The sign phase transition in the problem of interfering directed paths

We investigate the statistical properties of interfering directed paths in disordered media. At long distance, the average sign of the sum over paths may tend to zero (sign-disordered) or remain finite (sign-ordered) depending on dimensionality and the concentration of negative scattering sites $x$. We show that in two dimensions the sign-ordered phase is unstable even for arbitrarily small $x$ by identifying rare destabilizing events. In three dimensions, we present strong evidence that there is a sign phase transition at a finite $x_c > 0$. These results have consequences for several different physical systems. In 2D insulators at low temperature, the variable range hopping magnetoresistance is always negative, while in 3D, it changes sign at the point of the sign phase transition. We also show that in the sign-disordered regime a small magnetic field may enhance superconductivity in a random system of D-wave superconducting grains embedded into a metallic matrix. Finally, the existence of the sign phase transition in 3D implies new features in the spin glass phase diagram at high temperature.