics (PIMD), we show that the nuclear quantum effects (NQEs)influence the structures of H$_3$S and D$_3$S differently at finitetemperatures and the interval when H$_3$S possesses the symmetric high$T_{\text{c}}$ structure while D$_3$S does not is in agreement with, thoughtheir absolute values are lower than experiments. This is consistent with anearlier theoretical study using the stochastic self-consistent harmonicapproximation method in descriptions of the nuclei at 0 K.The remainingdiscrepancy can be substantially improved when the electronic structures arecalculated using a hybrid function. Our study presents a simple picture tointerpret the isotope dependent of $T_{\text{c}}$ and emphasizes the quantumnature in the high-pressure hydrogen sulfide system.

The quantum nature of the superconducting hydrogen sulfide at finite temperatures