# Determining the Superconducting Transition Temperatures of Liquids

## TL;DR

This paper introduces a non-perturbative method to calculate the superconducting transition temperatures of liquids, specifically applied to metallic hydrogen, revealing they can be superconducting in high-pressure regimes.

## Contribution

A novel non-perturbative approach linking electron-phonon coupling to T-matrix fluctuations for calculating $T_c$ in liquids.

## Key findings

- Metallic hydrogen liquids have $T_c$ above melting temperatures at high pressures.
- The method successfully estimates $T_c$ for metallic hydrogen.
- Superconductivity in metallic hydrogen liquids is predicted at pressures from 0.5 to 1.5 TPa.

## Abstract

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.