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arXiv:1610.08973 · Physical Review X

Global Phase Diagram of a Dirty Weyl Liquid and Emergent Superuniversality

Bitan Roy, Robert-Jan Slager, Vladimir Juričić

Published Sep 19, 2018• 82 citations

Abstract

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.

DOI: 10.1103/physrevx.8.031076

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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.

Weyl or Dirac fermions with the iconic linear energy-momentum relation and average density of states (ADOS)

ρ(E) \sim |E|^{d-1}

at energy

E

in

d

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>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=2

to

6

, here we show that

d=2

and

d=4

are the lower (

d_\ell

) and upper (

d_u

) 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 \geq 4

the associated correlation length exponent is found to be

ν\approx 0.5

(within numerical accuracy). However, the dynamic scaling exponent at the quantum critical point is pinned close to

z \approx d/2

(numerically) for any

d \geq 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

d_\ell

and

d_u

.

Three-dimensional Weyl semimetals (WSMs) constitute paradigmatic gapless topological phases whose nodal structure is stable against weak perturbations, including disorder. Two distinct ways to destroy the topology of the Weyl nodes are through pairwise annihilation in momentum space or by sufficiently strong disorder that eliminates the quasiparticle pole, driving the system into a non-Fermi-liquid diffusive-metallic state. In this work, we study the evolution of entanglement spectrum and entanglement entropy of a dirty WSM as it undergoes these two topological transitions. In the clean limit, the WSM topology is reflected by a locus of

ξ=1/2

eigenvalues of the reduced correlation matrix mirroring the structure of Fermi arcs. We show that disorder broadens this feature into a finite-width distribution, which disappears either through a gradual loss of spectral weight upon node annihilation or by melting into the background at the onset of metallicity. When tuning across the disorder-driven transition, the scaling of the Rényi entropies observed for weak disorder gradually breaks down as the system approaches the critical disorder strength.

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.

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.

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 quantum phase transition between two clean, noninteracting topologically distinct gapped states in three dimensions is governed by a massless Dirac fermion fixed point, irrespective of the underlying symmetry class, and this constitutes a remarkably simple example of superuniversality. For a sufficiently weak disorder strength, we show that the massless Dirac fixed point is at the heart of the robustness of superuniversality. We establish this by considering both perturbative and nonperturbative effects of disorder. The superuniversality breaks down at a critical strength of disorder, beyond which the topologically distinct localized phases become separated by a delocalized diffusive phase. In the global phase diagram, the disorder controlled fixed point where superuniversality is lost, serves as a multicritical point, where the delocalized diffusive and two topologically distinct localized phases meet and the nature of the localization-delocalization transition depends on the underlying symmetry class. Based on these features, we construct the global phase diagrams of noninteracting, dirty topological systems in three dimensions. We also establish a similar structure of the phase diagram and the superuniversality for weak disorder in higher spatial dimensions. By noting that

1/{r}^{2}

power-law correlated disorder acts as a marginal perturbation for massless Dirac fermions in any spatial dimension

d

, we have established a general renormalization group framework for addressing disorder driven critical phenomena for fixed spatial dimension

d>2

.