Determining the Superconducting Transition Temperatures of Liquids

We develop a non-perturbative approach for calculating the superconducting transition temperatures ($T_{c}$) of liquids. The electron-electron scattering amplitude induced by electron-phonon coupling (EPC), from which the effective pairing interaction can be inferred, is related to the fluctuation of the $T$-matrix of electron scattering induced by ions. By applying the relation, EPC parameters can be extracted from a path-integral molecular dynamics simulation. For determining $T_{c}$, the linearized Eliashberg equations are re-established in the non-perturbative context. We apply the approach to estimate $T_{c}$ of metallic hydrogen liquids. It indicates that metallic hydrogen liquids in the pressure regime from $0.5$ to $1.5\mathrm{\,TPa}$ have $T_{c}$ well above their melting temperatures, therefore are superconducting liquids.