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Physical Review Letters

Accurate Determination of the Quasiparticle and Scaling Properties Surrounding the Quantum Critical Point of Disordered Three-Dimensional Dirac Semimetals

Bo Fu, Wei Kang Zhu, Q. W. Shi, Qunxiang Li, Jinlong Yang, Zhenyu Zhang

Published Apr 3, 2017• 24 citations
Publisher

Abstract

Exploiting the enabling power of the Lanczos method in momentum space, we determine accurately the quasiparticle and scaling properties of disordered three-dimensional Dirac semimetals surrounding the quantum critical point separating the semimetal and diffusive metal regimes. We unveil that the imaginary part of the quasiparticle self-energy obeys a common power law before, at, and after the quantum phase transition, but the power law is nonuniversal, whose exponent is dependent on the disorder strength. More intriguingly, whereas a common power law is also found for the real part of the self-energy before and after the phase transition, a distinctly different behavior is identified at the critical point, characterized by the existence of a nonanalytic logarithmic singularity. This nonanalytical correction serves as the very basis for the unusual power-law behaviors of the quasiparticles and many other physical properties surrounding the quantum critical point. Our approach also allows the ready and reliable determination of the scaling properties of the correlation length and dynamical exponents. We further show that the central findings are valid for both uncorrelated and correlated disorder distributions and should be directly comparable with future experimental observations.

DOI
10.1103/physrevlett.118.146401

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Exploiting the enabling power of the Lanczos method in momentum space, we determine accurately the quasiparticle and scaling properties of disordered three-dimensional Dirac semimetals surrounding the quantum critical point separating the semimetal and diffusive metal regimes. We unveil that the imaginary part of the quasiparticle self-energy obeys a common power law before, at, and after the quantum phase transition, but the power law is nonuniversal, whose exponent is dependent on the disorder strength. More intriguingly, whereas a common power law is also found for the real part of the self-energy before and after the phase transition, a distinctly different behavior is identified at the critical point, characterized by the existence of a nonanalytic logarithmic singularity. This nonanalytical correction serves as the very basis for the unusual power-law behaviors of the quasiparticles and many other physical properties surrounding the quantum critical point. Our approach also allows the ready and reliable determination of the scaling properties of the correlation length and dynamical exponents. We further show that the central findings are valid for both uncorrelated and correlated disorder distributions and should be directly comparable with future experimental observations.
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 (zzz) and correlation length exponent ($\ensuremath{\nu}$) 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.
We study noninteracting systems with a power-law quasiparticle dispersion ${\ensuremath{\xi}}_{\mathbf{k}}\ensuremath{\propto}{k}^{\ensuremath{\alpha}}$ and a random short-range-correlated potential. We show that, unlike the case of lower dimensions, for $d>2\ensuremath{\alpha}$, there exists a critical disorder strength (set by the bandwidth), at which the system exhibits a disorder-driven quantum phase transition at the bottom of the band that lies in a universality class distinct from the Anderson transition. In contrast to the conventional wisdom, it manifests itself in, e.g., the disorder-averaged density of states. For systems in symmetry classes that permit localization, the striking signature of this transition is a nonanalytic behavior of the mobility edge, which is pinned to the bottom of the band for subcritical disorder and grows for disorder exceeding a critical strength. Focusing on the density of states, we calculate the critical behavior (exponents and scaling functions) at this novel transition, using a renormalization group, controlled by an $\ensuremath{\varepsilon}=d\ensuremath{-}2\ensuremath{\alpha}$ expansion. We also apply our analysis to Dirac materials, e.g., Weyl semimetals, where this transition takes place in physically interesting three dimensions.
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​.
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.
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.
We study the properties of the avoided or hidden quantum critical point (AQCP) in three-dimensional Dirac and Weyl semimetals in the presence of short range potential disorder. By computing the averaged density of states (along with its second and fourth derivative at zero energy) with the kernel polynomial method (KPM) we systematically tune the effective length scale that eventually rounds out the transition and leads to an AQCP. We show how to determine the strength of the avoidance, establishing that it is not controlled by the long wavelength component of the disorder. Instead, the amount of avoidance can be adjusted via the tails of the probability distribution of the local random potentials. A binary distribution with no tails produces much less avoidance than a Gaussian distribution. We introduce a double Gaussian distribution to interpolate between these two limits. As a result we are able to make the length scale of the avoidance sufficiently large so that we can accurately study the properties of the underlying transition (that is eventually rounded out), unambiguously identify its location, and provide accurate estimates of the critical exponents $\ensuremath{\nu}=1.01\ifmmode\pm\else\textpm\fi{}0.06$ and $z=1.50\ifmmode\pm\else\textpm\fi{}0.04$. We also show that the KPM expansion order introduces an effective length scale that can also round out the transition in the scaling regime near the AQCP.