# High-Temperature Superconductivity in the Lanthanide Hydrides at Extreme Pressures
## 2 Discussion
The changes highlighted above stem directly from a spectral weight transfer induced by many-body corrections (see Figure 3a,b). In DFT, the La system is described by a two-band system in absence of a long-range magnetic order. We note that DFT is a single Slater determinant approach, and hence cannot capture the role of paramagnetism, with an associated magnetic multiplet (fluctuating magnetic moment). Such effects typically induce a splitting of spectral features into satellites, as observed in Figure 3c,d, with a resultinglarge increase of the \(f\) character at the Fermi level. As sharp La features occur near the Fermi level, we emphasise that a higher level of theory is required to correctly capture the superconducting properties. For instance, in our calculations, the one-shot (DFT+DMFT) and full-charge self-consistent approach (DFT+DMFT+CSC) induces a small shift of the sharp La feature at the Fermi level, which in turn mitigates the \(f\) character increase at the Fermi level. Figure 1: Many-body corrections to the superconducting temperature. We report **(a)** phonon density of states for LaH\({}_{16}\); **(b)** phonon density of states for CeH\({}_{16}\); **(c,d)** Eliashberg function \(\alpha^{2}F(\omega)\) for LaH\({}_{16}\) and CeH\({}_{16}\); the spectral weight at the Fermi level is obtained at different levels of approximation: (i) DFT PBE (black line) and (ii) with the full-charge self-consistent formalism (DFT+DMFT+CSC, red line); (e,f) The superconducting temperature \(T_{c}\) obtained by the Allen and Dynes formalism for LaH\({}_{16}\) and CeH\({}_{16}\) at 250 GPa. We obtain a theoretical estimate for LaH\({}_{16}\) as \(T_{c}=166.2\) K by DFT and \(T_{c}=192.4\) K by DMFT, as well as a theoretical estimate for CeH\({}_{16}\) as \(T_{c}=69.7\) K by DFT and \(T_{c}=150.6\) K by DMFT. The full-charge self-consistent DMFT provides a large increase of the superconducting temperature, and the physical value of the Hund’s coupling for La and Ce is \(U=6\) eV and \(J=0.6\) eV. All calculations were performed in the P6/mmm phase of LaH\({}_{16}\) and CeH\({}_{16}\) at 250 GPa. The role of \(f\) orbitals appears to be very important for the superconducting properties in rare-earth hydrates [21]. That is why in the present paper, in addition to LaH\({}_{16}\) with the formally empty La \(f\) shell, we studied also cerium hydrate (CeH\({}_{16}\)) with one \(f\) electron in the atomic Ce configuration. We checked that both systems remain stable at pressures up to at least 250 GPa. We report in Figure 4 the DFT+DMFT+CSC framework applied to CeH\({}_{16}\) in the P6/mmm phase at 250 GPa. We attributed the decrease in \(T_{c}\) to a higher \(f\) occupation, which shifts the chemical potential away from the \(f\) spectral features present near the Fermi level (see Figure 4c,d). Furthermore, Figure 4b shows that there is no obvious sudden change at the Fermi level, but there is an obvious mutation at the Fermi level in Figure 4d, which is around 7(1/eV); the correction of DMFT greatly affects the Tc of Ce. Our findings point to a possible path for increasing \(T_{c}\) in lanthanide hydrides: an increase in the \(f\) character at the Fermi level in DFT+DMFT, which is associated with a lower degree of La-H and Ce-H covalency and a lower degree of hybridisation, which, in turn, is a marker for a higher superconducting temperature in these systems. Figure 2: Structural relaxation of clathrate lanthanides with many-body corrections. We report the structural relaxation of the LaH\({}_{16}\) (**a**,**c**) and prototype CeH\({}_{16}\) (**b**,**d**) compounds. All calculations are performed at 250 GPa. The volume density is obtained by the equation of state in DFT+DMFT+CSC that provides very similar results to PBE (not shown). Internal coordinates are relaxed with DFT+DMFT+CSC, building upon the recent implementation of DFT forces for ultra-soft pseudo-potentials. We report the forces and total energies obtained during the structural optimisation, respectively, in (**a**,**b**) for LaH\({}_{16}\) (CeH\({}_{16}\)).