Quantum Crystal Structure in the 250 K Superconducting Lanthanum Hydride
Ion Errea Francesco Belli Lorenzo Monacelli Antonio Sanna Takashi Koretsune Terumasa Tadano Raffaello Bianco Matteo Calandra Ryotaro Arita Francesco Mauri José A. Flores-Livas
cond-mat.supr-con cond-mat.mtrl-sci



  




                
         
      
          
      
        
       
        
          
         
  




The discovery of superconductivity at 200 K in the hydrogen sulfide system at large pressures <cit.> 
was a clear demonstration that hydrogen-rich materials can be high-temperature superconductors. 
The recent synthesis of  with a superconducting critical temperature () 
of 250 K <cit.> places these materials at the verge of reaching 
the long-dreamed room-temperature superconductivity. 
Electrical and x-ray diffraction measurements determined a weakly pressure-dependent  for  
between 137 and 218 gigapascals in a structure with a face-centered cubic (fcc) arrangement of La atoms <cit.>. 
Here we show that quantum atomic fluctuations stabilize in all this pressure range 
a high-symmetry  crystal structure consistent with experiments, which has a colossal electron-phonon 
coupling of λ∼3.5. 
Even if ab initio classical calculations neglecting quantum atomic vibrations 
predict this structure to distort below 230 GPa yielding a complex energy 
landscape with many local minima, the inclusion of quantum effects simplifies the energy landscape 
evidencing the  as the true ground state. The agreement between the calculated and experimental  
values further supports this phase as responsible for the 250 K superconductivity. 
The relevance of quantum fluctuations in the energy landscape found here questions many of the crystal structure 
predictions made for hydrides within a classical approach that at the moment guide the experimental 
quest for room-temperature superconductivity <cit.>.
Furthermore, quantum effects reveal crucial to sustain solids with extraordinary electron-phonon coupling that may otherwise be unstable <cit.>.


Quantum Crystal Structure in the 250 K Superconducting Lanthanum Hydride
    José A. Flores-Livas
    December 30, 2023
========================================================================








 

The potential of metallic hydrogen as a high- superconductor <cit.> 
was identified few years after the development of the Bardeen-Cooper-Schrieffer (BCS) theory, 
which explained superconductivity through the electron-phonon coupling mechanism. The main argument was that  can be maximized for light compounds due to their high vibrational frequencies. In view of the large pressures needed to metallize hydrogen <cit.>, chemical precompression with heavier atoms <cit.> 
was suggested as a pathway to decrease the pressure needed to reach metallicity and, thus, superconductivity. 
These ideas have bloomed thanks to modern ab initio crystal structure prediction methods based on density-functional theory (DFT) <cit.>. Hundreds of hydrogen-rich compounds have been predicted to be thermodynamically stable at high pressures and, by calculating the electron-phonon interaction parameters, their 's have been estimated <cit.>. The success of this symbiosis between DFT crystal structure predictions and  calculations is exemplified by the discovery of superconductivity in  at 200 K <cit.>. 
The prospects for discovering warm hydrogen-based superconductors in the next years are thus high, 
in clear contrast with other high- superconducting families such as cuprates or pnictides <cit.>, where the lack of a clear 
understanding of the superconducting mechanism hinders an in silico guided approach.   

DFT predictions in the La-H system proposed  to be thermodynamically stable against decomposition above 150 GPa. 
A sodalite type-structure with space group  and ∼280 K was suggested above ∼220 GPa 
(see Fig. <ref>), and a distorted version of it below with space group C2/m and a rhombohedral La sublattice <cit.>. 
By laser heating a lanthanum sample in a hydrogen-rich atmosphere within a diamond anvil cell (DAC), a lanthanum superhydride was synthesized right after <cit.>. 
Based on the unit cell volume obtained by x-ray diffraction, the hydrogen to lanthanum ratio was estimated to be between 9 and 12. 
An fcc arrangement of the La atoms was determined above ∼160 GPa, and a rhombohedral lattice below with R-3m space group for the La sublattice. 
Due to the small x-ray cross section of hydrogen, experimentally it is not possible to resolve directly the H sublattice. 
Early this year, evidences of a superconducting transition at 260 K  and 188 GPa were reported 
in a lanthanum superhydride <cit.>. These findings were confirmed and put in solid grounds 
few months later  by an independent group that measured a  of 250 K from 137 to 218 GPa in a structure with 
fcc arrangement of the La atoms and suggested a  stoichiometry <cit.>. 



Even if it is tempting to assign the record superconductivity to the  phase predicted previously <cit.>, there is a clear problem: 
the  structure is predicted to be dynamically unstable in the whole pressure range where a 250 K  was observed. 
This implies that this phase is not a minimum of the Born-Oppenheimer energy surface. 
Consequently, no  has been estimated for this phase in the experimental pressure range. 
Considering that quantum proton fluctuations symmetrize hydrogen bonds in the high-pressure X 
phase of ice <cit.> and in  <cit.>, 
this contradiction may signal a problem of the classical treatment of the atomic vibrations in the calculations. 
We show here how quantum atomic fluctuations completely reshape 
the energy landscape making the  phase the true ground state and the responsible 
for the observed superconducting critical temperature. 

We start by calculating with DFT the lowest enthalpy structures of  as a 
function of pressure with state-of-the-art crystal structure prediction methods <cit.>. 
The contribution associated with atomic fluctuations is not included, 
so that the energy just corresponds to the Born-Oppenheimer energy V(𝐑), where 𝐑 represents the position of atoms treated classically as simple points. As shown in Figure <ref>, different distorted phases of  are thermodynamically more stable than the  phase. Above ∼250 GPa all phases merge to the  symmetric phase. 
These results are in agreement with previous calculations <cit.>, 
even if we identify other possible distorted structures 
with lower enthalpy such as 
the  R-3m, C2 and P1 (not shown) phases.  
These phases not only imply a distortion of the H atoms, also show a La sublattice without 
an fcc arrangement, and thus should be detectable by x-ray. 
The fact that many structures are predicted 
underlines that the classical V(𝐑) energy surface is 
of a multifunnel structure tractable to many different saddle and local minima, as sketched in Figure <ref>. 

This picture completely changes when including the energy of quantum atomic fluctuations, 
the zero-point energy (ZPE). We calculate the ZPE within the stochastic self-consistent harmonic approximation (SSCHA) <cit.>. 
The SSCHA is a variational method that calculates the E(ℛ) energy of the system including 
atomic quantum fluctuations as a function of the centroid positions ℛ, 
which determine the center of the ionic wave functions. The calculations are performed without 
approximating the V(𝐑) potential, keeping all its anharmonic terms. 
We perform a minimization of E(ℛ) and determine the centroid positions at its minimum.
By calculating the stress tensor from E(ℛ) <cit.>, 
we relax the lattice parameters seeking for structures with isotropic stress conditions considering quantum effects. 
We start the quantum relaxation for both R-3m and C2 phases with the lattice that yields a classical isotropic 
pressure of 150 GPa and vanishing classical forces, i.e., calculated from V(𝐑). 
All quantum relaxations quickly evolve into the  phase. 
This suggests that the quantum E(ℛ) energy landscape is much simpler than the 
classical V(𝐑) as sketched in Figure <ref>. 
And that the sodalite symmetric  phase is the ground state for  in all the pressure range of interest. 
Quantum effects are colossal: reshaping the energy landscape and stabilizing structures by more than 60 meV per . 

Our results further confirm that the structure of  responsible for the 250 K superconductivity is .  
This is completely consistent with the fcc arrangement of La atoms found experimentally <cit.>. 
However, Geballe et al. <cit.> observed a rhombohedral distortion below ∼160 GPa, 
with an R-3m space group for the La sublattice and a rhombohedral angle of approximately 61.3°(c/a∼2.38 in the hexagonal representation). Our calculations show that this distortion is compatible with slight 
anisotropic stress conditions in the DAC. Indeed, performing a SSCHA minimization for our R-3m 
phase but keeping the rhombohedral angle fixed at 62.3°(the value that yields an isotropic pressure of 150 GPa 
at the classical level) the quantum stress tensor shows a 6% anisotropy between the diagonal direction and the perpendicular plane. 
This suggests that anisotropic conditions inside the DAC can produce the R-3m phase, 
while other experimental stress conditions could favor other crystal phases. 

The  phonon spectra calculated in the harmonic approximation from the Hessian of V(𝐑) 
show clear phonon instabilities in a broad region of the Brillouin zone (see Figure <ref>). 
These instabilities appear below ∼230 GPa. This is consistent with the fact that below this pressure 
many possible atomic distortions lower the enthalpy of this phase. On the contrary, as shown in Figure <ref>, 
when calculating the phonons from the Hessian of E(ℛ) <cit.>, 
which effectively captures the full anharmonicity of V(𝐑), no instability is observed. This confirms again 
that the  phase is a minimum in the quantum-energy landscape in the whole pressure range where 
a 250 K  was observed. 
While the  phase of  remains a minimum of E(ℛ) as low as ∼ 129 GPa, 
the case of  shows instabilities at 126 GPa, 
implying that at this pressure the  phase of LaD_10 distorts 
to a new phase (as suggested by Drozdov et al. <cit.>). 
Below this pressure we also predict that  composition is not 
longer thermodynamically stable 
and low-hydrogen compositions are likely to  occur.   



Flagrantly, the breakdown of the classical harmonic approximation for phonons makes impossible 
the estimation of  below ∼250 GPa in the  phase and questions all previous calculations <cit.>. 
Indeed, the anharmonic phonon renormalization remains huge also at 264 GPa (see Figure <ref>). 
On the contrary, with anharmonic phonons derived from the Hessian of E(ℛ) 
we can readily calculate the electron-phonon interaction 
and the superconducting  in the experimental range of pressure (120–210 GPa). 
The superconducting critical temperature is estimated fully ab initio  
–without any empirical parameter– by solving Migdal-Éliashberg (ME) equations and applying SuperConducting DFT (SCDFT). 
As shown in Figure <ref>, the numerical solutions of ME equations with anisotropic energy gap are almost 
on top of the experimental values. SCDFT values systematically show a slightly lower . 
Our reported values of  evidence the phonon-driven mechanism of superconductivity and confirm 
 in its  structure as responsible for the highest- up to date reported. 
Our calculations for  in the  phase are also in agreement with the experimental point reported. 
Despite the large anharmonic effects at play, the isotope coefficient α=- [ lnT_c(LaD_10) - lnT_c(LaH_10)] / ln 2 is close to 0.5 (0.43 around 160 GPa), the expected value in BCS theory, 
and it is in agreement with the experimentally reported α=0.46. 

We finally check  for the subtle rhombohedral distortion 
that could be induced by anisotropic stress conditions of pressure.  
Fixing the rhombohedral angle at 62.3° the obtained  
for the R-3m phase at 160 GPa is a 9% lower than for the . 
Thus, the observed weak pressure dependence of  is consistent with the absence of a rhombohedral distortion, 
as suggested by the x-ray data <cit.>. 
However, as argued above, undesired anisotropic stress conditions in the DAC can induce phase transitions. 
We thus believe that other experimental  measurements with lower values but around 200 K 
correspond to distorted structures induced by anisotropic conditions of pressure. 
In fact, we can safely rule out that compositions such as LaH_11, 
proposed to yield a high critical temperature <cit.>, 
is responsible for any sizable  (see Extended Data).   

In summary, this work demonstrates how quantum effects are of capital importance
in determining the ground state structures of superconducting hydrides,
challenging all current predictions and evidencing flaws in standard theoretical methods.
It also illustrates that quantum fluctuations are indispensable to sustain crystals 
with huge λ's (λ reaches a record value of 3.6 at 129 GPa for ) 
that be otherwise destabilized by the colossal electron-phonon interaction 
to distorted (low symmetry) structures reducing the electronic density 
of states at the Fermi level (see Extended Data) <cit.>.  
This is relevant since large λ is required to guarantee high- <cit.>, not simply light atomic masses. 















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Acknowledgements. 
This research was supported by the European Research Council (ERC) 
under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 802533); 
the Spanish Ministry of Economy and Competitiveness (FIS2016-76617-P);
NCCR MARVEL funded by the Swiss National Science Foundation; and 
grant-in-Aid for Scientific Research (No.16H06345, 18K03442 and No. 19H05825) 
from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
Computational resources were provided by 
the Barcelona Superconducting Center (project FI-2019-1-0031) and 
the Swiss National Supercomputing Center (CSCS).

Author contributions. 
The project was conceived by I.E. and J.A.F.-L. 
The SSCHA was developed by 
I.E., L.M., R.B., M.C. and F.M. In particular, R.B. developed the method to compute
the quantum energy Hessian and the anharmonic phonon dispersions, and L.M. developed the method to perform a quantum relaxation of the lattice parameters. 
I.E. and F.B. performed the SSCHA calculations. A.S. T.K., T.T., R.A. and J.A.F.-L. 
conducted studies on structure prediction and superconductivity. 
All authors contributed to the editing of the manuscript.

Competing interests. The authors declare no competing interests. 








                     Methods



Calculation details. 
First-principles calculations were performed within DFT and the generalized gradient approximation (GGA) as parametrized by Perdew, Burke, and Ernzerhof (PBE) <cit.>. Harmonic phonon frequencies were calculated within density functional perturbation theory (DFPT) <cit.> making use of the Quantum ESPRESSO code <cit.>. The SSCHA <cit.> minimization requires the calculation of energies, forces and stress tensors in supercells. These were calculated as well within DFT at the PBE level with Quantum ESPRESSO. For the final SSCHA populations, 1000 configurations were used to reduce the stochastic noise. In all these calculations we used ultrasoft pseudopotentials including 11 electrons for the La atoms, a plane-wave cut-off energy of 50 Ry for the kinetic energy and 500 Ry for the charge density.

In the harmonic phonon calculations for the  and R-3m phases, we used the primitive and rhombohedral lattices, respectively, with one  formula unit in the unit cell. A 20×20×20 Monkhorst-Pack shifted electron-momentum grid was used for these calculations with a Methfessel-Paxton smearing of 0.02 Ry. The DFT calculations performed for the SSCHA on supercells were performed on a coarser electron-momentum grid, which would correspond to a 12×12×12 grid in the unit cell. We explicitly verified that this coarser mesh yields a fully converged SSCHA gradient with respect to the electron-momentum grid, thus, not affecting the SSCHA minimization. The DFT supercell calculations for the SSCHA minimization on the C2 phase were performed keeping the same 𝐤-point density.

All phonon frequencies for 𝐪-points not commensurate with the supercell used in the SSCHA minimization were obtained by directly Fourier interpolating the real space force constants obtained in this supercell, which are calculated form the Hessian of E(ℛ). For the  phase the SSCHA calculation was performed both on a 2×2×2 and 3×3×3 supercell containing, respectively, 88 and 297 atoms. The phonon spectra shown in Figure <ref> for the  phase are obtained by Fourier interpolating directly the SSCHA energy Hessian force constants obtained in a 3×3×3 supercell. In Extended Data Figure <ref> we show that the phonon spectrum obtained interpolating directly the force constants in a 2×2×2 supercell yields similar results, indicating that the energy Hessian force constants are short-range and can be Fourier interpolated. Indeed, the  calculated with the 2×2×2 and 3×3×3 force constants for interpolating phonons only differs in approximately 3 K. Upon this, the SSCHA quantum structural relaxations in the R-3m and C2 phases were performed in supercells with 88 atoms. 

As shown in Ref. <cit.>, the Hessian of E(ℛ) is

    ∂^2 E(ℛ)/∂ℛ∂ℛ = Φ + (3)ΦΛ[ 1 - (4)ΦΛ]^-1(3)Φ.
 
Bold notation represents matrices and tensors in compact notation. In Eq. (<ref>),  Φ are the variational force constants of the SSCHA minimization, (n)Φ the quantum statistical averages taken with the SSCHA density matrix of the n-th order derivatives of V(𝐑), and Λ a tensor that depends on the temperature and  Φ. 1 is the identity matrix.  As we show in Extended Figure <ref>, setting (4)Φ=0 has a negligible effect on the phonons obtained from the Hessian defined in Eq. (<ref>). Therefore, (4)Φ is neglected throughout, and all superconductivity calculations in the  and R-3m phases are performed making use of the phonon frequencies and polarization vectors obtained from the Hessian of E(ℛ) with (4)Φ=0. We also estimated  with the phonon frequencies and polarization vectors obtained instead from Φ, resulting in a critical temperature 12 K lower within Allen-Dynes modified McMillan equation. This difference is small and within the uncertainty of the  calculation between SCDFT and anisotropic Migdal-Éliashberg calculations (see Figure <ref> and below).

The Éliashberg spectral function, which we used for the  calculations, is defined as 

    α^2 F(ω)     =    1/N_E_F∑_n𝐤, m𝐪,ν |g^ν_n𝐤,m𝐤+𝐪|^2 δ(ϵ_n𝐤-E_F)  
       ×   δ(ϵ_m𝐤+𝐪-E_F) δ(ω-ω_𝐪ν),

where N_E_F is the electronic density of states (DOS) at the Fermi energy (E_F), n and m are band indices, 𝐤 is a crystal momentum, ϵ_n𝐤 is a band energy, ω_𝐪ν is the phonon frequency of mode ν at wavevector 𝐪, and g^ν_n𝐤,m𝐤+𝐪 is the electron-phonon matrix element between a state n𝐤 and m𝐤+𝐪. We calculated α^2 F(ω) combining the SSCHA  phonon frequencies and polarization vectors obtained from the Hessian of E(ℛ) with the electron-phonon matrix elements calculated with DFPT. For the  and R-3m phases, the electron-phonon matrix elements were calculated in a 6×6×6 𝐪 point grid and a 40×40×40 𝐤 point grid. These were combined with the SSCHA phonons and polarization vectors obtained by Fourier interpolation to the 6×6×6 𝐪 point grid from the real space force constants coming from the Hessian of E(ℛ) in a 3×3×3 supercell for the  and in a 2×2×2 supercell for the R-3m. The Dirac deltas on the band energies are estimated by substituting them with a Gaussian of 0.004 Ry width. The calculated  α^2 F(ω) functions for the  phase are shown in Extended Figures <ref> and <ref>, while in Extended Figure <ref> we show the results for the R-3m phase.

Crystal phase diagram exploration.
To sample the enthalpy landscape of LaH_10 we employed the minima hopping method 
(MHM) <cit.>, which has been successfully used for 
global geometry optimization in a large variety of applications 
including superconducting materials such as H_3S, PH_3, and disilane  
at high pressure <cit.>. 
This composition was thoroughly explored with 1, 2, 3 and 4 formula units simulation cells. 
Variable composition simulations were also performed for other La-H compositions. 
Energy, atomic forces and stresses were evaluated at the DFT level with the GGA-PBE parametrization 
to the exchange-correlation functional. A plane wave basis-set with a high cutoff energy of 
900 eV was used to expand the wave-function together with the projector augmented wave (PAW) 
method as implemented in the Vienna Ab Initio Simulation Package vasp <cit.>. 
Geometry relaxations were performed with tight convergence criteria 
such that the forces on the atoms were less than 2 meV/Å and the stresses were less than 0.1 eV/Å^3. 
Extended Data Figure <ref> shows our calculated convex hull of enthalpy formation 
without considering the zero-point energy at 100, 150 and 200 GPa. 
Interestingly, there are many stable compositions in the convex hull. 
 becomes enthalpically stable at ∼125 GPa and remains in the convex well above 300 GPa. 
Below 150 GPa, R-3m and C2 phases () show unstable harmonic phonon at Γ, 
becoming saddle points of V(𝐑). 
However, harmonically one can find P1 stable structures (decreasing symmetry) by 
following the instability pattern (softening direction, i.e. along eigenvector polarization). 
P1 structures are degenerate in enthalpy within less 3 meV/ with respect C2.
Hence, we used the C2 as a representative of highly distorted structures for our study. 
 

Superconductivity calculations in the  phase.
Superconductivity calculations were performed within two different approaches that represent the state-of-the-art of ab initio superconductivity: Density functional theory for Superconductors (SCDFT) and the Éliashberg equations with full Coulomb interaction. 

SCDFT is an extension to DFT for a superconducting ground state <cit.>.
By assuming that the n k anisotropy in the electron-phonon coupling is negligible (see Ref. Lueders_SCDFT_PRB2005 for further details), the critical temperature is computed by solving an (isotropic) equation for the Kohn-Sham gap:

    Δ_s(ϵ)=𝒵(ϵ)Δ_s(ϵ)
    -∫ dϵ' 
    𝒦(ϵ,ϵ')tanh[β E(ϵ')/2]/2E(ϵ')Δ_s(ϵ'),

where ϵ is the electron energy and β the inverse temperature. 
The kernels 𝒦 and 𝒵 come from the exchange correlation functional of the theory <cit.> and depend on the properties of the pairing interactions: 
electron-phonon coupling and screened electron-electron repulsion. 
Eq. <ref> allows us to calculate  completely ab initio, without introducing an
empirical μ^* parameter (Coulomb pseudopotential). 
Dynamic effects on the Coulomb interaction (plasmon) were also tested and did not show any significant effect. 
In its isotropic form, the screened Coulomb interaction in SCDFT is accounted for by a function μ(ϵ,ϵ'), 
which is given by the average <cit.> 
RPA Coulomb matrix element on the iso-energy surfaces ϵ and ϵ' times the DOS at ϵ' (N(ϵ')):

    μ(ϵ,ϵ')=∑_n,m∬ d^3(k k') V^RPA_n k,m k'δ(ϵ-ϵ_n k)/N(ϵ_n k)δ(ϵ'-ϵ_m k').

The full energy dependence of the DOS is accounted in the calculations, while the electron-phonon coupling is described by the α^2 F(ω) of Eq. (<ref>).

The second approach we use to simulate the superconducting state is the anisotropic Éliashberg approach <cit.>. 
Here we include, together with the energy dependence of the electron DOS, the anisotropy of the electron-phonon coupling. 
The Green's function form of the Éliashberg equation <cit.> we solve is given as

    Σ_n 𝐤(i ω_i)    = 
        -1/N β∑_μ,𝐪,m V_mn^ ph(𝐪, iω_μ) G_m 𝐤+𝐪(iω_μ + iω_i),
    Δ_n 𝐤(i ω_i)    = 
        -1/N β∑_μ,𝐪,m{ V_mn^ ph(𝐪, iω_μ) + V_mn^ C(𝐪, iω_μ) }
       × | G_m 𝐤+𝐪(iω_μ + iω_i) |^2 Δ_m 𝐤+𝐪(iω_μ+iω_i).

Here, Σ_n 𝐤(iω_i) and Δ_n 𝐤(i ω_i) are the normal and anomalous self energy, and V_mn^ ph(𝐪, iω_μ) and V_mn^ C(𝐪, iω_μ) are the 𝐤-averaged phonon-mediated interaction and Coulomb interaction, respectively.
The explicit form of V_mn^ ph(𝐪, iω_μ) is given as 

    V_mn^ ph(𝐪, iω_μ) = ∑_ν | g_nm^ν(𝐪) |^2 D_ν(𝐪,iω_μ),

where |g_nm^ν(𝐪)|^2 is a 𝐤-averaged electron-phonon matrix element,

    |g_nm^ν(𝐪)|^2 = ∑_𝐤 |g^ν_n𝐤,m𝐤+𝐪|^2 δ(ϵ_n𝐤-E_F) δ(ϵ_m𝐤+𝐪-E_F)/∑_𝐤δ(ϵ_n𝐤-E_F) δ(ϵ_m𝐤+𝐪-E_F),

and D_ν(𝐪, iω_μ) is a free-phonon Green's function, D_ν(𝐪, iω_μ) = -2 ω_𝐪ν/(ω_μ^2 + ω^2_𝐪ν).
The electron-phonon matrix elements are calculated through a DFPT calculation with 6×6×6 𝐪 point grid, and are combined with the phonon frequencies and polarization vectors obtained by directly Fourier interpolating to this grid the force constants coming from the E(ℛ) Hessian in the 3×3×3 supercell. 
For the Coulomb interaction, V_mn^ C(𝐪, iω_μ) is approximated by 𝐤-averaged static Coulomb interaction within the random phase approximation, 1/N_k∑_𝐤 V^ RPA_m 𝐤, n 𝐤+𝐪(iω_μ = 0).
Using Eq. (<ref>), the Dyson equation was solved self-consistently and then Eq. (<ref>) was solved to estimate T_c with 36× 36 × 36 𝐤 point grid and 512 Matsubara frequencies.

In Extended Data Table <ref> we summarize all calculated 's within anisotropic ME and isotropic SCDFT. We also include the values obtained with McMillan equation and Allen-Dynes modified McMillan equation (μ^*=0.1). The calculated electron-phonon coupling constant, λ=2∫_0^∞dωα^2F(ω)/ω, 
and the logarithmic frequency average, 
=exp( 2/λ∫_0^∞dωα^2F(ω)/ωlogω), 
are also included in the table.

Quantum structural relaxations in the R-3m and C2 phases.
In Extended Figure <ref> we show the evolution of the pressure calculated along the different Cartesian directions for the R-3m throughout the SSCHA minimization but keeping the rhombohedral angle fixed at 62.3°. Thus, the centroid positions ℛ are optimized only considering the internal degrees of freedom of the R-3m phase. Even if at the classical level the stress is isotropic (within a 0.5%), after the SSCHA quantum relaxation an anisotropic stress of a 6%  is created between the z and x-y directions. The phonons obtained at the end of the minimization are shown in Extended Figure <ref>. Secondly, in Extended Figure <ref>, we show that starting from the result of this minimization but now relaxing also the lattice, the R-3m phase evolves into the  phase. It is clear how the pressure calculated with quantum effects  becomes isotropic when the rhombohedral angle becomes 60°, the angle corresponding to a fcc lattice in a rhombohedral description. Also it is evident that the Wyckoff positions of the R-3m phase evolve clearly into the  Wyckoff positions, which are summarized in Extended Data Table <ref>. 

In Extended Data Figure <ref> we show the evolution of the diagonal components of the pressure along the three different Cartesian directions for the monoclininc C2 when the lattice structure is relaxed with the SSCHA. The starting point is obtained by first performing a SSCHA relaxation of only internal atomic coordinates keeping the lattice parameters that yield an isotropic stress of 150 GPa. It is clear that quantum effects create an anisotropic stress if the lattice parameters are not modified. When the quantum relaxation of the lattice is performed, the lattice parameters are modified and an isotropic stress is recovered.

Extended Data Figure <ref> shows the structures of the R-3m and C2 phases obtained classically and after the quantum SSCHA relaxation. After the quantum relaxation, the symmetry of both structures is recognized as  with a tolerance of 0.001 Å for lattice vectors and 0.005 Å for ionic positions, consistent with the stochastic accuracy of the SSCHA. 
In the same Figure <ref>, 
the electronic density of states (DOS) as a function of pressure is plotted. 
Highly symmetric motif (Fm-3m) maximizes N_E_F, while in distorted structures 
(R-3m and C2) the occupation at the Fermi level 
is reduced by more than 20 %. This underlines that the classical distortions would lower N_E_F, reducing λ, as expected in a system destabilized by the electron-phonon interaction.

Transition temperatures from other La-H compositions.
Different compositions on the La-H phase diagram have been reported as thermodynamically stable. 
Presumably, the stabilization of these compositions and the measurement of different 's 
(see Drozdov et al. <cit.>) demonstrate that 
other stoichiometries are responsible for these measured 's.  
Notably, these 's appear substantially lower, for instance the values decrease from 250 K, to 215 K, 110 K 
and to 70 K. Experimentally there is not a clear correlation 
between sample preparation,  and pressure. 
In sample preparation of Drozdov et al. pressures can vary from 100 to 200 GPa 
(gradient inside the diamond anvil cell) and it was proposed that other 
stoichiometries (low-hydrogen content) are responsible for systematically lower 's. 

Conversely, in a later publication the same authors suggested other hydrogen rich system  
that is enthalpically competitive (LaH_11)  and possibly responsible for other high- phases. 
In order to explore this possibility, we did consider structure prediction runs 
with this stoichiometry and found crystalline structures that were previously 
reported in Ref. <cit.>. 
Extended Data Figure <ref> shows the structural motif and the 
corresponding phonons and α^2 F(ω) spectral function. 
We can rule out the possibility that high-, as measured in different samples,  
arises from LaH_11 in its P4/nmm (129) structure 
(lowest enthalpy structure for this composition at relevant experimental pressures).  
As seen in Extended Data Figure <ref>, this phase has a strong molecular crystal character, 
composed of H_2 units weakly interacting with La-lattice. 
This phase is indeed a poor metal with low occupation of electrons at the Fermi level 
due to its molecular character and cannot explain 70 K, or higher values of . 
Our estimated  with Allen-Dynes formula, harmonic phonons and using a μ^*=0.1 is 7 K at 100 GPa. 



10



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