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.
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.
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.
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.
In electronic band structure of solid state material, two band touching points with linear dispersion appear in pair in the momentum space. When they annihilate with each other, the system undergoes a quantum phase transition from three-dimensional Weyl semimetal (WSM) phase to a band insulator phase such as Chern band insulator (CI) phase. The phase transition is described by a new critical theory with a `magnetic dipole' like object in the momentum space. In this paper, we reveal that the critical theory hosts a novel disorder-driven quantum multicritical point, which is encompassed by three quantum phases, renormalized WSM phase, CI phase, and diffusive metal (DM) phase. Based on the renormalization group argument, we first clarify scaling properties around the band touching points at the quantum multicritical point as well as all phase boundaries among these three phases. Based on numerical calculations of localization length, density of states and critical conductance distribution, we next prove that a localization-delocalization transition between the CI phase with a finite zero-energy density of states (zDOS) and DM phase belongs to an ordinary 3D unitary class. Meanwhile, a localization-delocalization transition between the Chern insulator phase with zero zDOS and a renormalized Weyl semimetal (WSM) phase turns out to be a direct phase transition whose critical exponent ν=0.80±0.01. We interpret these numerical results by a renormalization group analysis on the critical theory.
We study the quantum phase diagram of a three dimensional noninteracting Dirac semimetal in the presence of either quenched axial or scalar potential disorder, by calculating the average and the typical density of states as well as the inverse participation ratio using numerically exact methods. We show that as a function of the disorder strength a half-filled (i.e., undoped) Dirac semimetal displays three distinct ground states, namely an incompressible semimetal, a compressible diffusive metal, and a localized Anderson insulator, in stark contrast to a conventional dirty metal that only supports the latter two phases. We establish the existence of two distinct quantum critical points, which respectively govern the semimetal-metal and the metal-insulator quantum phase transitions and also reveal their underlying multifractal nature. Away from half-filling the (doped) system behaves as a diffusive metal that can undergo Anderson localization only, which is shown by determining the mobility edge and the phase diagram in terms of energy and disorder.
The quantum phase transition between the three dimensional Dirac semimetal and the diffusive metal can be induced by increasing disorder. Taking the system of a disordered Z2 topological insulator as an important example, we compute the single particle density of states by the kernel polynomial method. We focus on three regions: the Dirac semimetal at the phase boundary between two topologically distinct phases, the tricritical point of the two topological insulator phases and the diffusive metal, and the diffusive metal lying at strong disorder. The density of states obeys a novel single parameter scaling, collapsing onto two branches of a universal scaling function, which correspond to the Dirac semimetal and the diffusive metal. The diverging length scale critical exponent ν and the dynamical critical exponent z are estimated, and found to differ significantly from those for the conventional Anderson transition. Critical behavior of experimentally observable quantities near and at the tricritical point is also discussed.
Pursuing complementary field-theoretic and numerical methods, we here paint the global phase diagram of a three-dimensional dirty Weyl system.The generalized Harris criterion, augmented by a perturbative renormalization-group analysis shows that weak disorder is an irrelevant perturbation at the Weyl semimetal (WSM)-insulator quantum-critical point.But, a metallic phase sets in through a quantum phase transition (QPT) at strong disorder across a multicritical point.The field-theoretic predictions for the correlation length exponent ν ¼ 2 and dynamic scaling exponent z ¼ 5=4 at this multicritical point are in good agreement with the ones extracted numerically, yielding ν ¼ 1.98 Æ 0.10 and z ¼ 1.26 Æ 0.05, from the scaling of the average density of states (DOS).Deep inside the WSM phase, generic disorder is also an irrelevant perturbation, while a metallic phase appears at strong disorder through a QPT.We here demonstrate that in the presence of generic but strong disorder, the WSM-metal QPT is ultimately always characterized by the exponents ν ¼ 1 and z ¼ 3=2 (to one-loop order), originating from intranode or chiralsymmetric (e.g., regular and axial potential) disorder.We here anchor such emergent chiral superuniversality through complementary renormalization-group calculations, controlled via ϵ expansions, and numerical analysis of average DOS across WSM-metal QPT.In addition, we also discuss a subsequent QPT (at even stronger disorder) of a Weyl metal into an Anderson insulator by numerically computing the typical DOS at zero energy.The scaling behavior of various physical observables, such as residue of quasiparticle pole, dynamic conductivity, specific heat, Grüneisen ratio, inside various phases as well as across various QPTs in the global phase diagram of a dirty Weyl liquid, are discussed.