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Quantum Multicritical Phenomena in Disordered Weyl Semimetal

Xunlong Luo, Baolong Xu, Tomi Ohtsuki, Ryuichi Shindou

Published Oct 2, 2017• 0 citations
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Abstract

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 $\nu=0.80\pm 0.01$. We interpret these numerical results by a renormalization group analysis on the critical theory.

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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\nu=0.80\pm 0.01ν=0.80±0.01. We interpret these numerical results by a renormalization group analysis on the critical theory.
In the electronic band structure of materials, two-band touching points with linear dispersion always appear in pairs in momentum space. When the pair annihilates, the system undergoes a quantum phase transition from a Weyl semimetal (WSM) to a band insulator (BI), which is described by a new critical theory. This paper reveals that the critical theory hosts a novel disorder-driven quantum multicritical point (QMCP), which is encompassed by three quantum phases: WSM, BI, and diffusive metal (DM). The authors clarify scaling theories around the QMCP as well as all the phase boundaries among these three phases. The scaling theory is general and easily extended to other systems.
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.
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.
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​.
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.
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.