Rare regions and avoided quantum criticality in disordered Weyl semimetals and superconductors
RRare regions and avoided quantum criticality indisordered Weyl semimetals and superconductors
J. H. Pixley a,b,c , Justin H. Wilson a a Department of Physics and Astronomy, Center for Materials Theory, Rutgers University,Piscataway, NJ 08854, USA b Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, NewYork, NY 10010. c Physics Department, Princeton University, Princeton, New Jersey 08544, USA
Abstract
Disorder in Weyl semimetals and superconductors is surprisingly subtle, at-tracting attention and competing theories in recent years. In this brief review,we discuss the current theoretical understanding of the effects of short-ranged,quenched disorder on the low energy-properties of three-dimensional, topologicalWeyl semimetals and superconductors. We focus on the role of non-perturbativerare region effects on destabilizing the semimetal phase and rounding the ex-pected semimetal-to-diffusive metal transition into a cross over. Furthermore,the consequences of disorder on the resulting nature of excitations, transport,and topology are reviewed. New results on a bipartite random hopping modelare presented that confirm previous results in a p + ip Weyl superconductor,demonstrating that particle-hole symmetry is insufficient to help stabilize theWeyl semimetal phase in the presence of disorder. The nature of the avoidedtransition in a model for a single Weyl cone in the continuum is discussed. Weclose with a discussion of open questions and future directions.
Contents1 Introduction 22 Model and set up 6
Email addresses: [email protected] (J. H. Pixley), [email protected] (Justin H. Wilson)
Preprint submitted to Elsevier February 8, 2021 a r X i v : . [ c ond - m a t . d i s - nn ] F e b Rare resonances in semimetals 11 p + ip Weyl superconductor . . . . . . . . . . . . . . . . . . . . 328.2 Beyond Linear Touching Points . . . . . . . . . . . . . . . . . . . 338.3 Random hopping model . . . . . . . . . . . . . . . . . . . . . . . 34
The massive Dirac equation, first shown to explain the relativistic electronby Dirac [1], describes a wide class of insulating, topologically non-trivial, solid-state materials [2, 3, 4]. These “topological insulators” have now been observedin a variety of weakly correlated two- and three-dimensional narrow gap semi-conductors. More recently, the focus has shifted due to the discovery of gap-less topological semimetals that are realized in the limit of a vanishing Diracmass that results in the valence and conduction bands touching at isolatedpoints in the Brillouin zone. While graphene [5] is a commonly known two-dimensional Dirac semimetal, it was only recently that three-dimensional Diracsemimetals where discovered in Cd As [6, 7, 8] and Na Bi [9, 10] as well asby doping the insulators Bi − x Sb x [11, 12, 13], BiTl(S − δ Se δ ) [14, 15], and(Bi − x In x ) Se [16, 17]. Breaking inversion or time-reversal symmetry lifts thetwo-fold degeneracy of the Dirac touching point converting a single Dirac pointinto two Weyl points. Three-dimensional Weyl semimetals have been clearlyidentified in the weakly correlated semiconductors with broken inversion symme-try [18, 19] TaAs, NbAs, TaP, and NbP [18, 20, 21, 22, 23, 24]. There are also sig-natures of strongly correlated Kondo-Weyl semimetals [25, 26] in Ce Bi Pd [27]2nd YbPtBi [28]. On the other hand, the observation of Weyl semimetals thatbreak time-reversal symmetry are much more rare with Mn Sn [29] being oneexample despite there being a number of proposed candidate materials with thepyrochlore iridates R Ir O being a prominent example [30, 31, 32].A nodal excitation spectrum extends well beyond electronic semimetals toalso include the neutral Bogolioubov-de Gennes (BDG) quasiparticles in super-conductors with non-trivial gap symmetry [33, 34, 35, 36]. In particular, gapsymmetries that induce line nodes in the superconducting gap, which intersectthe Fermi surface of the normal state at a finite number of points induces iso-lated nodal points in the BDG energy bands. For example, in three-dimensionsa p + ip superconductor can have Weyl points in its BDG band structure thatproduce gapless thermal Majorana excitations [37, 38, 39] that are the neutralanalog of an electronic Weyl semimetal.Weyl nodes of opposite chirality act as sources and sinks of Berry curva-ture, which endow they system with non-trivial topological properties. In time-reversal broken Weyl semimetals this can produce a non-zero anomalous Halleffect [40], whereas in inversion broken Weyl semimetals it can lead to a non-zerophoto-galvanic effect [41, 42] in addition to a large non-linear response in trans-port [43, 44]. The topological nature of Weyl semimetals can also be revealedby the observation of their gapless surface states [30] that exist along a line (i.e.an arc) connecting the projection of a pair of Weyl nodes of opposite helicityon the surface Brillouin zone (BZ). Topological Fermi arc surface states havenow been observed on the surface of Weyl semimetals in angle resolved photoemission and scanning tunneling microscopy experiments [21, 23, 45, 46, 47].While Weyl and massless Dirac fermions may not exactly exist in high-energy physics, their realization in the low-energy limit of an electronic band-structure has opened the possibility to observe exotic high-energy phenomenain solid-state materials. In particular, the axial anomaly [48, 49, 50] has con-sequences when massless Dirac or Weyl fermions are placed in parallel electricand magnetic fields in condensed matter experiments. In the lowest Landaulevel, this anomaly can be understood intuitively through a charge pumpingprocess between Weyl nodes of opposite chirality [50]. The observation of theaxial (or chiral) anomaly has been seen indirectly in a number of Dirac and Weylsemimetals via the observation of a negative magnetoresistance [51, 52, 53] forparallel electric and magnetic fields and a large (or even colossal) positive mag-netoresistance when the fields are perpendicular [54, 55, 56, 57, 58, 59]. Theseobservations are particularly remarkable since the connection to high energyphysics breaks down due to the existence of an underlying band structure witha bounded bandwidth and band-curvature effects [60].Moving away from the idealized band-structure limit, the lack of a hard-gap does not mean that topological protection, if it persists, cannot be relatedto an energy-gap protection vis-`a-vis disorder and interactions. The followingmanuscript reviews the current theoretical understanding of the properties andstability of the Weyl semimetal phase in the presence of short-range disorderwhile focusing on the non-perturbative rare-region effects. As materials, thesesystems inherently have disorder (e.g., impurity defects and vacancies); the type3f disorder (long- or short-ranged) depends on its source. For instance, long-ranged disorder can originate from Coulomb impurities [61, 62, 63] that locallydope the Weyl cone due to screening charge “puddles” that lead to a non-zerodensity of states and conductivity for any finite density of impurities. However,in the following review, we will focus on quenched short-ranged disorder thatarises due to dislocations, vacancies, and neutral impurities, focusing on andhighlighting rare-region effects. As we will see, this is a subtle non-perturbativeproblem in statistical physics and criticality, therefore interaction effects willnot be discussed in this review (apart mean-field assumptions leading to theformation of superconducting BDG quasiparticles).Theoretically, the problem of disordered Weyl semimetals dates back to thework of Fradkin in 1986 [64, 65] where he showed that the semimetal is pertur-batively stable to the inclusion of short-ranged disorder. This stability persists(perturbatively) up to a putative critical point were the density of states at theWeyl node becomes non-zero, which thus acts as an order-parameter for thetransition, see Fig. 1(a,b). This semimetal-to-diffusive metal quantum phasetransition was more accurately captured through a perturbative renormaliza-tion group (RG) calculation in d = 2 + (cid:15) dimensions [66]. Building upon this,Refs. [67, 68] generalized the nature of the transition to show how it arisesin arbitrary dimensions by allowing the nodal touching points to have an ar-bitrary power law. Unlike Anderson localization [69] (which occurs in thesemodels at a larger disorder strength), the primary indicator is the density ofstates [64]. This perturbative and field theory picture has since been refined[67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86] with abetter understanding for the theory at the critical point (though, as Ref. [86]points out, this understanding is still in question and ripe for further investiga-tion, particularly with regards to the correlation length exponent). Numerousnumerical simulations have found reasonable agreement with the perturbativeprediction of a quantum critical point with close to the expected dynamic crit-ical exponent [87, 88, 89, 90, 73, 91, 92, 78, 80, 93]. However, whether or notthe density of states remains truly non-zero in the supposedly stable semimetalphase, as in Fig. 1(b), was out of reach in early studies, where a large finite sizeeffect appears in the density of states at the Weyl node energy.The perturbative picture of a stable semimetal phase was challenged inRef. [94], which demonstrated that rare region effects could induce a non-zerodensity of states at the Weyl node and make the system diffusive. Moreover,these rare regions originate from uncharacteristically strong disorder strengths(statistically “rare”) producing a low-energy, quasi-bound wavefunction (i.e.falls off in a power law-fashion). However, initial numerical studies were notable to locate the rare states nor their effect on the low-energy density of states.In two-dimensional Dirac materials, on the other hand, similar rare-resonancesare reasonably well understood [95] and have been seen via scanning tunnel-ing microscopy on vacancies on the surface of graphite [96] and the d -wavehigh-temperature superconductor Bi Sr Ca(Cu − x Zn x ) O δ [97]. However, intwo-dimensions, disorder is a marginally relevant perturbation in the RG sense(see Sec. 3) and therefore, the rare resonances are a sub-leading effect on av-4 ρ ( E ) E/t
W/t = 0.400.500.600.700.800.901.001.10 ρ ( ) W/t l o g [ ρ ( )] W/t (a) (c)(b)
Figure 1:
Evolution of the disorder averaged density of states depicting a crossover due to an avoided quantum critical point . (a) Twist averaged density of state ρ ( E ) vs energy E showing the putative transition that is rounded into a crossover. (b) Thezero energy density states on a linear scale appears to depict a phase transition but on alog-scale (inset) it is non-zero but exponentially small at weak disorder (data is computed atfinite size L = 71 and KPM expansion order N C = 2048, see Sec. 5.2 for clarification). (c) Asummary schematic cross over diagram exemplifying the avoided transition, with each relevantscaling regime at weak disorder including a semimetal, a diffusive metal, and quantum criticalscaling that is rounded out by rare-region effects. From Ref. [101]. erage. In contrast, in three-dimensions the irrelevance of disorder implies thelow-energy theory should be dominated by rare-resonances. We remark in pass-ing that these rare states are the gapless analog of Anderson localized Lifshitzstates [98] that are well known to randomly fill in and soften the spectral bandgap. The instanton calculus [99, 67, 100] that is discussed below is directlyborrowed from the literature on Lifshitz states.A systematic procedure to isolate and study rare eigenstates in disorderedthree-dimensional Weyl materials was put forth in Ref. [101], which found excel-lent agreement with the predicted form of the non-perturbative, quasilocalizedeigenstates. It was also shown numerically that the density of states remains an analytic function of both disorder and energy at the Weyl node; the purportedperturbative transition is rounded into a crossover in the thermodynamic limitdue to a finite length scale induced by rare regions. Nonetheless, as already ob-served in a number of numerical studies the predicted quantum critical scalingappears over a finite energy window between E ∗ < E < Λ where E ∗ is the rareregion cross-over length scale that is non-zero in the presence of disorder and Λ isa high-energy cut-off where the linear approximation of the nodal crossing pointsbreaks down, see Fig. 1(c). Thus, the rare-region induced crossover was dubbedan avoided quantum critical point (AQCP). Naturally, these non-perturbativerare region effects could also destabilize the exotic topological properties of Weylsemimetals, something we will consider in detail in Sec. 7. As the theoreticaldescription of the perturbative quantum critical point and the rare region dom-inated limit are effectively expanding about two distinct “mean field” groundstate wavefunctions the precise microscopic derivation of an effective theory forthe AQCP has yet to be obtained. Ref. [102] has put forth a phenomenologicalfield theory description of the AQCP in terms of a gas of instantons interact-ing with a power law interaction that links the finite density of states with afinite-correlation length rounding out the putative transition.More recently, Buchhold et al. [103, 104] challenged the scenario of a AQCP5sing a fluctuation analysis of the instanton field theory that found an exactzero for the density of states in the semimetal phase. Such an analysis focusedon a single, linear, Weyl cone in the continuum and the previous numerics thatfound an AQCP, strictly speaking, did not apply. Following this, we put fortha numerical study of the nature of the avoidance in a single Weyl cone in thecontinuum [105]. The numerical results found a strong avoidance, inconsistentwith a stable semimetal phase.The purpose of this review is to present an exposition on the current un-derstanding of rare region effects in disordered Weyl semimetals. Our goal isto present the most straightforward approach to find rare regions in numericalsimulations as well as collect and discuss the various results that exist on theproblem across a range of microscopic models that all indicate that the pertur-bative QCP is rounded out into an avoided transition. The review therefore doesnot cover in depth the large body of RG literature that has developed aroundthe perturbative transition and instead points the interested reader to the rel-evant literature as needed. There are many open questions left to be explored,and we hope this review will aid other researchers to find, diagnose, and studynon-perturbative rare region effects in semimetals so that they may resolve someof these open issues. Our focus will be on the impact of non-perturbative rareregion effects on destabilizing the Weyl semimetal (or superconductor) phase(Sec. 5), the nature of the transport and low-energy quasiparticle excitations(Sec. 6), the stability of the topological Fermi arc surface states (Sec. 7.1), andthe persistence of the charge pumping process due to the axial anomaly (Sec. 7.2.New results on a random hopping model will be presented as additional evidencethat particle hole symmetry cannot help stabilize the Weyl semimetal phase inthe presence of disorder in Sec. 8.
2. Model and set up
In the following, we will be concerned with a broad class of models that canbe described as H = H Weyl + H disorder (1)where H Weyl describes the band structure that hosts linear touching pointsat low energy and H disorder contains random, short-ranged disorder. Near somemomenta K W , the band structure we are interested in has nodal linear touchingpoints H Weyl ≈ (cid:88) K W (cid:88) k ψ † k ( v ( k − K W ) · σ ) ψ k (2)where v is the velocity of the Weyl cone, σ is a vector of the Pauli matri-ces, and ψ k is a two-component spinor of annihilation operators. The resultinglow-energy dispersion is given by E ( k ) ≈ ± v | k | . However, this approxima-tion cannot hold across the entire Brillouin zone of the lattice. First, for bothsimplified and DFT derived models of Weyl semimetals, there can be significantband curvature away from the band touching. Even at the band-touching point,isotropy is not guaranteed, and furthermore, the Weyl cone can “tilt” leading6o Type-II Weyl semimetals [102] with a finite Fermi surface at the nodal en-ergy. Since Type-II Weyl semimetals have a finite density of states at the nodalenergy, they will not host the disorder physics which we are reviewing in thispaper. And finally, by the Fermion doubling theorem [106, 107, 108] every lat-tice model will have an even number of Weyl cones. In Sec. 9, we discuss howwe can exactly isolate a single node numerically and observe the same physics.One of the main observables that we will focus on is the low energy densityof states, which is defined as ρ ( E ) = 1 L (cid:88) i δ ( E − E i ) (3)where E i are the low energy eigenvalues and L is the linear system size. Afocus of this review is understanding the stability of the Weyl semimetal phaseto disorder. One aspect of this question boils down to whether or not thedensity of states at the Weyl node remains precisely zero in the presence ofweak disorder, as scaling theory would predict [64, 65, 66]. In the clean limit,for a three-dimensional Dirac or Weyl semimetal the scaling of the low-energydensity of states goes like ρ ( E ) ∼ v E , (4)and central to this review is how ρ (0) is modified in the presence of disorder.The particular form the of H disorder depends on the physical problem, butone of the most studied problems is potential disorder. In this case, H disorder = (cid:88) r ψ † r V ( r ) ψ r , (5)where the potential V ( r ) is short-range correlated (cid:104) V ( r ) V ( r (cid:48) ) (cid:105) dis = f ( | r − r (cid:48) | )and averages to zero, where f ( r ) (cid:28) O ( e − r/ξ ) for r (cid:29) ξ and some ξ . In latticemodels, we will often take (cid:104) V ( r ) V ( r (cid:48) ) (cid:105) dis = W δ rr (cid:48) .In the absence of particle hole symmetry, it is important to remove theleading perturbative finite size effect. For a random potential V ( r ) drawn froma normal distribution with zero mean and standard deviation W , the leadingperturbative correction to the energy is E = L − (cid:80) r V ( r ) ∼ W/L / (randomsign) that produces a random broadening to the energy levels. To eliminatethis finite size effect, the potential can be shifted so that each sample has arandom potential that sums exactly to zero. We denote the shifted by potentialas ˜ V ( r ) = V ( r ) − L − (cid:80) r V ( r ). The simplest model to consider only has one parameter in the bandstructure(the hopping) that gives rise to three-dimensional Weyl fermions with timereversal symmetry on a simple cubic lattice. We will use this and the model inthe following section to introduce and describe the numerical results, then laterdiscuss other models and their differences. This model is defined as H Weyl = (cid:88) r ,µ = x,y,z (cid:0) it µ ψ † r σ µ ψ r +ˆ µ + H . c (cid:1) , (6)7here ψ r is a two-component Pauli spinor, the σ µ are the Pauli matrices with µ = x, y, z . The clean dispersion for t x = t y = t z = t/ E ( k ) = ± t (cid:113) sin( k x ) + sin( k y ) + sin( k z ) , (7)and the model has eight Dirac points at the time reversal invariant momentain the Brillioun zone. The band structure has time reversal symmetry butlacks inversion. In the presence of potential disorder, this model belongs to theGaussian Orthogonal Ensemble (GOE) of random matrices. This is due to anexplicit symmetry in the model described by ψ r → ( − x + z σ y ψ r . (8)Despite the model having the anti-unitary symmetry iσ y K (where K is complexconjugation in the real space basis), this model does not belong to a symmetryclass with T = −
1. To understand why, once we project this model into thesubspaces described by the symmetry (8), we obtain the Hamiltonian˜ H Weyl = − (cid:88) r [ t x ( − x + z c † r c r +ˆ x + t y ( − x + z c † r c r +ˆ y + t z c † r c r +ˆ z ] , (9)and ˜ H dis = (cid:80) r c † r V ( r ) c r where c r is no longer a spinor. This Hamiltonian makesit clear that the time-reversal operator for this model has T = +1. Moreover,the time reversal symmetry can be removed by putting twisted boundary con-ditions on each sample of size L , accommodating the twist with a gauge trans-formation amounts to t µ = t exp( iθ µ /L ) / − π < θ µ < π . By taking a twistof θ = ( π, π, π ) on even L we can maximize the finite size gap (= 2 √ π/L ))in our simulations. Another commonly studied model that we will discuss has broken time re-versal symmetry that allows the Weyl points to occur at arbitrary points in theBZ. This model is defined as H = (cid:88) r , ˆ ν (cid:0) ψ † r ( t ν σ z + t (cid:48) ν σ ν ) ψ r +ˆ ν + H . c (cid:1) − (cid:88) r ψ † r ( mσ z ) ψ r (10)where the hopping parameters are t ν = t/ ν = x, y, z and t (cid:48) ν = t (cid:48) / ν = x, y and zero otherwise. The band structure is determined by the massterm m and the hopping coefficients. The dispersion of the model is given by E ( k ) = ± (cid:115) t (cid:48) (sin( k x ) + sin( k y ) ) + [ t (cid:88) ν cos( k ν ) − m ] . (11)Setting t (cid:48) = t , the band structure hosts a regime with 4 Weyl points for | m | < t at K W = (0 , π, ± K ) , ( π, , ± K ) where K = arccos( m/t ), and 2 Weyl pointsfor t < | m | < t at K W = (0 , , ± K ) where K = arccos m/t −
2. At the8ransition between these regions at | m | = t, t an anisotropic nodal dispersionappears that for | m | = 3 t goes like E ( k ) ≈ ± t (cid:113) ( k z / + ( k x + k y ) at lowenergy. The dispersion at | m | = t is similar but there are 3 anisotropic nodalpoints at K N = ( π, , , (0 , π, , (0 , , π ). Interestingly, the anisotropic nodesgives rise to a low energy density of states ρ ( E ) ∼ | E | / and the renormaliza-tion group (see Secs. 3.2 and 8.2) will also predict the existence of a putativesemimetal-to-diffusive metal QCP and is an excellent case to compare with aWeyl node. In the presence of potential disorder, this model belongs to theGUE ensemble in the diffusive metal phase. At this point, it is useful to describe the theoretical and numerical impli-cations of twisted boundary conditions . We will limit the discussion here toone-dimension for ease, but the discussion easily generalizes to higher dimen-sions and even beyond square lattices.In the strictest sense, twisted boundary conditions for the single particlewave function in one-dimension (denoted x ) of size L identifies φ ( x + L ) = φ ( x ) e iθ (12)with a twist θ , which within our lattice model can be implemented via a gaugetransformation ψ x → e − iθx/L ψ x , (13)where x is the coordinate over which the twisted boundary conditions are imple-mented, and we have restored periodic boundary conditions via this gauge trans-formation. Physically, this is equivalent to threading a magnetic flux throughthe sample, breaking time-reversal symmetry. However, there is an alternativeway to understand these boundary conditions in terms of a time-reversal sym-metric system. If one considers an infinite system which is periodic with period L , then the Hamiltonian H = (cid:80) x,x (cid:48) ψ † x (cid:48) t x (cid:48) ,x ψ x , where we assume t x (cid:48) ,x = 0 when x (cid:48) − x ≥ L/
2, can be block diagonalized using Bloch’s theorem, and the resultis H k = L − (cid:88) x,x (cid:48) =0 e − ik ( x − x (cid:48) ) ψ † x (cid:48) t x (cid:48) ,x ψ x , (14)where k ∈ [0 , π/L ). If we identify k = θ/L , then this is precisely the gauge-transformed Hamiltonian with twisted boundary conditions. As such, the use oftwisted boundary conditions can be equivalently viewed as randomly samplingpoints in the Brillouin zone of an infinite system with a supercell of size L .
3. Perturbative Transition
Through the use of the self-consistent Born approximation and large- N tech-niques, one can identify both the perturbative stability of the semimetallic phaseand the existence of a purported transition [64, 65]. This method, however, does9ot give the correct critical theory (it is only correct in a large- N approxima-tion, see Ref. [109] for a full discussion), and more sophisticated techniques areneeded [66]. We nonetheless review it here since it gives one of the simplestmeans to understand the perturbative stability and the existence of criticality. Near a Weyl node, the first-quantized Hamiltonian takes the form H = − iv σ · ∇ + V ( r ) , (15)where V ( r ) is a disorder potential with (cid:104) V ( r ) (cid:105) dis = 0 and (cid:104) V ( r ) V ( r (cid:48) ) (cid:105) dis =Ω δ ( r − r (cid:48) ) (regularized by a momentum space cutoff Λ which is often suppliedby lattice regularization).The disorder-averaged Green’s function can be written in terms of a self-energy Σ as G ( k , ω ) = 1 ω − v k · σ − Σ( k , ω ) = (cid:28) ω − v k · σ − V ( r ) (cid:29) dis . (16)At second-order in V ( r ) and self-consistently, we can writeΣ( ω ) = Ω (cid:90) | k (cid:48) | < Λ d k (cid:48) (2 π ) ω − Σ( ω )( ω − Σ( ω )) − | k (cid:48) | . (17)At ω = 0 this equation has an obvious solution of Σ(0) = 0 which describesthe stability of the semimetal to disorder, but as we increase Ω, an imaginarysolution Σ(0) = iγ appears whenΩ > Ω c = 1 / (cid:90) | k (cid:48) | < Λ d k (2 π ) / | k (cid:48) | , (18)indicating a finite density of states at the nodal energy and a diffusive metal.This is also our first indication of a critical point Ω c .This is perhaps the simplest way to construct the self-consistent Born ap-proximation for this problem. For more detailed accounts see Refs. [110, 111,112]. With a one-loop renormalization group analysis [66] one can identify a tran-sition and obtain its critical properties. Up to order O (Ω ) using a d = 2 + (cid:15) expansion one finds the renormalization flow equations (after converting to thedimensionless Ω → ΩΛ / (2 π v )) d Ω d log(Λ) = (2 − d )Ω + 2Ω ,dvd log(Λ) = v ( z − − Ω) . (19)10n particular, in three-dimensions ( d = 3) the unstable fixed point is perturba-tively accessible and is located at Ω c = (cid:15)/ z = 1+ (cid:15)/ ν = 1 /(cid:15) . Taking (cid:15) = 1 (as appropriate for d = 3)these analytic predictions give z = 3 / ν = 1. While existing numericalwork finds agreement with the one-loop prediction for the dynamical exponent,the correlation length exponent has been harder to pin down .In the absence of non-perturbative effects, scaling theory can be appliedto extract the critical exponents from data on the density of states [87]. Thedynamical exponent can be found by the scaling of the density of states awayfrom the nodal energy (at Ω = Ω c ) ρ ( E, Ω c ) ∼ | E | d/z − (20)while the scaling of ρ (0) and its 2 n derivatives can give us both critical expo-nents [87, 113] | ρ (2 n ) (0) | ∼ | δ | − ( z (2 n +1) − d ) ν , (21)where δ = (Ω − Ω c ) / Ω c parametrizes the distance from the putative criticalpoint. The critical scaling form in energy is then ρ ( E, Ω) ∼ | δ | ν (3 − z ) f ± ( E | δ | − νz ) , (22)where f ± ( x ) are scaling functions for positive and negative δ . This analysis willbe useful in Sec. 5.2 after we find and suppress rare region effects to reveal thecritical behavior in these models before the avoidance rounds it out.Field theory descriptions and RG calculations of the transition in variousforms have gone well beyond what have detailed here. As the current reviewfocuses on the avoidance of the transition and we will not cover them in thisreview but point the reader to the relevant literature in Ref. [109].
4. Rare resonances in semimetals
To capture rare regions in Dirac and Weyl semimetals, a natural startingpoint is to treat them as sufficiently dilute so that we can ignore there in-teraction and replace each rare region with a strong potential well. Such anapproach works well in describing similar rare resonances in two-dimensionalDirac semimetals [95]. The quintessential way in which each rare region is thendescribed is via an exact solution to Eq. (15) with V ( r ) = V Θ( r − R ) [94], aspherical potential of strength V and radius R , replacing the rare region with asimple but strong potential. There are solutions for this potential at all energies E ; first, with spherical symmetry total angular momentum is a good quantum We suspect that this is due to past studies using the density of states at the Weyl nodeor the conductivity as an indicator of the transition, but this is problematic as they are bothexpected to always be non-zero due to rare region effects. Instead, the use of the secondderivative of the density of states provides an unbiased indicator as discussed in Sec. 5.2.
50 100 150 200 250 300 350
Realization . . . . . . . . E / t Weyl statesRare state r − − − − (cid:31) ψ † ψ ψ b i n (r) r r -1.9 r -1.6 sample 1sample 2 (a) (b) (c) Figure 2:
Rare eigenstates and a simple way to find them . (a) A demonstration offinding rare eigenstates by studying the distribution of the square of low-energy eigenvaluescomputed with Lanczos on H as a function of the disorder realization for the Hamiltonian H in Eq. (10) at weak disorder W = 0 . t and system size L = 18. The perturbative Weylstates form a dense band of states that make up a peak in the density of states (as shownin Fig. 4) whereas the rare states occur at lower energy below the peak filling in the finitesize gap (induced by the twist). (b) The corresponding rare eigenstate state labelled in (a)displaying its power law decay from its maximal value with a power law | ψ ( r ) | ∼ /r . ingood agreement with the theoretical expectation in Eq. (28). (c) Two distinct rare statesfound at low energy and weak disorder for W = 0 . t , L = 25, with power law exponents = 1 . . number, so we label eigenstates with radial quantum number n , total angularmomentum j , and J z quantum number j z such that ψ E ; n,j,j z ( r, θ, φ ) = f E,j ( r ) χ + j,j z ( θ, φ ) + ig E,j ( r ) χ − j,j z ( θ, φ ) (23)the spinors χ ± j,j z are eigenstates of σ · L with total angular momentum quantumnumbers previously defined and orbital angular momenta (cid:96) ± = j ∓ . The fullsolution of these equations is in Refs. [94, 103, 104] but for our purposes it isimportant to note that when E = 0, bound-state solutions exist for particularvalues (when j = 1 /
2, then V R = pπ with integer p ) and has f ,j ( r ) ∼ r − ( j +3 / , (24)for r (cid:29) R and g ,j ( r > R ) = 0. These solutions still exist for modifications ofthe potential away from a square well, and when the condition for their existenceis not quite satisfied (when slightly off-resonance), there is still an effect on thedensity of states (though if it effects the density of states exactly at the nodalenergy is disputed Refs. [103, 104, 114] at the time of this writing). At weak disorder and small L we can loosely break spectrum into two parts,perturbatively dressed Weyl states and non-perturbative rare states. This sep-aration has been argued to be parametrically enhanced in lower dimensionalmodels with a modified power law dispersion [116].The irrelevance of disorder implies that at weak disorder and finite L , themajority of the low energy eigenstates will be perturbatively renormalized Weylstates. In particular, applying perturbation theory yields perturbed energies12hat go like E ∼ /L (i.e. they still disperse linearly in momentum) with abroadening at weak disorder like ∼ W /L . In addition, the perturbed planewave states ψ pert k , ± ( r ) = ψ k , ± ( r ) + ψ k , ± ( r ) + . . . , (25)acquire an L independent contribution of random sign and magnitude W , ψ k , ± = 1 L / e i k · r φ k , ± , (26) ψ k , ± ( r ) ∼ W (random sign) , (27)where φ k , ± is a normalized two-component spinor as shown in Ref. [101]. Asshown in Fig. 3(a), this is in good agreement with numerical results at weakdisorder.In the model represented by Eq. (6), Weyl points exist at the time-reversalsymmetric points in the Brillouin zone. Thus for even system sizes L , we canchoose a twist θ = ( π, π, π ) to move the Weyl states maximally away in energy.In general, with careful analysis one can identify where the Weyl points are inthe mini-Brillouin zone defined by the twisted boundary conditions and movethem as far from zero energy as possible with a clever combination of systemsize and twists θ . This is useful for identifying states that develop due to arare resonance: they should be weakly modified via a twist in the boundaryconditions. In fact we can see in Fig. 2(a) an example of this where the typical(plane-wave-like) states all live in a narrow band (of width W /L ) around themean energy, and rare states are filling in the finite-size gap induced by thetwisted boundary conditions. As we predicted by Eq. (24), we expect thesestates to be quasibound, and in all likelihood they come from the smallest totalangular momentum j = 1 /
2, implying [94] ψ rare ( r ) ∼ | r − r | , (28)about some point r . Indeed, this is the case, such as in Fig. 3(b). The nu-merically observed rare states across the models in Eqs. (6) and (10) all show apower law decay, Fig. 2(b,c), with power law values ranging from ∼ . − . V ( r ) much larger than its surrounding values.Another clear distinction between perturbative and rare states can be seenin “twist dispersions” (i.e. determining how the energy eigenvalues vary under atwist). The perturbatively dressed states have Weyl cones in the dispersion witha reduced velocity v − v ∼ − W . The rare states on the other hand stronglyreconstruct the band structure and they disperse very weakly away from thecrossing points. It is also possible to find samples with multiple rare regions ina single sample. For example, as shown in Ref. [101] and Fig. 3(c), two rarestates produces a pair of quasilocalized states that have a non-zero tunnelingmatrix element between them. 13 a) (b) (c) Figure 3:
Eigenstates at weak disorder viewed through the projected probabilitydensity (cid:80) z | ψ E ( x, y, z ) | for the model in Eq. (6) . (a) An example of a perturbativestate at low energy E taken from the first Weyl peak at W = 0 . t and L = 25. The stateis delocalized across the system and the twist of θ x = π/ x -direction. (b) A rare state obtained for a disorder strength W = 0 . t and a (c) bilocalizedrare state with two rare regions for disorder strength W = 0 . t , both are values of W arebelow the avoided transition. Figures reproduced form Ref. [101].
5. Density of states
In the following section we focus on the effects of rare regions on the lowenergy density of states. First, we present a derivation of the density of statesusing instanton calculus and then its numerical estimate showing it remains non-zero for any disorder strength. This brings us to a measure of the avoidance, aswell as a phenomenological theory for the AQCP and a systematic way to tunethe avoidance length scale.
The issue of density of states was raised within the first paper on the subject,Ref. [64]. Formally, one can write the density state for the Weyl equation,Eq. (15) setting v = 1, as ρ V ( E ) = 1 L (cid:88) n δ ( E − E n ) (29)where ( − i σ · ∇ + V ( r )) ψ n = E n ψ n . This can be written as a functional integral ρ V ( E ) = 1 L (cid:90) D [ ψ ( r ) , χ ( r ) , Υ] e i (cid:82) d rχ † ( r )( E + i σ ·∇− V ( r )) ψ ( r )+ i Υ[( (cid:82) d rψ † ( r ) ψ ( r )) − . (30)Determining the disorder averaged quantity ρ ( E ) then for a Gaussian randomfield with (cid:104) V ( r ) V ( r (cid:48) ) (cid:105) = W K ( r − r (cid:48) ) can be accomplished also with a functionalintegral ρ ( E ) = (cid:90) D [ V ] e − W (cid:82) d r d r (cid:48) V ( r ) K − ( r − r (cid:48) ) V ( r (cid:48) ) ρ V ( E ) , (31)where K − is defined such that (cid:82) d y (cid:48) K − ( y − y (cid:48) ) K ( y (cid:48) − y (cid:48)(cid:48) ) = δ ( y − y (cid:48)(cid:48) ).Taking the saddle point approximation of this averaged quantity at E = 0 thenamounts to solving the non-linear integro-differential equation (cid:20) − i σ · ∇ − χ W (cid:90) d r (cid:48) K ( r − r (cid:48) ) ψ † ( r (cid:48) ) ψ ( r (cid:48) ) (cid:21) ψ ( r ) = 0 , (32)14 . − . . . . E/t − − − − − − − − ρ ( E ) L = 30 , W/t = 0 . − . − . . . . E/t − − − − − − − − ρ ( E ) Normalized, N R = 10 Unnormalized L = 30 , W/t = 0 . N R = 40 N R = 80 N R = 160 Figure 4:
Density of states ρ ( E ) as a function of energy E displaying low-energyperturbative states . (Left) The use of periodic boundary conditions (and certain systemsizes) can place a zero energy Weyl state in the spectrum that becomes broadened by disorder.This produces a large finite size effect in the zero energy density of states that obscures therare region contribution that is orders of magnitude smaller at weak disorder. (Right) Usinga twist to move the Dirac state to a non-zero energy introduces a finite size gap of order ∼ /L . The low energy peaks are refereed to as “perturbative Weyl peaks” that are composedof Weyl eigenstates that have been perturbatively renormalized by the disorder. The use ofnormalized random vectors (or random phase vectors) allows for an accurate determinationof the zero energy density of states. From Ref. [39]. for normalized solutions (with χ allowed to be freely chosen). As a mean-field,this will lead to a change in the density of states δρ (0) = 1 L e − χ (cid:82) d r d r (cid:48) ψ † ( r ) ψ ( r ) K ( r − r (cid:48) ) ψ † ( r (cid:48) ) ψ ( r (cid:48) ) . (33)As it turns out, there are solutions to (32) that resemble the rare resonancespreviously discussed in Sec. 4 (see Ref. [94] for the full analysis). Within somesuitable approximations, given some correlation length of the disorder ξ andstrength W , the change of the density of states for a single rare resonance willfollow δρ (0) ∼ L e − C ξ W for some constant C while if there are a finite density ρ of these, we expect δρ (0) ∼ ρ e − C ξ W , (34)for some constant C , at the level of the saddle-point approximation for a single rare state. This picture is complicated by work that includes fluctuations ontop of the mean-field (in Refs. [103, 104]) and finds an extensive number ofzero-modes which drive δρ (0) → The following results are obtained using numerically exact approaches thatutilize sparse matrix vector multiplication to reach large system sizes beyondthat accessible within exact diagonalization. In particular, the density of states15 l og ρ ( ) (t/W) (a) KPMLanczos ρ ( ) W/t (b)
LanczosL=70 no twist90 no twist110 no twist30 twist- π
40 twist- π
50 twist- π
60 twist- π Figure 5:
Rare region contribution to the density of states in the model in Eq. (6).(a) The log of the zero energy density of states computed using the KPM and Lanczos (at asmall but finite energy) averaged over a large number of samples at fixed twist values displayingan excellent fit to the rare region form in Eq. (35) for over four orders of magnitude. (b)Comparing the zero energy density of states with periodic boundary conditions and even L that produces a large finite size effect due to perturbative Dirac states (see Fig. 5). This finitesize effect masks the rare region contribution that is converged in system size deep into thesemimetal regime (below the AQCP) and several orders of magnitude below the Weyl peakartifacts . Instead the density of states is always non-zero, though exponentially small at weakdisorder. Taken from Ref. [101]. (DOS) is computed using the kernel polynomial method (KPM) that expandsthe density of states using Chebychev polynomials to a finite order N C [117].The calculation of the DOS using KPM involves a stochastic trace, which toresolve the exponentially small rare region contribution needs to be done asaccurately as possible. This can be achieved by evaluating the stochastic traceusing either random phase vectors [118, 119] or random vectors with a fixednormalization [120] (see Fig. 4), either of which reduces the noise in the signalby orders of magnitude allowing for the rare region contribution to be properlyestimated well below the avoided quantum critical point.Focusing on the model in Eq. (6) with a twist of θ = ( π, π, π ) (to push theWeyl states as far away from E = 0 as possible similar to Fig. 4). The relevantquestion is now whether rare states fill in the finite size gap in an extensivemanner that is L -independent. The average density of states in the semimetalregime has a rare region contribution that fills in the finite size gap and is L -independent. At higher energies the Weyl peaks round into Weyl shoulders thatare still spaced like ∼ /L . Extracting the L -independent contribution of thelow energy density of states is shown in Fig. 5 and compared with the results ofperiodic boundary conditions that host a clean Dirac state at zero energy. Therare eigenstates thus contribute a non-zero but exponentially small DOSlog ρ (0) ∼ − ( t/W ) (35)in good agreement with the theoretical prediction of Ref. [94] and Eq. (34).The presence of a non-zero DOS at the Weyl node suggests that the semimetal-to-diffusive metal QCP has been rounded out into a crossover. In order to testfor the existence of a disorder driven quantum critical point, it is essential to16etermine if the density of states remains an analytic function of energy. Theanalytic properties of the density of states can be determined by directly eval-uating its energy derivatives. As the transition is avoided, in the presence ofrandomness, the DOS in the vicinity of the Weyl node energy remains analytic,namely ρ ( E ) = ρ (0) + 12! ρ (cid:48)(cid:48) (0) E + 14! ρ (4) (0) E + . . . . (36)While the coefficients of this expansion can in principle be estimated using a fitthere is a weak amount of rounding introduced by the fit that is not intrinsic tothe problem. Therefore, it is more accurate to use the KPM expansion of ρ ( E )to obtain the exact expression for the derivatives ρ (cid:48)(cid:48) (0) and ρ (4) (0) in terms ofa similar Chebyshev expansion, the explicit expression for ρ (cid:48)(cid:48) (0) can be foundin Ref. [113].The analytic nature of the DOS according to Eq. (36) is computed withKPM and shown in Fig. 6(a), demonstrating that ρ (cid:48)(cid:48) (0) has a broad maximumfor increasing disorder strength. The location of the peak provides an accurateand unbiased estimate of the avoided QCP. We stress that the DOS at zeroenergy cannot be used as an indicator of the transition as it is always non-zero, and we suspect that a similar issue plagues the DC conductivity, though acareful numerical analysis of the rare region contribution to transport remainsto be done. Focusing on ρ (cid:48)(cid:48) (0), we saturate the height of the peak as we increasethe system size and KPM expansion order as shown in Fig. 6. As ρ (cid:48)(cid:48) (0) doesnot diverge, we conclude that the DOS remains an analytic function of energy,and the perturbative transition is rounded out into a cross over. Motivated by the success of the instanton saddle-point point approxima-tion capturing the rare region contribution to the density of states Gurarie inRef. [102] treated the system at weak disorder as a gas of instantons interactingwith a pair potential in a power-law like form (due to the quasi-localized rarestates) that goes as ∼ a /r where a is the characteristic size of a rare state.The resulting effective field theory produces a single particle Green’s functionthat is analytic with a finite correlation length that is given by ξ ∼ /ρ (0) ∼ A e a ( t/W ) , (37)here A and a are non-universal constants. This result confirms that the non-zerodensity of states at the Weyl node produces a finite length scale, albeit expo-nentially large at weak disorder, which rounds out the perturbative transition.As the value of the density of states and its dependence on the disorder strengthdepends strongly on the type of disorder distribution chosen, it is an interestingto understand how to tune the avoidance length scale ξ , which we now turn to. In order to access the properties of the AQCP we need to first remove therounding of the transition by suppressing rare events and obtain an accurate17stimate of the location of the avoided transition (which is in fact difficult as it isa cross-over). First, it is possible to tune the probability to generate rare eventsrather naturally through varying the disorder distribution P [ V ]. As shown inRef. [113], tuning the tails of P [ V ] allows us to control the strength of theavoidance as measured by the size of ρ (cid:48)(cid:48) (0). It was found that binary disorderhas a rather strong suppression of rare events and so interpolating betweenGaussian and binary via a double Gaussian distribution is a rather nice choiceof P [ V ] to control the strength of the avoidance. As shown in Fig. 6 (b), aswe tune from Gaussian to binary ρ (cid:48)(cid:48) (0) goes from exhibiting a strong avoidancewith a highly rounded peak and a small value to significantly sharpen as rareevents are suppressed. Importantly, we can use the location of the maximum ofthe peak in ρ (cid:48)(cid:48) (0) as an unbiased, accurate estimate of the avoided transition.We stress that any estimate of the location of the AQCP based on the densityof states ρ (0) or the conductivity at the Weyl node will be plagued by the factthat they are always non-zero due to rare regions and neither have a truly sharpchange in the thermodynamic limit. However, there have been several attemptsin the literature to estimate the location of the AQCP using where the densityof states looks like it is lifting off of zero on a linear scale, which tends to greatlyunderestimate W c that leads to in inaccurate estimate of ν .In the limit of binary disorder we aptly posed to estimate the critical prop-erties of the AQCP before it becomes rounded out by rare regions at the lowestenergy scales. Using binary disorder we estimate the critical exponents ν and z from the data in Fig. 7 using the critical scaling form in d = 3 dimensions[87, 113] of the zero energy density of states and its 2 n derivatives as we de-scribed in Sec. 3 (near the critical point we can replace Ω with W , though wenote here Ω ∝ W ). In addition, within the KPM the finite expansion orderleads to an infrared energy scale that goes like δE ∼ /N C and thus we canuse N C scaling in the manner we do finite E scaling to obtain similar scalingfunctions. For example, if we consider the scaling of the density of states at theAQCP W = W c at sufficiently large L we have ρ ( E, W = W c , N C ) ∼ N − d/zC g ( EN C ) (38)where g ( x ) is an unknown scaling function. We stress that due to the avoidednature of the transition these scaling forms are only applicable in the energyregime E (cid:29) E ∗ where E ∗ is non-zero energy scale which is induced by the rareregions of the random potential.The collapse of the numerical data, shown in Fig. 7(a), works over a fulldecade until the avoidance leads to deviations from the critical scaling form.The power law nature of the zero-energy density of states and the second deriva-tive above and below the transition respectively are shown in Fig. 7(b), whichdisplays a critical power law as in Eq. (21) over a full decade until the avoid-ance rounds it out at small δ due to the non-zero DOS. In Figs. 7(c) and (d), wecollapse the data in terms of the finite energy scaling forms in Eq. (22) aboveand below the transition. The collapsed data is compared with the cross overfunctions that are obtained by integrating the RG equations in Eq. (19), which18 ρ '' ( ) W/t N C =2 , L=312 , 312 , 312 , 712 , 131 ρ '' ( ) W/t box σ =10.750.50.250 -2-1.6-1.2-0.8-0.4 0 0.7 0.8 0.9 1 ρ ( ) ( ) x10 W/t (a) (b)(c) (d)
Figure 6:
The strength of avoidance . (a) Converging the peak in ρ (cid:48)(cid:48) (0) for the case ofGaussian disorder with increasing system size and KPM expansion order, which demonstratesthat the density of states remains an analytic function of disorder and energy. (b) The strengthof the avoidance can be tuned by varying the tails of the probability distribution; here we usea double Gaussian distribution with mean ± W √ − σ and a standard deviation σW so thattuning σ controls the tails of the two Gaussians. As σ → ± W ) where the avoidance is the weakest. (c) ρ (cid:48)(cid:48) (0) as a function ofdisorder for various values of σ , demonstrating it tunes the strength of the avoidance. We alsocompare to the standard box distribution uniformly sampled between [ − W/ , W/
2] which hasa much weaker avoidance than Gaussian but not as sharp as binary disorder. (d) The fourthderivative showing the avoidance can also be tuned in the higher derivatives of the low-energydensity of states. Taken from Ref. [113]. display remarkable agreement (after adjusting the two bare RG energy scales);see Ref. [73] for the explicit cross-over functions.
6. Excitations and Transport
We now turn to the nature of the single particle excitation spectrum andtransport properties within linear response.
The excitation spectrum of the system can be extracted through the analyticstructure of the single particle Green’s function. Averaging over disorder, the19 -1 N C d / z - ρ ( E ) N C E/t N C , L (a) , 312 , 312 , 312 , 712 , 1212 , 1512 , 181 -3 -2 -1 -2 -1 ρ ( E ) E/t -3 -2 -1 -2 -1 | δ | (b) ρ (0), W>W c ρ ''(0), W W/t=0.7000.7250.7500.7750.8000.8200.8300.8400.850 -3 -2 -1 | δ | ν ( z - d ) ρ ( E ) | δ | - ν z E/t (d) W/t=0.8800.890.900.910.920.940.960.981.001.051.10 Figure 7: The perturbative critical scaling of the DOS is clearly revealed usingbinary disorder to suppress rare region effects for the model in Eq. (6) . (a) FiniteKPM expansion order scaling of the energy dependence of the DOS at the avoided transitionwith binary disorder utilizing the scaling form in Eq. (38), which shows a good collapse.(Inset) Unscaled density of states showing the power-law scaling at the transition ρ ( E ) ∼| E | d/z − with z = 1 . ± . 05 for increasing N C . (b) The power-law scaling of the zeroenergy density of states (and its second derivative) above (and below) the AQCP showing apower-law dependence with a fit to Eq. (21) (dashed lines) that yields a correlation length ν = 1 . ± . 06 from ρ (0) and ν = 1 . ± . 08 from ρ (cid:48)(cid:48) (0). Collapsing the data followingthe finite energy scaling form in Eq. (22) is shown in (c) and (d) for W < W c and W > W c respectively. The dashed lines are the cross-over functions computed by integrating the RGequations and are explicitly presented in the Appendix in Ref. [73]. Results from Ref. [113]. retarded Green’s function — expanded in the vicinity of single-particle poles —takes the form G ± ( k , ω ) ≈ Z ± ( k ) ω − E ± ( k ) + iγ ± ( k ) (39)where Z ± ( k ) is the quasiparticle residue, E ± ( k ) is the single particle spectrumin the valence and conduction band basis (labelled ± ), and γ ± ( k ) = 1 /τ ± ( k ) isthe damping rate or inverse quasiparticle lifetime.To get a sense of how rare regions can alter the excitation spectrum webegin by reviewing the T -matrix results from Ref. [121]. If we work in thedilute impurity limit of a collection of spherically symmetric square wells thepotential takes the form V ( r ) = (cid:80) N imp j =1 λ j Θ( b − | r − R j | ) for N imp impuri-ties with a fixed width b of random strength λ j taken from a Gaussian dis-tribution P [ λ ]. The full Green function is obtained through Dyson’s equation20 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 γ ( k = ) /t (t/W) (a) KPMT-matrix 0 0.04 0.08 0.12 0.16 0.2 0.5 1 1.5 2W/t Z ( k = ) W/t (b) T-matrixKPMRG 0 0.4 0.8 1.2 1.6 0 1 2 3 4 Figure 8: The nature of single-particle excitations at the Dirac node at k = 0 forthe model in Eq. (6). (a) The exponentially small damping at weak disorder following thenon-perturbative rare region form in Eq. (43). (b) The quasiparticle residue at the Dirac noderemains finite across the AQCP as demonstrated in the exact numerics with Gaussian disorderthat we compare with the RG that misses the rare-regions and the T-matrix which misses theperturbative transition. (Inset) The quasiparticle residue out to large disorder. Results takenfrom Ref. [121]. G ( k , ω ) − αβ = G (0) αβ ( k , ω ) − αβ − Σ αβ ( k , ω ) where G (0) αβ ( k , ω ) is the free Green. Inthe dilute impurity limit ( i.e. N imp /V (cid:28) V ) the self-energy ismomentum independent and given byΣ αβ ( ω ) = δ αβ N imp V (cid:90) dλP [ λ ] T ( λ ) ( k , k ) | v | k | = E (40)where T ( λ ) ( k , k ) denotes the T -matrix of for scattering off of a single impurity ofstrength λ and P [ λ ] denotes the normal distribution used to sample λ j . For eachrealization of λ , the T -matrix is computed analytically by solving the quantummechanical scattering problem off a single well. The T -matrix is given by T ( λ ) ( k , k ) = − πvk (cid:88) j (2 j + 1) e iδ j − ik (41)and it is solely determined by the scattering phase shift δ j (in the angularmomentum channel j ) that is known exactly for this problem (see Ref. [94] forthe explicit expression). As a result, the leading long-wave length contributionarises from only considering the j = 1 / E ± ( k ) , γ ± ( k ) , Z ± ( k ) atboth the perturbative and non-perturbative level from the simplified problemof a single impurity.To develop a complete solution of the problem, in Ref. [121] the single particleproperties where extracted by computing the Green’s function with the KPMthat was compared with the T -matrix approximation. At weak disorder atthe Weyl node ( K W ) the average low energy dispersion remains linear withmomentum E ± ( k ) = ± v ( W ) | k − K W | , (42)21here the Weyl velocity v ( W ) is reduced (perturbatively like − W ). On theother hand, the damping rate is exponentially small in disorder, namely for k ≈ K W γ ( k ) ≈ γ ( K W ) + B | k − K W | γ ( K W ) = Ae − a ( t/W ) , (43)which agrees well with a T -matrix approximation as shown in Fig. 8(a). While inthis regime Z ( k ) ≈ const . and 0 < Z ( k ) < 1, the notion of Weyl quasiparticleexcitation exists, but they are no longer sharp due to the non-zero lifetime.Namely for sharp excitations to exist at the Weyl node requires | E ± ( k ) | (cid:29) γ ( k )as k → K W so that the (effectively) Lorentzian peaks in the spectral functionat E ± ( k ) have negligible overlap. However, due to rare regions we necessarilyhave γ ( K W ) (cid:29) | E ± ( k ) | for k ≈ K W as | E ± ( k ) | → z = 3 / ν = 1. In addition, the anomalous dimension of the AQCP (denoted as η )is defined via the Green function asRe G (0 , ω ) ∼ ω (1+ η ) /z . (44)where the power law only holds for frequencies above the rare region energy scale( ω (cid:29) E ∗ ). Numerical calculations using binary disorder to suppress rare regionshave obtained η = 0 . ± . 04, which is in good agreement with a modified RGscheme from Ref. [66] that yields η = 1 / 8. As shown in Fig. 9(b) we can collapsethe data using a finite KPM ( N C ) scaling formRe G (0 , ω = 0 , N C ) ∼ N (1+ η ) /zC h ( ωN C ) , (45)where h ( x ) is an unknown scaling function for well over two decades. We close this section by discussing the diffusive transport properties of disor-dered Weyl semimetals that is mediated by the rare regions and will not reviewthe work on the perturbative transition [68, 75]. Currently, demonstrating these22 A + ( k , ω ) ω /t L=25W/t=0.5 k=3k , R3k , T2k , R2k , Tk , Rk , T (a) -1 N C ( + η ) / z G ' ( , ω ) - N C ω /t N C =2 -3 -2 -1 -2 -1 G ' ( , ω ) - ω /t (b) Figure 9: Dynamical scaling of single-particle excitations in momentum space forthe model in Eq. (6). (a) The single-particle spectral function for individual samples atdifferent momentum k = 2 π/L comparing a typical (i.e. perturbative) sample and a raresample with a non-perturbative state, which produces a non-zero overlap near ω = 0 for anymomentum due to the power-law nature of the wavefunction. (b) The critical nature of theGreen’s function at the Dirac node ( k = 0) and binary disorder at the AQCP with excellentdata collapse in terms of the finite KPM scaling form in Eq. (45). (Inset) The real part ofthe inverse Green’s function at k = 0 at the AQCP showing a clear power-law scaling inagreement with Eq. (44) until it rounds out at the lowest energy due to the avoidance. FromRef. [121]. effects numerically is an open problem, despite the fact that hints of the avoidedtransition have been seen in scaling of the conductance [88, 89] and wavepacketdynamics [93]. In particular, Ref. [88] identified the perturbative semimetalregime and the diffusive metal phase in the average conductance, however nosigns of rare region induced diffusion where found in the semimetal regime.Nonetheless, signs of critical scaling have been observed in the behavior of theaverage conductance as a function of disorder strength and Fermi energy [89] aswell as a decreasing wavepacket velocity near the AQCP [93].As shown in Ref. [94] at sufficiently weak disorder there are three expectedregimes in energy for the transport properties at linear response as seen in inFig. 10. At high energies the behavior is dominated by the self-consistent Bornapproximation of Sec. 3. At the lowest and intermediate energy scales, tunnelingbetween rare regions dominate the behavior leading to diffusive transport anda finite DC conductivity. Identifying these signatures in numerics remains apressing open question.Ref. [122] has built upon the previous T -matrix approach making it selfconsistent to compute the DC conductivity. Solving the self-consistent equationsyields a general result for the DC conductivity for an arbitrary scattering rate γ ( µ ) at the chemical potential µ , that is given by σ dc ( µ ) = σ µ + γ ( µ ) γ ( µ ) , (46)where σ > µ = 0 and the conductivity is determined by the scattering rate (and23 igure 10: The three distinct scaling regimes of the conductivity σ (solid blue line)and diffusion constant D (red dashed line) as a function of the chemical potential µ due to rare regions. At high energies µ > µ ∼ v / ρ (0) / b / /W the self consistentborn described in Sec. 3 works well. In the intermediate regime v ρ (0) b ∼ µ < µ < µ ∼ v / ρ (0) / and the low energy regime µ < µ the density of states is dominated by rareregions. In the low energy regime transport is mediated by tunneling between rare regions[as in the bi-quasi-localized wavefunction in Fig. 3 (c)] that produces a diffusion constant D = l /τ where l is the typical hopping length and τ is the inverse damping at the Weylnode which is exponentially large at weak disorder. This approach misses the presence of theAQCP and so it is absent from these scaling regimes. Reprinted figure with permission fromNandkishore et al. Phys. Rev. B 89, 245110 (2014) [94]. Copyright (2014) by the AmericanPhysical Society. σ ), which is exponentially small but non-zero at weak disorder due to rareregions as shown in Eq. (43) and Fig. 8. Thus the DC conductivity is expectedbe non-zero for any non-zero disorder strength and rare regions have convertedthe Weyl semimetal to a diffusive metal. In future work, it will be worthwhileto test these predictions numerically. 7. Rare regions and the topological properties of Weyl semimetals The effects of rare regions on the topological properties of Dirac and Weylsemimetals remains a pressing question of ongoing research. Here, we brieflyreview work on the effects of rare regions on Fermi arc surface states and theaxial anomaly. It will be very exciting to build on these results to understandthe effects of rare regions on the nature of quantum oscillations in thin filmsand the negative magnetoresistance that has been used to infer the existence ofthe axial anomaly. The Fermi arc surface states are robust to surface disorder [123]. However,while not robust to bulk disorder, the surface states still survive to some extentwith bulk disorder: perturbatively becoming power-law [124] and dissolving intothe bulk [125] hybridizing with rare states [115]. Recent work [126] further showsthat a perturbative transition on the surface can occur, separate from the bulk24 urface chiral velocity (a) (b) − . − . . . . k y /π − − E n ( k y , k z = ) − . − . . . . k z /π − − E n ( k y = , k z ) (c) Figure 11: Fermi arc surface states in the model defined in Eq. (10) for two Weylnodes with open boundary conditions along the x -direction. (a) A schematic of thesurface Brillouin zone displaying the projection of the Weyl points onto the surface and theFermi arc of surface states connecting them. Their chiral velocity is denoted with a greenarrow. (b) and as a function of k z for k y = 0 (c) demonstrating the surface Fermi arcdispersion in Eq. (50). The surface states of the opposing surfaces are shown in red and blueand the bulk states are in black. Taken from Ref. [115]. (which rare events probably also make into an avoided criticality). We reviewhere how the surface character persists despite these facts.The interplay of Fermi arc surface states and rare regions can be most easilystudied in the model in Eq. (10) with two Weyl points and a single Fermi arcbetween them on particular surfaces. For the present model this represents aFermi arc along a straight line between the projection of the two Weyl pointsonto either the x − z or y − z surface, see Fig. 11 (a). In the clean limit of themodel in Eq. (10) (setting t = t (cid:48) ) we obtained the exact surface eigenstates tobe (for opening the boundary along x ) ψ S ( x, y, z ) = 1 L e i ( yk y + zk z ) f S ( x ) φ, (47) f S ( x ) = (cid:112) − λ λ x − (48)where the spinor φ T = (1 , − / √ 2, we have defined λ ( k y , k z ) = m/t − cos( k y ) − cos( k z ) , (49)and λ ≥ ψ S ∼ e − x/ξ ( k ⊥ ) where ξ = 1 / ln λ and k ⊥ = ( k y , k z ) is the momentum on the surface. Thedispersion of these Fermi arc surface states is E S ( k ⊥ ) = t sin( k y ) (50)and the opposing surface have states that disperse with the opposite chirality.The dispersion computed with open boundaries in the x -direction along k y (for k z = 0) and along k z (for k y = 0) are shown in Fig. 11(b) and (c), respectively,showing excellent agreement with Eq. (50). Even though these states only dis-perse along k y , the results presented in this section are qualitatively insensitiveto any non-zero curvature along the arc that will make them disperse along both k y and k z . 25 y l n ρ ( x , y ) − − − x y l n ρ ( x , y ) − − − x y l n ρ ( x , y ) − − − x y l n ρ ( x , y ) − − − . . . . . . φ z / π . . . . . E / t . . . . W e i g h t o n x = s u r f a ce . . . . . φ y / π . . . . . . . E / t . . . . W e i g h t o n r a r e s t a t e (a)(b) (c) (d)(e) (f) c de f Figure 12: Fermi arc surface states hybridizing with perturbative and rare bulkstates. Energy eigenvalues as a function of the twist in the boundary condition (i.e. twistdispersions) versus a twist along z for a perturbative state (a) and a twist along y for a rarestate (b), in both cases periodic boundary conditions shown in green. In (a) we identify aperturbative hybridization between a bulk Weyl state and a topological Fermi arc surface statethat is shown in (c), where as in (b) we see that the linearly dispersing Fermi arc surface statehas hybridized with the weakly dispersing rare state. Corresponding hybridized wavefunctionsare shown in (d), (e), (f) as labelled in (b) demonstrating that they are linear combinationsof Fermi arc surface states and quasilocalized rare states. Taken from Ref. [115]. These surface states produce a metallic density of states. Upon introduc-ing disorder, these surface states broaden and develop a non-zero quasiparticlelifetime both perturbatively and non-perturbatively. Despite being quasi-two-dimensional they do not localize in the presence of weak disorder. Within per-turbation theory these surface states remain bound to the surface in a power lawfashion [124]. Non-perturbative rare states in the bulk however hybridize witheach state along the arc as they are broad in k − space due to the 1 /r powerlaw tail (as discussed in Sec. 6.1). As a result, the extensive bulk DOS fullyhybridizes with the surface states in the large L limit [115]. The surface-bulkhybridization due to rare regions is shown in Fig. 12.Despite the surface-bulk hybridization the chiral velocity of the states re-mains remarkably robust. In the clean limit the chiral velocity is defined as v c = ∂ k ⊥ E S ( k ⊥ ) and at zero energy is given by v c = t ˆ y , thus we can focus onthe y -component in the following. In the presence of disorder, we generalize thedefinition of the chiral velocity in terms of derivative of a twist in the boundarycondition ( φ y ), which is given by v c,S = Tr S (0) ( ∂H/∂φ y | φ y =0 ) (51)where Tr S ( n ) is a trace over the sheet (in the y − z plane) n with n = 0 beingone of the two surfaces, and − e∂H/∂φ y | φ y =0 = J y is the current operator alongthe y -direction. Generalizing this notion, we define a chiral velocity per sheet26 x − . − . . . . v c ( x , E = ) W/t = . . . . . . . 02 4 60 . . . (a) (b) W/t . . . . . . v c , S L = v S − − Figure 13: The chiral velocity of Fermi arc surface states in the presence of disorder. (a) The surface chiral velocity as a function of disorder strength W for various system sizesas well as its value extracted from the pole of the surface Green function (in red). (Inset)Surface chiral velocity on a log-scale showing that it remains non-zero even upon entry into theAnderson insulating phase, which it does by forming local current loops within the localizationlength. (b) The average chiral velocity per sheet as a function of x as in Eq. (52), showingthat it becomes random in the bulk and remains on the edges to large disorder. Taken fromRef. [115]. of the system S ( x ) in a manner that can be computed with the KPM to yield v c ( x, E ) = Tr S ( x ) ( J y δ ( E − H ))Tr S ( x ) ( − eδ ( E − H )) , (52)for the Hamiltonian H . Data for the surface chiral velocity as a function ofdisorder strength is shown in Fig. 13(a). Interestingly, while on a linear scaleit appears that v c,S goes to zero at the Anderson localization transition in thebulk (that occurs near W l /t ≈ . − . So far our analysis has focused on Weyl semimetals in the presence of dis-order. We apply a constant magnetic field in the z -direction to the model inEq. (10), which we include by Peierls substitution t y (cid:55)→ t y e − iBx , t (cid:48) y (cid:55)→ t (cid:48) y e − iBx , (53)for all sites. In order take periodic boundary conditions in the x and y directionswe choose a gauge where t x (cid:55)→ t x e − iBL x y , t (cid:48) x (cid:55)→ t (cid:48) x e − iBL x y (54)27 igure 14: Magnetic field effects on rare state and band structure. (a) A rare statein the presence of an orbital magnetic field of various strengths parameterized by the fluxΦ for multiplies of the flux quanta Φ survives and does not affect the short range natureof the power law decay, though the longer range nature of the wavefunction is modified bythe field. At larger fields, when the magnetic length is of similar size as the rare region weexpect this behavior to change, which is an interesting open question. (b) For a twist alongthe z -direction (Eq. (10)) without a magnetic field, we clearly observe the two Weyl cones inthe twist dispersion in the folded band structure. (c) In the presence of a magnetic field twochiral Landau levels with opposite velocity develop (green and blue) along the direction ofthe magnetic field emanating out of the two Weyl points with different helicities. Taken fromRef. [127]. for the boundary hopping terms between the first and last sites (i.e. x = 1 and x = L x , where L x is the system size in the x -direction). (Note that we aresetting the lattice constant a = 1, as well as (cid:126) = e = 1). Due to the periodicboundary conditions in x and y we can only access integer values of the magneticflux quanta Φ = h/e .To demonstrate that rare eigenstates survive the addition of a magnetic fieldwe first find a rare state with B = 0 and then “turn on” the field as shown inFig. 14(a). This demonstrates that the power-law bound rare eigenstates survivethe addition of a magnetic field, which is quite natural as their local nature willonly be affected by fields with a magnetic length of comparable size. We seein Fig. 14(a) that the rare state is modified at larger distances but the shortdistance power law tail remains robust for this range of magnetic fields.The lowest Landau levels of the time-reversal broken Weyl model in Eq. (10)are chiral, emanating out of each Weyl node that disperse along the directionof the applied magnetic field (in this case the z -direction). To probe the chiralLandau levels in the absence of translational symmetry we apply a twist ( φ z )along the z -direction and study how the low energy eigenstates near the chiralLandau levels disperse as a function of φ z . In the clean limit as shown in Fig. 14(c) this produces two Landau levels of opposite chirality emanating out of theoriginal Weyl points that can be seen in Fig. 14 (b).We can further deform the model to access the physics of a single cone byapplying a non-local potential. In particular, we add a momentum dependentpotential U ( k ) to the model in Eq. (10) (in the parameter space of two Weylnodes), where U ( k ) = (cid:40) , for 0 ≤ k z < π,U , for π ≤ k z < π. (55)28 igure 15: A single Weyl cone realized in the lattice model in Eq. 10 and rarestates in the presence of an orbital magnetic field Φ . The single Weyl node is onlyeffective as we have added the non-local potential in Eq. (55) with U = 2 t to “push” theother Weyl node to high energy. (a) The dispersion curve of the single chiral Landau levelin the clean limit, the arrows denote the chiral band consistent with Eq. (56). (b) A samplewith a rare eigenstate and a magnetic field corresponding to one flux quanta Φ = Φ . Therare state is weakly dispersing and it hybridizes with the chiral Landau level. Nonetheless,the topological charge pumping process following Eq. (55) remains satisfied. (c) The samerare sample with two flux quanta Φ = 2Φ , now the rare state only disrupts the chiral natureof the band for one of the chiral Landau levels. (d) A sample with two rare regions thatproduces a bi-quasi-localized wavefunction that is able to hybridize with both of the chiralLandau levels but does not destroy the topological nature of the bands. The color indicates ξ L = ( (cid:80) r | ψ r ( E ) | ) − / which is a localization length; when ξ L ∼ L the state is maximallydelocalized. Taken from Ref. [127]. and by focusing on low energies we can ignore the other cone that has beenpushed to much higher energies that we take to be U = 2 t . As a result, inthe presence of a magnetic field at low energy we are now effectively probing asingle Weyl point as shown in Fig. 15(a).In the following we will distinguish between chiral bands, where the topologi-cal charge pumping process can occur, from non-chiral bands where this has beendestroyed by examining the velocity of the each eigenvalue E n , v ( φ ) = dE n /dφ (where φ is the twist along z , dropping the subscript in this section) throughone pumping cycle. Namely, the bands are chiral if δE = (cid:90) π v ( φ ) dφ = E (2 π ) − E (0) (cid:54) = 0 , (56)29 igure 16: A rare state in the presence of an orbital magnetic field for the model inEq. (10) with two Weyl nodes .(a) For one flux quanta Φ = Φ a rare state hybridizes thetwo chiral Landau levels at low energy near the Weyl node destroying the topological chargepumping process. (b) For two flux quanta Φ = 2Φ and one rare state one renormalized chiralLandau level remains. Taken from Ref. [127]. whereas trivial bands will always have δE = 0.Focusing on the limit of a single Weyl node and a field strength correspondingto one flux quanta, as shown in Fig. 15(a), we see the non-local potential hassuccessfully pushed the other Weyl point to higher energy. As found in Fig. 15(b)the non-dispersing band due to a single rare region hybridizes with the chiralLandau level. However, it does so in manner that retains the chiral nature ofthe band as defined in Eq. (56) by wrapping around the mini-BZ to continuethe charge pumping process and the axial anomaly survives. In other words,the presence of a rare state for a single Weyl node only slows down the chargepumping process but does not destroy it. It is interesting to then thread twoflux quanta through the same rare sample as shown in Fig. 15(b), where the rarestate produces one staircase while the other chiral Landau level persists almostunaffected (apart from near their hybridization). In Fig. 15(d) we consider twoflux quanta and a sample with two rare regions [similar to Fig. 3(c)] and noweach rare state hybridizes with the two Landau levels while still allowing thecharge pumping process to survive.However, if we remove the non-local potential so that we are considering aphysically relevant model with two Weyl points at the same energy then therare state hybridizes the two distinct chiral Landau levels and destroys thetopological charge pumping, as shown in Fig 16(a). While this also occursperturbatively due to internode scattering, the rare state induces this effect atmuch lower energy at weak disorder (e.g., notice where the two chiral Landaulevels level repel each other at φ = π at much higher energy then the rare stateis occurring at). On the other hand, as we increase the flux quanta, other chiralLandau levels will be induced and therefore a single rare state is not sufficientto hybridize all four chiral Landau levels. Instead one pair survives that allowsfor the chiral charge pumping to persist for this finite size sample, Fig 16 (b).However, in the thermodynamic limit the number of flux quanta scale as ∼ L or − µ Λ E avoided QCPQCP Localized E=0 states ThBI ρ ( E ) ∼ A ( E ) e − B ( E ) Quantum Critical ρ ( E ) ∼ | E | α < α ≤ . ThSM2 ρ ( E ) ∼ E Quantum Critical ρ ( E ) ∼ | E | ThDM ρ ( E ) ∼ const . µ/t W/t ThDM AI ThBI (a) (b) Figure 17: Phase diagram of a disordered three-dimensional p + ip superconductorin Eq. (57) . (a) As rare regions destabilize the vanishing density of states of the p + ip superconductor the phase diagram at the center of the band ( E = 0) has only two phasesat finite disorder that are a diffusive metal and Anderson insulating phase (that is smoothlyconnected to the band insulator). The orange lines denote the cross over boundaries between2 and 4 Weyl nodes (varying µ ) and the AQCP (as a function of W ). (b) Phase diagram interms of the average density of states across the band (or Anderson) insulator to diffusive metalphase transition and the cross over at the avoided transition. Note that the true Andersonlocalization transition shown here with a QCP label is not critical in the average density ofstates as it is made up of localized Lifshitz states [100]. Figures taken from Ref. [39]. and the density of rare states scales as ∼ L and thus rare regions will inevitablydestroy the charge pumping process for any number flux quanta.In the limit of larger disorder strength so that the model is sufficiently deepin the diffusive metal, phase a non-linear sigma model analysis is appropriate[71, 72]. In this limit there is no longer any sense of “band” to talk about,yet the chiral anomaly and the charge pumping process survives in the limitof a single Weyl node. While this study has discussed the microscopic effectof rare regions on the chiral charge pumping process it has not resolved thenature of this will appear in the transport and the magneto-resistance. It willbe interesting to explore these questions in future work. 8. Particle-hole symmetry We now turn to two models that have an exact particle-hole symmetry. Thefirst of which is the superconducting analog of Eq. (10), and thus potentialdisorder enters as a particle-hole symmetric term realizing a model in class Dthat was studied in detail in Ref. [39]. The second model we consider is abipartite random hopping model of the form that appears in Eq. (6) that fallsinto class BDI. The results obtained for the random hopping model are new,being reported here for the first time. These problems represent two distinctexamples of disordered Weyl semimetals in chiral symmetry classes of purely off-diagonal matrices where the zero energy state in the center of the band playsa special role. In the models we consider, the Weyl node energy sits preciselyat the symmetric band center to see if particle-hole symmetry can somehow“protect” the Weyl state from disorder. However, as we will see below rareregions still dominate the behaviour of the Weyl node and the avoidance is onlyenhanced by the chiral symmetry. 31 y (cid:31) z | ψ ( x ) | dz . . 00 11 1818 (a) r − − | ψ b i n ( r ) | | ψ bin | ∼ r − . (b) Figure 18: Rare eigenstate in the p + ip superconductor in Eq. (57) . The projectedprobability density of the wavefunction in the x − y plane displaying a quasi-localized rarestate. (b) The rare state displays a clear power law decay from its maximal value with anexponent in reasonable agreement with the analytic expectation. Despite the presence ofparticle-hole symmetry the nature of the rare states look qualitatively similar to the case ofpotential disorder. Taken from Ref. [39]. p + ip Weyl superconductor In Ref. [39], a three-dimensional model for a p + ip superconductor wasinvestigated in the presence of disorder. Here, we work under the assumptionof a non-zero pairing amplitude that we do not determine self-consistently. Themodel Hamiltonian is given by H SC = (cid:88) r , ˆ ν (cid:104) t c † r +ˆ ν c r + i ∆ ˆ ν c † r +ˆ ν c † r + h . c . (cid:105) + (cid:88) r ( V ( r ) − µ ) c † r c r , (57)where ˆ ν = ± ˆ x, ± ˆ y, ± ˆ z , the p x + ip y superconducting gap is given by ∆ ˆ x =∆ , ∆ ˆ y = − i ∆ , ∆ ˆ z = 0, and 2∆ is the maximum size of the superconductinggap for the clean model. In the absence of disorder we can rewrite Eq. (57)using Nambu spinors Ψ r = ( c r c † r ) T and the clean part of model is then givenby Eq. (10) (with ψ r replaced with Ψ r ) and this results in the particle holesymmetric disorder H disorder = (cid:88) r Ψ † r V ( r ) τ z Ψ r (58)where τ z is the z -Pauli matrix in Nambu space.The phase diagram of this model is shown in Fig. 17(a) in the space of µ − W . This model allows for a much larger multitude of clean band structuresto consider (such as anisotropic nodal points at | µ | = 2 t ). In addition to nodalsuperconductors in the clean limit and a diffusive thermal metal of the BDGexcitations at finite disorder, this model also hosts a thermal band insulator ofBDG quasiparticles at large µ , which in the presence of disorder is smoothlyconnected to a thermal Anderson insulating phase that sets in at large disorderstrength. The orange vertical line marks the anisotropic semimetal point thatseparates the 4 and 2 Weyl node regimes. The horizontal and diagonal orangeline marks the avoided transition that eventually merges with the band insula-tor transition. The transition between the diffusive thermal metal and thermal32 . . . . . . . . W/t − − − − − − − − ρ ( ) L = 18 , Weyl E = 0 state L = 30 , Weyl E = 0 state L = 60 , Weyl E = 0 state L = 18 , no Weyl E = 0 state L = 30 , no Weyl E = 0 state L = 60 , no Weyl E = 0 state . . . . . ( t/W ) − − − − − − − − ρ ( ) L = 18 L = 30 L = 60 e − a/W (a) (b) Figure 19: Zero energy DOS for the p + ip model in Eq. (57) . Utilizing twistedboundary conditions allows us to compute the rare region contribution to the density of stateswhich is converged in system size and orders of magnitude below the case of periodic boundaryconditions with L that produces a large finite size effect due to the Weyl peak at zero energy.(b) The converged zero energy density of states plotted on a log scale versus ( t/W ) displayingthat the rare region scaling form works over several orders of magnitude of the DOS. Takenfrom Ref. [39]. Anderson insulator is a true Anderson localization transition of the BDG quasi-particles that will appear in level statistics and wavefunction properties but isnot observable in the density of states. While perturbatively a disorder driventransition is also predicted along the boundary between the diffusive thermalmetal and thermal band insulator, this is also rounded out into a crossover.This enriches the phenomena of the avoided transition into Fig. 17(b).Focusing on the limit of four Weyl cones (see Fig. 17), the model possessesan AQCP due to rare regions. In Fig. 18, we show a rare state that is power lawbound with an exponent close to the analytic expectation in Eq. (35). In Fig. 19we show the computed rare region contribution to the DOS being well fit bythe rare region form in Eq. (35) over several overs of magnitude. The densityof states remains an analytic function and the peak in ρ (cid:48)(cid:48) (0) is saturated.A major distinction between the current limit and potential disorder is once ρ (0) is of order O ( t ), an antilocalization peak appears in the low energy DOS.This can be captured within a non-linear sigma model analysis once the stiffness(i.e. ρ (0)) is of order one; it yields an energy dependent DOS that goes like ρ (0) − ρ ( E ) ∼ (cid:112) | E | (59)near E = 0 [37]. It is a natural question to ask: What the impact of rare regions is on othernodal band structures beyond just Dirac points that also maintain the pertur-bative irrelevance of disorder? There are now a large number of examples oftopological semimetals [128, 129, 130] and it is an exciting question to studythe effects of rare regions on their low energy excitations. One clear exampleof this is the anisotropic nodal band structure at the transition between a Weylsemimetal and a band insulator, i.e. by traversing the phase diagram in Fig. 1733 ρ ’’ ( ) µ /t L=31, N C =2 31, 2 39, 2 71, 2 Figure 20: Rounding out the non-analytic behavior in the DOS across the diffusivemetal-to-band insulator transition in the p + ip superconductor. The anisotropic nodaldispersion produces a non-analytic DOS ρ ( E ) ∼ | E | / with a divergent ρ (cid:48)(cid:48) (0) ∼ | µ − µ I | − / that is rounded out by rare regions that produce a smooth and analytic average DOS; as onecan see since ρ (cid:48)(cid:48) (0) is saturated in L and N c . Taken from Ref. [39]. (a) in µ at W = 0. At this location the low energy density of states goes like ρ ( E ) ∼ | E | / and is non-analytic (see Sec. 2.2) and disorder is perturbativelyirrelevant, and the critical theory is sketched in Fig. 17(b). In particular, as wetune µ in the clean limit, we have ρ (cid:48)(cid:48) (0) ∼ | µ − µ I | − / . As shown in Fig. 20,this non-analytic behavior in the DOS is rounded out by rare regions and thetransition between the Weyl semimetal and the band insulator is no longer sharpin the average DOS.On the other hand, the transition between the Weyl semimetal and the bandinsulator is indeed a sharp transition with regards to the nature of the wave-function. States in the band insulator are exponentially localized Lifshitz statesthat round out the sharpness of the spectral band gap. However, transportand wave function probes (such as the inverse participation ratio or the typicaldensity of states) are sensitive to this realizing a true quantum phase separatinga diffusive metal and an Anderson insulator. Due to the anisotropic nodal bandstructure and the irrelevance of disorder, an avoided quantum phase transitionalso appears here in the average DOS, see Figs. 17 and 20. A simpler model with particle-hole symmetry can be realized with a randomhopping model with only a single term in the model. This will allow us to inter-polate between weak and strong randomness as well directly study the interplayof rare regions and the “pile up” of low energy states due to antilocalizationeffects.In this section, we present new results on the model in Eq. (6) with a randomhopping, namely H RH Weyl = (cid:88) r ,µ = x,y,z (cid:0) it µ ( r ) ψ † r σ µ ψ r +ˆ µ + H . c . (cid:1) , (60)34 ρ ( E ) E -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 ρ ( ) (1/W) N C =2 , L=1112 , 1112 , 1112 , 111extrapolated in N C (a) (b) (c) ρ '' ( ) W N C =2 extrapolated in N C Figure 21: Results demonstrating a strong avoidance in the random hoppingmodel in Eq. (60) . Here we focus on a large system size of L = 111 using the KPM toeliminate any finite size effects at these KPM expansion orders. (a) The density of states ρ ( E ) as a function of energy E for various disorder strengths starting from W = 0 . W = 1 (green) computed for a KPM expansion order N C = 2 averaged over 1000 sampleseach with a random twist in the boundary condition. We see a clear anti-localization peak[with the form given in Eq. (59)] appear at low energy roughly when the density of statesappears to lift off from zero on this linear scale. (b) The zero energy density of states versus(1 /W ) for various expansion orders as well as the value extrapolated in N C showing a clearrare region scaling consistent with Eq. (35) over almost five orders of magnitude (fit shown asa grey dashed line). (c) The second derivative of the density of states ρ (cid:48)(cid:48) (0) versus disorderstrength W to probe the analytic properties of the density of states. The antilocalizationpeak is a separate nonanalyticity that is generated when the zero-energy density of statesis sufficiently large. Here, we see that leads to an additional rounding of the peak in ρ (cid:48)(cid:48) (0)signalling an analytic density of states at the avoided transition. For larger disorder strengthsthe antilocalization peak takes over, causing a massive, negative ρ (cid:48)(cid:48) (0). where t µ ( r ) is taken to be a random variable along each bond (labeled by r and µ ). We parametrize t µ ( r ) with mean √ − W and standard deviation W so that the limit of purely random hopping is accessible at W = 1. Using thesymmetries of the hopping model derived in Sec. 2.1 [see Eq.(8)] the currentbipartite random hopping model falls into the chiral orthogonal BDI Altland-Zirnbauer class [131].The average DOS as a function of energy of the random hopping model isshown in Fig. 21(a) for several values of W . The appearance of the antilocal-ization peak (described by Eq. (59) [132]) clearly appears right after the AQCP(which occurs near W c ≈ . N C -independent estimate of ρ (0) and ρ (cid:48)(cid:48) (0) weutilize the recent analysis we put forth in Ref. [105] that utilizes the convolutionof the Jackson kernel with the analytic scaling form of the density of states inEq. (36) to obtain ρ N C (0) = ρ (0) + 12 ρ (cid:48)(cid:48) (0) (cid:18) π ∆ N C (cid:19) + · · · (61)where ρ N C (0) is the KPM data and ∆ is the bandwidth of the model. The resultof the extrapolation is shown in Fig. 21(b) and (c) as grey circles demonstratingthe density of states is converged across this range of W .Coming to the nature of the avoidance we show results for ρ (cid:48)(cid:48) (0) in Fig. 21(c)35or various KPM expansion orders that are converged at this system size ( L =111). Distinct from our previous discussions, due to antilocalization effects ρ (cid:48)(cid:48) (0)changes sign and becomes negative for W (cid:38) . 55. The peak at the AQCP isstrongly rounded out; while it initially sharpens for increasing N C , it eventuallydecreases due to antilocalization effects. In order to try and extract an N C -independent estimate we extrapolate the zero energy density of states acrossthe range of N C = 2 − using Eq. (61) to obtain the grey circles indicatinga very strong rounding in the large N C limit. In summary, our results areconsistent with random hopping also destabilizing the Weyl semimetal phaseand an interesting interplay between the perturbative transition, rare regions,and antilocalization effects lead to an even stronger rounding of ρ (cid:48)(cid:48) (0) . 9. A single Weyl node in the continuum The numerical models studied thus far included band curvature and an evennumber of Weyl nodes in the Brillouin zone. It could reasonably be arguedthat scattering between opposite chirality nodes and/or band curvature effectscould produce rare resonances that we see. Additionally, work by Buchholdet al. [103, 104] has suggested that ρ (0) = 0 identically due to individual rareresonances and in any reasonable disorder scheme. As a consequence, criticalitycould be restored. It is thus important to study the perfectly linear, single Weylnode to determine if avoided criticality persists, and in this section we showexplicitly that it does, suggesting rare resonances as described in Sec. 4 play asignificant role in the physics as we have described.The single Weyl node Hamiltonian in the continuum takes the form H = − iv σ · ∇ + V ( r ) (62)where the potential is Gaussian distributed with zero mean and (cid:104) V ( r + R ) V ( r ) (cid:105) = W e − R /ξ . (63)Without loss of generality we set v = 1 and ξ = 1 in the following. To sim-ulate this model, a discretization of momentum space is required. However,the discretized grid or “lattice” is arbitrary and our results should not dependon it. Therefore, we consider two different discretizations of the continuum inmomentum space the simplest being a simple cubic lattice and the second is aface centered cubic (FCC) lattice (which provides the most dense sphere pack-ing in 3D). As demonstrated in Refs. [85, 105] it is possible to still implementsparse matrix-vector techniques by working with vectors in momentum spaceand whenever we act with the potential (that is defined in real space) we usea series of fast Fourier transforms (and their inverse). For a matrix of size N this increases the computation cost from N to N (log N ) and the matrix-vectorapproaches remain fast.While there are prior numerical results in the limit of a single Weyl cone,no rare states were identified. An example of a rare state found using the36 − − − − −0.08 −0.06 −0.04 −0.02 0.00−0.10−0.050.000.050.10 All points are doubly degenerate E δV (a) (b) (c) Figure 22: A rare state found in the fcc model in Eq. (62) with system size L = 25 √ π/ √ W = 0 . E = 0 . r = n a + n a + 22 a a = (1 , − , a = ( − , , a = (1 , , − | ψ ( r ) | ∼ /r . . (c) Adding a spherical potential to the disorderpotential at the maximum of | ψ ( r ) | with radius two and strength δV , we can tune the rarestate energy through zero energy. Taken from Ref. [105]. FCC discretization is shown in Fig. 22(a). This displays the expected powerlaw going like ψ rare ( r ) ∼ / | r − r | . (Fig. 22(b)) in excellent agreement withEq. (28). The energy of this state is E = 0 . N C → ∞ extrapolated ρ (cid:48)(cid:48) (0) which further appears saturated with systemsize in Fig. 23(a) and a similarly extrapolated in N C and converged in L ρ (0)in accordance with Eq. (35). Thus, our results demonstrate the AQCP survivesin the limit of a single Weyl cone. This also directly suggests that a singleWeyl cone is unstable to infinitesimal disorder due to the appearance of a finitedensity of states. 10. Discussion In this review we have examined numerical results on rare region effectsin Dirac and Weyl semimetals. While it is possible to tune the probability ofgenerating rare events we discussed how it is not possible to exactly remove rareevents in the presence of randomness. We have shown a great deal of numericalevidence in the presence and absence of particle hole symmetry, as well as in theBdG spectrum that disorder generically destroys the semimetal phase in three-dimensions and the putative transition is avoided, which is rounded out into acrossover. It will be interesting to see how these effects appear in transport.In addition to the non-zero conductivity expected at the Weyl node the nature37 . . . W . . . ρ (cid:31)(cid:31) ( ) Λ = √ π , π √ , π √ W/W c 01 1 2( W/W c ) − − − − − − − − − ρ ( ) / W c Λ ξ = √ π, π √ , π √ W/W c . . (a) (b) Figure 23: The avoidance survives a single, perfectly linear Weyl cone. Fit-ting ρ (0) as a polynomial in 1 /N C allows us to extract ρ (0) and ρ (cid:48)(cid:48) (0) as explained inRef. [105]; the results at different sizes and momentum space cutoffs are presented. Mul-tiple sizes are shown at a given cutoff: light-to-dark gray curves represent different sizes L Λ √ /π = 32 √ , √ , √ , √ L = 160 √ π/ √ ρ (cid:48)(cid:48) (0) is drift-ing as a function of the cutoff, but the size of the avoidance remains the same. In fact, thedrift due to cutoff can be understood by a simple generalization of the self-consistent Born ap-proximation [105]. (Inset) shows the data normalized by the peak of ρ (cid:48)(cid:48) (0), where the avoidedcritical point W c occurs. (b) ρ (0) fits the rare region form well. Other system size data isomitted since it lies directly on top of the shown curve. The gray curves represents a fit tothe rare region equation (35). Inset is the same data on a linear scale. Taken from Ref. [105]. of the anomalous Hall effect in the presence of rare regions in disordered timereversal broken Weyl semimetals [133, 93] and the photogalvanic effect [134] ininversion broken Weyl semimetals are both worthy of further investigation.The effects of rare eigenstates on the topological properties of Weyl semimet-als have also been discussed. The destabilization of Fermi arc surface states dueto hybridizing with non-perturbative rare eigenstates was demonstrated in con-junction with a remarkably robust surface chiral velocity. In the presence ofmagnetic fields rare eigenstates survive and destroy the chiral charge pumpingprocess that arises due to the axial anomaly in the presence of parallel electricand magnetic fields. Interestingly, in the limit of a single Weyl cone this chargepumping process survives non-perturbative rare region effects. In future work,it is important to connect these observations with dynamical transport proper-ties in the presence of electro-magnetic fields [135, 136] to understand the roleof effects of disorder on the chiral lifetime.Last, we investigated the limit of a single Weyl cone with a linear dispersionnumerically. This study was motivated by recent field theoretic results that haveargued the Weyl semimetal phase is stable to rare region effects. However, theexact numerical simulations presented in Sec. 9 is at odds with this prediction asthey are consistent with an avoided quantum criticality that implies the singleWeyl node is unstable to disorder. These discrepancies between the field theoryexpectations and the numerics should be of central interest in future studies(Ref. [114] offers a one resolution, for instance).A wide open direction of research is developing a theoretical understandingof the role of interactions in the presence of rare region effects. Not only doesthis pertain to strongly correlated Weyl semimetals but also the formation of38he Kondo-Weyl semimetal ground state may be affected. It will be exciting todetermine the interplay of these effects in future work.We close with a brief discussion of a connection between this avoided crit-icality three-dimensions [137] and the magic-angle effect in the band-structureof twisted bilayer graphene in two-dimensions [138]. By replacing the randompotential by a three-dimensional, quasiperiodic potential, rare events are re-moved. As a result, a true quantum phase transition between a Weyl semimetaland a diffusive metal can be stabilized [139]. As the Weyl semimetal phase isstable, the low energy scaling of the density of states remains ρ ( E ) ∼ v − E and we can extract the effective velocity from ρ (cid:48)(cid:48) (0) ∝ /v . At finite energy,the quasiperiodic potential opens gaps, which forms minibands that contributeto the renormalization of the velocity. On approach to the transition as shownin Ref. [137], ρ (cid:48)(cid:48) (0) diverges signalling a non-analytic density of states and thevelocity goes to zero like v ∼ | W − W c | β/ with β ≈ 2. In the diffusive metalphase, level statistics are random matrix theory like and plane-wave eigenstatesat the Weyl node delocalize in momentum space. Extending these models totwo dimensions shows that these transitions survive in the form of a semimetalto metal phase transition. By applying perturbation theory from twisted bilayergraphene yields a magic-angle condition in the velocity that coincides with theeigenstate phase transition. Thus, rather remarkably, this transition is funda-mentally related to the magic-angle effect in twisted bilayer graphene. 11. Acknowledgements We thank Yang-Zhi Chou, Sankar Das Sarma, Sarang Gopalakrishnan, Pal-lab Goswami, David Huse, Junhyun Lee, Rahul Nandkishore, Leo Radzihovsky,Gil Refeal, and Jay Sau for various collaborations and discussions related tothe work reviewed here, and we thank Bitan Roy, Bj¨orn Sbierski, and SergeySyzranov for useful comments on an early draft as well as insightful discussions.We would also like to thank Alexander Altland, Peter Armitage, AlexanderBalatsky, Michael Buchhold, Sudip Chakravarty, Matthew Foster, Victor Gu-rarie, Silke Paschen, Thomas Searles, and Weida Wu for useful discussions. Thiswork is supported by NSF CAREER Grant No. DMR-1941569. The authors ac-knowledge the Beowulf cluster at the Department of Physics and Astronomy ofRutgers University, and the Office of Advanced Research Computing (OARC) atRutgers, The State University of New Jersey (http://oarc.rutgers.edu), for pro-viding access to the Amarel cluster and associated research computing resourcesthat have contributed to the results reported here. The Flatiron Institute is adivision of the Simons Foundation. References [1] P. A. Dirac, The quantum theory of the electron, in: Proc. R. Soc. Lond.A, Vol. 117, The Royal Society, 1928, pp. 610–624. doi:10.1098/rspa.1928.0023 . 392] M. Z. Hasan, C. L. Kane, Colloquium: Topological insulators, Rev. Mod.Phys. 82 (4) (2010) 3045–3067. doi:10.1103/RevModPhys.82.3045 .[3] M. Z. Hasan, J. E. Moore, Three-Dimensional Topological Insulators,Annu. Rev. Condens. Matter Phys. 2 (1) (2011) 55–78. doi:10.1146/annurev-conmatphys-062910-140432 .[4] B. A. Bernevig, T. L. Hughes, Topological Insulators and TopologicalSuperconductors, Princeton University Press, 2013.[5] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K.Geim, The electronic properties of graphene, Rev. Mod. Phys. 81 (1)(2009) 109–162. doi:10.1103/RevModPhys.81.109 .[6] M. Neupane, S.-Y. Xu, R. Sankar, N. Alidoust, G. Bian, C. Liu, I. Be-lopolski, T.-R. Chang, H.-T. Jeng, H. Lin, et al., Observation of a three-dimensional topological Dirac semimetal phase in high-mobility Cd As ,Nat. Commun. 5 (2014) 3786. doi:10.1038/ncomms4786 .[7] S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolotnyy, B. B¨uchner, R. J.Cava, Experimental realization of a three-dimensional Dirac semimetal,Phys. Rev. Lett. 113 (2) (2014) 027603. doi:10.1103/PhysRevLett.113.027603 .[8] Z. Liu, J. Jiang, B. Zhou, Z. Wang, Y. Zhang, H. Weng, D. Prabhakaran,S. Mo, H. Peng, P. Dudin, et al., A stable three-dimensional topologicalDirac semimetal Cd As , Nat. Mater. 13 (7) (2014) 677–681. doi:10.1038/nmat3990 .[9] Z. Liu, B. Zhou, Y. Zhang, Z. Wang, H. Weng, D. Prabhakaran, S.-K.Mo, Z. Shen, Z. Fang, X. Dai, et al., Discovery of a three-dimensionaltopological Dirac semimetal, Na Bi, Science 343 (6173) (2014) 864–867. doi:10.1126/science.1245085 .[10] S.-Y. Xu, C. Liu, S. K. Kushwaha, R. Sankar, J. W. Krizan, I. Belopolski,M. Neupane, G. Bian, N. Alidoust, T.-R. Chang, et al., Observation ofFermi arc surface states in a topological metal, Science 347 (6219) (2015)294–298. doi:10.1126/science.1256742 .[11] B. Lenoir, M. Cassart, J.-P. Michenaud, H. Scherrer, S. Scherrer, Trans-port properties of Bi-RICH Bi-Sb alloys, J. Phys. Chem. Solids 57 (1)(1996) 89–99. doi:10.1016/0022-3697(95)00148-4 .[12] A. Ghosal, P. Goswami, S. Chakravarty, Diamagnetism of nodal fermions,Phys. Rev. B 75 (11) (2007) 115123. doi:10.1103/PhysRevB.75.115123 .[13] J. C. Y. Teo, L. Fu, C. L. Kane, Surface states and topological invariants inthree-dimensional topological insulators: Application to Bi − x Sb x , Phys.Rev. B 78 (4) (2008) 045426. doi:10.1103/PhysRevB.78.045426 .4014] S.-Y. Xu, Y. Xia, L. Wray, S. Jia, F. Meier, J. Dil, J. Osterwalder, B. Slom-ski, A. Bansil, H. Lin, et al., Topological phase transition and textureinversion in a tunable topological insulator, Science 332 (6029) (2011)560–564. doi:10.1126/science.1201607 .[15] T. Sato, K. Segawa, K. Kosaka, S. Souma, K. Nakayama, K. Eto, T. Mi-nami, Y. Ando, T. Takahashi, Unexpected mass acquisition of Diracfermions at the quantum phase transition of a topological insulator, Nat.Phys. 7 (11) (2011) 840–844. doi:10.1038/nphys2058 .[16] M. Brahlek, N. Bansal, N. Koirala, S.-Y. Xu, M. Neupane, C. Liu,M. Z. Hasan, S. Oh, Topological-metal to band-insulator transition in(Bi − x In x ) Se thin films, Phys. Rev. Lett. 109 (18) (2012) 186403. doi:10.1103/PhysRevLett.109.186403 .[17] L. Wu, M. Brahlek, R. V. Aguilar, A. Stier, C. Morris, Y. Lubashevsky,L. Bilbro, N. Bansal, S. Oh, N. Armitage, A sudden collapse in the trans-port lifetime across the topological phase transition in (Bi − x In x ) Se ,Nat. Phys. 9 (7) (2013) 410–414. doi:10.1038/nphys2647 .[18] S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, B. Wang,N. Alidoust, G. Bian, M. Neupane, C. Zhang, et al., A Weyl Fermionsemimetal with surface Fermi arcs in the transition metal monopnictideTaAs class, Nat. Commun. 6 (2015) 7373. doi:10.1038/ncomms8373 .[19] H. Weng, C. Fang, Z. Fang, B. A. Bernevig, X. Dai, Weyl SemimetalPhase in Noncentrosymmetric Transition-Metal Monophosphides, Phys.Rev. X 5 (1) (2015) 011029. doi:10.1103/PhysRevX.5.011029 .[20] S.-Y. Xu, N. Alidoust, I. Belopolski, Z. Yuan, G. Bian, T.-R. Chang,H. Zheng, V. N. Strocov, D. S. Sanchez, G. Chang, et al., Discovery of aWeyl fermion state with Fermi arcs in niobium arsenide, Nat. Phys. (2015)748–754 doi:10.1038/nphys3437 .[21] S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang,R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, et al., Discovery of a Weylfermion semimetal and topological Fermi arcs, Science 349 (6248) (2015)613–617. doi:10.1126/science.aaa9297 .[22] S.-Y. Xu, I. Belopolski, D. S. Sanchez, C. Zhang, G. Chang, C. Guo,G. Bian, Z. Yuan, H. Lu, T.-R. Chang, P. P. Shibayev, M. L. Prokopovych,N. Alidoust, H. Zheng, C.-C. Lee, S.-M. Huang, R. Sankar, F. Chou, C.-H.Hsu, H.-T. Jeng, A. Bansil, T. Neupert, V. N. Strocov, H. Lin, S. Jia, M. Z.Hasan, Experimental discovery of a topological Weyl semimetal state inTaP, Sci. Adv. 1 (10) (2015) e1501092. doi:10.1126/sciadv.1501092 .[23] B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard,X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, H. Ding,Experimental discovery of Weyl semimetal TaAs, Phys. Rev. X 5 (3)(2015) 031013. doi:10.1103/PhysRevX.5.031013 .4124] C.-L. Zhang, Z. Yuan, Q.-D. Jiang, B. Tong, C. Zhang, X. C. Xie, S. Jia,Electron scattering in tantalum monoarsenide, Phys. Rev. B 95 (8) (2017)085202. doi:10.1103/PhysRevB.95.085202 .[25] H.-H. Lai, S. E. Grefe, S. Paschen, Q. Si, Weyl–Kondo semimetal in heavy-fermion systems, Proc. Natl. Acad. Sci. U.S.A. 115 (1) (2018) 93–97. doi:10.1073/pnas.1715851115 .[26] P.-Y. Chang, P. Coleman, Parity-violating hybridization in heavy Weylsemimetals, Phys. Rev. B 97 (15) (2018) 155134. doi:10.1103/PhysRevB.97.155134 .[27] S. Dzsaber, X. Yan, M. Taupin, G. Eguchi, A. Prokofiev, T. Shiroka,P. Blaha, O. Rubel, S. E. Grefe, H.-H. Lai, Q. Si, S. Paschen, Gi-ant spontaneous Hall effect in a nonmagnetic Weyl-Kondo semimetal,arXiv:1811.02819 [cond-mat] (Oct. 2019). arXiv:1811.02819 .[28] C. Y. Guo, F. Wu, Z. Z. Wu, M. Smidman, C. Cao, A. Bostwick,C. Jozwiak, E. Rotenberg, Y. Liu, F. Steglich, H. Q. Yuan, Evidencefor Weyl fermions in a canonical heavy-fermion semimetal YbPtBi, Nat.Commun. 9 (1) (2018) 4622. doi:10.1038/s41467-018-06782-1 .[29] K. Kuroda, T. Tomita, M.-T. Suzuki, C. Bareille, A. A. Nugroho,P. Goswami, M. Ochi, M. Ikhlas, M. Nakayama, S. Akebi, R. Noguchi,R. Ishii, N. Inami, K. Ono, H. Kumigashira, A. Varykhalov, T. Muro,T. Koretsune, R. Arita, S. Shin, T. Kondo, S. Nakatsuji, Evidence formagnetic Weyl fermions in a correlated metal, Nat. Mater. 16 (11) (2017)1090–1095. doi:10.1038/nmat4987 .[30] X. Wan, A. M. Turner, A. Vishwanath, S. Y. Savrasov, Topologicalsemimetal and Fermi-arc surface states in the electronic structure of py-rochlore iridates, Phys. Rev. B 83 (20) (2011) 205101. doi:10.1103/PhysRevB.83.205101 .[31] W. Witczak-Krempa, G. Chen, Y. B. Kim, L. Balents, Corre-lated quantum phenomena in the strong spin-orbit regime, Annu.Rev. Condens. Matter Phys. 5 (1) (2014) 57–82. doi:10.1146/annurev-conmatphys-020911-125138 .[32] P. Goswami, B. Roy, S. Das Sarma, Competing orders and topology in theglobal phase diagram of pyrochlore iridates, Phys. Rev. B 95 (8) (2017)085120. doi:10.1103/PhysRevB.95.085120 .[33] A. J. Leggett, A theoretical description of the new phases of liquid He,Rev. Mod. Phys. 47 (2) (1975) 331–414. doi:10.1103/RevModPhys.47.331 .[34] C.-R. Hu, Midgap surface states as a novel signature for d xa - x b -wavesuperconductivity, Phys. Rev. Lett. 72 (10) (1994) 1526–1529. doi:10.1103/PhysRevLett.72.1526 .4235] G. E. Volovik, The Universe in a Helium Droplet, OUP Oxford, 2009.[36] A. P. Schnyder, P. M. R. Brydon, Topological surface states in nodalsuperconductors, J. Phys. Condens. Matter 27 (24) (2015) 243201. doi:10.1088/0953-8984/27/24/243201 .[37] T. Senthil, M. P. A. Fisher, Quasiparticle localization in superconductorswith spin-orbit scattering, Phys. Rev. B 61 (14) (2000) 9690–9698. doi:10.1103/PhysRevB.61.9690 .[38] P. Goswami, L. Balicas, Topological properties of possible Weyl supercon-ducting states of URu Si , arXiv:1312.3632 [cond-mat, physics:hep-th](Dec. 2013). arXiv:1312.3632 .[39] J. H. Wilson, J. H. Pixley, P. Goswami, S. Das Sarma, Quantum phasesof disordered three-dimensional Majorana-Weyl fermions, Phys. Rev. B95 (15) (2017) 155122. doi:10.1103/PhysRevB.95.155122 .[40] A. A. Burkov, Anomalous Hall effect in Weyl metals, Phys. Rev. Lett.113 (18) (2014) 187202. doi:10.1103/PhysRevLett.113.187202 .[41] P. Goswami, G. Sharma, S. Tewari, Optical activity as a test for dynamicchiral magnetic effect of Weyl semimetals, Phys. Rev. B 92 (16) (2015)161110. doi:10.1103/PhysRevB.92.161110 .[42] Q. Ma, S.-Y. Xu, C.-K. Chan, C.-L. Zhang, G. Chang, Y. Lin, W. Xie,T. Palacios, H. Lin, S. Jia, P. A. Lee, P. Jarillo-Herrero, N. Gedik, Directoptical detection of Weyl fermion chirality in a topological semimetal, Nat.Phys. 13 (9) (2017) 842–847. doi:10.1038/nphys4146 .[43] L. Wu, S. Patankar, T. Morimoto, N. L. Nair, E. Thewalt, A. Little, J. G.Analytis, J. E. Moore, J. Orenstein, Giant anisotropic nonlinear opticalresponse in transition metal monopnictide Weyl semimetals, Nat. Phys.13 (4) (2017) 350–355. doi:10.1038/nphys3969 .[44] F. de Juan, A. G. Grushin, T. Morimoto, J. E. Moore, Quantized circu-lar photogalvanic effect in Weyl semimetals, Nat. Commun. 8 (1) (2017)15995. doi:10.1038/ncomms15995 .[45] H. Inoue, A. Gyenis, Z. Wang, J. Li, S. W. Oh, S. Jiang, N. Ni, B. A.Bernevig, A. Yazdani, Quasiparticle interference of the Fermi arcs andsurface-bulk connectivity of a Weyl semimetal, Science 351 (6278) (2016)1184–1187. doi:10.1126/science.aad8766 .[46] N. Xu, H. M. Weng, B. Q. Lv, C. E. Matt, J. Park, F. Bisti, V. N. Strocov,D. Gawryluk, E. Pomjakushina, K. Conder, N. C. Plumb, M. Radovic,G. Aut`es, O. V. Yazyev, Z. Fang, X. Dai, T. Qian, J. Mesot, H. Ding,M. Shi, Observation of Weyl nodes and Fermi arcs in tantalum phosphide,Nat. Commun. 7 (1) (2016) 11006. doi:10.1038/ncomms11006 .4347] Q.-Q. Yuan, L. Zhou, Z.-C. Rao, S. Tian, W.-M. Zhao, C.-L. Xue, Y. Liu,T. Zhang, C.-Y. Tang, Z.-Q. Shi, Z.-Y. Jia, H. Weng, H. Ding, Y.-J. Sun,H. Lei, S.-C. Li, Quasiparticle interference evidence of the topologicalFermi arc states in chiral fermionic semimetal CoSi, Sci. Adv. 5 (12) (2019)eaaw9485. doi:10.1126/sciadv.aaw9485 .[48] S. L. Adler, Axial-vector vertex in spinor electrodynamics, Phys. Rev.177 (5) (1969) 2426–2438. doi:10.1103/PhysRev.177.2426 .[49] J. S. Bell, R. Jackiw, A PCAC puzzle: π → γγ in the σ -model, Il NuovoCimento A 60 (1) (1969) 47–61. doi:10.1007/BF02823296 .[50] H. B. Nielsen, M. Ninomiya, The Adler-Bell-Jackiw anomaly and Weylfermions in a crystal, Phys. Lett. B 130 (6) (1983) 389–396. doi:10.1016/0370-2693(83)91529-0 .[51] H.-J. Kim, K.-S. Kim, J.-F. Wang, M. Sasaki, N. Satoh, A. Ohnishi,M. Kitaura, M. Yang, L. Li, Dirac versus Weyl fermions in topological in-sulators: Adler-Bell-Jackiw anomaly in transport phenomena, Phys. Rev.Lett. 111 (24) (2013) 246603. doi:10.1103/PhysRevLett.111.246603 .[52] T. Liang, Q. Gibson, M. N. Ali, M. Liu, R. J. Cava, N. P. Ong, Ultrahighmobility and giant magnetoresistance in the Dirac semimetal Cd As , Nat.Mater. 14 (3) (2015) 280–284. doi:10.1038/nmat4143 .[53] Q. Li, D. E. Kharzeev, C. Zhang, Y. Huang, I. Pletikosi´c, A. V. Fedorov,R. D. Zhong, J. A. Schneeloch, G. D. Gu, T. Valla, Chiral magnetic effectin ZrTe , Nat. Phys. 12 (6) (2016) 550–554. doi:10.1038/nphys3648 .[54] D. T. Son, B. Z. Spivak, Chiral anomaly and classical negative mag-netoresistance of Weyl metals, Phys. Rev. B 88 (10) (2013) 104412. doi:10.1103/PhysRevB.88.104412 .[55] S. A. Parameswaran, T. Grover, D. A. Abanin, D. A. Pesin, A. Vish-wanath, Probing the chiral anomaly with nonlocal transport in three-dimensional topological semimetals, Phys. Rev. X 4 (3) (2014) 031035. doi:10.1103/PhysRevX.4.031035 .[56] E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, Chiral anomaly, dimen-sional reduction, and magnetoresistivity of Weyl and Dirac semimetals,Phys. Rev. B 89 (8) (2014) 085126. doi:10.1103/PhysRevB.89.085126 .[57] A. A. Burkov, Chiral anomaly and diffusive magnetotransport in Weylmetals, Phys. Rev. Lett. 113 (24) (2014) 247203. doi:10.1103/PhysRevLett.113.247203 .[58] P. Goswami, J. H. Pixley, S. Das Sarma, Axial anomaly and longitudi-nal magnetoresistance of a generic three-dimensional metal, Phys. Rev. B92 (7) (2015) 075205. doi:10.1103/PhysRevB.92.075205 .4459] J. Cano, B. Bradlyn, Z. Wang, M. Hirschberger, N. P. Ong, B. A. Bernevig,Chiral anomaly factory: Creating Weyl fermions with a magnetic field,Phys. Rev. B 95 (16) (2017) 161306. doi:10.1103/PhysRevB.95.161306 .[60] M. M. Vazifeh, M. Franz, Electromagnetic response of Weyl semimetals,Phys. Rev. Lett. 111 (2) (2013) 027201. doi:10.1103/PhysRevLett.111.027201 .[61] V. M. Galitski, S. Adam, S. Das Sarma, Statistics of random voltagefluctuations and the low-density residual conductivity of graphene, Phys.Rev. B 76 (24) (2007) 245405. doi:10.1103/PhysRevB.76.245405 .[62] H. Beidenkopf, P. Roushan, J. Seo, L. Gorman, I. Drozdov, Y. S. Hor,R. J. Cava, A. Yazdani, Spatial fluctuations of helical Dirac fermions onthe surface of topological insulators, Nat. Phys. 7 (12) (2011) 939–943. doi:10.1038/nphys2108 .[63] B. Skinner, Coulomb disorder in three-dimensional Dirac systems, Phys.Rev. B 90 (6) (2014) 060202. doi:10.1103/PhysRevB.90.060202 .[64] E. Fradkin, Critical behavior of disordered degenerate semiconductors. I.Models, symmetries, and formalism, Phys. Rev. B 33 (5) (1986) 3257–3262. doi:10.1103/PhysRevB.33.3257 .[65] E. Fradkin, Critical behavior of disordered degenerate semiconductors. II.Spectrum and transport properties in mean-field theory, Phys. Rev. B33 (5) (1986) 3263–3268. doi:10.1103/PhysRevB.33.3263 .[66] P. Goswami, S. Chakravarty, Quantum criticality between topologicaland band insulators in 3+1 dimensions, Phys. Rev. Lett. 107 (19) (2011)196803. doi:10.1103/PhysRevLett.107.196803 .[67] S. V. Syzranov, V. Gurarie, L. Radzihovsky, Unconventional localizationtransition in high dimensions, Phys. Rev. B 91 (3) (2015) 035133. doi:10.1103/PhysRevB.91.035133 .[68] S. V. Syzranov, L. Radzihovsky, V. Gurarie, Critical transport in weaklydisordered semiconductors and semimetals, Phys. Rev. Lett. 114 (16)(2015) 166601. doi:10.1103/PhysRevLett.114.166601 .[69] F. Evers, A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80 (4)(2008) 1355–1417. doi:10.1103/RevModPhys.80.1355 .[70] S. V. Syzranov, P. M. Ostrovsky, V. Gurarie, L. Radzihovsky, Criti-cal exponents at the unconventional disorder-driven transition in a Weylsemimetal, Phys. Rev. B 93 (15) (2016) 155113. doi:10.1103/PhysRevB.93.155113 .[71] A. Altland, D. Bagrets, Effective field theory of the disordered Weylsemimetal, Phys. Rev. Lett. 114 (25) (2015) 257201. doi:10.1103/PhysRevLett.114.257201 . 4572] A. Altland, D. Bagrets, Theory of the strongly disordered Weyl semimetal,Phys. Rev. B 93 (7) (2016) 075113. doi:10.1103/PhysRevB.93.075113 .[73] J. H. Pixley, P. Goswami, S. Das Sarma, Disorder-driven itinerant quan-tum criticality of three-dimensional massless Dirac fermions, Phys. Rev.B 93 (8) (2016) 085103. doi:10.1103/PhysRevB.93.085103 .[74] B. Roy, S. Das Sarma, Erratum: Diffusive quantum criticality in three-dimensional disordered Dirac semimetals [Phys. Rev. B , 241112(R)(2014)], Phys. Rev. B 93 (11) (2016) 119911. doi:10.1103/PhysRevB.93.119911 .[75] B. Roy, V. Juriˇci´c, S. Das Sarma, Universal optical conductivity of adisordered Weyl semimetal, Sci. Rep. 6 (1) (2016) 32446. doi:10.1038/srep32446 .[76] S. V. Syzranov, V. Gurarie, L. Radzihovsky, Multifractality at non-Anderson disorder-driven transitions in Weyl semimetals and other sys-tems, Ann. Phys. (N. Y.) 373 (2016) 694–706. doi:10.1016/j.aop.2016.08.012 .[77] T. Louvet, D. Carpentier, A. A. Fedorenko, On the disorder-driven quan-tum transition in three-dimensional relativistic metals, Phys. Rev. B94 (22) (2016) 220201. doi:10.1103/PhysRevB.94.220201 .[78] B. Sbierski, K. S. C. Decker, P. W. Brouwer, Weyl node with randomvector potential, Phys. Rev. B 94 (22) (2016) 220202. doi:10.1103/PhysRevB.94.220202 .[79] T. Louvet, D. Carpentier, A. A. Fedorenko, New quantum transition inWeyl semimetals with correlated disorder, Phys. Rev. B 95 (1) (2017)014204. doi:10.1103/PhysRevB.95.014204 .[80] X. Luo, B. Xu, T. Ohtsuki, R. Shindou, Quantum multicriticality indisordered Weyl semimetals, Phys. Rev. B 97 (4) (2018) 045129. doi:10.1103/PhysRevB.97.045129 .[81] X. Luo, T. Ohtsuki, R. Shindou, Unconventional scaling theory indisorder-driven quantum phase transition, Phys. Rev. B 98 (2) (2018)020201. doi:10.1103/PhysRevB.98.020201 .[82] I. Balog, D. Carpentier, A. A. Fedorenko, Disorder-driven quantumtransition in relativistic semimetals: Functional renormalization via theporous medium equation, Phys. Rev. Lett. 121 (16) (2018) 166402. doi:10.1103/PhysRevLett.121.166402 .[83] B. Roy, R.-J. Slager, V. Juriˇci´c, Global phase diagram of a dirty Weylliquid and emergent superuniversality, Phys. Rev. X 8 (3) (2018) 031076. doi:10.1103/PhysRevX.8.031076 .4684] E. Brillaux, D. Carpentier, A. A. Fedorenko, Multifractality at the Weyl-semimetal–diffusive-metal transition for generic disorder, Phys. Rev. B100 (13) (2019) 134204. doi:10.1103/PhysRevB.100.134204 .[85] B. Sbierski, C. Fr¨aßdorf, Strong disorder in nodal semimetals: Schwinger-Dyson–Ward approach, Phys. Rev. B 99 (2) (2019) 020201. doi:10.1103/PhysRevB.99.020201 .[86] B. Sbierski, S. Syzranov, Non-Anderson critical scaling of the Thoulessconductance in 1D, Ann. Phys. (N. Y.) 418 (2020) 168169. doi:10.1016/j.aop.2020.168169 .[87] K. Kobayashi, T. Ohtsuki, K.-I. Imura, I. F. Herbut, Density of statesscaling at the semimetal to metal transition in three dimensional topo-logical insulators, Phys. Rev. Lett. 112 (1) (2014) 016402. doi:10.1103/PhysRevLett.112.016402 .[88] B. Sbierski, G. Pohl, E. J. Bergholtz, P. W. Brouwer, Quantum transportof disordered Weyl semimetals at the nodal point, Phys. Rev. Lett. 113 (2)(2014) 026602. doi:10.1103/PhysRevLett.113.026602 .[89] B. Sbierski, E. J. Bergholtz, P. W. Brouwer, Quantum critical exponentsfor a disordered three-dimensional Weyl node, Phys. Rev. B 92 (11) (2015)115145. doi:10.1103/PhysRevB.92.115145 .[90] J. H. Pixley, P. Goswami, S. Das Sarma, Anderson localization and thequantum phase diagram of three dimensional disordered Dirac semimetals,Phys. Rev. Lett. 115 (7) (2015) 076601. doi:10.1103/PhysRevLett.115.076601 .[91] S. Liu, T. Ohtsuki, R. Shindou, Effect of disorder in a three-dimensionallayered Chern Insulator, Phys. Rev. Lett. 116 (6) (2016) 066401. doi:10.1103/PhysRevLett.116.066401 .[92] S. Bera, J. D. Sau, B. Roy, Dirty Weyl semimetals: Stability, phasetransition, and quantum criticality, Phys. Rev. B 93 (20) (2016) 201302. doi:10.1103/PhysRevB.93.201302 .[93] K. Kobayashi, M. Wada, T. Ohtsuki, Ballistic transport in disorderedDirac and Weyl semimetals, Phys. Rev. Research 2 (2) (2020) 022061. doi:10.1103/PhysRevResearch.2.022061 .[94] R. Nandkishore, D. A. Huse, S. L. Sondhi, Rare region effects dominateweakly disordered three-dimensional Dirac points, Phys. Rev. B 89 (24)(2014) 245110. doi:10.1103/PhysRevB.89.245110 .[95] T. O. Wehling, A. M. Black-Schaffer, A. V. Balatsky, Dirac materials,Adv. Phys. 63 (1) (2014) 1–76. doi:10.1080/00018732.2014.927109 .4796] M. M. Ugeda, I. Brihuega, F. Guinea, J. M. G´omez-Rodr´ıguez, Missingatom as a source of carbon magnetism, Phys. Rev. Lett. 104 (9) (2010)096804. doi:10.1103/PhysRevLett.104.096804 .[97] S. H. Pan, E. W. Hudson, K. M. Lang, H. Eisaki, S. Uchida, J. C. Davis,Imaging the effects of individual zinc impurity atoms on superconductivityin Bi Sr CaCu O δ , Nature 403 (6771) (2000) 746–750. doi:10.1038/35001534 .[98] P. Van Mieghem, Theory of band tails in heavily doped semiconductors,Rev. Mod. Phys. 64 (3) (1992) 755–793. doi:10.1103/RevModPhys.64.755 .[99] M. Suslov, Density of states near an Anderson transition in four-dimensional space: Lattice model, J. Exp. Theor. Phys. 79 (2) (1994)307. doi:10.1134/1.558140 .[100] S. Yaida, Instanton calculus of Lifshitz tails, Phys. Rev. B 93 (7) (2016)075120. doi:10.1103/PhysRevB.93.075120 .[101] J. H. Pixley, D. A. Huse, S. Das Sarma, Rare-region-induced avoided quan-tum criticality in disordered three-dimensional Dirac and Weyl semimet-als, Phys. Rev. X 6 (2) (2016) 021042. doi:10.1103/PhysRevX.6.021042 .[102] V. Gurarie, Theory of avoided criticality in quantum motion in a randompotential in high dimensions, Phys. Rev. B 96 (1) (2017) 014205. doi:10.1103/PhysRevB.96.014205 .[103] M. Buchhold, S. Diehl, A. Altland, Vanishing density of states in weaklydisordered Weyl semimetals, Phys. Rev. Lett. 121 (21) (2018) 215301. doi:10.1103/PhysRevLett.121.215301 .[104] M. Buchhold, S. Diehl, A. Altland, Nodal points of Weyl semimetalssurvive the presence of moderate disorder, Phys. Rev. B 98 (20) (2018)205134. doi:10.1103/PhysRevB.98.205134 .[105] J. H. Wilson, D. A. Huse, S. Das Sarma, J. H. Pixley, Avoided quan-tum criticality in exact numerical simulations of a single disordered Weylcone, Phys. Rev. B 102 (10) (2020) 100201. doi:10.1103/PhysRevB.102.100201 .[106] H. Nielsen, M. Ninomiya, A no-go theorem for regularizing chiral fermions,Phys. Lett. B 105 (2) (1981) 219–223. doi:10.1016/0370-2693(81)91026-1 .[107] H. Nielsen, M. Ninomiya, Absence of neutrinos on a lattice: (I). Proofby homotopy theory, Nucl. Phys. B 185 (1) (1981) 20–40. doi:10.1016/0550-3213(81)90361-8 . 48108] H. Nielsen, M. Ninomiya, Absence of neutrinos on a lattice: (II). Intuitivetopological proof, Nucl. Phys. B 193 (1) (1981) 173–194. doi:10.1016/0550-3213(81)90524-1 .[109] S. V. Syzranov, L. Radzihovsky, High-dimensional disorder-driven phe-nomena in Weyl semimetals, semiconductors, and related systems,Annu. Rev. Condens. Matter Phys. 9 (1) (2018) 35–58. doi:10.1146/annurev-conmatphys-033117-054037 .[110] Y. Ominato, M. Koshino, Quantum transport in a three-dimensionalWeyl electron system, Phys. Rev. B 89 (5) (2014) 054202. doi:10.1103/PhysRevB.89.054202 .[111] A. Sinner, K. Ziegler, Corrections to the self-consistent Born approxi-mation for Weyl fermions, Phys. Rev. B 96 (16) (2017) 165140. doi:10.1103/PhysRevB.96.165140 .[112] J. Klier, I. V. Gornyi, A. D. Mirlin, From weak to strong disorder in Weylsemimetals: Self-consistent Born approximation, Phys. Rev. B 100 (12)(2019) 125160. doi:10.1103/PhysRevB.100.125160 .[113] J. H. Pixley, D. A. Huse, S. Das Sarma, Uncovering the hidden quantumcritical point in disordered massless Dirac and Weyl semimetals, Phys.Rev. B 94 (12) (2016) 121107. doi:10.1103/PhysRevB.94.121107 .[114] J. P. S. Pires, B. Amorim, A. Ferreira, ˙I. Adagideli, E. R. Mucciolo,J. M. V. P. Lopes, Randomness determines the fate of quantum critical-ity in 3D Dirac semimetals, arXiv:2010.04998 [cond-mat, physics:physics,physics:quant-ph] (Nov. 2020). arXiv:2010.04998 .[115] J. H. Wilson, J. H. Pixley, D. A. Huse, G. Refael, S. Das Sarma, Do thesurface Fermi arcs in Weyl semimetals survive disorder?, Phys. Rev. B97 (23) (2018) 235108. doi:10.1103/PhysRevB.97.235108 .[116] M. G¨arttner, S. V. Syzranov, A. M. Rey, V. Gurarie, L. Radzihovsky,Disorder-driven transition in a chain with power-law hopping, Phys. Rev.B 92 (4) (2015) 041406. doi:10.1103/PhysRevB.92.041406 .[117] A. Weiße, G. Wellein, A. Alvermann, H. Fehske, The kernel polyno-mial method, Rev. Mod. Phys. 78 (1) (2006) 275–306. doi:10.1103/RevModPhys.78.275 .[118] R. Alben, M. Blume, H. Krakauer, L. Schwartz, Exact results for a three-dimensional alloy with site diagonal disorder: Comparison with the co-herent potential approximation, Phys. Rev. B 12 (10) (1975) 4090–4094. doi:10.1103/PhysRevB.12.4090 .[119] T. Iitaka, T. Ebisuzaki, Random phase vector for calculating the traceof a large matrix, Phys. Rev. E 69 (5) (2004) 057701. doi:10.1103/PhysRevE.69.057701 . 49120] A. Hams, H. De Raedt, Fast algorithm for finding the eigenvalue dis-tribution of very large matrices, Phys. Rev. E 62 (3) (2000) 4365–4377. doi:10.1103/PhysRevE.62.4365 .[121] J. H. Pixley, Y.-Z. Chou, P. Goswami, D. A. Huse, R. Nandkishore,L. Radzihovsky, S. Das Sarma, Single-particle excitations in disorderedWeyl fluids, Phys. Rev. B 95 (23) (2017) 235101. doi:10.1103/PhysRevB.95.235101 .[122] T. Holder, C.-W. Huang, P. M. Ostrovsky, Electronic properties of disor-dered Weyl semimetals at charge neutrality, Phys. Rev. B 96 (17) (2017)174205. doi:10.1103/PhysRevB.96.174205 .[123] O. Shtanko, L. Levitov, Robustness and universality of surface states inDirac materials, Proc. Natl. Acad. Sci. U.S.A. 115 (23) (2018) 5908–5913. doi:10.1073/pnas.1722663115 .[124] E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, P. O. Sukhachov, Origin ofdissipative Fermi arc transport in Weyl semimetals, Phys. Rev. B 93 (23)(2016) 235127. doi:10.1103/PhysRevB.93.235127 .[125] R.-J. Slager, V. Juriˇci´c, B. Roy, Dissolution of topological Fermi arcsin a dirty Weyl semimetal, Phys. Rev. B 96 (20) (2017) 201401. doi:10.1103/PhysRevB.96.201401 .[126] E. Brillaux, A. A. Fedorenko, Fermi arcs and surface criticality in dirtyDirac materials, arXiv:2009.12138 [cond-mat] (Sep. 2020). arXiv:2009.12138 .[127] J. Lee, J. H. Pixley, J. D. Sau, Chiral anomaly without Landau levels:From the quantum to the classical regime, Phys. Rev. B 98 (24) (2018)245109. doi:10.1103/PhysRevB.98.245109 .[128] Y. Xu, F. Zhang, C. Zhang, Structured Weyl points in spin-orbit coupledfermionic superfluids, Phys. Rev. Lett. 115 (26) (2015) 265304. doi:10.1103/PhysRevLett.115.265304 .[129] B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, C. Felser, R. J. Cava,B. A. Bernevig, Beyond Dirac and Weyl fermions: Unconventional quasi-particles in conventional crystals, Science 353 (6299) (2016) aaf5037. doi:10.1126/science.aaf5037 .[130] B. J. Wieder, Y. Kim, A. M. Rappe, C. L. Kane, Double Dirac semimetalsin three dimensions, Phys. Rev. Lett. 116 (18) (2016) 186402. doi:10.1103/PhysRevLett.116.186402 .[131] A. Altland, M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopicnormal-superconducting hybrid structures, Phys. Rev. B 55 (2) (1997)1142–1161. doi:10.1103/PhysRevB.55.1142 .50132] R. Gade, Anderson localization for sublattice models, Nucl. Phys. B398 (3) (1993) 499–515. doi:10.1016/0550-3213(93)90601-K .[133] H. Shapourian, T. L. Hughes, Phase diagrams of disordered Weyl semimet-als, Phys. Rev. B 93 (7) (2016) 075108. doi:10.1103/PhysRevB.93.075108 .[134] E. J. K¨onig, H.-Y. Xie, D. A. Pesin, A. Levchenko, Photogalvanic effectin Weyl semimetals, Phys. Rev. B 96 (7) (2017) 075123. doi:10.1103/PhysRevB.96.075123 .[135] A. A. Burkov, Dynamical density response and optical conductivity intopological metals, Phys. Rev. B 98 (16) (2018) 165123. doi:10.1103/PhysRevB.98.165123 .[136] B. Cheng, T. Schumann, S. Stemmer, N. P. Armitage, Probing chargepumping and relaxation of the chiral anomaly in a Dirac semimetal,arXiv:1910.13655 [cond-mat] (Oct. 2019). arXiv:1910.13655 .[137] J. H. Pixley, J. H. Wilson, D. A. Huse, S. Gopalakrishnan, Weyl semimetalto metal phase transitions driven by quasiperiodic potentials, Phys. Rev.Lett. 120 (20) (2018) 207604. doi:10.1103/PhysRevLett.120.207604 .[138] Y. Fu, E. J. K¨onig, J. H. Wilson, Y.-Z. Chou, J. H. Pixley, Magic-angle semimetals, npj Quantum Mater. 5 (1) (2020) 1–8. doi:10.1038/s41535-020-00271-9 .[139] V. Mastropietro, Stability of Weyl semimetals with quasiperiodic disor-der, Phys. Rev. B 102 (4) (2020) 045101. doi:10.1103/PhysRevB.102.045101doi:10.1103/PhysRevB.102.045101