On the disorder-driven quantum transition in three-dimensional relativistic metals

The Weyl semimetals are topologically protected from a gap opening against weak disorder in three dimensions. However, a strong disorder drives this relativistic semimetal through a quantum transition towards a diffusive metallic phase characterized by a finite density of states at the band crossing. This transition is usually described by a perturbative renormalization group in $d=2+\varepsilon$ of a $U(N)$ Gross-Neveu model in the limit $N \to 0$. Unfortunately, this model is not multiplicatively renormalizable in $2+\varepsilon$ dimensions: An infinite number of relevant operators are required to describe the critical behavior. Hence its use in a quantitative description of the transition beyond one-loop is at least questionable. We propose an alternative route, building on the correspondence between the Gross-Neveu and Gross-Neveu-Yukawa models developed in the context of high energy physics. It results in a model of Weyl fermions with a random non-Gaussian imaginary potential which allows one to study the critical properties of the transition within a $d=4-\varepsilon$ expansion. We also discuss the characterization of the transition by the multifractal spectrum of wave functions.