Figure 11
The upper panel (a) shows the S-H bond length in the pressure range between 120 and 180 GPa where without the hydrogen ZPM, the R3m structure was expected to be stable with the S-H bond splitting into a long (blue squares) and short (violet squares) sulfur-hydrogen bond. The amplitude of the calculated ZPM of the S-H bond is indicated by the red error bars3,4. The S-H amplitude of the zero point motion, ZPM, is larger than the S-H splitting in the range 130–180 GPa therefore in this pressure range the ZPM stabilizes the 
structure4 in agreement with experiments2. The panel (b) shows the Fermi energy EF4 in the small hole Fermi pocket at Γ as a function pressure. The top of this band crosses the chemical potential at 130 GPa giving the Lifshitz transition of type 1 for the appearing of a new Fermi surface. The position of the vHs E2 remains below the chemical potential but it remains in the energy range of pairing interaction. Panel (c) shows the variation of the experimental isotope coefficient calculated from data in ref. 2 which shows a divergence from 0.3 at 180 GPa to 1.5 at 135 GPa, which is not predicted by the BCS theory. The critical temperature decreases toward zero with a decrease of about 60 K in a range of 30 GPa, beteen 160 GPa and 130 GPa which is not predicted by the BCS theory. Both phenomena are predicted by the general theory of multigap superconductivity near a Lifshitz transition3,4.
