The sign phase transition in the problem of interfering directed paths

TL;DR

This paper studies the phase transition of the sign of interfering directed paths in disordered media, revealing a critical point in 3D and instability in 2D, with implications for magnetoresistance, superconductivity, and spin glasses.

Contribution

It demonstrates the existence of a sign phase transition in 3D and analyzes the instability of the sign-ordered phase in 2D, providing new insights into disordered systems.

Findings

In 2D, the sign-ordered phase is unstable for any negative scattering site concentration.

In 3D, a finite critical concentration $x_c$ marks the sign phase transition.

The sign transition affects physical phenomena like magnetoresistance and superconductivity.

Abstract

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.