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arXiv:1605.02009 · Physical review. B./Physical review. B

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

Thibaud Louvet, David Carpentier, Andrei A. Fedorenko

Published Dec 5, 2016• 45 citations
arXivHTMLPDF

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 $d=2+\ensuremath{\varepsilon}$ of a $U(N)$ Gross-Neveu model in the limit $N\ensuremath{\rightarrow}0$. Unfortunately, this model is not multiplicatively renormalizable in $2+\ensuremath{\varepsilon}$ 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 $d=4\ensuremath{-}\ensuremath{\varepsilon}$ expansion. We also discuss the characterization of the transition by the multifractal spectrum of wave functions.

DOI
10.1103/physrevb.94.220201

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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+\ensuremathεd=2+\ensuremath{\varepsilon}d=2+\ensuremathε of a U(N)U(N)U(N) Gross-Neveu model in the limit N\ensuremath→0N\ensuremath{\rightarrow}0N\ensuremath→0. Unfortunately, this model is not multiplicatively renormalizable in 2+\ensuremathε2+\ensuremath{\varepsilon}2+\ensuremathε 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\ensuremath−\ensuremathεd=4\ensuremath{-}\ensuremath{\varepsilon}d=4\ensuremath−\ensuremathε expansion. We also discuss the characterization of the transition by the multifractal spectrum of wave functions.
In the presence of randomness, a relativistic semimetal undergoes a quantum transition towards a diffusive phase. A standard approach relates this transition to the U(N) Gross-Neveu model in the limit of N→0. We show that the corresponding fixed point is infinitely unstable, demonstrating the necessity to include fluctuations beyond the usual Gaussian approximation. We develop a functional renormalization group method amenable to include these effects and show that the disorder distribution renormalizes following the so-called porous medium equation. We find that the transition is controlled by a nonanalytic fixed point drastically different from that of the U(N) Gross-Neveu model. Our approach provides a unique mechanism of spontaneous generation of a finite density of states and also characterizes the scaling behavior of the broad distribution of fluctuations close to the transition. It can be applied to other problems where nonanalytic effects may play a role, such as the Anderson localization transition.
A Weyl semimetal is a three-dimensional topological gapless phase. In the presence of strong enough disorder it undergoes a quantum transition towards a diffusive metal phase whose universality class depends on the range of disorder correlations. Similar to other quantum transitions driven by disorder, the critical wave functions at the semimetal--diffusive-metal transition exhibit multifractality. Using renormalization group methods we study the corresponding multifractal spectrum as a function of the range of disorder correlations for generic disorder including random scalar and vector potentials. We also discuss the relation between the geometric fluctuations of critical wave functions and the broad distribution of the local density of states (DOS) at the transition. We derive a new scaling relation for the typical local DOS and argue that it holds for other disorder-driven transitions in which both the average and typical local DOS vanish on one side of the transition. As an illustration we apply it to the recently discussed unconventional quantum transition in disordered semiconductors with power-law dispersion relation near the band edge.
One could tune a topological double-Weyl semimetal or a topological triple-Weyl semimetal to become a topologically trivial insulator by opening a band gap. This kind of quantum phase transition is characterized by the change of certain topological invariant. A new gapless semimetallic state emerges at each topological quantum critical point. Here we perform a renormalization group analysis to investigate the stability of such critical points against perturbations induced by random scalar potential and random vector potential. We find that the quantum critical point between double-Weyl semimetal and band insulator is unstable and can be easily turned into a compressible diffusive metal by any type of weak disorder. The quantum critical point between triple-Weyl semimetal and band insulator flows to a stable strong-coupling fixed point if the system contains a random vector potential merely along the z-axis, but becomes a compressible diffusive metal when other types of disorders exist.
Weyl or Dirac fermions with the iconic linear energy-momentum relation and average density of states (ADOS) ρ(E)∼∣E∣d−1ρ(E) \sim |E|^{d-1}ρ(E)∼∣E∣d−1 at energy EEE in ddd spatial dimensions, constitute a unique setup to study the disorder-driven semimetal-to-metal quantum phase transition (QPT) between ballistic (realized for weak disorder) and diffusive (stabilized at stronger disorder) quasiparticles. Such a QPT takes place only for d>2d>2d>2 and falls beyond the realm of the Anderson metal-to-insulator transition. From numerically computed ADOS (using the kernel polynomial method) in dirty Weyl systems in d=2d=2d=2 to 666, here we show that d=2d=2d=2 and d=4d=4d=4 are the lower (dℓd_\elldℓ​) and upper (dud_udu​) critical dimensions for such an unconventional QPT, respectively, as suggested from the solution of quasiparticle lifetime within the self-consistent Born approximation. Consequently, for d≥4d \geq 4d≥4 the associated correlation length exponent is found to be ν≈0.5ν\approx 0.5ν≈0.5 (within numerical accuracy). However, the dynamic scaling exponent at the quantum critical point is pinned close to z≈d/2z \approx d/2z≈d/2 (numerically) for any d≥3d \geq 3d≥3, which is shown to be an exact result from a field-theoretic renormalization group calculation. Therefore, Weyl semimetal-to-metal QPTs can be studied field theoretically around both dℓd_\elldℓ​ and dud_udu​.
Disordered noninteracting systems in sufficiently high dimensions have been predicted to display a non-Anderson disorder-driven transition that manifests itself in the critical behavior of the density of states and other physical observables. Recently, the critical properties of this transition have been extensively studied for the specific case of Weyl semimetals by means of numerical and renormalisation-group approaches. Despite this, the values of the critical exponents at such a transition in a Weyl semimetal are currently under debate. We present an independent calculation of the critical exponents using a two-loop renormalization-group approach for Weyl fermions in $2\ensuremath{-}\ensuremath{\varepsilon}$ dimensions and resolve controversies currently existing in the literature.
Dirac and Weyl fermions appear as quasiparticle excitations in many different condensed-matter systems. They display various quantum transitions which represent unconventional universality classes related to the variants of the Gross-Neveu model. In this paper we study the bosonized version of the standard Gross-Neveu model---the Gross-Neveu-Yukawa theory---at three-loop order, and compute critical exponents in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions for a general number of fermion flavors. Our results fully encompass the previously known two-loop calculations, and agree with the known three-loop results in the purely bosonic limit of the theory. We also find the exponents to satisfy the emergent superscaling relations in the limit of a single-component fermion, order by order up to three loops. Finally, we apply the computed series for the exponents and their Pad\'e approximants to several phase transitions of current interest: metal-insulator transitions of spin-1/2 and spinless fermions on the honeycomb lattice, emergent supersymmetric surface field theory in topological phases, as well as the disorder-induced quantum transition in Weyl semimetals. Comparison with the results of other analytical and numerical methods is discussed.