Anharmonic theory of superconductivity in the high-pressure materials

Electron-phonon superconductors at high pressures have displayed the highest values of critical superconducting temperature $T_c$ on record, now rapidly approaching room temperature. Despite the importance of high-$P$ superconductivity in the quest for room-temperature superconductors, a mechanistic understanding of the effect of pressure and its complex interplay with phonon anharmonicity and superconductivity is missing, as numerical simulations can only bring system-specific details clouding out key players controlling the physics. Here we develop a minimal model of electron-phonon superconductivity under an applied pressure which takes into account the anharmonic decoherence of the optical phonons. We find that $T_c$ behaves non-monotonically as a function of the ratio $\Gamma/\omega_0$, where $\Gamma$ is the optical phonon damping and $\omega_0$ the optical phonon energy at zero pressure and momentum. Optimal pairing occurs for a critical ratio $\Gamma/\omega_0$ when the phonons are on the verge of decoherence ("diffuson-like" limit). Our framework gives insights into recent experimental observations of $T_c$ as a function of pressure in the complex BCS material TlInTe$_2$.