Negative superfluid density: Mesoscopic fluctuations and reverse of the supercurrent through a disordered Josephson junction

We calculate the Josephson coupling energy $U_J\left(\phi \right)$ [related to the supercurrent $I=(2e/ħ{)dU}_J/d\phi ]$ for a disordered normal metal between two superconductors with a phase difference $\phi .$ We demonstrate that the symmetry of the scattering matrix of noninteracting quasiparticles in zero magnetic field implies that $U_J\left(\phi \right)$ has a minimum at $\phi =0.\mathrm{}$ A maximum (that would lead to a π junction or negative superfluid density) is excluded for any realization of the disorder.