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$\backslash$mathrm\$\backslash$,TPa\\$ have \$T\_\c\\$ well above their melting temperatures, therefore are superconducting liquids.

We derive the definition of the Berry phase for adiabatic transport of a composite Fermion (CF) in a half-filled composite Fermi-liquid (CFL). It is found to be different from that adopted in previous investigations by Geraedts et al. With the definition, the numerical evaluation of the Berry phase becomes robust and free of extraneous phase factors. We show that the two forms of microscopic wave-functions of the CFL, i.e., the Jain-Kamilla type wave function and the standard CF wave function, yield different distributions of the Berry curvature in the momentum space. For the former, the Berry curvature has a continuous distribution inside the Fermi sea and vanishes outside, whereas for the latter, the Berry curvature is uniform in the whole momentum space. To facilitate an analytic derivation for the latter, we reveal a simple structure of standard CF wave functions by establishing their connections to the Segal-Bargmann transform. We conclude that the CF with respect to both the microscopic wave-functions is not a massless Dirac particle.