Discovery of Weyl fermion semimetals and topological Fermi arc states
M. Zahid Hasan, Su-Yang Xu, Ilya Belopolski, Shin-Ming Huang
DDiscovery of Weyl fermion semimetals andtopological Fermi arc states
M. Zahid Hasan ∗ ,
1, 2
Su-Yang Xu, Ilya Belopolski, and Shin-Ming Huang
3, 1 Laboratory for Topological Quantum Matter and Spectroscopy (B7),Department of Physics, Princeton University,Princeton, New Jersey 08544, USA Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Department of Physics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan (Dated: February 24, 2017)
Abstract
Weyl semimetals are conductors whose low-energy bulk excitations are Weyl fermions, whereastheir surfaces possess metallic Fermi arc surface states. These Fermi arc surface states are pro-tected by a topological invariant associated with the bulk electronic wavefunctions of the material.Recently, it has been shown that the TaAs and NbAs classes of materials harbor such a stateof topological matter. We review the basic phenomena and experimental history of the discov-ery of the first Weyl semimetals, starting with the observation of topological Fermi arcs and Weylnodes in TaAs and NbAs by angle and spin-resolved surface and bulk sensitive photoemission spec-troscopy and continuing through magnetotransport measurements reporting the Adler-Bell-Jackiwchiral anomaly. We hope that this article provides a useful introduction to the theory of Weylsemimetals, a summary of recent experimental discoveries, and a guideline to future directions.Keywords: topological phase of matter, topological insulator, quantum Hall effect, Chern num-ber, topological invariant, topological phase transition, Weyl materials ∗ Email: [email protected] a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b . INTRODUCTION The rich correspondence between high-energy particle physics and low-energy condensedmatter physics has been a source of insights throughout the history of modern physics .It has led to important breakthroughs in many aspects of fundamental physics, such asthe Planck constant and blackbody radiation, the Pauli exclusion principle and magnetism,as well as the Anderson-Higgs mechanism and superconductivity, which in turn helped de-velop materials with useful applications. In the past decade, the discovery of massless Diracfermions in graphene and on the surface of topological insulators has taken center stagein condensed matter science and has led to novel considerations of the Dirac equation incrystals . The Dirac equation, proposed in 1928 by Paul Dirac, represents a founda-tional unification of quantum mechanics and special relativity in describing the nature ofthe electron. Its solutions suggest three distinct forms of relativistic particles: the Dirac,Majorana, and Weyl fermions. Only one year later, in 1929, Hermann Weyl pointed outthat the Dirac equation without mass naturally gives rise to a simpler Weyl equation,whosesolutions are associated with massless fermions of definite chirality, particles known as Weylfermions . Weyls equation was intended as a model of elementary particles, but in nearly86 years, no candidate Weyl fermions have ever been established in high-energy experiments.Neutrinos were once thought to be such particles but were later found to possess a smallmass. Recently, emergent, quasiparticle analogs of Weyl fermions have been discovered incertain electronic materials that exhibit strong spinorbit coupling and topological behavior.Just as Dirac fermions emerge as signatures of topological insulators , in certain types ofsemimetals, such as tantalum or niobium arsenides, electrons behave like Weyl fermions.In 1937, the physicist Conyers Herring considered the conditions under which electronicstates in solids have the same energy and crystal momentum even in the absence of anyparticular symmetry . Near such accidental band touching points, the low-energy excita-tions are described by the Weyl equation . In recent times, these touching points havebeen studied theoretically in the context of topological materials and are referred to as Weylpoints, and the quasiparticles near them are the emergent Weyl fermions . In thesecrystals, the quantum mechanical wave function of an electron state acquires a geometric orBerry phase when tracing out a closed loop in momentum space. This Berry phase is identi-cal to that acquired by an electron tracing out a closed loop in real space in the presence of2 magnetic monopole. In the same way that magnetic monopoles correspond to sources orsinks of magnetic flux, a Weyl semimetal hosts momentum space monopoles that correspondto sources or sinks of Berry curvature. These Berry curvature monopoles are precisely theWeyl points of a Weyl semimetal. Furthermore, the chirality of the Weyl fermion corre-sponds to the chiral charge of the Weyl point. In a Weyl semimetal, the chirality associatedwith each Weyl node can be understood as a topologically protected charge, hence broaden-ing the classification of topological phases of matter beyond insulators. Remarkably, Weylnodes are extremely robust against imperfections in the host crystal and are protected bythe crystal’s inherent translational invariance. The real-space Weyl points are associatedwith chiral fermions, and in momentum space they behave like magnetic monopoles. Thefact that Weyl nodes are related to magnetic monopoles suggests that they will be foundin topological materials that are in the vicinity of a topological phase transition. Like atopological insulator, a Weyl semimetal hosts topological surface states arising from a bulktopological invariant. However, while the surface states of a topological insulator have aFermi surface that consists of closed curves in momentum space, a Weyl semimetal hostsan exotic, anomalous surface-state band structure containing topological Fermi arcs, whichform open curves that terminate on bulk Weyl points . Theory has suggested that insystems where inversion or time-reversal symmetry is broken, a topological insulator phasenaturally allows a phase transition to a Weyl semimetal phase. Building on these ideas, manyresearchers, including the Princeton University group, used ab initio calculations to predictcandidate materials and perform angle-resolved photoemission spectroscopy (ARPES) todetect bulk Weyl points and surface Fermi arcs in TaAs and its cousins . ARPES isan ideal tool for studying such a topological material, as known from the extensive body ofwork on topological insulators . The ARPES technique involves shining light on a materialand measuring the energy, momentum, and spin of the emitted photoelectrons, both fromthe surface and the bulk. This allows for the direct observation of both bulk Weyl pointsand Fermi arc surface states. Weyl semimetals further give rise to fascinating phenomena intransport, including a chiral anomaly in the presence of parallel electric and magnetic fields,a novel anomalous Hall response, and exotic surface-state quantum oscillations. Even moreexotic effects may arise in the presence of superconductivity, where Weyl semimetals maygive rise to quasiparticles exhibiting non-Abelian statistics, potentially providing a platformto realize novel effects in spintronics or a new type of topological qubit .3n this review, we survey the experimental discovery of the first Weyl semimetal in TaAs.We first provide some key elements of the theory of Weyl semimetals. Then, we offer ahistory and some intuition for Weyl semimetals from the point of view of a search problem.Next, we review the key theoretical and experimental works of the discovery of the firstWeyl semimetal in TaAs by ARPES. We also discuss the discovery of Weyl semimetals inthe other compounds of the TaAs family, namely NbAs, TaP, and NbP. As an applicationof the ARPES results, we discuss observation of the chiral anomaly in TaAs by transport.We further mention briefly, but in no way attempt to discuss exhaustively, a wide rangeof closely related topics, including: inversion-breaking Weyl semimetals beyond the TaAsclass, topological line node semimetals, strongly Lorentz violating Type II Weyl semimetals,magnetic Weyl semimetals, and Weyl superconductors. These additional directions will nodoubt continue to enrich a fascinating and rapidly developing field of research. II. KEY CONCEPTS
Before reviewing the experimental discovery of Weyl fermion semimetals, we present afew key concepts useful to understand what follows. We do not by any means attempt toprovide a complete review of the current theoretical understanding of Weyl semimetals.We present an explicit theoretical model for a Weyl semimetal. We consider a two-bandBloch Hamiltonian in three dimensions, h ( k ) = a ( k ) σ x + b ( k ) σ y + c ( k ) σ z a ( k ) = t (cos( k x ) + cos( k y ) + cos( k z ) − cos( k ) − , (1) b ( k ) = t sin( k y ), c ( k ) = t sin( k z )The lattice constant is set to unity. Here, the σ are the Pauli matrices, k = ( k x , k y , k z ) isthe crystal momentum, t is a hopping parameter and h ( k ) is the 2 × k ± W = ( ± k , , a ( k ), b ( k ) and c ( k )simultaneously vanish. At these special k ± W the two bands are degenerate.We can expand the Hamiltonian in the vicinity of the k ± W to derive a low-energy effectivemodel. We note that a ( k ) ≈ ( k − k ± W ) · ∇ a ( k ± W ) and similar for b ( k ) , c ( k ). We define4 ± ≡ k − k ± W and we find that the low-energy model gives a cone-shaped dispersion, h ± ( p ) = v x p ± x σ x + v y p ± y σ y + v z p ± z σ z (2) v x = − t sin( ± k ), v y = v z = t Our result is tantamount to the Weyl Hamiltonian of particle physics , H W ( p ) = p · σ (3)However, we note that unlike H W ( p ), our effective theory h ± ( p ) shows a Weyl fermionwhich has different Fermi velocities in different momentum directions, and so is not isotropic.This behavior is quite reasonable because Lorentz invariance is not required in low-energyeffective theories in crystals. Next, we consider the effect of adding small, arbitrary perturba-tions to h ( k ), taking a (cid:48) ( k ) = a ( k ) + ∆ a ( k ) and similar for b ( k ) , c ( k ). Will such perturbationsgap out the Weyl cones? To lowest order, we find a (cid:48) ( k ) ∼ ( k − k ± W ) · ∇ a (cid:48) ( k ± W ) + ∆ a ( k ± W ) =( k − ( k ± W + ∆ k ± )) · ∇ a (cid:48) ( k ± W ), where ∆ k ± in general will depend on all three of the perturbingfunctions ∆ a ( k ), ∆ b ( k ) and ∆ c ( k ). We see that the low-energy Hamiltonian remains gapless,but that the Weyl points move around in momentum space as we perturb the system.The local stability of the Weyl points is related to the Chern number, a topologicalinvariant well-known from the integer quantum Hall effect. The Chern number χ is definedas, χ = 12 π (cid:90) (cid:90) dk x dk y ˆ z · Ω, Ω = ∇ k × A , A = − i (cid:104) k, n |∇ k | k, n (cid:105) (4)Here, A is the Berry connection and Ω is the Berry curvature. In the case of the integerquantum Hall effect, the system is two-dimensional, so we integrate over the entire Brillouinzone. To find the Hall conductivity of a quantum Hall state, we further sum the result overall occupied bands n . In the case of a Weyl semimetal, the system is three-dimensional,so we must choose some closed two-dimensional manifold within the bulk Brillouin zoneand calculate the Chern number on that manifold. If we choose a small spherical manifoldenclosing the Weyl point, we find that the Weyl point is associated with a Chern number χ = ±
1. We refer to this χ as the chiral charge of the Weyl point. The chiral chargemeasures the Berry flux through the spherical manifold, analogous to Gauss’s law in classicalelectrodynamics. In the same way as for an electric charge, the chiral charge is quantizedand the Chern number on any manifold depends only on the enclosed chiral charge.5t is clear that Weyl points arise generically in two-band models. In particular, theunderlying mathematics does not care about the basis of the Hamiltonian. Typically, aWeyl semimetal refers to a normal electron system, but the basis can also be a Bogoliubovspinor, describing a Weyl superconductor, or it may consist of bosonic particles, describing,for instance, a photonic Weyl semimetal. At the same, it is important to note that mostnormal electron crystals have both time reversal and inversion symmetry, which requires allbands to be everywhere doubly-degenerate. In such a system, a band crossing correspondsto a four-fold degeneracy, requiring a description by a four-band model. However, in a four-band model, there are too many Hamiltonian matrix elements that must be simultaneouslyset to zero, so band crossings do not arise generically. In this way, the experimental studyof Weyl semimetals has long been held back because nature prefers to maintain both timereversal and inversion symmetries.If Weyl points are topologically stable, we might ask how they can be created or de-stroyed. While an individual Weyl point cannot be gapped out within band theory by smallperturbations, a large perturbation can cause Weyl points of opposite chirality to annihilateeach other, leaving the system gapped. If we imagine slowly applying a large perturbationto h ( k ), we will find that the Weyl points move around in momentum space until the systemarrives at a critical point where the Weyl points sit on top of each other. A further pertur-bation can then cause the system to gap out. Conversely, Weyl points can only be createdthrough such a critical point, in sets of equal and opposite chiral charge. An equivalentstatement is that the net chiral charge in the entire Brillouin zone is always zero. It is alsopossible to gap a Weyl point by going beyond band theory. If h ( k ) is taken to describe anormal electron system, then superconductivity may gap the Weyl points by breaking U (1)charge conservation symmetry. A charge density wave or disorder may also allow scatteringdirectly between the k ± W , gapping the Weyl cones by breaking translation symmetry. Inthis sense, Weyl points are protected by charge and translation symmetries. By contrast,the Dirac points in graphene , topological insulators and Dirac semimetals typically re-quire not only the symmetries implied within band theory, but further rely on time reversalsymmetry, inversion symmetry or other crystal space group symmetries. We see that Weylpoints are uniquely robust in that they are topologically stable within band theory withoutany additional symmetries.The integer quantum Hall effect is associated with gapless chiral edge modes guaranteed6y the Chern number. What boundary states are guaranteed by Chern numbers in Weylsemimetals? We consider two-dimensional slices of the bulk Brillouin zone, as shown inFig. 1 c-e . Any slice not containing the Weyl points is gapped, and we can calculate a Chernnumber for that slice. In this way, we can view the three-dimensional band structure as aset of two-dimensional slices with a tuning parameter, k x . As we scan k x , the bulk band gapcloses and reopens and we tune the system through a topological phase transition, changingthe Chern number. The critical slices k x = ± k are the slices containing a Weyl point. Wecan similarly partition the surface states of a Weyl semimetal into one-dimensional edges,as shown in the surface Brillouin zone square in Fig. 1 d . Edges associated with a non-zeroChern number in the bulk will host gapless chiral edge modes, Fig. 1 e , while edges with azero Chern number are gapped, Fig. 1 c . We can see that these one-dimensional edge statesthen assemble into a sheet of surface states which terminate at the surface projections of theWeyl points, forming a topological Fermi arc surface state. On a constant-energy cut of thesurface band structure, the Fermi arc forms an open, disjoint curve. Much like the Diraccone surface state of a three-dimensional Z topological insulator, the Fermi arc of a Weylsemimetal is anomalous in the sense that it cannot arise in any isolated two-dimensionalsystem, but only on the two-dimensional boundary of a three-dimensional bulk. However,the Fermi arc arguably provides the most dramatic example to date of an anomalous bandstructure, because unlike the Dirac cone surface state or any traditional two-dimensionalband structure, the constant-energy contours do not even form closed curves. III. SEARCH FOR MATERIALS
The paramount experimental challenge to the discovery of the first Weyl semimetal wasfinding suitable material candidates. It was necessary to find compounds available in high-quality single crystals with Weyl fermions and Fermi arcs accessible under reasonable mea-surement conditions. In the five years or so preceding the discovery of TaAs, roughly from2011 to 2015, the fundamental theory of Weyl semimetals was essentially available and pow-erful ARPES systems were ready to tackle the problem. The missing link was the lack ofmaterial candidates. In this section, we discuss the particular insights relating to material7earch that bridged theory to experiment and directly opened the experimental study ofWeyl semimetals. We also highlight some open problems along these directions.It was perhaps a historical accident that the community initially focused on time-reversalbreaking Weyl semimetals. After Murakami’s work showing a connection between Weylsemimetals and topological insulators , Wan et al. proposed in 2011 the first material can-didate for a Weyl semimetal in a family of magnetic pyrochlore iridates, R Ir O , where R is a rare-earth element . In this crystal, the Ir atoms form a sublattice of corner-sharingtetrahedra and the authors argued that the material prefers an all-in/all-out configuration ofmagnetic moments on the tetrahedra, breaking time-reversal symmetry . Next, theoreticalanalysis showed that, by increasing the on-site Coulomb interaction strength, the systemexhibits a transition from a magnetic metal to a Mott insulator. In between, there is anintermediate phase, which was found to be a Weyl semimetal. Wan et al. also made anexplicit connection between Weyl semimetals and the integer quantum Hall (IQH) state.Specifically, they proposed the idea of calculating Chern numbers on two-dimensional mani-folds in the three-dimensional Brillouin zone of a bulk material. In a Weyl semimetal, therewould be manifolds with nonzero Chern numbers. As in the IQH effect, the one-dimensionaledge of the two-dimensional manifold would protect chiral edge states and these would as-semble together to form Fermi arc surface states on the two-dimensional surface of a Weylsemimetal. The specific material proposal and theoretical advances spurred considerableinterest in Weyl semimetals. However, attempts to realize a Weyl semimetal in R Ir O in experiment were met with significant challenges. The all-in/all-out magnetic order isunder debate in experiments . Also, while metal-insulator transitions were observed inEu Ir O and Nd Ir O and transport and optical behaviors were roughly consistent witha semimetal or a narrow band-gap semiconductor , the results overall were inconclusive.At the same, ARPES measurements are lacking, possibly because it has been difficult togrow large, high-quality single crystals or because it has been difficult to prepare a flatsample surface for measurement.Shortly following the proposal for a time-reversal breaking Weyl semimetal in R Ir O ,Burkov and Balents proposed, also in 2011, an engineered time-reversal breaking Weylsemimetal in a heterostructure built up from topological insulator and magnetic layers .They provided a simple tight-binding model which explicitly includes the necessary ingredi-ents for a Weyl semimetal and shows only two Weyl points, providing the “hydrogen atom”8f a Weyl semimetal in theory. They also point out that their model shows a non-quantizedanomalous Hall conductivity proportional to the separation of Weyl points in momentumspace. Despite its theoretical significance, this model has remained unrealized in experiment.It requires a topological insulator and a trivial insulator with suitable lattice match thatcan both be grown by a thin film technique such as molecular beam epitaxy. In addition,magnetism must somehow be incorporated into the system, either by doping or perhaps byreplacing the trivial insulator with a ferromagnet. Studying the emergent band structure ofsuch a system in experiment would also be a formidable challenge.Similar in spirit was a subsequent work by Bulmash et al. , who in 2014 proposed engineer-ing a time-reversal breaking Weyl semimetal by doping a topological insulator . Specifically,they proposed starting with HgTe, a three-dimensional topological insulator, tuning it tothe critical point for a band inversion by Cd doping and then introducing a magnetic orderthrough Mn doping, giving Hg − x − y Cd x Mn y Te. Again, this proposal was quite reasonablein that it introduced the key ingredients for a Weyl semimetal in an explicit way throughdoping. Like the proposal by Burkov and Balents, the resulting system would also have onlytwo Weyl points. However, despite the conceptual elegance of this work, ARPES measure-ments on Hg − x − y Cd x Mn y Te were unsuccessful. One reason might be that disorder fromdoping degraded sample quality too severely.With the arrival of topological Dirac semimetals in Na Bi and Cd As , the commu-nity also considered magnetic doping of a Dirac semimetal as a way to realize a time-reversalbreaking Weyl semimetal, although there are no formal published proposals. These ideaswere close to the work by Bulmash et al. , but now the Dirac semimetal state was achievedintrinsically and only the time-reversal breaking needed to be implemented by doping. In-deed, these proposals faced difficulties similar to the case of Hg − x − y Cd x Mn y Te. Specifically,it was challenging to grow high-quality single crystals with magnetic doping which gave riseto a useful magnetic order. Another concern is that the spin-splitting from a magnetic dop-ing may be too small to produce Weyl points above available experimental resolution andspectral linewidth in an ARPES measurement.It was gradually appreciated that an intrinsic material may be easier to study than anengineered or doped system. However, finding an intrinsic time-reversal breaking Weylsemimetal presents its own challenges. The original proposal for R Ir O is a case in point,because the magnetic order is difficult to predict from calculation and has not been conclu-9ively shown in experiment. Also, it is unclear whether the value of the correlation parameter U appropriate for the real material actually places the system in the Weyl semimetal phase.In 2011, the same year of the proposal for R Ir O and the topological insulator multilayer,Xu et al . proposed a Weyl semimetal in HgCr Se . Unlike R Ir O , HgCr Se is known tobe an intrinsic ferromagnet with a Curie temperature, T C (cid:39)
120 K, which is easily accessiblein experiment. ARPES experiments on this system were not successful, perhaps becausecrystals are not of sufficiently high quality. However, recently, evidence for half-metallicitywas reported in transport experiments, which may lead to renewed interest in this material .Another interesting property is that the bands responsible for the band inversion arise fromHg and Se, while the ferromagnetism is associated with Cr. This makes the band inversionin ab initio calculation robust to changes in U . Nonetheless, one concern is that the cubicstructure of HgCr Se allows many magnetization axes, which may favor the formation ofsmall magnetic domains. It remains a considerable experimental challenge to create largemagnetic domains in situ , as well as to understand whether a Weyl semimetal could beshown in an ARPES experiment which averages over many magnetic domains. Nonetheless,the proposal for HgCr Se offers an approach to searching for new candidates. In particular,it may be fruitful to study systems with an experimentally well-known magnetic order andwhere the Weyl semimetal state is robust to free parameters in the ab initio calculation.As the experimental complications of time-reversal breaking Weyl semimetals were ap-preciated, it was understood that breaking inversion symmetry may provide an easier routeto the first Weyl semimetal. In retrospect, this should be rather clear. While the effectof breaking inversion symmetry and time-reversal symmetry is mathematically similar atthe level of a single-particle Hamiltonian, these two symmetries relate to deeply differentphenomena in any real material. Inversion symmetry breaking is a property of a crystalstructure, which can be measured directly by X-ray diffraction and presents no particularcomplications to ab initio calculation. By contrast, magnetism is a correlated phenomenonwhich is extremely difficult to reliably predict from first principles, challenging to under-stand in experiment and difficult to accurately capture in an ab initio calculation even ifthe magnetic order is experimentally known. Additional complications arise because exper-iments must be carried out below the magnetic transition temperature and a measurementmay average over many small magnetic domains. By contrast, large inversion breaking do-mains have been observed in inversion breaking materials. A well known example is the bulk10ashba material BiTeI, where the bulk Rashba splitting due to inversion breaking can bedirectly observed in ARPES. The idea that inversion breaking systems are simpler reignitedthe search for Weyl semimetals and led to the prediction and experimental observation ofthe first Weyl semimetal in TaAs.Before discussing the theoretical prediction of TaAs, we note that despite the early focuson time-reversal breaking Weyl semimetals, there was some interest in inversion breakingsystems. In 2012, Singh et al. considered the tunable topological insulators TlBi(S − x Se x ) and TlBi(S − x Te x ) and proposed to engineer a Weyl semimetal by alternating between Seand Te from layer to layer, which would break inversion symmetry . While this proposalwas directly motivated by the recent experimental success in realizing a topological phasetransition in TlBi(S − x Se x ) , the expected separation of the Weyl points in the pro-posed system fell below available experimental resolution in ARPES. Hal´asz and Balentsproposed in 2012 an inversion-breaking Weyl semimetal in a HgTe/CdTe heterostructure,inspired by a superlattice model similar to the time-reversal breaking model in a topologicalinsulator multilayer . This proposal remains unrealized, perhaps for reasons similar to theearlier proposal by Burkov and Balents, that there are considerable experimental challengesto studying the emergent band structure of such a superlattice. Lastly, in 2014, Liu andVanderbilt proposed realizing an inversion breaking Weyl semimetal by tuning an inversionbreaking topological insulator through a topological phase transition, exactly as first dis-cussed by Murakami . Specifically, they predict a Weyl semimetal in LaBi − x Sb x Te andLuBi − x Sb x Te for x ∼ . − .
9% and x ∼ . − . x . Nonetheless, such tunable inversion breakingsystems are of considerable interest because they would exhibit a topological phase tran-sition to a Weyl semimetal, which to date is unrealized. Realizing such a system mayrequire us to understand why the Weyl semimetal phase is so narrow in LaBi − x Sb x Te andLuBi − x Sb x Te and whether this behavior is natural.The material search for intrinsic inversion breaking Weyl semimetals was based on a broadfoundation of known crystal structures. These have been accumulated by X-ray diffractionexperiments performed over the course of a century or more of research in physics andchemistry and which have been cataloged in databases such as the Inorganic Crystal Struc-ture Database of FIZ Karlsruhe. Certain catalogs of magnetic compounds also exist with11easurements of magnetic transition temperatures or magnetic structures. These sourcesprovided a starting point for material searches. It is worth noting that nature prefers tomaintain both inversion and time-reversal symmetry, severely limiting from the outset thenumber of possible material candidates for both kinds of Weyl semimetals.Although ab initio calculations for inversion-breaking systems are simpler, the calculationis still challenging because Weyl points typically occur at arbitrary k points rather than athigh symmetry points or on high symmetry lines. As a result, for each compound, it isnecessary to calculate the band structure at all k points throughout the bulk Brillouin zoneto demonstrate or exclude the existence of Weyl points. A large spin-orbit coupling ispreferred to produce a large spin-splitting. A closely-related requirement is that the Weylpoints should be well-separated. Materials which also crystallize in an inversion-symmetricstructure are excluded. The Weyl points should be very near the Fermi level and, forfuture transport experiments, it is preferable to find systems without irrelevant pocketsat the Fermi level. The materials conditions such as cleavability and large single domaincrystallinity that make ARPES measurements feasible were also considered to narrow downthe candidate list further. In 2014, TaAs emerged as a promising candidate satisfyingthese ARPES criteria . Shortly thereafter, in early 2015, the observation of bulk Weylfermions and topological Fermi arcs were finally reported in TaAs, demonstrating the firstWeyl fermion semimetal .With the recipe for an inversion-breaking Weyl semimetal better understood, many otherinversion-breaking Weyl semimetals were proposed . Here, we do not attempt to surveythese later inversion-breaking Weyl semimetal candidates. These systems will no doubt leadto many fascinating directions of research in the future. IV. EXPERIMENTAL REALIZATION OF WEYL SEMIMETALS
Following independent theoretical predictions by the Princeton and Institute of Physics,Chinese Academy of Sciences (IOP) groups , the discovery of the first Weyl semimetal inTaAs followed quickly also by Princeton and IOP independently. Experimental methodsto demonstrate a topological origin of Fermi arc were shown earlier by Xu et.al., . Bothbulk Weyl cones and topological Fermi arc surface states were directly observed by ARPES12n bulk TaAs single crystals, demonstrating a Weyl semimetal at the same time in the bulkand on the surface and in excellent agreement with calculation. . Equivalently, a nonzeroChern number was directly measured from the Fermi arcs. These observations confirmedthe bulk-boundary correspondence between Weyl cones and topological Fermi arcs. Thediscovery of the first Weyl semimetal in TaAs not only opened a new chapter in the study oftopological phases of matter, but also validated the shift of the community from time-reversalbreaking candidates to inversion breaking candidates. The identification of TaAs was quicklyfollowed by the discovery of Weyl semimetals in NbAs and TaP , with additional workon NbP . Below, we review the experimental discovery of Weyl semimetals by ARPESin TaAs and its cousins. We also discuss general criteria for demonstrating a Weyl semimetalin ARPES, which arose from studies of these compounds . A. Inversion symmetry breaking Weyl semimetal in TaAs
TaAs crystallizes in a body-centered tetragonal lattice system. The lattice constantsare a = 3 .
437 ˚A and c = 11 .
656 ˚A, and the space group is I md ( a/ , a/ , δ ) , δ (cid:39) c/
12. Figure 2 a shows a schematic illustration of TaAs’s crystal lattice.Crucially, the lattice breaks inversion symmetry, allowing it to realize a Weyl semimetal.The crystal can be viewed as a stack of square lattice layers of Ta or As (Fig. 2 b ). A surveyof the band structure shows that TaAs is a semimetal (Fig. 2 c ). Further first-principlesband structure calculations show that TaAs is a Weyl semimetal . Specifically, in thepresence of spin-orbit coupling, the bands are non-degenerate everywhere in k space exceptat the Kramers’ points due to the breaking of inversion symmetry. These non-degeneratebands intersect at 24 discrete points, the Weyl points, in the bulk Brillouin zone (Fig. 2 d ).On the (001) surface of TaAs, the 24 Weyl nodes project onto 16 points in the surfaceBrillouin zone. Eight projected Weyl points near the surface Brillouin Zone boundary ( ¯ X and ¯ Y points) have a projected chiral charge of ±
1. The other eight projected Weyl points,close to midpoints of the ¯Γ − ¯ X ( ¯ Y ) lines, have a projected chiral charge of ±
2. Figure 2 e shows the calculated surface state Fermi surface. Near the midpoint of each ¯Γ − ¯ X ( ¯ Y ) line,there is a crescent-shaped feature, which consists of two curves that join each other at thetwo end points. The two curves of the crescent are two Fermi arcs, and the two end points13orrespond to the projected Weyl points with projected chiral charge ± . The Weyl points and Weyl conesin TaAs were accessed by ARPES measurements at soft X-ray photon energies, which aresensitive to the bulk states. Soft X-ray data showed that the Fermi surface of TaAs consistsof discrete points at specific incident photon energies and binding energies, at generic pointsin the Brillouin zone (Fig. 3 a ). Away from these discrete points in binding energy, thebands dispersed linearly in both the in-plane momenta, k x and k y (Figs. 3 b,c ) and the out-of-plane momentum k z (Figs. 3 d ). The observation of linearly-dispersing band crossings atgeneric points in the bulk Brillouin zone demonstrated Weyl points and Weyl nodes in TaAs,showing that TaAs is a Weyl semimetal.Independently of the bulk-sensitive soft X-ray ARPES measurements, surface-sensitiveARPES measurements at vacuum ultraviolet photon energies showed Fermi arcs on the (001)surface of TaAs. The ARPES measured Fermi surface showed three dominant features: abowtie-shaped feature centered at the ¯ X point, a elliptical feature centered at the ¯ Y point,and a crescent-shaped feature near the midpoint of each ¯Γ − ¯ X ( ¯ Y ) line (Fig. 4 a ). The resultswere in excellent agreement with the ab initio calculation (Fig. 2 e ). More extensive APRESmeasurements of the crescent showed that it consists of two segments joined together at theirendpoints (Fig. 4 b ), strongly suggesting Fermi arcs. Ab initio calculation showed topologicalFermi arcs in close agreement with the crescent observed in ARPES, showing that Fermiarcs were observed. A non-zero Chern number was also directly observed, providing anotherdemonstrating of Fermi arcs from the ARPES data, see discussion below. Finally, ARPESdata also showed that the terminations of the Fermi arcs coincide with the projections ofthe Weyl nodes, which demonstrated the bulk-boundary correspondence principle (Fig. 4 d ),further confirming the observation of a Weyl semimetal in TaAs . Later measurements byspin-resolved ARPES showed the spin texture of the Fermi arcs . In calculation, the spintexture was found to wind against the dispersion of the Fermi arc, so that if we traverse theFermi arc in a clockwise way, the spin texture winds in a counter-clockwise way (Fig. 4 c ).This spin texture was directly measured by spin-resolved ARPES and was found to be inexcellent agreement with the ab initio calculation (Fig. 4 e,f ). The relationship of the spintexture to the topological invariants of the bulk is not understood. It appears that there isno obvious constraint placed on the Fermi arc spin texture by the chiral charge of the Weyl14oints. We note that no spin-resolved soft X-ray ARPES end-station is currently available,which has hindered the measurement of the spin texture of the Weyl points and Weyl cones.This has prevented the direct experimental measurement of the chiral charge in a Weylsemimetal. To directly measure the chiral charge, either the appropriate ARPES systemmust become available or a new material is needed where the bulk band structure can beaccessed by vacuum ultraviolet ARPES.A non-zero Chern number in TaAs was directly measured from an ARPES spectrumof the surface states . The authors considered a cylindrical tube in the bulk Brillouinzone enclosing two Weyl points of the same chiral charge. The Chern number on this twodimensional manifold is +2 (Fig. 5 a ). As a result, the one-dimensional edge of the cylindricalslice hosts protected chiral edge states which cross the bulk band gap and have net chirality+2 (Fig. 5 b ). In this way, a Chern number can be directly measured in an ARPES spectrumof the surface state band structure by counting the net chirality of surface state crossings on aclosed loop in momentum space (Fig. 5 c ). It was found in ARPES that the one-dimensionalband structure on the loop showed two edge states of the same chirality at the Fermi level,in agreement with the topological theory (Fig. 5 d ). The direct measurement of a non-zeroChern number in TaAs by ARPES provides yet another independent demonstration thatTaAs is a Weyl semimetal. B. Criteria for topological Fermi arcs in Weyl semimetals
The case of TaAs showed that topological Fermi arcs do not always appear as disjointarcs. Specifically, in TaAs, the Fermi arcs appeared in pairs which together formed a closedcontour. As a result, the signature of a Fermi arc in the experiments on TaAs was not adisjoint arc but rather a Chern number or a surface state kink, as discussed above. Moregenerally, it was directly relevant for experiment to understand the ways that a topologicalFermi arc can arise in a material. As far as is currently understood, there are four distinctcriteria for topological Fermi arcs . All signatures are in principle experimentally accessiblein an ARPES measurement of a surface state band structure. Each signature alone, observedin any set of surface state bands, is sufficient to show that a material is a Weyl semimetal,1. Disjoint arc : Any surface state constant-energy contour with an open curve is a Fermiarc and demonstrates a Weyl semimetal.15.
Kink off a rotation axis : A Weyl point is characterized by chiral charge n , equal tothe Chern number on a small spherical manifold enclosing the Weyl point in the bulkBrillouin zone, illustrated by the small sphere in Fig. 5(e) . For a Weyl point ofchiral charge | n | > . Off a rotation axis, such a kink guarantees a Weyl semimetal.3. Odd number of curves : For projected chiral charge | n | >
1, the constant-energy con-tours may consist entirely of closed curves and the kink may be below experimentalresolution, so the constant-energy contour appears everywhere smooth. However, if | n | is odd, the constant-energy contour will consist of an odd number of curves, so atleast one curve must be a Fermi arc.4. Non-zero Chern number : Consider any closed loop in the surface Brillouin zone wherethe bulk band structure is everywhere gapped and, at some energy, add up the signsof the Fermi velocities of all surface states along this curve, with +1 for right moversand − . A non-zero sum on at least oneloop demonstrates a Weyl semimetal, provided the loop is chosen to be contractibleon the torus formed by the surface Brillouin zone.Note that while (1), (2) and (3) describe properties of a constant-energy slice of the Fermisurface, the counting argument (4) requires a measurement of the dispersion. We note alsothat criterion (4) allows us to determine all bulk topological invariants and Weyl points ofa material by studying only its surface states.One caveat in this rather formal analysis is that in a real ARPES experiment we neverrigorously measure only the surface or bulk states, but rather some combination of the two.There are, furthermore, many experimental scenarios where a surface state unrelated toFermi arcs can na¨ıvely satisfy one of the criteria. For instance, the photoemission crosssection of certain bands or certain regions of a band may be suppressed under particular16easurement conditions. This effect can give rise to an apparently disjoint contour or anapparent Chern number in a completely topologically trivial material. In addition, a kinkwill always be smeared out in an experiment, so that it can be challenging to distinguishbetween a kink and a smooth but rapidly dispersing band. Therefore, an ab initio calcula-tion remains crucial for ARPES studies of new Weyl semimetal candidates. However, it isequally unacceptable to simply show a general agreement between an ARPES spectrum anda calculation. We note that a number of early works claimed to show a Weyl semimetal byARPES in TaAs and NbP by measuring a Chern number, but those measurements dependedon resolving Fermi arcs which clearly fell below any available experimental resolution or spec-tral linewidth . As a result, those works essentially used a rough, overall agreementbetween calculation and experiment to claim a Weyl semimetal. A reasonable standard fordetection of a Weyl semimetal in ARPES is to consider a set of surface states which can beconfirmed in calculation to be topological Fermi arcs and which can be clearly identified inan ARPES spectrum. These Fermi arcs will not always show up as disjoint arcs, but theymust satisfy at least one of the four criteria discussed here. C. Weyl fermion transport and signatures of the chiral anomaly
It is known in quantum field theory that quantum fluctuations can violate classical conser-vation laws, a phenomenon called a quantum anomaly . Perhaps the best-studied exampleis the chiral anomaly associated with Weyl fermions . Historically, the chiral anomalywas crucial in understanding a number of important aspects of the Standard Model of par-ticle physics, such as the triangle anomaly associated with the decay of the neutral pion π , Refs. . Despite having been discovered more than 40 years ago, quantum anomaliesremained solely in the realm of high-energy physics.The discovery of Weyl fermion semimetals provides a natural route to realizing the chiralanomaly in condensed matter physics. Consider a Weyl semimetal with a particular con-figuration of Weyl points of positive and negative chiral charge in the bulk Brillouin zone.Parallel magnetic and electric fields can pump electrons between Weyl cones of oppositechirality, giving rise to a population imbalance between Weyl cones of positive and nega-tive chiral charge. This means that the numbers of Weyl fermions of a given chirality arenot separately conserved . The key observable consequence of the chiral anomaly in17 condensed matter system is that the longitudinal resistance is predicted to decrease as afunction of an external magnetic field, giving rise to a negative longitudinal magnetoresis-tance (LMR). Except for this natural platform, Dirac semimetals, a class of materials thathost Dirac fermion quasiparticles, may similarly give rise to a negative LMR under externalmagnetic field. In that case, the magnetic field not only directly produces a chiral anomalybut additionally serves the purpose of splitting the Dirac cone into a pair of Weyl cones ofopposite chirality by breaking time-reversal symmetry. It is worth noting that under suchconditions, extra caution is needed, because in real materials a magnetic field may havemany effects other than a simple Zeeman splitting of the band structure .The negative LMR was directly detected in electrical transport experiments on the TaAsfamily (Figs. 6 a-c ) and a number of Dirac semimetals . Other supporting evidenceincludes: (1) The negative LMR was prominent only in the geometry of parallel electricand magnetic fields. This is consistent with the (cid:126)E · (cid:126)B term in the chiral anomaly formu-lation (Fig. 6 c ). (2) The negative LMR did not depend on the electrical current directionwith respect to the crystalline axis (Figs. 6 a,b ). However, these data are not conclusive.A number of other effects can also lead to a negative LMR in metals , and some ofthem may show the same systematic dependences as described above. Hence, to achievean unambiguous proof, further study is needed. One particular phenomenon that may pro-vide stronger evidence is the dependence of the LMR on chemical potential. Because Weylpoints are monopoles of Berry curvature, the magnitude of negative LMR is expected tofollow a dramatic 1 /E dependence as the Fermi level moves away from the energy of theWeyl points. This E F dependence (Fig. 6 d ) can distinguish the chiral anomaly from othernegative LMR effects and, therefore, provides a clearer demonstration of the chiral anomalydue to Weyl fermions. V. OUTLOOK
It has been less than a year since the experimental realization of the first Weyl semimet-als, and the field has already evolved dramatically. One topic of recent interest is therealization in a material of a strongly Lorentz-violating Weyl fermion. Traditionally, stud-ies of Weyl fermions in quantum field theory were concerned with applications to particle18hysics, where Lorentz symmetry is respected. However, low-energy effective field theoriesin crystals need not satisfy Lorentz invariance, providing a richer variety of allowed theories.In particular, the form of the dispersion of a Weyl fermion in a crystal has more freedomthan in particle physics. Recently, it was pointed out that this freedom allows for a noveltype of Weyl fermion where the Weyl cone is tipped over on its side . Such stronglyLorentz-violating, or Type II, Weyl fermions allow us to study, in table-top experiments,exotic Lorentz-violating theories that are beyond the Standard Model. They also open upexperimental opportunities for studying novel spectroscopic and transport phenomena spe-cific to Type II Weyl fermions. Such phenomena include a chiral anomaly associated with atransport response that depends strongly on the direction of the electric field, an antichiraleffect of the chiral Landau level, a modified anomalous Hall effect, and emergent Lorentzinvariance arising from electronelectron interactions . To date, Type II Weyl fermionshave only been suggested in W − x Mo x Te and observed in LaAlGe . Therefore, it isof importance to continue the study of Type II Weyl semimetals.Because the TaAs family exhibits twelve pairs of Weyl points, it is of some interest tofind simpler materials with fewer Weyl points. Material searches are under way to find the“hydrogen atom” versions of Weyl semimetals with the minimum number of Weyl pointspossible, either four Weyl points in inversion-breaking Weyl semimetals or two Weyl pointsin magnetic Weyl semimetals. An additional challenge is to tune the materials so thatthe Fermi level is close to the Weyl points, ideally within 5-10 meV, without irrelevantpockets near the Fermi level, so that the Weyl fermionic excitations constitute the dominanttransport channel. Moreover, the T -breaking Weyl semimetals and Weyl superconductors(both T -breaking and I -breaking) can give rise to a wide range of novel properties. Tounderstand the interplay between the electronic interaction and the topological state inWeyl fermion semimetals, T -breaking magnetic Weyl materials could be crucial becausemagnets already harbor strong interactions. From a purely mathematical point of view,interacting Weyl phases further broaden the classification of topological phases of matter.Exploring the nontrivial spin polarization properties of interacting Weyl materials couldreveal a rich phase diagram. In a loose sense, the spin texture is approximately proportionalto the Berry flux and, hence, projects like monopoles or antimonopoles near the Weyl nodesin momentum space. This opens up opportunities for applications in spintronics. RealizingWeyl superconductors would be another exciting frontier. The Majorana surface states of19 Weyl superconductor can be potentially used for topological qubits. Given the rapiddevelopment of the field, it is also quite possible that in a few years, the most excitingfrontier will be something not projected here. VI. ACKNOWLEDGMENTS
The authors thank Adam Kaminski, Arun Bansil, BaoKai Wang, Bingbing Tong, ChengGuo, Chenglong Zhang, Chi-Cheng Lee, Chi Zhang, Chuang-Han Hsu, Daixiang Mou, DanielS. Sanchez, Donghui Lu, Fangcheng Chou, Fumio Komori, Guang Bian, Guoqing Chang,Hai-Zhou Lu, Hao Zheng, Hong Lu, Horng-Tay Jeng, Hsin Lin, J. D. Denlinger, Jie Ma,Junfeng Wang, Kenta Kuroda, Koichiro Yaji, Lunan Huang, Madhab Neupane, MakotoHashimoto, Mykhailo L. Prokopovych, Nan Yao, Nasser Alidoust, Pavel P. Shibayev, RamanSankar, Shik Shin, Shuang Jia, Shun-Qing Shen, Sungkwan Mo, Takeshi Kondo, Tay-RongChang, Titus Neupert, Vladimir N. Strocov, Xiao Zhang, Yun Wu, Zhujun Yuan and ZiquanLin for collaborations. Work at Princeton University by S.-Y.X. and M.Z.H. is supportedby the US Department of Energy under Basic Energy Sciences Grant No. DOE/BES DE-FG-02-05ER46200 and Grant No. DE-AC02-05CH11231 at the Advanced Light Source atLawrence Berkeley National Laboratory (LBNL), and Princeton University funds. M.Z.H.acknowledges Visiting Scientist user support from LBNL and partial support from the Gor-don and Betty Moore Foundation under Grant No. GBMF4547/Hasan. Shin-Ming Huangacknowledges visiting scientist support from Princeton University. Wilczek, F. Why are there Analogies between Condensed Matter and Particle Theory?
Phys.Today , 11-13 (1998). Anderson, P. W.
Basic Notions of Condensed Matter Physics , (Addison Wesley, 1984) Geim, A. K. & Novoselov, K. S. The rise of graphene.
Nature Mater. , 183-191 (2007). Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators.
Rev. Mod. Phys. , 3045-3067(2010). Hasan, M. Z. & Moore, J. E. Three-Dimensional Topological Insulators.
Ann. Rev. Cond. Mat.Phys. , 55-78 (2011). Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors.
Rev. Mod. Phys. ,1057-1110 (2011). Volovik, G. E.
The Universe in a Helium Droplet (Clarendon Press, Oxford, 2009). Turner, A. M. & Vishwanath, A. Beyond band insulators: topology of semi-metals and inter-acting phases. arXiv:1301.0330. Hasan, M. Z., Xu, S.-Y. and Neupane, M. Topological Insulators, Topological Semimetals,Topological Crystalline Insulators, and Topological Kondo Insulators, in
Topological Insulators,Fundamentals and Perspectives (eds Ortmann, F., Roche, S. and Valenzuela, S. O.) (JohnWiley & Sons, 2015). Vafek, O. & Vishwanath, A. Dirac Fermions in Solids
Ann. Rev. Cond. Mat. Phys. , 83 (2014). Hasan, M. Z., Xu, S.-Y. and Bian, G.,
Phys. Scr.
T164 Weyl, H. Elektron und gravitation. I.
Z. Phys. , 330-352 (1929). Herring, C. Accidental Degeneracy in the Energy Bands of Crystals,
Physical Review ,365-373 (1937). Abrikosov, A. A. & Beneslavskii, S. D. Some properties of gapless semiconductors of the secondkind,
Journal of Low Temperature Physics , 141-154 (1971). Nielsen, H. B. & Ninomiya, M. The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal,
Physics Letters B , , 389-396 (1983). Murakami, S. Phase transition between the quantum spin Hall and insulator phases in 3D:emergence of a topological gapless phase.
New J. Phys. , 356 (2007). Wan, X., Turner, A. M., Vishwanath, A. & Savrasov, S. Y. Topological Semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates.
Phys. Rev. B , 205101(2011). Yang, K.-Y., Lu, Y.-M. and Ran,Y. Quantum Hall effects in a Weyl semimetal,Possible appli-cation in pyrochlore iridates
Phys. Rev. B , 075129 (2011). Burkov, A. A. & Balents, L. Weyl semimetal in a topological insulator multilayer.
Phys. Rev.Lett. Burkov, A. A., Hook,M. D. & Balents L. Topological nodal semimetals.
Phys. Rev. B ,235126 (2011). Xu, G. et al.
Chern semi-metal and quantized anomalous Hall effect in HgCr Se . Phys. Rev.Lett. Singh, B., Sharma, A., Lin, H., Hasan, M. Z., Prasad, R., and Bansil, A. Topological electronicstructure and Weyl semimetal in the TlBiSe class of semiconductors. Phys. Rev. B , 115208(2012). Xu, S.-Y, Xia, Y., et al.
Topological phase transition and texture inversion in a tunable topo-logical insulator.
Science , 560 (2011). Balents, L. Weyl electrons kiss.
Physics , 36 (2011). Vishwanath, A. Where the Weyl Things Are,
Physics , 84 (2015). Huang, S. M., Xu, S.-Y, et al.
A Weyl fermion semimetal with surface Fermi arcs in thetransition metal monopnictide TaAs class.
Nature Commun. , 7373 (2015). (submitted in2014). Weng, H., et al.
Weyl semimetal phase in non-centrosymmetric transition metal monophos-phides.
Phys. Rev. X , 011029 (2015). Xu, S.-Y. et al.
Discovery of a Weyl Fermion semimetal and topological Fermi arcs.
Science , 613-617 (2015). Xu, S.-Y. et al.
Observation of Fermi arc surface states in a topological metal (DegenerateWeyl points).
Science , 294-298 (2015). Xu, S.-Y. et al.
Discovery of a Weyl Fermion semimetal state in NbAs.
Nature Phys. ,748-754 (2015). Xu, S.-Y. et al.
Experimental discovery of a topological Weyl semimetal state in TaP.
ScienceAdvances , e1501092 (2015). Lv, B. Q. et al.
Experimental Discovery of Weyl Semimetal TaAs.
Phys. Rev. X , 031013(2015). Lv, B. Q. et al.
Observation of Weyl nodes in TaAs.
Nature Phys. , 724-727 (2015). Lu, L. et al.
Experimental observation of Weyl points.
Science , 622-624 (2015). Belopolski, I. et al.
Criteria for Directly Detecting Topological Fermi Arcs in Weyl Semimetals.
Phys. Rev. Lett. , 066802 (2016) Meng, T. & Balents, L. Weyl superconductors.
Phys. Rev. B , 1054504 (2012). Bednik, G., Zyuzin, A. A. & Burkov, A. A. Superconductivity in Weyl semimetals.
Phys. Rev.B. , 035153 (2015). Li, Y. & Haldane, F. D. M. Topological nodal Cooper pairing in doped Weyl seimetals. Preprintat http://arxiv.org/abs/1510.01730 (2015). Tafti, F. F., Ishikawa, J. J., McCollam, A., Nakatsuji, S. & Julian, S. R. Pressure TunedInsulator to Metal Transition in Eu Ir O . Phys. Rev. B , 205104 (2012). Ueda, K. et al.
Variation of charge dynamics in the course of metal-insulator transition forpyrochlore-type Nd Ir O . Phys. Rev. Lett. , 136402 (2012). Shapiro, M. C. et al.
Structure and magnetic properties of the pyrochlore iridate Y Ir O . Phys. Rev. B , 214434 (2012). Liu, H. et al.
Magnetic order, spin dynamics and transport properties of the pyrochlore iridateY Ir O . Solid State Commun. , 1 (2012). Sushkov, A. B. et al.
Optical evidence for a Weyl semimetal state in pyrochlore Eu Ir O . Phys. Rev. B , 241108 (2015). Bulmash, D., Liu, C.-X. & Qi, X.-L. Prediction of a Weyl semimetal inHg − x − y Cd x Mn y Te.
Phys. Rev. B , 081106(R) (2014). Phys. Rev. Lett. , 087002(2015). Liu, Z. K. et al.
Discovery of a Three-Dimensional Topological Dirac Semimetal, Na Bi.
Science , 864 (2014). Xu, S.-Y. et al.
Lifshitz transition and Van Hove singularity in a three-dimensional topologicalDirac semimetal.
Phys. Rev. B , 075115 (2015). Neupane, M. et al . Observation of a three-dimensional topological Dirac semimetal phase inhigh-mobility Cd As . Nat. Commun. , 3786 (2014). Liu, Z. K. et al.
A stable three-dimensional topological Dirac semimetal Cd As . Nat. Mat. , 677 (2014). Borisenko, S. et al . Experimental Realization of a Three-Dimensional Dirac Semimetal.
Phys.Rev. Lett. , 027603 (2014). Guan, T. et al.
Evidence for Half-Metallicity in n -Type HgCr Se . Phys. Rev. Lett. , 087002(2015). Hal´asz G. B. & Balents, L. Time-reversal invariant realization of the Weyl semimetal phase.
Phys. Rev. B , 035103 (2012). Liu, J. & Vanderbilt, D. Weyl semimetals from noncentrosymmetric topological insulators.
Phys. Rev. B , 155316 (2014). Shekhar, C. et al.
Extremely large magnetoresistance and ultrahigh mobility in the topologicalWeyl semimetal candidate NbP,
Nature Phys. , 645-649 (2015). Wang, Z. et al.
Helicity protected ultrahigh mobility Weyl fermions in NbP.
Phys. Rev. B ,121112(R) (2015). Huang, S.-M. et al.
New type of Weyl semimetal with quadratic double Weyl fermions.
PNAS , 1180-1185 (2016). Xu, S.-Y. et al.
Discovery of Lorentz-violating Weyl fermion semimetal state in LaAlGe mate-rials. arXiv:1603.07318. Chang, G. et al.
Theoretical prediction of magnetic and noncentrosymmetric Weyl fermionsemimetal states in the R -Al- X family of compounds ( R =rare earth, Al=aluminium, X =Si,Ge). arXiv:1604.02124. Soluyanov, A. A. et al.
Type-II Weyl semimetals.
Nature , 495-498 (2015). Chang, T.-R. et al.
Prediction of an arc-tunable Weyl Fermion metallic state in Mo x W − x Te . Nature Commun. , 10639 (2016). Sun, Y. et al.
Prediction of the Weyl semimetal in the orthorhombic MoTe . Phys. Rev. B ,161107(R) (2015). Chang, G. et al. , A strongly robust Weyl fermion semimetal state in Ta S . Sci. Adv. ,e1600295 (2015). Koepernik, K. et al. , TaIrTe : A ternary type-II Weyl semimetal. Phys. Rev. B , 201101(R)(2016). Aut´es, G. et al. , Robust Type-II Weyl Semimetal Phase in Transition Metal Diphosphides X P ( X = Mo, W). Preprint at http://arxiv.org/abs/1603.04624 (2016). Liu, C.-C et al.
Weak Topological Insulators and Composite Weyl Semimetals: β -Bi X ( X =Br, I). Phys. Rev. Lett. , 066801 (2016). Hosur, P. & Qi, X. Recent developments in transport phenomena in Weyl semimetals.
Comp.Rend. Phy. , 857 (2013). Grushin, A. G. Consequences of a condensed matter realization of Lorentz violating QED inWeyl semi-metals.
Phys. Rev. D , 045001 (2012). Bergholtz, E. J. et al . Topology and Interactions in a Frustrated Slab: Tuning from WeylSemimetals to
C >
Phys. Rev. Lett. , 016806 (2015). Trescher, M. et al . Quantum transport in Dirac materials: Signatures of tilted and anisotropicDirac and Weyl cones.
Phys. Rev. B , 115135 (2015). Beenakker, C. Tipping the Weyl cone.
Journal Club for Condensed Matter Physics , posted ugust, 2015. Zyuzin, A. A. & Tiwari, R. P. Anomalous Hall Effect in Type-II Weyl Semimetals. Preprint athttp://arxiv.org/abs/1601.00890 (2016). Isobe, H. & Nagaosa, N. Coulomb interaction effect in tilted Weyl fermion in two dimensions.
Phys. Rev. Lett. , 116803 (2016). Yang, L. et al.
Weyl Semimetal in non-Centrosymmetric Compound TaAs.
Nature Phys. ,728-732 (2015). Xu, N. et al.
Observation of Weyl nodes and Fermi arcs in tantalum phosphide
Nat. Commun. , 11006 (2016). Liu, Z. K. et al.
Evolution of the Fermi surface of Weyl semimetals in the transition metalpnictide family.
Nat. Mat. (27) (2016). Xu, D. F. et al.
Observation of Fermi arcs in non-centrosymmetric Weyl semimetal candidateNbP.
Chinese Phys. Lett. , 107101 (2015). Souma, S. et al.
Direct Observation of Nonequivalent Fermi-Arc States of Opposite Surfacesin Noncentrosymmetric Weyl semimetal NbP. Preprint at http://arxiv.org/abs/1510.01503(2015). Lv, B. Q. et al.
Observation of Fermi-Arc Spin Texture in TaAs.
Phys. Rev. Lett. , 217601(2015). Xu, S.-Y. et al.
Spin Polarization and Texture of the Fermi Arcs in the Weyl Fermion SemimetalTaAs.
Phys. Rev. Lett. , 096801 (2016). Bertlmann, R. A.
Anomalies in Quantum Field Theory. (International Series of Monographson Physics, 2011). Adler, S. Axial-Vector Vertex in Spinor Electrodynamics.
Physical Review , 2426-2438(1969). Bell, J.S. & Jackiw, R. A PCAC puzzle: π → γγ in the σ -model. Il Nuovo Cimento A , 47(1969). Duval, C., Horvath, Z., Horvathy, P. A., Martina, L. & Stichel, P. C. Berry phase correctionto electron density in solids and “exotic” dynamics.
Mod. Phys. Lett. B , 373 (2006). Fukushima, K., Kharzeev, D. E. & Warringa, H. J. Chiral magnetic effect.
Phys. Rev. D ,074033 (2008). Son, D. T. & Spivak, B. Z. Chiral anomaly and classical negative magnetoresistance of Weyl etals. Phys .Rev. B , 104412 (2013). Burkov, A. A. Chiral Anomaly and Diffusive Magnetotransport in Weyl Metals.
Phys. Rev.Lett. , 247203 (2014). Xiong, J. et al.
Evidence for the chiral anomaly in the Dirac semimetal Na Bi.
Science ,413-416 (2015). Zhang, C. et al.
Detection of chiral anomaly and valley transport in Dirac semimetals. Preprintat http://arxiv.org/abs/1504.07698 (2015). Kim, H.-J. et al.
Dirac versus Weyl Fermions in Topological Insulators: Adler-Bell-JackiwAnomaly in Transport Phenomena.
Phys. Rev. Lett. , 246603 (2013). Ritchie, L. et al.
Magnetic, structural, and transport properties of the Heusler alloys Co MnSiand NiMnSb.
Phys. Rev. B , 104430 (2003). Pippard, A. B.
Magnetoresistance in metals.
Cambridge University Press, 1989. Hu, J., Rosenbaum, T. F. & Betts, J. B. Current Jets, Disorder, and Linear Magnetoresistancein the Silver Chalcogenides.
Phys. Rev. Lett. , 186603 (2005). Argyres, P. N. & Adams, E. N. Longitudinal Magnetoresistance in the Quantum Limit.
Phys.Rev. , 900 (1956). Sugihara, K., Tokumoto, M., Yamanouchi, C. & Yoshihiro, K. Longitudinal Magnetoresistanceof n-InSb in the Quantum Limit.
J. Phys. Soc. Jpn. , 109-115 (1976). Kikugawa et al.
Realization of the axial anomaly in a quasi-two-dimensional metal. Preprintat http://arxiv.org/abs/1412.5168 (2014). Goswami, P. Pixley, J. H. & Das Sarma, S. Axial anomaly and longitudinal magnetoresistanceof a generic three dimensional metal.
Phys. Rev. B , 075205 (2015). Huang, X. et al.
Observation of the chiral anomaly induced negative magneto-resistance in 3DWeyl semi-metal TaAs.
Phys. Rev. X , 031023 (2015). Zhang, C.-L. et al.
Signatures of the Adler-Bell-Jackiw chiral anomaly in a Weyl semimetal.
Nature Commun. , 10735 (2016). Ghimire, N. J. et al.
Magnetotransport of single crystalline NbAs,
Journal of Physics: Con-densed Matter , 152201 (2015). Luo, Y. et al.
A novel electron-hole compensation effect in NbAs.
Phys. Rev. B , 205134(2015) Moll, P. J. W. et al.
Magnetic torque anomaly in the quantum limit of the Weyl semi-metal bAs. Preprint at http://arxiv.org/abs/1507.06981 (2015). Yang, X. et al.
Chiral anomaly induced negative magneoresistance in topological Weylsemimetal NbAs. Preprint at http://arxiv.org/abs/1506.03190 (2015).
Zhang, C. et al.
Large magnetoresistance over an extended temperature regime in monophos-phides of tantalum and niobium,
Phys. Rev. B
2, 041203 (2015).
Zhang, C. et al.
Quantum Phase Transitions in Weyl Semimetal Tantalum Monophosphide.Preprint at http://arxiv.org/abs/1507.06301 (2015).
Du, J. et al . Sci. China Phys. Mech. Astron. , 657406 (2016). Arnold, F. et al . Nat. Commun . , 11615 (2016). Jia S., Xu S.-Y. and Hasan M. Z. Weyl Semimetals, Fermi Arcs and Chiral Anomalies (A ShortReview). arXiv:1612.00416.
Suzuki, T. et al . Nat. Phys. , 1119-1123 (2016). reakingor symmetry ba (cid:2239) Fermi arc (cid:2239) IG. 1:
Weyl fermions on a lattice. a,
One way to create a Weyl semimetal is to start witha massless Dirac fermion in a system with both time reversal and inversion symmetries. Such aDirac fermion corresponds to the intersection of two doubly-degenerate bands and can be realized,for instance, at the critical point of a topological phase transition between a normal insulator anda topological insulator. b, By breaking time reversal or inversion symmetry, the Dirac fermionsplits into a pair of Weyl fermions of opposite chiralities. Each Weyl fermion is a monopole oranti-monopole of Berry curvature and, equivalently, is associated with a Chern number. TheChern number guarantees the existence of a topological Fermi arc surface state that connects theprojections of the Weyl points in the surface Brillouin zone. c-e,
Weyl semimetals are characterizedby Chern numbers, as in the integer quantum Hall effect. For instance, we can consider a systemwith two Weyl points of chirality ± k x , d . When a slice is swept through a Weyl point, the two-dimensionalsystem undergoes a topological phase transition and the Chern number changes by ±
1. For sliceswith a Chern number ν = 0, the one-dimensional edge of the two-dimensional slice is gapped, c ,while slices with a Chern number ν = +1 host a protected gapless chiral edge mode, e . The Fermiarc can be understood as arising from all of the chiral edge states assembled together into a surfacestate. Ta(Nb) As(P) c W2 W1 k z ( π / c ) k x (2π/ a )-0.5 0 0.5 k y (2π/ a )-0.5 0 0.5 bd e Γ (cid:1850)(cid:3364) (cid:1851)(cid:3364)(cid:1839)(cid:3365) Γ Σ N Σ Z Γ X Γ(cid:3364)
FIG. 2:
The crystal and electronic structures of the Weyl semimetal TaAs. a,
Body-centered tetragonal structure of TaAs. The crystal lattice lacks an inversion center. b, Scanningtunneling microscopy (STM) topographic image of the (001) surface of TaAs, revealing a squarelattice. c, Survey of the band structure of TaAs. At this level, the system is quite simple, withonly two bands, a conduction and valence band, in the vicinity of the Fermi level, which approacheach other on the Σ − N − Σ line. d, The conduction and valence bands cross each other, formingnodal crossing points in k -space where the bulk energy gap vanishes. This panel shows the nodalcrossing points in the bulk Brillouin zone. In the absence of spin-orbit coupling, the nodal crossingpoints are nodal-lines, i.e. 1D rings, on the k x = 0 mirror plane, M x , and two nodal-lines on the k y = 0 mirror plane, M y . In the presence of spin-orbit coupling, each nodal-line vaporizes intosix 0D nodal points, the Weyl nodes. The Weyl nodes are denoted by small circles. Black andwhite show the opposite chiral charges of the Weyl points. We denote the 8 Weyl nodes located onthe k z = 2 π/c plane as W and the other 16 nodes away from this plane as W . e, Theoreticallycalculated (001) surface-state Fermi surface. Adapted with modifications from Ref. . (cid:1863) (cid:3052) (cid:2024) / (cid:1853) (cid:1863) (cid:3051) (cid:2024)/(cid:1853) (cid:1831) (cid:2886) e V
012 -0.5 0 0.5 a LowHigh (cid:1863) || (cid:1344) (cid:2879)(cid:2869) Weyl (cid:3397) -1 -0.5 0 0.5 1 bc d
Weyl (cid:3398) (cid:1863) (cid:2884) (cid:2024)/(cid:1855)(cid:1863) || (cid:2024)/(cid:1853) (cid:1831) (cid:2886) e V (cid:1831) (cid:2886) e V FIG. 3:
Weyl fermions in TaAs. a,
ARPES-measured k x , k y bulk Fermi surface of TaAs.The Fermi surface consists of discrete points that arise from the Weyl nodes. b, In-plane energydispersion ( E B − k (cid:107) ) that goes through a W Weyl node. A linear dispersion is clearly observed,consistent with the Weyl fermion cone. c, In-plane energy dispersion ( E B − k (cid:107) ) that goes througha pair of W Weyl nodes. d, Out-of-plane energy dispersion ( E B − k ⊥ ) that goes through two W Weyl nodes with the same k x , k y but different k z value. Adapted with modifications from Ref. . X YMk x (Å -1 ) a b -0.2-0.4-0.6 -0.1 0.0 0.1 k x (Å -1 ) k y ( Å - ) c k x k y k z k x k y d k y ( Å - ) e E B (eV) S p i n po l a r i za ti on -0.8 E B (eV) f S p i n po l a r i za ti on IG. 4:
Topological Fermi arcs in TaAs. a,
ARPES spectrum of the surface state bandstructure of TaAs near the Fermi level, E F . The green dotted lines denote the boundaries of thesurface Brillouin zone. b, High resolution ARPES Fermi surface of the double Fermi arcs thatarise from a pair of projected W Weyl nodes. The k -space region of this map is indicated by theblue box in panel (a). c, Theoretically calculated Fermi surface of the double Fermi arcs that arisefrom the same pair of projected W Weyl nodes. We indicate schematically the spin texture of theFermi arcs. d, Lower box: illustration of four W Weyl nodes in the bulk Brillouin zone, two ofeach chirality. Upon projection on the (001) surface Brillouin zone, the two +1 and the two − ±
2. Top surface: ARPESspectrum of the two Fermi arcs connecting the projected Weyl nodes. The black and white circlesin panels (b-d) show the projected W Weyl nodes with opposite chiralities. e,f,
Spin polarizationalong the k x direction, as measured in spin-resolved ARPES at two points on the Fermi arcs, asindicated by the red and orange dots in (c). Adapted with modifications from Refs. . X YMk x (Å -1 ) k y ( Å - ) E B ( e V ) k loop (Å -1 ) c d k x k y k z CBVB E B k loop k loop a b Chiral edge statesChern number = 2 e Chiral edge states IG. 5:
Bulk-boundary correspondence in the Weyl semimetal TaAs. a,
Illustration of aChern number in TaAs. The blue and red dots are Weyl points of opposite chiralities. We considera cylindrical tube extending through the bulk Brillouin zone and enclosing two Weyl points of thesame chirality. There is a net enclosed chiral charge of +2, so the Chern number on this two-dimensional slice of the Brillouin zone is +2. b, On the one-dimensional boundary of the tube, theChern number protects two chiral edge states, which make up one slice of the topological Fermiarcs. c, We can directly demonstrate that TaAs is a Weyl semimetal from the surface state bandstructure as measured in ARPES by drawing a loop in the surface Brillouin zone and countingthe crossings to show a nonzero Chern number. The loop is shown by the dotted black line. d, The surface states along the loop. We find two edge states of the same chirality, showing a Chernnumber of +2. Note that for a conventional electron or hole pocket, such a counting argumentwill always give 0. (e) The four criteria for a topological Fermi arc. (1) A disjoint contour. (2) Aclosed contour with a kink. (3) No kinks within experimental resolution, but an odd set of closedcontours. (4) An even number of contours without kinks, but net non-zero chiral edge modes.Adapted with modifications from Refs. . bc d e FIG. 6:
Signatures of the Adler-Bell-Jackiw chiral anomaly in TaAs. a,b,
Longitudinalmagnetoresistance (LMR) at T = 2 K, for two samples a1 and c2, Ref. . See again Ref. forcomplete data on additional samples a3, c4 and a5 discussed in that study. The green curves are fitsto the LMR in the semiclassical regime. c, Magnetoconductivity as a function of the angle betweenthe (cid:126)E and (cid:126)B fields. The y axes of panels (a-c) are the change of the resistivity with respect tothe zero-field resistivity, ∆ ρ = ρ ( B ) − ρ ( B = 0), or the magnetoconductivity as determined from∆ ρ . d, Dependence of the chiral coefficient C W , appropriately normalized by the other fittingcoefficients, on chemical potential, E F . Remarkably, the observed scaling behavior is 1 /E , asexpected from the dependence of the Berry curvature on chemical potential in the simplest modelof a Weyl semimetal, Ω ∝ /E . e, A cartoon illustrating the chiral anomaly in TaAs . The chiralanomaly leads to an axial charge pumping, for (cid:126)E · (cid:126)B (cid:54) = 0. This causes a population imbalancebetween Weyl cones of opposite chiralities. The charge pumping reaches an equilibrium with theaxial charge relaxation, characterized by a timescale τ a , Refs. . Note that the axial chargerelaxation time τ a can be directly obtained from the observed negative LMR data through thechiral coefficient C W = e τ a / π (cid:126) g ( E F ), as discussed elsewhere . Adapted with modificationsfrom Ref. ..