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

Abstract

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 + ε of a U (N ) Gross-Neveu model in the limit N → 0. Unfortunately, this model is not multiplicatively renormalizable in 2+ε 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 − ε expansion. We also discuss the characterization of the transition by the multifractal spectrum of wave functions.

Gross-Neveu-Yukawa model at three loops and Ising critical behavior of Dirac systems

“…Various of these transitions have been suggested to belong to the universality class defined by the chiral transition appearing in the 3D GrossNeveu (GN) model [7] well-known in the context of highenergy physics and conformal field theories [17,18]. This applies, in particular, to the interaction-induced transition toward a charge density wave of electrons on the 2D honeycomb lattice that breaks the (Ising) sublattice symmetry [7], or the disorder-driven transition toward a diffusive metal in a 3D Weyl semi-metal [15]. …”

New quantum transition in Weyl semimetals with correlated disorder

“…Indeed, we have recently showed that the consistent description of the transition with SR disorder beyond two loops involves an infinite number of relevant operators generated by the RG flow. To overcome this obstacle we proposed an alternative way based on a 4 − ε expansion [21]. However, generalization of this approach to the case of LR disorder is a non trivial task which remains to be done.…”

“…In particular, the transition is characterized through the critical exponents ν and z: at this transition, the correlation length diverges as ξ ∼ |∆ − ∆ c | −ν where ∆ c is the critical disorder strength, while the dynamic critical exponent z is defined trough the scaling between energy and momenta at the transition ω ∼ k z . Besides, it was found that exactly at the transition the wave functions and local DOS exhibit multifractality similar to the Anderson transition but with different universality classes [21,23].…”

“…(2) corresponds to the short-range (SR) disorder since it becomes the Dirac δ-function in real space: the impact of such uncorrelated disorder has been studied previously in Refs. [15][16][17][18][19][20][21][22]. The strength of the long-range (LR) correlated disorder is given by ∆ 2 .…”

“…Dirac versus Weyl semimetals or number of cones [10][11][12][13][14]. This transition has been recently intensively studied both numerically [15][16][17][18] as well as analytically using 2 + ε [19,20] and 4 − ε [21,22] expansions. In particular, the transition is characterized through the critical exponents ν and z: at this transition, the correlation length diverges as ξ ∼ |∆ − ∆ c | −ν where ∆ c is the critical disorder strength, while the dynamic critical exponent z is defined trough the scaling between energy and momenta at the transition ω ∼ k z .…”