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The quantum nature of the superconducting hydrogen sulfide at finite temperatures

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 isotope substitution of hydrogen (H) by deuterium (D), however, shows an anomalous $T_{\text{c}}$ decrease of $\sim$100 K at 140 to 160 GPa, much larger than the Bardeen-Cooper-Schrieffer theory prediction. Using ab initio path-integral molecular dynamics (PIMD), we show that the nuclear quantum effects (NQEs) influence the structures of H$_3$S and D$_3$S differently at finite temperatures and the interval when H$_3$S possesses the symmetric high $T_{\text{c}}$ structure while D$_3$S does not is in agreement with, though their absolute values are lower than experiments. This is consistent with an earlier theoretical study using the stochastic self-consistent harmonic approximation method in descriptions of the nuclei at 0 K.The remaining discrepancy can be substantially improved when the electronic structures are calculated using a hybrid function. Our study presents a simple picture to interpret the isotope dependent of $T_{\text{c}}$ and emphasizes the quantum nature in the high-pressure hydrogen sulfide system.