Disorder-driven itinerant quantum criticality of three-dimensional massless Dirac fermions

Abstract

Progress in the understanding of quantum critical properties of itinerant electrons has been hindered by the lack of effective models which are amenable to controlled analytical and numerically exact calculations. Here we establish that the disorder driven semimetal to metal quantum phase transition of three dimensional massless Dirac fermions could serve as a paradigmatic toy model for studying itinerant quantum criticality, which is solved in this work by exact numerical and approximate field theoretic calculations. As a result, we establish the robust existence of a non-Gaussian universality class, and also construct the relevant low energy effective field theory that could guide the understanding of quantum critical scaling for many strange metals. Using the kernel polynomial method (KPM), we provide numerical results for the calculated dynamical exponent (z) and correlation length exponent (ν) for the disorder-driven semimetal (SM) to diffusive metal (DM) quantum phase transition at the Dirac point for several types of disorder, establishing its universal nature and obtaining the numerical scaling functions in agreement with our field theoretical analysis.

Rare-Region-Induced Avoided Quantum Criticality in Disordered Three-Dimensional Dirac and Weyl Semimetals

“…Interestingly, the one-loop perturbative renormalization group (RG) calculations of the critical exponents for the proposed SM to DM QCP are consistent with the CCFS inequality (since ν ¼ 1, Refs. [22,23]) as, in fact, are the two-loop RG calculations [26,39] and all numerical estimates in the literature [24,25,32,33,35,36]; therefore, it is not a priori obvious that rare region effects should change the universality of this transition. Given the field theoretic RG analyses and the large body of direct numerical studies of the disorder-driven SM-DM QCP, finding the various critical exponents and identifying the critical coupling, as well as the apparent consistency between the theoretical (and numerical) correlation exponent with the CCFS inequality, it seems reasonable to assume that the rare regions arising out of nonperturbative disorder effects do not change the nature of the QCP in any substantial manner.…”

“…The goal of the current work is to settle this question definitively. Although the disordered Dirac-Weyl systems have been theoretically studied very extensively in the literature [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39], essentially all of this work, except for a very recent one (Ref. [27]), study the properties of the disorder-driven SM-DM quantum phase transition, taking it for granted that such a disorder-induced QCP indeed exists in three dimensions following the predictions of the perturbative field theory.…”

“…More recently, the properties of this proposed QCP have been calculated in a renormalization-group treatment of the problem [23], which has now been extended to two loops [26,39]. The field theory of the QCP can be constructed in terms of an interacting "Q 4 theory" (similar to a ϕ 4 theory for magnetism but now Q is the replicated matrix field) strongly coupled to massless Dirac fermions [33], while for the Weyl case due to topological considerations, a separate field theory has been derived [29,38]. Tuning the clean model away from the Dirac or Weyl limit by varying the power law of the dispersion relation [30,31], this transition has been shown to occur even in some one-dimensional models [34] (akin to a long-range Ising model).…”

“…Because of the invariable presence of disorder in all solid-state materials, there has been a substantial amount of theoretical activity studying the effect of disorder on noninteracting Dirac and Weyl fermions [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. Focusing on the undoped (i.e., Fermi energy at E ¼ 0) Dirac point (i.e., the band touching point), the quadratically vanishing density of states at zero energy [ρðEÞ ∼ E 2 ] associated with the linear three-dimensional energy band dispersion places these problems in a different class than that of a conventional metal with a parabolic energy dispersion and a nonzero Fermi energy.…”

“…We choose a relatively simple model that has been shown [28,33] to exhibit a sharp SM to DM transition (or crossover), without the additional complications of mass terms. Using (separately) Lanczos [45,46] and the kernel polynomial method (KPM) [47], we provide definitive numerical evidence for the existence of two distinct types of low-jEj eigenstates in the three-dimensional undoped (i.e., Fermi level at the band touching point taken to be zero energy) Dirac-Weyl systems for weak disorder strengths.…”