Ferroelectric quantum critical point in superconducting hydrides: The case of H$_3$S

Abstract

H$_3$S sulfur hydride has been widely investigated for its high superconducting critical temperature $T_c$ of 203 K at about $p_c = 155$ GPa. Despite being the precursor of superconducting hydrides, a detailed picture of its phase diagram in an extended temperature and pressure range is still missing. To determine it with inclusion of both thermal and quantum effects, we carry out path integral molecular dynamics combined to a MACE neural network potential trained on BLYP density functional theory configurations. The resulting H$_3$S phase diagram is characterized by the ferroelectric transition between the Im$\bar{3}$m and R3m phases, which originates from a quantum critical point (QCP) located at $p_\mathrm{QCP} \approx 134$ GPa. We show that the experimental $T_c$ peak falls into a paraelectric region of large nuclear quantum fluctuations above the ferroelectric QCP, as measured by local phonon Green's functions resolved in imaginary time, where fluctuating dipole moments are at play. We study the critical behavior of the system in the proximity of the QCP by a finite-size scaling analysis, showing that it belongs to the 4D Ising universality class. We finally discuss its implications for the superconducting state.

Figures (7)

• Figure 1: H_3S phase diagram, computed by PIMD using a MACE-MLIP trained with a DFT-BLYP dataset for $L=4$: ferroelectric transition (gray points, dashed line); classical molecular dynamics (MD) transition (red empty circles, solid line); extrapolated QCP (filled purple circle) and ferroelectric line (black squares, solid line); experimental superconducting $T_c$ (green diamonds)Drozdov_2015Einaga_2016. The colormap refers to the $\sigma^2_{g(\beta/2)}/\sigma^2_{g(0)}$ ratio plotted in Fig. \ref{['fig:order_parameters']}(d); it measures the relative intensity of the retarded fluctuations of local moments. Data are interpolated using a weighted sum of radially symmetric basis functions. Blue and red shaded areas are the ferroelectric and paraelectric phases, respectively.
• Figure 2: Order parameter and local observables for $L=4$. Panel (a): Order parameter for the ferroelectric transition. Full and empty symbols are the absolute and the real values of the order parameter, $\Delta_{\textrm{abs}}$ and $\Delta$, respectively. The variance of $\Delta$ is in panel (b). Different temperatures are reported: 50 K (black triangles), 100 K (pink diamonds), 150 K (cyan circles), 200 K (orange squares) and 300 K (green reversed triangles). The vertical dashed lines indicate the transition pressures. Panel (c): Proton centroids distribution along the S-S direction for several pressures at $T=50$ K. Thick black and red curves indicate the ferroelectric transition and the unimodal-to-bimodal variation, respectively. Panel (d): $\sigma^2_{g(\beta/2)}/\sigma^2_{g(0)}$ ratio, where $\sigma^2_{g(\tau)}$ is the variance of the local Green function in imaginary time. Dashed lines indicate its maximum.
• Figure 3: Soft optical mode frequencies as a function of pressure at different temperatures computed for $L=4$ from PIMD phononsMorresi2021 (see also the SMSM). Colors and symbols are the same as Fig. \ref{['fig:order_parameters']}(a)-(b). Dashed vertical lines indicate the ferroelectric transition.
• Figure 4: Finite-size scaling of $\Delta$ at $T=50$ K (black triangles), 67 K (blue circles) and 100 K (red stars), and supercell sizes chosen to keep $L_\tau/L$ constant. We use $g[u L^{1/\nu}]$ as a functional form, where $u = \frac{V -V_c}{V_c}$, with $V_c$ the critical volume. Inset: the critical exponent $\nu$ is obtained by minimizing the cost function $C(\nu)$ defined in the text.
• Figure :