Rare region effects dominate weakly disordered 3D Dirac points

TL;DR

This paper demonstrates that rare region effects cause a nonzero density of states at weak disorder in 3D Dirac systems, eliminating the Dirac semimetal phase and influencing transport properties.

Contribution

It reveals that nonperturbative rare region effects dominate, leading to a finite density of states at any nonzero disorder, challenging previous perturbative assumptions.

Findings

Rare regions induce a nonzero density of states at all disorder levels.

Transport near the Dirac point is diffusive, dominated by hopping between rare resonances.

No true Dirac semimetal phase exists at finite disorder due to rare region effects.

Abstract

We study three-dimensional Dirac fermions with weak finite-range scalar potential disorder. In the clean system, the density of states vanishes quadratically at the Dirac point. Disorder is known to be perturbatively irrelevant, and previous theoretical work has assumed that the Dirac semimetal phase, characterized by a vanishing density of states, survives at weak disorder, with a finite disorder phase transition to a diffusive metal with a non-vanishing density of states. In this paper we show that nonperturbative effects from rare regions, which are missed by conventional disorder-averaged calculations, instead give rise to a nonzero density of states for any nonzero disorder. Thus, there is no Dirac semimetal phase at non-zero disorder. The results are established both by a heuristic scaling argument and via a systematic saddle point analysis. We also discuss transport near the Dirac point. At the Dirac point, we argue that transport is diffusive, and proceeds via hopping between rare resonances. As one moves in chemical potential away from the Dirac point, there are interesting intermediate-energy regimes where the rare regions produce scattering resonances that determine the DC conductivity. We derive a scaling theory of transport near disordered 3D Dirac points. We also discuss the interplay of disorder with attractive interactions at the Dirac point, and the resulting granular superconducting and Bose glass phases. Our results are relevant for all 3D systems with Dirac points, including Weyl semimetals.