Fermions in a random medium

TL;DR

This paper investigates the effects of quenched disorder and short-range interactions on directed polymers in 1+d' dimensions, revealing phase behaviors and critical exponents relevant to disordered media.

Contribution

It introduces a continuum field theory for directed polymers in disordered media, analyzing the relevance of interactions and deriving critical exponents in various dimensions.

Findings

Attractive forces create bound states with localization length scaling as |g|^{- u_ot}

Repulsive forces lead to mutual avoidance with a pair distribution function scaling as |r|^{ heta}

In 2D, the exponents are approximately rac{1}{0.8} and 2.4 respectively.

Abstract

We study the continuum field theory for an ensemble of directed polymers r_i (t) in 1+d' dimensions that live in a medium with quenched point disorder and interact via short-ranged pair forces g \Psi (r_i - r_j). In the strong-disorder (or low-temperature) regime, such forces are found to be relevant in any dimension d' below the upper critical dimension for a single line. Attractive forces generate a bound state with localization length \xi_\perp \sim |g|^{-\nu_\perp}; repulsive forces lead to mutual avoidance with a pair distribution function P(r_i - r_j) \sim |r_i - r_j|^\theta reminiscent of interacting fermions. In the experimentally important dimension d' = 2, we obtain \nu_\perp \approx 0.8 and \theta \approx 2.4 .