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arXiv:1308.3953 · Physical Review Letters

Density of States Scaling at the Semimetal to Metal Transition in Three Dimensional Topological Insulators

Koji Kobayashi, Tomi Ohtsuki, Ken‐Ichiro Imura, Igor F. Herbut

Published Jan 7, 2014• 173 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.112.016402

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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.
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 reconsider the phase diagram of a three-dimensional Z2{\mathbb{Z}}_{2}Z2​ topological insulator in the presence of short-ranged potential disorder, with the insight that nonperturbative rare states destabilize the noninteracting Dirac semimetal critical point separating different topological phases. Based on our numerical data on the density of states, conductivity, and wave functions, we argue that the putative Dirac semimetal line is destabilized into a diffusive metal phase of finite extent due to nonperturbative effects of rare regions. We discuss the implications of these results for past and current experiments on doped topological insulators.
Identifying unconventional quantum phase transitions is one of the most fundamental subjects in quantum physics. To this end, critical exponents in disorder-driven quantum phase transitions in Weyl semimetals and symmetry-protected topological phases have been extensively studied in recent years. In this Rapid Communication, we provide precise critical exponent of the Anderson metal-insulator transition in three-dimensional (3D) orthogonal class with particle-hole symmetry, class CI, as $\ensuremath{\nu}=1.16\ifmmode\pm\else\textpm\fi{}0.02$. We further study disorder-driven quantum phase transitions in the 3D nodal line Dirac semimetal model, which belongs to class BDI, and estimate the critical exponent as $\ensuremath{\nu}=0.80\ifmmode\pm\else\textpm\fi{}0.02$. From a comparison of the exponents, we conclude that a disorder-driven reentrant insulator-metal transition from the topological insulator phase in the class BDI to the metal phase belongs to the same universality class as the Anderson transition in the 3D class BDI. We also argue that small disorder drives the nodal line Dirac semimetal in the clean limit to the metal.
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​.
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.