commentary Weyl semimetals, Fermi arcs and chiral anomalies Shuang Jia, Su-Yang Xu and M. Zahid Hasan Physicists have discovered a new topological phase of matter, the Weyl semimetal, whose surface features a non-closed Fermi surface whereas the low-energy in the bulk emerge as Weyl . A brief review of these developments and perspectives on the next steps forward are presented.

eyl semimetals are semimetals these lines, highlighting the experimental of this line of research for materials or metals whose techniques and matching materials discovery were thoroughly discussed in Wexcitation is the Weyl , a conditions that are necessary for realizing ref. 14, where the search was based on the particle that played a crucial role in quantum these research directions. Inorganic Crystal Structure Database of FIZ feld theory but has not been observed as a Karlsruhe27, which systematically records the fundamental particle in vacuum1–24. Weyl Topological material search lattice structure of crystals that have been fermions have defnite , either Although the theory of the Weyl semimetal synthesized over the course of a century. By lef-handed or right-handed. In a Weyl has been around for a long time in various following this approach and calculating the semimetal, the can be understood as forms1–4, its discovery had to wait until band structure of materials that are likely a topologically protected chiral charge. Weyl recent developments. Tis is because to be semimetals, many experimentally nodes of opposite chirality are separated in experimental realization requires appropriate feasible Weyl semimetal candidates have momentum space and are connected only simulation and characterizations of been identifed10,14,28,29,34. through the crystal boundary by an exotic materials. Historically, the frst two materials Te theoretical prediction of Weyl non-closed surface state, the Fermi arcs. predicted to show the efect, the pyrochlore semimetal states in the TaAs class of

Remarkably, Weyl fermions are robust while iridates R2Ir2O7 (ref. 4) and the magnetically compounds was independently reported in carrying currents, giving rise to exceptionally doped superlattice6, were both time-reversal- 2015 by two groups14,17. Te experimental high mobilities. Teir spins are locked to breaking (magnetic) materials. Perhaps realization of the frst Weyl semimetal in their momentum directions, owing to their infuenced by the early work, for a long while TaAs followed soon afer15,18. Both bulk character of momentum-space magnetic the community continued to focus on time- Weyl cones and topological Fermi-arc monopole confguration. Because of the reversal-breaking materials as candidates for surface states were directly reported by , the presence of parallel Weyl semimetals 7,10. Tese candidates were photoemission, providing a topological electric and magnetic felds can break the extensively studied by many experimental proof 15 based on methods shown earlier16. apparent conservation of the chiral charge, groups. Unfortunately, their eforts were not Te negative magnetoresistance, which making a Weyl metal, unlike ordinary non- successful in either demonstrating Fermi serves as an early signature of the chiral magnetic metals, more conductive with arcs or isolating the Weyl quasiparticles. anomaly, was reported from transport an increasing magnetic feld. Tese new Later, it was realized that the ‘time-reversal- experiments in the TaAs family of topological phenomena beyond topological breaking’ route to fnding suitable materials materials19,20. Tis family of materials was insulators make new physics accessible and faces a number of obstacles such as the then experimentally shown to include suggest potential applications, despite the strong correlations in magnetic materials, NbAs and TaP (refs 10,21–24). A set of early stage of the research25–46. the destruction of sample quality upon experimental criteria was developed for In this Commentary, we will review magnetic doping, and the issue of magnetic directly detecting topological Fermi arcs key experimental progress and present an domains in photoemission experiments. in Weyl semimetals without the need for outlook for future directions of the feld. On the other hand, for a long period, detailed comparison with band-structure We aim to expound our perspectives on the inversion-breaking Weyl materials were calculation24. At present, the TaAs key results and the experimental approaches relatively unexplored. Following work on a family remain the only topological Weyl currently used to access the new physics, as semimetallic state located at a topological semimetals that have been clearly confrmed well as their limitations. Moreover, although phase transition point12, Singh and in experiments, independent of comparison most of the current experiments focus on collaborators13 discussed this possibility, but to band-structure calculations10,11. the discovery of new Weyl materials and they considered compositional fne-tuning of Te discovery of TaAs also established demonstration of novel Weyl physics such as spin–orbit strength in TlBi(S1−xSex)2, which a feasible materials method to search for the chiral anomaly, it is becoming clear that is challenging to fabricate. Looking for Weyl new Weyl semimetals that are more likely a crucial step forward is to develop schemes semimetal candidates in naturally occurring to be experimentally realizable. Shortly afer for achieving quantum control of the new inversion-breaking non-centrosymmetric TaAs, a number of new Weyl semimetal physics by electrical and optical means. We single crystals can, however, avoid the candidates along the same line of thinking discuss some theoretical proposals along difculties described above. Te advantages were proposed.

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a b a 10 E 20 0 B –10 cm) cm)

μΩ E B μΩ –20 (

10 ( ρ ρ c ∆ ∆ –30 –40 0 –50 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 –2.0 –1.0 0 1.0 2.0 B (T) B (T)

c d e E B 2 10 c2 20 ϕ )

–1 a1 a WAL 3

C c cm 1 4 –1 Ta 10 3/2 c Ω

c B W (m C 2 σ

∆ –2 0 10 10 As a a5 b 10–4 –100 –50 0 50 100 –10 0 10 20 30 40

ϕ (°) EF (eV) with respect to the Weyl node

Figure 1 | Signatures of Adler–Bell–Jackiw chiral anomaly in TaAs. a,b, Longitudinal magnetoresistance data for TaAs. The red lines are the data, and the green lines are the theoretical fi ts. a and c are the lattice vectors along the in-plane and out-of-plane directions, respectively, of the TaAs tetragonal lattice. Background subtraction is discussed in ref. 20. c, The tetragonal crystal lattice of TaAs. d, Magnetoresistance data as a function of the angle between the electric (E) and magnetic (B) fi elds. e, Dependence of the chiral coef cient CWAL, normalized by other fi tting coef cients, on chemical potential, EF. Remarkably, the observed 2 2 scaling behaviour is 1/EF , as expected from the dependence of the Berry curvature on chemical potential in the simplest model of a Weyl semimetal, Ω ∝ 1/EF .

CWAL is the coef cient describing the magnitude of the 3D weak anti-localization (WAL) ef ect. BC is a critical fi eld that characterizes a crossover between the

WAL regime and the negative magnetoresistance regime. a1, a3, a5, c2 and c4 are dif erent samples measured in the experiments. These samples have slightly dif erent chemical potential levels because they were grown under slightly dif erent conditions. Adapted from ref. 20, Nature Publishing Group.

Primary examples include Ta3S2, SrSi2, chirality that are separated in momentum chiral anomaly, diverges as the chemical Mo1−xWxTe 2 and LaAlGe. In particular, SrSi2 space. T is process violates the conservation potential approaches the energy of the Weyl realizes a quadratic double Weyl state29, of chiral charge and leads to an axial charge node (Fig. 1e), consistent with its relation while Mo1−xWxTe 2 (refs 30–33,35), LaAlGe current, making a Weyl semimetal more to the Berry curvature f eld. T ese transport (ref. 34) and Ta3S2 (ref. 28) are the ‘type-II’ conductive in an increasing magnetic f eld results, in combination with angle-resolved Weyl semimetals, where the Weyl fermion that is parallel to the electric f eld. photoemission spectroscopy (ARPES) manifests as a strongly Lorentz-violating, A clear identif cation of a Weyl semimetal data modelled to be consistent with them, tilted-over cone in the band structure30. with Fermi arcs in TaAs paved the way demonstrated the existence of the chiral Experimental evidence of the bulk Weyl for the detection of the chiral anomaly. As anomaly in TaAs driven by Weyl fermions. state has been shown in LaAlGe (ref. 34) and shown in Fig. 1a,b, the magnetoresistance It is worth noting that the chiral anomaly

Mo1−xWxTe 2 (refs 35,36). Δρ decreases with increasing magnetic can also arise in a Dirac semimetal because f eld for a f nite range of f elds. T e the Dirac fermions can split into pairs of Transport response and chiral anomaly observed correct negative longitudinal Weyl fermions of opposite chirality under an Weyl fermions are expected to exhibit magnetoresistance serves as an important external magnetic f eld37. various forms of quantum anomalies f rst signature of the chiral anomaly. In In most metals, magnetoresistance such as a topological Fermi arc and chiral addition, a number of supporting pieces is known to be positive; so a negative anomaly. T e chiral anomaly is important of evidence were reported in ref. 20. magnetoresistance, especially a longitudinal in understanding some structures of the T ese include the following. (1) T ere one, is quite rare. But the chiral anomaly of particle physics which is a sharp dependence of the negative is not the only possible origin. It may arise is based on quantum f eld theory. A magnetoresistance on the angle between the through the giant magnetoresistance ef ect well-known case is the triangle anomaly electric and magnetic f elds (Fig. 1d). (2) T e in magnetic materials. Also, with a high associated with the decay of the neutral presence of the negative magnetoresistance electronic mobility, it can occur through pion. Weyl semimetals provide an electronic does not depend on the direction of the a purely classical geometric ef ect, the route to realizing the chiral anomaly in E f eld with respect to the crystalline axis. current-jetting. Furthermore, the negative condensed matter. Since the Weyl nodes (3) T e negative magnetoresistance shows magnetoresistance due to the chiral anomaly are separated in momentum space, parallel a strong dependence on the chemical is a Berry curvature ef ect. However, a magnetic and electric f elds can pump potential EF. T e chiral coef cient, which generic band structure without Weyl electrons between Weyl nodes of opposite gives a measure of the magnitude of the nodes can carry non-zero Berry curvature

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a 1 b c Bulk FS with nodes Weyl dispersion 0.0 0.0 ) a /

π 0 0.4 (eV) ( B

y 0.4 E k

0.8

–1 0.8 0 1 –0.2 0 0.2 50 51 52 53 54 –1 kx (π/a) ky (Å ) kz (π/c)

d e f Non-closed fermi arcs –0.4 Surface FS Chiral edge modes (C = 2) 1.5 ) –1

0.0 Å

( –0.5 y

1.0 k

0.5 0.4 –0.6 )

–1 Y Å ( y Weyl fermion nodes k g Γ –0.4 0.0 0.8 )

–0.5 –1

Å –0.5

1.2 ( y k

–1.0 X + – 1.6 –0.6 0.0 0.5 1.0 0.0 0.4 0.8 1.2 –0.1 0.0 0.1 –1 –1 –1 kx (Å ) k|| (Å ) kx (Å )

Figure 2 | Observation of Weyl fermions and topological Fermi arcs. a, ARPES-measured kx, ky bulk-band Fermi surface (FS) projection of TaAs. The Fermi surface consists of discrete 0D points that arise from the Weyl nodes. b, In-plane energy dispersion (E−ky) that goes through a pair of Weyl nodes of opposite chirality. c, Out-of-plane energy dispersion (E−kz) that goes through a pair of Weyl nodes. Panels b and c show that the bands disperse linearly away from the Weyl nodes along both the in-plane and the out-of-plane directions. d, Surface-state Fermi surface of TaAs. We focus on the crescent-shaped feature near the midpoint — — of the Γ − Χ line. The Weyl semimetallic nature of TaAs can be demonstrated from the surface-state band structure as measured in ARPES by drawing a loop (dashed black line) in the surface Brillouin zone and counting the crossings to show a non-zero Chern number15. e, The surface states along the loop: two edge states of the same chirality seen in the momentum-space cuts of the ARPES data reveal a Chern number of 2 for this projection. ARPES methods for measuring Chern numbers are elaborated in refs 15,24,35. Note that for a conventional electron or hole pocket, such a counting argument should result in a null (0) value. f, Surface-state Fermi surface map at the k-space region corresponding to the terminations of the crescent Fermi arcs. g, Bulk Fermi surface map at the k-space region corresponding to the W2 Weyl nodes. Adapted from ref. 15, AAAS.

as long as time-reversal or ZrTe5 is that of a gapped semiconductor systematically tunes the Berry curvature inversion symmetry is broken. Whether with a small (~50 meV) gap, not a Dirac feld efectively. Specifcally, Fig. 1e shows such a generic band structure with non-zero semimetal. Tis suggests that a thorough that the magnitude of the chiral anomaly 2 Berry curvature yet without Weyl nodes theoretical understanding of the origins diverges over 1/EF . Tis demonstrates a can lead to a negative magnetoresistance of the negative magnetoresistance in Berry curvature monopole behaviour, and is not theoretically understood. A this material is lacking. Considering the thus constitutes a strong signature of the recent experiment reported negative complexity of the phenomenon, in order topological Weyl fermion physics. magnetoresistance in ZrTe5 (ref. 38). Tere, to achieve a convincing case, one needs the interpretation was that ZrTe5 is a Dirac to provide data that are sensitive to the Weyl fermions and Fermi arc topology semimetal, and the splits into unique properties of Weyl fermions and ARPES serves as a decisive method of pairs of Weyl fermions under the external their systematic relation to the Berry demonstrating the Weyl semimetal state magnetic efect. Subsequent photoemission curvature feld. At present, only TaAs experimentally, since it can directly observe and scanning tunnelling microscopy shows the dependence on the position of bulk Weyl fermions and surface Fermi experiments39,40 clearly revealed, however, the chemical potential with respect to the arcs in the electronic band structure, thus that the robust electronic ground state of energy of the Weyl node (see Fig. 1e) that probing the bulk boundary correspondence

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a b

VNL S VSD S

d Δμ z ISD z y x ky L >> d Lg Ld

N N kx

E c d × σ σ+ +

EF ky E k

VH kx

e–

e– I

Figure 3 | Electronic and optical control of Weyl fermions. a, Schematics of a non-local electrical transport device that uses the axial current arising from the chiral anomaly. VNL, non-local voltage; VSD, source–drain voltage. b, On the surface of a Weyl semimetal, electrons exhibit an unusual path in real (z) and momentum (kx − ky) space under an external magnetic fi eld along the z direction. Not all paths are shown in the fi gure. c, An intense circularly polarized light can break time-reversal symmetry and lead to a non-zero Hall conductivity through the Floquet ef ect in the absence of an external magnetic fi eld. Inset: the solid (open) circles show the Weyl nodes in the absence (presence) of the light. d, Circularly polarized light can generate a large photogalvanic current in a Weyl semimetal that breaks inversion symmetry and any mirror symmetry. The inset shows the optical selection rule of right-handed circularly polarized light (σ+) from the lower band to the upper band of the Weyl cone. Figure based on refs 41–43,45.

essential for proving a topological state of ky and kz (Fig. 2b,c). (3) T e band crossings level, as seen in this momentum-space cut matter, as demonstrated for topological are not located at a high-symmetry point in the ARPES data (Fig. 2e). T erefore the insulators11,25,26. Here we review key ARPES or along a high-symmetry line but rather Chern number is C = 2 (refs 15,24,35). T is observations that led to an unambiguous are observed at generic k points of the demonstrates that Weyl semimetals, in some bulk–boundary topological demonstration Brillouin zone (Fig. 2a–c). T e surface Fermi aspects, feature momentum-space properties of the Weyl semimetal state in TaAs. T is arcs are demonstrated by the following that are higher-dimensional analogues of demonstration methodology is a useful key observations in the experimental data. quantum Hall-like states but without external guideline for future research in this f eld, as (4) T e ARPES-measured Fermi surface magnetic f elds10. An intrinsic IQH state is it can be more generally applied to discover of TaAs (Fig. 2d) reveals a crescent-shaped not possible in 3D (ref. 11). T e observation Weyl or related topological states, or new feature consisting of two curves that join at of a non-zero Chern number in momentum topological quasiparticles in other materials. their two end points. (5) T e surface band space provides a clear and unambiguous A Weyl node is a crossing point between structure along a k loop that encloses the demonstration of the topological origin of the bulk conduction and valence band. T us end point of the crescent feature exhibits the Fermi arcs. T e details of this Chern detection of a Weyl node requires the ability two chiral edge states, thus realizing a Chern number method using ARPES are reported to measure states along all three momentum- number (C) of +2 (Fig. 2e), in analogy with in refs 10,15,35,46. (6) Finally, ARPES data space directions, kx, ky and kz. T e kz the integer quantum Hall (IQH) chiral showed that the terminations of the Fermi resolution is achieved by tuning dif erent edge states. A Chern number is the band- arcs coincide with the projections of the incident photon energies. T e existence of structure invariant (momentum-space Weyl nodes, demonstrating the topological bulk Weyl cones and Weyl nodes in TaAs wavefunction winding index) that describes correspondence principle between bulk is shown by the following experimental the topology of the IQH state. It counts the and boundary (Fig. 2f,g) and therefore the observations in Fig. 2a–c. (1) T e bulk number of the protected edge states which materials’ topologically non-trivial nature. conduction and valence bands touch at zero- is also proportional to the Hall conductivity. A few remarks are in order. T e dimensional (0D) isolated points (Fig. 2a). In contrast to the IQH case with a single demonstration in TaAs only requires the (2) T ese two bands disperse linearly away protected edge state (C = 1), we now have above-described ARPES data along with from the crossing point along all three of kx, two chiral edge states crossing the Fermi established results in topological band

NATURE MATERIALS | VOL 15 | NOVEMBER 2016 | www.nature.com/naturematerials 1143 commentary

theory. It does not rely on an agreement As shown in Fig. 3a, in this device it is Spectroscopy (B7), Department of Physics, Princeton between the details of band dispersions and proposed to have a thin and long ribbon of a University and at the Princeton Institute for Science frst-principles band-structure calculations. Weyl semimetal sample. On the lef, parallel and Technology of Materials (PRISM), Princeton, Such an experimental demonstration of magnetic and electric felds lead to the axial New Jersey 08544, USA. topological invariance is robust and reliable. current, which can drif to the right and e-mail: [email protected]; By contrast, a number of recent ARPES be detected in the form of a spontaneous [email protected] works have attempted to prove the existence voltage/current without applying an electric of topological Fermi arcs solely based on the feld. 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Te large photocurrent can 43. Chan, C.-K., Lee, P. A., Burch, K. S., Han, J. H. & Ran, Y. Phys. Rev. Lett. 116, 026805 (2016). fermion) materials such as PbTaSe2 (ref. 46). be used for fabricating high-sensitivity and 44. Wang, Y.-H. et al. Science 342, 453–457 (2013). Here, we highlight the potential broad-bandwidth infrared detectors, and for 45. Chan, C.-K., Lindner, N. H., Refael, G. & Lee, P. A. Preprint at application of Weyl semimetals in devices. light-harvesting schemes in solar cells. http://arxiv.org/abs/1607.07839 (2016). It will be important to achieve systematic But these are only guidelines for potential 46. Bian, G. et al. Nat. Commun. 7, 10556 (2016). control over Weyl physics by electrical and applications. Te plethora of unusual physics Acknowledgements optical means, so that these phenomena can harboured by Weyl materials may lead to We thank I. Belopolski, S.-M. Huang, G. Bian, N. Alidoust potentially be harnessed in device settings. something far more exotic that we have not and M. Neupane for comments, and D. Haldane, 41 A recent theoretical report proposed yet imagined. ❐ I. Klebanov and E. Witten for discussion as a part of an electrical device that can make use of Princeton Summer School on New Insights Into Quantum current due to the chiral anomaly. Te chiral Shuang Jia is at the International Center for Matter as a part of Prospects in Teoretical Physics Program at IAS. S.J. is supported by the National Basic Research Quantum Materials, School of Physics, Peking charge pumping efect (Fig. 1f) leads to a Program of China (Grant No. 2014CB239302 and No. current, the axial current, which is believed University and at the Collaborative Innovation 2013CB921901). Work at Princeton by S.-Y.X and M.Z.H. to be nearly dissipationless. Any relaxation Center of Quantum Matter, Beijing 100871, is supported by the US Department of Energy under of the axial current requires scattering from China. Su-Yang Xu is at the Laboratory for Basic Energy Sciences (Grant No. DOE/BES DE-FG-02- one Weyl node to the other, which occupies Topological Quantum Matter and Spectroscopy 05ER46200 and No. DE-AC02-05CH11231 at Advanced Light Source at LBNL) and Princeton University funds. a limited volume of the parameter space as it (B7), Department of Physics, Princeton University, M.Z.H. acknowledges Visiting Scientist user support from involves a large specifc momentum transfer New Jersey 08544, USA. M. Zahid Hasan is at the Lawrence Berkeley National Laboratory, PRISM, and partial vector that connects the two Weyl nodes. Laboratory for Topological Quantum Matter and support from the Moore Foundation.

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