Abstract
Hydrogen compounds are peculiar as the quantum nature of the proton can crucially affect their structural and physical properties. A remarkable example are the high-pressure phases 1,2 of H 2 O, where quantum proton fluctuations favor the symmetrization of the H bond and lower by 30 GPa the boundary between the asymmetric structure and the symmetric one 3 . Here we show that an analogous quantum symmetrization occurs in the recently discovered 4 sulfur hydride superconductor with the record superconducting critical temperature T c = 203 K at 155 GPa. In this system, according to classical theory 5-9 , superconductivity occurs via formation of a structure of stoichiometry H 3 S with S atoms arranged on a body-centered-cubic (bcc) lattice. For P 175 GPa, the H atoms are predicted to sit midway between two S atoms, in a structure with Im3m symmetry. At lower pressures the H atoms move to an off-center position forming a short H−S covalent bond and a longer H· · · S hydrogen bond, in a structure with R3m symmetry 5-9 . X-ray diffraction experiments confirmed the H 3 S stoichiometry and the S lattice sites, but were unable to discriminate between the two phases 10 . Our present ab initio density-functional theory (DFT) calculations show that the quantum nuclear motion lowers the symmetrization pressure by 72 GPa. Consequently, we predict that the Im3m phase is stable over the whole pressure range within which a high T c was measured. The observed pressure-dependence of T c is closely reproduced in our calculations for the Im3m phase, but not for the R3m phase. Thus, the quantum nature of the proton completely rules the superconducting phase diagram of H 3 S.
The discovery of high-temperature superconductivity
Crystal structure in the conventional bcc cell of the R3m and Im3m phases. In the R3m the H−S covalent bond of length d1 is marked with a solid line and the longer H· · · S hydrogen bond of d2 length with a dotted line. In the Im3m phase d1 = d2.
in compressed hydrogen sulfide 4 has generated intense interest over the last year, and has led to a number of theoretical studies aimed at understanding the phase diagram of the H-S system as well as the origin of the astonishingly high T c observed 5-9,12-15 . The overall consensus is that H 2 S, the only stable compound formed by hydrogen and sulfur at ambient conditions, is metastable at high pressures and its decomposition gives rise to several H-S compounds. High-T c superconductivity is believed to occur in a structure with the H 3 S stoichiometry, and is considered to be conventional in nature, i.e., mediated by the electron-phonon interaction 4, 5, 7, 9, [12] [13] [14] [15] . Alternatives to conventional superconductivity have also been discussed 16 . According to structural predictions 5-9 , H 3 S adopts a rhombohedral R3m form between approximately 112 and 175 GPa, and a cubic Im3m at higher pressures. As shown in Fig. 1 , the R3m phase is characterized by covalently bonded SH 3 units with a covalent arXiv:1512.02933v1 [cond-mat.supr-con] 9 Dec 2015

Hydrogen compounds are peculiar as the quantum nature of the proton can crucially affect their structural and physical properties. A remarkable example are the high-pressure phases 1, 2 of H 2 O, where quantum proton fluctuations favor the symmetrization of the H bond and lower by 30 GPa the boundary between the asymmetric structure and the symmetric one 3 . Here we show that an analogous quantum symmetrization occurs in the recently discovered 4 sulfur hydride superconductor with the record superconducting critical temperature T c = 203 K at 155 GPa. In this system, according to classical theory [5] [6] [7] [8] [9] , superconductivity occurs via formation of a structure of stoichiometry H 3 S with S atoms arranged on a body-centered-cubic (bcc) lattice. For P 175 GPa, the H atoms are predicted to sit midway between two S atoms, in a structure with Im3m symmetry. At lower pressures the H atoms move to an off-center position forming a short H−S covalent bond and a longer H· · · S hydrogen bond, in a structure with R3m symmetry [5] [6] [7] [8] [9] . X-ray diffraction experiments confirmed the H 3 S stoichiometry and the S lattice sites, but were unable to discriminate between the two phases 10 . Our present ab initio density-functional theory (DFT) calculations show that the quantum nuclear motion lowers the symmetrization pressure by 72 GPa. Consequently, we predict that the Im3m phase is stable over the whole pressure range within which a high T c was measured. The observed pressure-dependence of T c is closely reproduced in our calculations for the Im3m phase, but not for the R3m phase. Thus, the quantum nature of the proton completely rules the superconducting phase diagram of H 3 S.
The discovery of high-temperature superconductivity
Crystal structures of the competing phases.
Crystal structure in the conventional bcc cell of the R3m and Im3m phases. In the R3m the H−S covalent bond of length d1 is marked with a solid line and the longer H· · · S hydrogen bond of d2 length with a dotted line. In the Im3m phase d1 = d2.
in compressed hydrogen sulfide 4 has generated intense interest over the last year, and has led to a number of theoretical studies aimed at understanding the phase diagram of the H-S system as well as the origin of the astonishingly high T c observed [5] [6] [7] [8] [9] [12] [13] [14] [15] . The overall consensus is that H 2 S, the only stable compound formed by hydrogen and sulfur at ambient conditions, is metastable at high pressures and its decomposition gives rise to several H-S compounds. High-T c superconductivity is believed to occur in a structure with the H 3 S stoichiometry, and is considered to be conventional in nature, i.e., mediated by the electron-phonon interaction 4, 5, 7, 9, [12] [13] [14] [15] . Alternatives to conventional superconductivity have also been discussed 16 . According to structural predictions 5-9 , H 3 S adopts a rhombohedral R3m form between approximately 112 and 175 GPa, and a cubic Im3m at higher pressures. As shown in Fig. 1 Energetics. EBO, E vib , and E = E vib + EBO curves are shown as a function of the reaction coordinate Q that transforms the Im3m structure (Q = 0) into the R3m structure (Q = 1), as well as the relative coordinate x that measures the off-centering of the H atoms, defined as x = (d2 − a/2)/(a/2), where d2 is the distance of the hydrogen bond and a the lattice parameter (see Fig. 1 ). Note that E vib and E are plotted considering the left axis, while EBO considering the right axis. Crystal symmetry implies that E(Q) = E(−Q), so that the curves can be fitted to polynomials with only even terms. This guarantees that the transition is second-order according to Landau theory 11 . Results are presented for two different volumes of the primitive bcc lattice; V = 97.85a H−S bond of length d 1 . Each of these H atoms is bonded to the next S atom by a hydrogen H· · · S bond of length d 2 . The Im3m phase, in contrast, has full cubic symmetry, with d 1 = d 2 so that each H atom resides midway between the two S atoms, as shown in Fig. 1 . The R3m structure is nevertheless very close to cubic symmetry, for example, the DFT-relaxed R3m structure, which represents the minimum of the Born-Oppenheimer energy surface (BOES), has a rhombohedral angle of 109.49
• at ≈150 GPa, compared to 109.47
• for a perfect bcc lattice. We have verified that imposing a cubic angle on the R3m structure has a negligible effect on the energy difference between the R3m and Im3m structures. Consequently, we assume a cubic lattice for both phases in the following.
The bond-symmetrizing second-order transition from R3m to Im3m occurs at 175 GPa according to our static lattice calculations. At this pressure, our harmonic phonon calculations show that a Γ-point optical phonon of the high-symmetry Im3m phase becomes imaginary, implying that Im3m is at a saddle point of the BOES between 112 and 175 GPa, while the R3m phase lies at the minimum. Crystal symmetry guarantees that the transition is of second-order type. As occurs in the highpressure ice X phase 3,17,18 and other hydrogenated compounds 19 , the quantum nature of the proton can radically alter the pressure at which the second-order phase transition occurs and, in the present case, can strongly affect the stability of the R3m phase below 175 GPa. Determining the stability ranges of these phases therefore requires the inclusion of vibrational zero-point energy (ZPE) alongside the static BOES energy. However, the presence of imaginary phonon frequencies hinders calculations of the ZPE, since the quasi-harmonic approximation breaks down, and anharmonicity becomes a crucial ingredient.
To elucidate the role of anharmonicity and quantum effects in the pressure range in which the record T c was observed, we make use of the stochastic self-consistent harmonic approximation (SSCHA) 20, 21 . The variational SSCHA method was devised for calculating the free energy and phonon spectra while fully incorporating quan- tum and anharmonic effects, and it is therefore perfectly suited for our purpose. All of the calculations presented here are performed at 0 K. Primitive cells for the R3m and Im3m structures contain 4 atoms (1 S atom and 3 H atoms), and therefore a particular nuclear configuration can be described by a 12-dimensional vector R containing the atomic coordinates. In the classical limit the ZPE is neglected and the energy of a nuclear configuration R is given by the DFT Born-Oppenheimer energy E BO (R). In the SSCHA, the ZPE is accounted for by approximating the nuclear wave-function by a Gaussian centered on a centroid coordinate R c , which denotes the average and most probable position of the nuclei. For a given R c , the width of the Gaussian is obtained by a variational minimization of the expectation value of the sum of the nuclear potential and kinetic energies. In the following analysis it is convenient to split the SSCHA total energy E(R c ) into static and anharmonic-vibrational-ZPE contributions: E(R c ) = E BO (R c ) + E vib (R c ).
We study the energy landscape E(R c ) along the line defined by R c (Q) = R Im3m + Q(R R3m − R Im3m ), where R Im3m and R R3m are, respectively, the coordinates corresponding to the saddle point and minimum of the BOES, representing the two different symmetries. Here, Q is a real number describing the reaction coordinate, so that at Q = 0 the centroids are located at the atomic positions of Im3m, and at Q = 1 at the atomic positions of R3m. Hence, Q measures the off-centering of the hydrogen nuclear wave-function and can be associated with the relative coordinate x = (d 2 − a/2)/(a/2) that quantifies the length of the H· · · S hydrogen bond with respect to the symmetric position (a is the lattice parameter). We analyze the curve E(R c (Q)) ≡ E(Q) for a fixed primitive bcc unit cell. As shown in Fig. 2(a) for a cell volume of 97.85a 3 0 the E BO (Q) curve has a shallow double-well structure favoring the R3m structure by only 5.6 meV/H 3 S. However, after adding the E vib (Q) energy calculated with the SSCHA, the full E(Q) curve shows a clear minimum at Q = 0, which favors the Im3m structure. At a larger volume of 102.11 a 3 0 , which corresponds to a pressure of around 130 GPa, the minimum of E(Q) is also at Q = 0, despite the fact that the BornOppenheimer well in E BO (Q) becomes deeper, as shown in Fig. 2(c) . Repeating these calculations for D 3 S, we find that the Im3m structure is the most favorable once the ZPE has been included. We therefore conclude that the quantum nature of the nuclei symmetrizes the hydrogen bond and leads to a proton wave-function centered at the atomic positions of Im3m for both H 3 S and D 3 S. To eliminate the possibility that the energy minimum occurs beyond the R c (Q) line studied, we performed an unconstrained SSCHA minimization, optimizing both the width of the Gaussians and the R c centroid positions. The results of this minimization show again that, within stochastic error, the centroid position obtained corresponds to the Im3m structure, in which the H−S covalent and H· · · S hydrogen bond distances equalize, leading to symmetric hydrogen bonds.
The difference between the vibrational energies of R3m and Im3m as a function of the x coordinate is weakly dependent on volume. This allows us to interpolate E(x) in a wide volume range and estimate the pressure at which the proton wave-function shifts away from the centered position. Our calculations show that this symmetry breaking occurs at 103 GPa in H 3 S and 115 GPa in D 3 S (see Fig. 3 ). The higher transition pressure in D 3 S is due to weaker quantum effects. This isotope effect is similar to the one observed in the ice VII/ice X transition 22 . Considering that below 112 GPa the R3m phase is expected to transform into a very different C2/c phase consisting of isolated H 2 S and H 2 molecules with H 3 S stoichiometry 8 , R3m-H 3 S might not be formed. However, D 3 S may adopt the R3m structure at pressures below the transition to the Im3m phase.
The quantum proton symmetrization has an enormous impact on the phonon spectra of H 3 S. As mentioned earlier, and shown in Fig. 4 , the phonon spectra of Im3m-H 3 S have several imaginary modes in the harmonic approximation below 175 GPa. This is analogous to ice X, which has only real positive phonon frequencies once the classical limit predicts symmetrization of the hydrogen bond [23] [24] [25] . On the contrary, the corresponding anharmonic SSCHA phonon spectra show well-behaved phonon dispersion relations with positive frequencies in the pressure range of interest (Fig. 4) . The anharmonic renormalization of the phonon energies is huge, especially for the H-S bond-stretching modes in which H atoms move towards the neighboring S atoms, which are precisely those modes which drive the second-order phase transition between the Im3m and R3m phases. Therefore, the proximity to the second-order quantum phase transition is the origin of the strong anharmonicity.
While the bond symmetrization in ice X occurs in an insulating system, H 3 S is metallic and the symmetrization strongly affects the superconductivity. Indeed, the calculation of the electron-phonon coupling and the superconducting T c lend further support to the suggestion that Im3m-H 3 S yields the record T c . We use Wannier interpolated electron-phonon matrix elements in our calculations 26 and estimate T c solving the isotropic MigdalEliashberg equations. The phonon frequencies and polarizations that enter the electron-phonon calculations are calculated using the SSCHA. The results obtained for the Im3m structure using anharmonic phonon frequencies agree well with experimental measurements of T c for H 3 S and D 3 S and correctly capture the observed increase in T c with decreasing pressure. We also find an isotope coefficient α = −[lnT c (D 3 S) − lnT c (H 3 S)]/ln2 for H→D substitution of α = 0.35 at 210 GPa and α = 0.40 at 155 GPa in good agreement with experiment (see Fig. 4(c) ). The electron-phonon coupling constant λ, which scales with the phonon frequencies as ∝ 1/ω 2 , is enhanced with decreasing pressure due to the overall softening of the phonon modes. This explains the smooth decrease of T c with increasing pressure. Between approximately 130 and 150 GPa the increase in λ is compensated by the decrease in the average phonon frequency and T c saturates. For the R3m structure we also present SSCHA calculations keeping the centroids at the Q = 1 position. We find a rapid drop of T c with decreasing pressure as in previous harmonic calculations 13 , in stark contrast to the experimental data. Therefore, the observed high-T c superconductivity cannot be explained by H 3 S in the R3m phase. The sudden drop in T c measured for D 3 S below 150 GPa 4 , which is not present in H 3 S, could arise from the need for higher temperatures to anneal the D 3 S sample or could indicate the symmetry breaking that we predicted at 115 GPa. Indeed, the predicted transition pressure depends on the choice of the exchange correlation functional. Even if our choice of the PBE exchangecorrelation functional 27 appears appropriate, based on agreement between the experimentally observed equation of state 10 and DFT calculations, we cannot exclude a small error on the transition pressure.
METHODS
Supercell calculations for the SSCHA 20,21 and linear response calculations 28 were performed within DFT and the generalized gradient approximation functional 27 as implemented in the Quantum ESPRESSO 29 code. We used ultrasoft pseudopotentials 30 , a plane-wave cutoff energy of 60 Ry for the kinetic energy and 600 Ry for the charge density. The charge density and dynamical matrices were calculated using a 32
3 Monkhorst-Pack shifted electron-momentum grid for the unit cell calculations. This mesh was adjusted accordingly in the supercell calculations. The electron-phonon coupling was calculated by using electron and phonon momentum grids composed of up to 42 × 42 × 42 randomly displaced points in the Brillouin zone. The isotropic Migdal-Eliashberg equations were solved using 512 Matsubara frequencies and µ * = 0.16. The SSCHA calculations were performed using a 3×3×3 supercell for both H 3 S and D 3 S in the Im3m phase, yielding dynamical matrices on a commensurate 3×3×3 q-point grid. The difference between the harmonic and anharmonic dynamical matrices in the 3×3×3 phonon momentum grid was interpolated to a 6×6×6 grid. Adding the harmonic matrices to the result, the anharmonic dynamical matrices were obtained on a 6×6×6 grid. These dynamical matrices were used for the anharmonic electron-phonon coupling calculation. The SS-CHA calculations for Q = 1 were performed with a 2×2×2 supercell. For consistency, the vibrational energies presented in Fig. 2 were also calculated using a 2×2×2 supercell. The electron-phonon calculations for Q = 1 were, however, performed with the SSCHA dynamical matrices interpolated to a 6×6×6 grid from the 2×2×2 mesh.
The E vib (Q) curves in Fig. 2 were obtained as follows. E vib was calculated for Q = 0 and Q = 1 with the SSCHA. With the SSCHA calculation at Q = 1, we extracted dE vib dQ (Q = 1) with no further computational effort. Considering that the derivative of the curve at Q = 0 vanishes by symmetry, we can get straightforwardly a potential of the form E vib (Q) = A+BQ 2 +CQ 4 . The E vib fit curves presented in Fig. 2 were obtained in this way. The extra point obtained at Q = 0.5 for H 3 S at V = 97.85a 3 0 (see Fig. 2(a) ) confirmed the validity of the fitting procedure. The E BO (Q) BOES energies were calculated for many Q points yielding an accurate fitting curve. Fig. 3 was obtained using a polynomial interpolation of the BOES in the volume range shown and adding the E 

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