Review of experiments on the chiral anomaly in Dirac-Weyl semimetals
RReview of experiments on the chiral anomaly in Dirac-Weyl semimetals
N. P. Ong and Sihang Liang
Department of Physics, Princeton University, Princeton, NJ 08544 (Dated: October 20, 2020)We provide a review of recent experimental results on the chiral anomaly in Dirac/Weyl semimet-als. After a brief introduction, we trace the steps leading to the prediction of materials that featureprotected 3D bulk Dirac nodes. The chiral anomaly is presented in terms of charge pumping be-tween the chiral Landau levels of Weyl fermions in parallel electric and magnetic fields. The relatedchiral magnetic effect and chiral zero sound are described. Current jetting effects, which presentmajor complications in experiments on the longitudinal magnetoresistance, are carefully analyzed.We describe a recent test that is capable of distinguishing these semiclassical artifacts from intrinsicquantum effects. Turning to experiments, we review critically the longitudnial magnetoresistanceexperiments in the Dirac/Weyl semimetals Na Bi, GdPtBi, ZrTe and TaAs. Alternate approachesto the chiral anomaly, including experiments on non-local transport, thermopower, thermal con-ductivity and optical pump-probe response are reviewed. In the Supplement, we provide a briefdiscussion of the chiral anomaly in the broader context of high energy physics and relativistic quan-tum field theory, as well the anomaly’s starring role at the nexus of quantum physics and differentialgeometry. I. INTRODUCTION
In the past 2 decades, research on Dirac electrons insemimetals has grown rapidly into a major field of ac-tivity, especially in topological quantum matter. Ini-tially, the focus was on the protected two-dimensional(2D) Dirac states in graphene and on the surfaces oftopological insulators (TIs). In 2011-2012, the publica-tion of Refs. [1–5] caused a sea change that shifted re-search from TIs towards Dirac/Weyl semimetals. Thesearch culminated in the discovery of the Dirac semimet-als, Na Bi [6] and Cd As , followed by the Weyl semimet-als TaAs and NbAs (Sec. II). The realization of 3D Diracstates in semimetals revived intense interest in the chiralanomaly [7, 8] and its observability in crystals [9–13].In quantum field theory (QFT), an anomaly [14–16]is the breaking of a classically allowed symmetry whenquantum effects are turned on (see Supplementary In-formation Sec. VIII). The chiral (or axial) anomaly –the first example discovered [7, 8] – remains the mostinvestigated experimentally. In the massless limit, theLagrangian of Dirac fermions splits into two indepen-dent parts describing right-handed and left-handed mass-less fermions. Conservation of chiral symmetry impliesthat the two populations N R and N L are separately con-served. However, coupling of the fermions to a vectorgauge field (the electromagnetic field) breaks the chiralsymmetry, resulting in the appearance of the anomalyterm. In 1968 the difficulty of calculating the decay rateof the neutral pion π was resolved by the discovery ofthe triangular anomaly diagram by Adler [7] and Belland Jackiw [8]. Subsequently, anomalies have appearedin diverse phenomena occuring at vastly different energyscales (see Supplementary Information for details). In1983, Nielsen and Ninomiya [9] proposed that the chi-ral anomaly should be observable in crystals as a large,negative longitudinal magnetoresistance (LMR).The discovery of Dirac and Weyl semimetals led to a surge of experiments investigating the LMR (Sec. V).The findings fall into two groups with distinct features.In the semimetals Na Bi and GdPtBi, which have lowcarrier mobilities, the negative LMR is very large (25to 500%) and monotonic, settling down to a constant inhigh magnetic field B . In the second group, comprised ofthe Weyl semimetals TaAs and NbAs, which have veryhigh mobilities, the LMR is strongly non-monotonic; itis weakly negative (0.1 to 2%) in small B but becomespositive above 1 T. Inconsistencies soon raised strongconcerns that the observed LMR in the second group isan artifact caused by B -induceed inhomogeneous currentflow known as current jetting (Sec. IV). Recently, Liang et al. [17] proposed a test that distinguishes the intrinsicLMR from current jetting artifacts. The test confirmsthat the negative LMR is intrinsic in the first group butartifactual in the second (closely similar to LMR resultsin bulk Bi and InAs). We review the results in Sec. V,as well as other experimental approaches to the chiralanomaly (Sec. VI). A recent review of Dirac semimetalscomplementary to our review is Ref. [18].The acronyms used are ABJ (Adler Bell Jackiw)ARPES (angle-resolved photoemission spectroscopy),CME (chiral magnetic effect), CVD (chemical vapor de-position), CZS (chiral zero sound), IS (inversion symme-try), LL0 (Landau level with n = 0), LMR (longitudi-nal magnetoresistance), QCD (quantum chromodynam-ics), QFT (quantum field theory), SdH (Shubnikov deHaas), TI (topological insulator), TRI (time-reversal in-variance), and TRIM (time-reversal invariant momenta). II. SYMMETRY PROTECTION AND WEYLNODES
Initially, the search for materials with protected 3DDirac and Weyl states [1–5] focussed on the two symme-tries, time-reversal invariance (TRI) and inversion sym- a r X i v : . [ c ond - m a t . s t r- e l ] O c t metry (IS). These are sufficient to protect 3D Dirac nodesthat are located at the time-reversal invariant momenta(TRIM) K (which satisfy K = − K + G with G a recip-rocal lattice vector). The predicted materials, notably β -crystobalite BiO [5], are unfortunately unstable.[To illustrate protection by TRI at a TRIM K , we takeΨ K and Φ K to be 2-spinor eigenstates at K that aretime-reversed partners. Then by the Kramers theorem,they must be orthogonal, i.e. (cid:104) Ψ K , Φ K (cid:105) = 0. If V isa weak potential that is invariant under TR, the matrixelement (cid:104) Ψ K , V Φ K (cid:105) also vanishes by the same argument.Hence TRI protects the node at K against gap formationinduced by V .]Subsequently, it was realized that including point-group symmetry confers protection of Dirac nodes any-where along a symmetry axis (e.g. the rotation axis of C n , with n = 3 , , et al. [6]) that the two semimet-als Na Bi [6] and Cd As are Dirac semimetals protectedby symmetry under C and C rotations, respectively.In the case of Na Bi (Sec. V A), the workhorse for thechiral anomaly, Wang et al. showed that, with the choiceof basis for the 4-spinorˆΨ = ( | S, + (cid:105) , | P, + (cid:105) , | S, −(cid:105) , | P, −(cid:105) ) T (1)(where [ · · · ] T indicates transpose, and ± refer to the z component of J z ), the linearized Hamiltonian close to theDirac node at k D + splits into two 2 × H ( q ) = v F βq z q + q − − βq z βq z − q − − q + − βq z , (2)where β < q = k − k D + .The upper and lower blocks are compactly written as H − = v F ( q x τ x − q y τ y + βq z τ z ) ,H + = v F ( − q x τ x − q y τ y + βq z τ z ) , (3)where { τ i } are Pauli matrices acting in the orbital sub-space ( S, P ). H ∓ represent Weyl nodes of chirality χ = ∓ ( χ is thedeterminant of the velocity matrix ˜ V ± /v F defined by therelation H ± = q · ˜ V ± · τ .)In zero magnetic field, the two Weyl nodes coincide in k space. If TRI is broken in finite B , the Zeeman energyleads to their separation. A characteristic of Weyl statesis that the n = 0 Landau level is chiral. As discussedin Sec. III A, in an applied E , the chiral anomaly ispredicted for carriers occupying the n = 0 level.A detailed symmetry analysis of space group symme-tries that can protect 3D Dirac nodes has been reportedby Yang and Nagaosa [19]. With the choice for the in-version operator P = τ z , they show how the distincteigenvalues of the point-group rotation C n protect theDirac nodes for n = 3 (Na Bi) and n = 4 (Cd As ). Thedistinct values of the eigenvalues implies that all matrix elements formed between states in the conduction and va-lence bands must vanish. If P = τ x (the case of β -BiO ),the Dirac node is pinned at a TRIM. The remaining case P = has not yet led to a candidate. III. LANDAU QUANTIZATION AND CHIRALANOMALYA. Chiral Anomaly
Following the publication of Ref. [1], several ap-proaches for detecting the chiral anomaly were pro-posed [3, 10–13]. We review the approach that exploitsthe chiral nature of the lowest Landau level in 3D bulkWeyl fermions.In a strong magnetic field B (cid:107) ˆz , the bulk electronicstates in a conventional semimetal are quantized intoLandau levels (LL) indexed by n = 0 , , , · · · . For each ofthe LLs, the energy disperses as E ( n, k z ) = ( (cid:126) k z ) / m z where m z is the effective mass for dispersion (cid:107) ˆz .A distinguishing feature of Weyl nodes is that the LLwith n = 0 (hereafter called LL0) is chiral, i.e. E (0 , k z ) = ± v F k z (Fig. 1a). For the Weyl node with χ = +1, theslope is positive, i.e. the velocity v (cid:107) B , whereas for χ = − v (cid:107) − B . Hence, when B exceeds B Q (the field atwhich the chemical potential µ enters the LL0), we havetwo independent populations N L and N R of massless left-and right-moving massless fermions. Because they do notintermix, their respective current densities J R and J L areindependently conserved. Hence the total current density J = J L + J R and difference current density J = J L − J R are also conserved, viz. ∂ t ρ + ∇ · J = 0 , ∂ t ρ + ∇ · J = 0 , (4)where ρ and ρ are the respective charge densities (su-perscripts reflect the chirality matrix γ (Sec. VIII)).This represents chiral symmetry of massless fermions inthe LL0.Application of an electric field E (cid:107) B (cid:107) ˆz causes thepopulation of the branch moving down the potential in-cline (say N R ) to increase while the uphill branch de-creases at the same rate – the chiral symmetry is brokenby coupling to E and B . The “pumping rate” dN /dt isthe product of the 2D density of states D = L / (2 π(cid:96) B )of the LL0 and the rate of increase of available statesdriven by E , dn/dt = ( L/ π ) dk z /dt (where L is thesample volume and (cid:96) B = (cid:112) (cid:126) /eB the magnetic length).We have1 L dN dt = 12 π(cid:96) B π eE (cid:126) = e π (cid:126) E · B ≡ A . (5)The anomaly term A acts like a source term that ruinsconservation of the current J , which we express as ∂ t ρ + ∇ · J = A , (6)where ρ = N /L . Equation 6, which represents thechiral anomaly in a semimetal, implies the emergence ofa new axial charge current that strongly enhances theconductivity σ zz if the axial relaxation time τ A greatlyexceeds τ , the conventional transport lifetime. Theenhancement is observable as a large, negative LMR( B (cid:107) E ). B. Chiral Magnetic Effect and Zero Sound
Although they share a common origin, the chiralanomaly is distinct from the “chiral magnetic effect”(CME) [20, 21] which predicts that a magnetic field B applied to a Weyl semimetal (with E = ) spontaneouslygenerates an axial charge current (cid:107) B or − B , dependingon the chirality χ . In condensed matter, the appearanceof a DC (transport) current driven by an applied B act-ing alone violates the laws of thermodynamics. Franz andcollaborators [22] have shown that the CME vanishes incalculations that are properly regularized. The CME byitself cannot exist in magnetoresistance experiments.The CME may have experimental consequences whenthe system is periodically driven off equilibrium at highfrequencies. Song and Dai [23] have predicted that, ina Weyl semimetal with 2 or more pairs of Weyl nodes,there exists a collective mode in which each Weyl FS ex-ecutes a breathing mode with a specific phasing betweennodes. Unlike a plasmon, this mode displays a gapless,linear dispersion because local charge densities and cur-rents rigorously vanish everywhere at all times. They callthe collective mode chiral zero sound (CZS). The sim-plest example is sketched in Fig. 1d. The 2 pairs of Weylnodes are symmetrically arrayed in the k x - k y plane with B (cid:107) k z . As shown, nodes with positive chirality ( χ = 1,blue) disperse with velocity v (cid:107) B , whereas nodes with χ = − v (cid:107) − B .At an instant in the breathing cycle, the local chemicalpotential µ loc lies above E F in the 2 nodes drawn withlarger k F in Fig. 1d, whereas µ loc lies below E F forthe nodes with smaller k F . The occupation factor ineach node is indicated by thick lines in the accompanyingdispersion sketches. With this phasing, it is clear thatthe charge currents cancel pairwise between the 4 nodes.Deviations of the charge density from equilibrium alsocancel. The cancellations allow the mode to propagateas an acoustic wave. The CZS may contribute stronglyto both the heat capacity and the thermal conductivityat low T (Sec. VI C). IV. CURRENT JETTING ARTIFACTS
In high-mobility semimetals, longitudinal magnetore-sistance experiments are greatly complicated by artifactscaused by current jetting. We assume that the cur-rent density (cid:104) J (cid:105) (spatially averaged over the sample) is (cid:107) B (cid:107) ˆx . The cyclotronic motion of the carriers in the y - z plane causes the transverse conductivities σ yy and σ zz to decrease as 1 / ( µ e B ) , whereas the longitudinalconductivity σ xx is unaffected. The anisotropy results inpronounced concentration of J into a narrow jet alignedwith B and a corresponding reduction at the edges par-allel to ˆx . Hence, the voltage drop detected at the edgedecreases, masquerading as a negative LMR even though σ xx is unchanged.In the low-mobility semimetals Na Bi and GdPtBiwith µ e ∼ /Vs, the effects of currentjetting are still observable, but they introduce a relativelyweak distortion that can be corrected for. However, inthe Weyl semimetals TaAs, NbAs and NbP, all of whichhave very high carrier mobilities ( µ e > /Vs),the fractional change in the observed negative LMR sig-nal is typically very small (0.1 to 2 %) and confined toweak B ( ± B , the MR becomes stronglypositive. Frequently, changing the voltage contact place-ments reverses the sign of the low- B LMR, consistentwith an extrinsic origin for the observed LMR.Using numerical simulations of J ( x ) in a longitudinal B , Liang et al. [17] devised a “squeeze” test that revealswhen current jetting artifacts present a serious concern.The test is based on simultaneous measurements of themaximum and minimum values attained by J x ( x, y, z )in the y - z plane. The sample is cut in the shape of athin, square plate with side a (cid:29) c , the thickness (Fig.1b). Small current injection pads (of diameter d (cid:39) c )are placed in the centers of two opposing edges. The line( (cid:107) ˆx ) joining the pads is called the spine. Numerical sim-ulations reveal that, under current jetting, J x is stronglypeaked along the spine and suppressed at the edges (Fig.1c). Using local voltage contact pairs, the minimum andmaximum values of J x are measured vs. B to yield the ef-fective resistances R edge ( B ) and R spine ( B ), respectively.Liang et al. showed that, in Na Bi and GdPtBi, R edge and R spine both decrease with increasing B , which veri-fies that the decrease in ρ xx is intrinsic and current jettingeffects are subdominant (Secs. V A and V B).In contrast, R edge and R spine in high-mobility Bi showdivergent field profiles; even at 100 K, R edge decreases tovalues very close to zero while R spine increases steeply bya factor of 100 (see Fig. 2 in Ref. [17]). The divergenttrends are artifacts arising from strong focussing of thejet along the spine. The test has also been applied toTaAs (Sec. V D).A rule of thumb is to compare B Q with B cyc , the fieldat which µ e B exceeds ∼
5. If B cyc < B Q , the steepgrowth of current-jetting artifacts effectively precludesreliable observation of any intrinsic LMR in the quantumlimit. V. CHIRAL ANOMALY INMAGNETORESISTANCEA. The Dirac semimetal Na Bi The Dirac semimetal Na Bi crystallizes in the hexag-onal P /mmc phase ( D h ). The conduction band isprimarily derived from the Na-3 s states while the upper-most valence band is derived from Bi-6 p x,y states [6]. Thestrong spin-orbit coupling (SOC) causes the Na-3 s bandto lie below Bi-6 p x,y by ∼ | P, ± (cid:105) above the light-hole band | P, ± (cid:105) . The resulting band crossings lead to Dirac nodesat k ± D = (0 , , ± . π/c ) along the z axis (Γ- A ). Becausethe S and P bands transform under C with different irre-ducible representations and TRI is preserved, the nodesare protected against gap formation. At energies nearthe node energy E , it is sufficient to retain the states [6] | S + , ± (cid:105) , | P − , ± (cid:105) (7)which are bonding and antibonding combinations of or-bitals centered at the two Na ions (for S ) and Bi ions (for P ) with parity eigenvalues ± .As described in Sec. II, the breaking of TRI in applied B splits each Dirac node into 2 Weyl nodes of oppo-site chirality χ . Under Landau quantization, the Landaulevel at n = 0 (LL0) is strictly chiral, dispersing either (cid:107) B or − B depending on χ . The existence of only 2Dirac nodes with E very close to E F and the absenceof other (spectator) bands at E F make Na Bi a very at-tractive platform to search for the chiral anomaly despiteits hyper-sensitivity to moist air.Notably, in samples with low Na vacancies, E F liesjust below E . As T decreases from 300 K, the resistiv-ity ρ rises monotonically by a factor of ∼
20 to saturate at21 mΩcm below 20 K, with Hall density n H ∼ × cm − and mobility µ e ∼ /Vs [24]. The dis-tinctly non-metallic profile is also observed in GdPtBi.The weak SdH oscillations observed vs. B indicate thatthe LL0 is entered at ∼ B is aligned with (cid:104) J (cid:105) (cid:107) ˆx , negative LMR be-comes apparent at ∼
100 K (Fig. 2a). At 4.5 K, ρ xx undergoes a 6-fold decrease before saturating above 8T, consistent with the appearance of the chiral anomaly(Xiong et al. [24]). The low mobility (2,600 cm /Vs)implies that current jetting artifacts should only appearabove 11 T [17]. Comparison of the observed σ xx ∼ B in weak B with the Son-Spivak expression yields an es-timate of the axial relaxation time τ a ∼ τ , thetransport lifetime at B = 0.By tilting B in the x - z plane (at an angle θ to E ) as well as in the azimuthal x - y plane (angle φ ),Xiong et al. observed that the conductance enhancement∆ σ xx ( B, θ, φ ) assumes the form of a collimated “plume”with axis parallel to (cid:104) J (cid:105) (Fig. 2b). Displayed as a po-lar plot, the angular width at 2 T is closer to the form ∆ σ xx ∼ cos θ (or cos φ ) instead of cos θ .Applying the squeeze test (Sec. IV) to Na Bi, Liang etal. observed [17] that both resistances R spine and R edge decrease monotonically with increasing B (applied (cid:107) (cid:104) J (cid:105) ),confirming that the negative LMR is intrinsic. However,the curve of R spine ( B ) consistently lies above R edge ( B ).Moreover, R spine ( B ) displays a broad minimum near 10T followed by a gradual increase at larger B as shownin (Fig. 2c) (this reflects the competition between theintrinsic LMR and current jetting effects which are seenabove 10 T). Both features are striking evidence that,despite the low mobility, current jetting effects can stillproduce observable distortions. Comparing the measuredcurves against numerical simulations, Liang et al. showedthat it is possible to remove the distortions to extractthe intrinsic curve R int ( B ), which is sandwiched between R spine ( B ) and R edge ( B ) (Fig. 2d). The inferred R int ( B )reveals that the chiral anomaly leads to a 10-fold de-crease in ρ xx , which implies that the axial lifetime τ A is10 × longer than the Drude lifetime τ at B = 0. Thisis currently the most reliable measurement of the ratio τ A /τ by dc transport. As seen in Fig. 2d, R int settlesdown to a B -independent value in the LL0 ( B > σ xx ( θ, φ ). Extension to the more elab-orate angular MR experiments in tilted B has not beenattempted. B. The Half-Heusler GdPtBi
The unit cell of the half-Heusler GdPtBi is comprisedof Pt-Gd tetrahedra arrayed in the zincblende structure.The low-lying states involve only the Bi 6 p and Pt 4 s bands | j, m j (cid:105) = | , ± (cid:105) and | , ± (cid:105) , which are 4-folddegenerate at energy E at the Γ point. In zero H , thelattice symmetry T d together with TRI protects the 4-fold degeneracy [25, 26]. In addition to being air-stable,GdPtBi has the advantage that a slight off-stoichiometryduring growth (or doping with Au) can shift E F frombelow E to above.A magnetic field B lifts the 4-fold degeneracy via theZeeman energy [26]. The larger Zeeman gap in | , ± (cid:105) (3 × that in | , ± (cid:105) ) leads to band crossings that defineWeyl nodes separated in k space. Hirschberger et al. [25]observed a large negative LMR when H is aligned with (cid:104) J (cid:105) .The curves of ρ xx vs. B at fixed T measured with B (cid:107) (cid:104) J (cid:105) (Fig. 3a) bear a close resemblance to those inNa Bi. As T is lowered from 200 K, the LMR begins todisplay a prominent negative trend at 125 K. The curveat 6 K displays a steep decrease of ρ xx by a factor of5 between 0 and 14 T. The angular dependence of thecurves of ρ xx ( B ) was mapped out in detail for B tiltedat angle θ out of the x - y plane (Fig. 3b) and angledwithin the x - y plane. Again, the enhanced conductivityappears as a broad plume centered around the axis with B (cid:107) (cid:104) J (cid:105) .The tunability of E F by doping provides an impor-tant test of the chiral anomaly that could not be done inNa Bi. By investigating 16 samples with E F on eitherside of the Weyl node energy (in zero H ), Hirschberger et al. [25] demonstrated that the LMR is strikingly large( ρ (9 T ) /ρ (0) = 0 .
1) for samples with E F close to the Weylnode energy. As E F is moved away from the node en-ergy (as determined by the weak-field Hall effect), theLMR signal is gradually suppressed. These trends areconsistent with the chiral anomaly.Liang et al. [17] applied the squeeze test to GdPtBi andobserved that both R spine and R edge decreased mono-tonically with increasing B ( R spin ( B ) is shown in Fig.3c). The test confirmed that the negative LMR is in-trinsic, just as in Na Bi. As shown in Fig. 3d, thenumerically extracted intrinsic resistance R int ( B ) con-tinues to decrease at the highest applied B (the LL0 isreached at ∼
25 T). To date, Na Bi and GdPtBi pro-vide the firmest evidence (based on LMR) for the chiralanomaly in semimetals.
C. The layered semimetal ZrTe The semimetal ZrTe crystallizes in a layered structurewith space group Cmcm ( D h ). Within each a - c layer,prismatic chains of ZrTe run parallel to the a axis (theneedle axis), with adjacent chains linked by additionalTe ions. The quasi-2D layers are stacked along the b axisby van der Waals interaction to form the 3D structure.At Γ, band inversion of orbitals from Te p states leads toa massive Dirac node above a very small gap.Roughly concurrent with the LMR experiment onNa Bi [24], negative LMR was also reported in ZrTe byLi et al. [27], and attributed to the chiral magnetic effect(Fig. 4a). Subsequent experiments on ZrTe have led toa bewildering array of transport behavior, caused by thesensitivity of the unusually small carrier population to Tevacanies and variation of the chemical potential µ with T . In early samples grown by chemical vapor transportwith high densities of Te vacancies, the resistivity profile ρ a vs. T displays a large peak at T p (varying from 60to 160 K). Angle-resoved photoemission measurements(Fig. 4b) on a sample with T p = 135 K show that µ shifts considerably with T [28]. At 2 K, µ lies at thebottom of the conduction band close to Γ. As T is raisedto 255 K, µ crosses the small gap (50-80 meV) to endup near the top of its valence band (at T p , µ is mid-gap, in agreement with the profile of ρ a ). Evidence for atemperature-driven topological transition from a strongTI to weak TI occuring at T p (138 K) has been obtainedfrom infrared spectroscopy [29].The Te vacancy density is lower in flux-grown crys-tals. As T decreases from 300 to 2 K, ρ a rises mono-tonically to approach saturation below ∼
10 K. ARPESmeasurements [30] show that, at 17 K, µ lies close tothe top of the valence band ( µ does not enter the band gap in the sample studied). The existence of a mod-erately large ( ∼ B (cid:107) (cid:104) J (cid:105) (cid:107) a [30]. However, when B was tilted at an angle θ exceeding 1.5 ◦ , the negativeLMR vanished as shown in Fig. 4c (with θ defined inthe inset). In addition, the LMR monitored at θ = 0changed to a positive sign when B was increased above 3T. Both the high sensitivity to θ and the sign change arenot understood. Unlike Na Bi and GdPtBi, the squeezetest to eliminate current jetting artifacts has yet to beperformed. The high mobility (60,000 cm /Vs at 2 K),which makes B cyc (cid:28) B Q (0.83 vs. 4 T), suggests thatcurrent jetting effects may be dominant in ZrTe .Apart from the LMR behavior, ZrTe displays a richassortment of transport features at 2 K engendered bythe Berry curvature Ω . As shown in Fig. 4d, the Hall re-sistivity ρ yx displays a large anomalous Hall effect (AHE)which has been mapped out over the entire solid angleof the vector H by Liang et al. [30]. A very interestingfeature is the emergence of a true, anomalous planar Halleffect that reverses sign with the in-plane H . The signreversal with H (Onsager behavior) distinguishes it fromthe so-called “planar Hall effect” engendered by domainwall anisotropy in magnetic thin films.Under uniaxial stress applied (cid:107) a , ρ a in flux-growncrystals initially decreases to a attain a minimum at astrain (cid:15) min ∼ .
12 % and then rises quadratically athigher strain. Mutch et al. [31] interpret this unusual be-havior as evidence for a sign-change of the mass term m in the Dirac cone arising from a topological phase tran-sition from a strong TI to a weak TI phase. A negativeLMR that changes sign above ∼ θ was also observed in the two TI phases. D. Weyl Semimetals
Unlike in the Dirac semimetals, the space group of theWeyl semimetals, TaAs, NbAs, TaP and NbP, lacks in-version symmetry. Hence each Dirac node is already splitinto isolated Weyl nodes in zero B . TaAs belongs to thespace group I md . There exist 24 Weyl nodes, with 8located on the k z = 0 plane and 16 lifted off the plane.The large number of nodes combined with the very highmobilities (100,000 to 150,000 cm /Vs) make LMR datadifficult to analyze. Initially, observations of a small,negative LMR in weak H were interpreted as the chi-ral anomaly (Fig. 5a) [32, 33]. However, subsequentstudies [34, 35] found that the LMR feature is fragile,changing sign if the voltage contacts are rearranged.The application of the squeeze test to TaAs and NbPdemonstrated [17] that current jetting distortions onsetat a field B cyc ∼ B Q = 7.04 T ( B Q is de-termined by the quantum oscillations). Figure 5b showsthat, as B is increased from zero, R spine increases verysteepy whereas R edge falls to values below the detectionlimit before B Q is attained (the divergent profiles areclosely similar to those in Bi). Based on these findings,Liang et al. conclude that LMR experiments cannot beused to confirm the chiral anomaly in TaAs and NbP.Alternate techniques to detect the anomaly-induced cur-rents are described below. VI. COMPLEMENTARY EXPERIMENTSA. Non-local Transport
Parameswaran et al. [12] have proposed a non-localtransport experiment to detect the chiral anomaly inDirac/Weyl semimetals. The test assumes that an unbal-ance in the electrochemical potentials µ R,LEC of two Weylnodes (labelled R and L ) is established by injection of apump current at one end ( x = 0) of a long thin-film sam-ple in an applied probe field B p . The electro-chemical po-tential difference, δµ EC ( x ) = µ REC ( x ) − µ LEC ( x ), decaysexponentially along the sample’s length as δµ EC ( x ) = δµ EC (0) e − x/(cid:96) v . The diffusion length (cid:96) v is given by (cid:96) v = √ Dτ v , where D is the carrier diffusion constantand τ v is the intervalley scattering lifetime (Fig. 6a). Theelectro-chemical potential unbalance may be detected bypairs of voltage contacts at various points along the sam-ple (Fig. 6b). Importantly, the chiral current directionof either Weyl node depends on the local direction of themagnetic field. Hence, if one could apply a detection field B d at the voltage contacts distinct from B p , one couldverify if the detected signal has a sign dictated by B d · B p .In Ref. [36], Zhang et al. employed a focused ion beamto fabricate multiple pairs of probes on a CVD grownthin film Cd As sample to measure the non-local sig-nal (Fig. 6c). They showed that the measured non-localsignal has contributions from both Ohmic diffusion (theconventional current) and the polarization diffusion aris-ing from the charge pumping (Figs. 6d,e,f). They foundthat the valley polarization diffusion length is ∼ × theconventional Ohmic diffusion length (Fig. 6g). Becauseof the short length scales, however, the crucial test of ap-plying an independent B d at the voltage contacts couldnot be carried out. B. Thermopower
The thermopower S xx provides a probe of the LL0. In-stead of the usual quadratic dispersion along H (cid:107) ˆx , wenow have a strictly 1D linear dispersion. By the Mottformula, the flat density of states causes strong suppres-sion of S xx . As noted, the lowest Landau level (LL0)of Weyl fermions is chiral. This striking feature impliesthat the density of states D ( E ) is nominally indepen-dent of energy E , in contrast with the divergent form D n ∼ ( E − E n ) − at higher LLs (with E n = ( n + ) (cid:126) ω c ).The thermopower vanishes if particle-hole symmetryexists at the chemical potential. Using the Mott relation, S xx = ( π k B T / e )[ ∂ ln σ xx /∂E ], we have S xx ∼ ∂ D n /∂E if the E dependence of the carrier lifetime is negligible. Hence S xx is expected to be strongly suppressed in thechiral LL0.Hirschberger et al. [25] investigated the strong vari-ation of the thermopower S xx ( B, T ) in GdPtBi vs B .When B is aligned (cid:107) (cid:104) J Q (cid:105) (the applied thermal currentdensity), S xx is observed to decrease monotonically by afactor of 6 as B is increased from 0 to 14 T in a samplewith B Q ∼
25 T (Fig. 7a). The steep decrease is consis-tent with the approach to the strongly suppressed valueas µ approaches the quantum limit.Unlike in Na Bi, a complication in GdPtBi is thatthe Weyl nodes have to be created by applying a fi-nite B [25, 26]. The Weyl states and their characteristicLandau levels appear only when B exceeds B p ∼ α xx = S xx /ρ xx ,which relates the charge current to the applied gradientvia J x = α xx ( −∇ T ). The field profile of α xx displays aprominent peak at B p , which marks the onset of the Weylregime ( B > B p ). Regardless of this distinction, α xx isalso observed to decay in magnitude when B exceeds B p .The absence of Weyl fermions in GdPtBi in the low-field region 0 < B < B p precludes comparison of the mea-sured curves with weak-field Boltzmann-equation calcu-lations of α xx ( B ) reported in Refs. [37–39]. The sup-pression of S xx by B has also been observed in Cd As by Jia et al. [40]. Curves of S xx vs. B (with B (cid:107) (cid:104) J Q (cid:105) )are shown in Fig. 7b. C. Thermal Conductivity and chiral zero sound
In a recent measurement of the thermal conductivity κ xx vs. B in TaAs, Xiang et al. [41] observed mag-netic quantum oscillations in κ xx with remarkably largeamplitude when J Q is aligned with B (Figs. 8a). Theoscillations have the same period as (but are antiphasedwith) the SdH oscillations in the conductivity σ xx . Thepeak-to-peak amplitude ∆ κ xx ∼
12 W/Km of the largestoscillation is 3.4 × the zero- B value κ (Figs. 8b). Theoverall values of κ xx are 50-100 × larger than predictedby the standard Wiedemann-Franz law. After eliminat-ing several potential causes, the authors identify the oscil-lations as arising from the propagation of the CZS modepredicted by Song and Dai [23] (Sec. III B). They showthat both the magnitude and phase of the oscillationscan be fitted to the CZS model. D. Optical pump-probe experiment
Jadidi et al. [42] have employed pump-probe mea-surements at terahertz frequencies to investigate chiralcharge pumping and measure carrier relaxation in theWeyl semimetal TaAs in applied B . An intense “pump”pulse at frequency 3.4 THz with optical field E pump (cid:107) B is employed to generate photo-excited carriers. Thelow photon energy (14 meV) ensures that only carriersvery close to the Weyl nodes are excited. The result-ing changes to the reflection coefficient are detected bya weak probe pulse. By varying the delay time betweenpump and probe, they infer the relaxation time of thecarriers (Fig. 9). In addition to the usual hot-carriercontribution (which relax very rapidly), they observeda long-lived metastable response that persists beyond 1ns in the presence of B (Fig. 9a). The metastable re-sponse is interpreted as evidence for the axial current(Fig. 9b). As a test, they verified that the metastableresponse is present only when E pump is aligned parallelto B , and vanishes when E pump is perpendicular to B .The metastable axial anomaly signal is found to be linearin B (inset in Fig.9c). VII. PERSPECTIVE
In this review, we surveyed experiments on the chiralanomaly in condensed matter physics. Our discussion,however, falls short of conveying the importance of theanomalies in quantum field theory. The experiments herejoin a very long thread of anomaly topics that extendsthrough meson physics to quark physics and gravitation.In the Supplementary Information, we provide a longerdiscussion of the crucial discovery of Adler and Bell andJackiw of the chiral anomaly in the context of the decayof π . We describe how the anomaly A wrecks the con-servation of the axial current J . At the level of quarks,the (non-Abelian) anomaly solves the U A (1) problem andleads to the θ -vacuum. At a higher level of abstraction,we refer to Fujikawa’s path-integral formulation whichrelates A to the chiral zero-modes of massless fermions.The anomaly has a starring role in the long mathematicalodyssey from the Gauss-Bonnet theorem to the Atiyah-Singer index theorem. We direct readers to several refer-ences on this rich subject. VIII. SUPPLEMENTARY INFORMATION:CHIRAL ANOMALY IN QFT
We briefly survey the discovery of the chiral anomalyin quantum field theory (QFT). Introductory discussionsof anomalies are found in Peskin and Schroeder [14] andCheng and Li [43]. More advanced treatments are givenby Weinberg [45], Nakahara [15] and Bertlmann [16]. Pe-skin and Schroeder [14], Cheng and Li [43], and Aitchi-son and Hey [44] are excellent references for QCD phe-nomenology. Recent brief reviews of anomalies are givenby Adler [46] and Jackiw [47]. Concepts in differential ge-ometry are reviewed in Nakahara [15] and Frankel [51].
A. Chiral symmetry of massless fermions
The Lagrangian describing the free Dirac fermion is L = ¯Ψ( iγ µ ∂ µ − m ) ˆΨ , (1)where m is the fermion mass and ˆΨ is a 4-spinor (with¯Ψ ≡ ˆΨ † γ ). The set of 4 × γ µ , are γ , γ , γ , γ . Here, we focus on the chirality matrix γ = γ ≡ iγ γ γ γ which anticommutes with γ µ , viz. { γ , γ µ } = 0 , ( µ = 0 , , , . (2)In the Dirac (or Bjorken-Drell) representation, γ is di-agonal whereas γ is block off-diagonal.Weyl observed that, in the limit m →
0, the La-grangian L becomes ( L L + L R ), which describes inde-pendent “left” and “right” massless populations. In thislimit, it is more convenient to adopt the chiral represen-tation in which γ is diagonal, viz. γ = (cid:20) − ˆ1 00 ˆ1 (cid:21) . Theleft- and right-handed spinors ˆΨ
L,R are the eigen-spinorsof γ , with eigenvalues χ ≡ ±
1, viz. γ ˆΨ L = ( −
1) ˆΨ L , γ ˆΨ R = (+1) ˆΨ R . (3)In the chiral representation, we haveˆΨ L = (cid:18) ˆ u L (cid:19) , ˆΨ R = (cid:18) u R (cid:19) , (4)where ˆ u L,R are 2-spinors. The labels L and R warrant acomment. In photons, right- and left-handedness refer tothe locking of the photon’s spin (cid:126)σ parallel or anti-parallelto its momentum p . This actually refers to the eigenval-ues h of the helicity operator (cid:126)σ ˆ · p . For massless fermions,we have χ = h for fermions. However, χ = − h for an-tifermions. Hereafter, ‘left’ and ‘right’ are understood aslabels for the eigenstates of γ , rather than the helicity.It is helpful to regard ˆΨ L,R as produced by the projec-tion operators P ± = (1 ± γ ) satisfying P ± = P ± and P − P + = 0, viz.ˆΨ L = (1 − γ )2 ˆΨ , ˆΨ R = (1 + γ )2 ˆΨ . (5)Projection yields (1 − γ ) γ γ µ (1 + γ ) = 0. This ver-ifies Weyl’s observation that, if m → L is the sum ofindependent L and R terms, viz. L = i ¯Ψ L γ µ ∂ µ ˆΨ L + i ¯Ψ R γ µ ∂ µ ˆΨ R . (6)By Noether’s theorem, symmetries of the Lagrangian L lead to conservation laws. With m = 0 in Eq. 1, L isinvariant under the 2 global transformationsˆΨ = (cid:18) ˆ u L ˆ u R (cid:19) → (cid:18) ˆ u L ˆ u R (cid:19) e iθ , (cid:18) ˆ u L ˆ u R (cid:19) → (cid:18) ˆ u L e iθ ˆ u R e − iθ (cid:19) . (7)In the first transformation ( ˆΨ → e iθ ˆΨ), the 2-spinors ˆ u L and ˆ u R are rotated in isospin space by the same angle θ ,while in the second ( ˆΨ → e iθγ ˆΨ), the rotations are ofopposite signs (axial rotation).The two types of rotations are starting points for dis-cussing the symmetry properties of massless fermions.Using the chiral representation, we either rotate thephase of ˆ u L and ˆ u R in unison, or in opposite directions(the latter is achieved by the chirality matrix γ in theexponent). If the Lagrangian is invariant under both op-erations, we have conservation of the vector current j µ and the axial (or chiral) current j µ defined by j µ = ¯Ψ γ µ ˆΨ , j µ = ¯Ψ γ µ γ ˆΨ . (8)With ˆΨ = ˆΨ L + ˆΨ R , j µ simplifies to the sum of vectorcurrents associated with L and R states (projection re-moves the cross terms ¯Ψ R γ µ ˆΨ L and ¯Ψ L γ µ ˆΨ R ). Hencewe have j µ = ¯Ψ L γ µ ˆΨ L + ¯Ψ R γ µ ˆΨ R ≡ j µL + j µR . (9)The sum of the currents in Eq. 8 leads to j µ + j µ = ¯Ψ γ µ (1 + γ ) ˆΨ = 2 j µR , (10)whereas the difference yields j µ − j µ = ¯Ψ γ µ (1 − γ ) ˆΨ = 2 j µL . (11)Equations 10 and 11 then reveal the simple relation j µ = j µR − j µL . (12)The axial current measures the difference of the chargeand currents between the L and R populations; it is cen-tral to the chiral anomaly in Dirac/Weyl semimetals.As mentioned, we have the two conservation laws ∂ µ j µ = ∂ρ∂t + ∇ · J = 0 , (13) ∂ µ j µ = ∂ρ ∂t + ∇ · J = 0 , (14)with j µ ≡ ( ρ, J ) T and j µ ≡ ( ρ , J ) T .These conservation laws are valid at the classical level.The process of quantization (brought about by couplingto vector gauge fields) destroys the conservation of j µ ,with experimental consequences. This constitutes thechiral anomaly. B. Adler-Bell-Jackiw anomaly
The chiral anomaly was discovered in the process of un-derstanding conservation properties of the axial currentand its role in pion decay [7, 8, 46, 47]. The π mesons( π + , π , π − ) T are the lightest of all hadrons. Withoutany hadronic state to decay into, the primary channelsare then leptonic (to electrons and muons); this resultsin a relatively long lifetime (26 ns) for the charged pi-ons π ± . Remarkably, neutral pions π decay 300 mil-lion times faster (lifetime 8.4 × − s) via the QED pro-cess π → γ (this channel is forbidden for π ± becauseof charge conservation). The struggle to calculate theamplitude M ( π → γ ) set the stage for the chiralanomaly. As pions are nearly “massless”, the problemseemed tailor-made for the powerful current algebra tech-niques then in vogue, but these techniques yielded zerofor M ( π → γ ) (the Veltman-Sutherland result). In1968, Adler and Bell and Jackiw (ABJ) independentlyidentified the correct process. The axial current is notconserved. Instead the conservation is violated by asource term A , viz. ∂ µ j µ = A , (15)with A (the Abelian anomaly term) given by A = e π ε µναβ F µν F αβ , (16)where ε µναβ is the antisymmetric tensor and F µν theelectromagnetic field tensor. The quantity A defined inEq. 16 is identical to the expression from charge pumpingbetween chiral Landau levels (Eq. 5 in main text).The lifetime of π calculated from A achieved impres-sive agreement with experiment provided one assumedthat quarks come in 3 colors (this was an early evidencefor color). The ABJ triangle feynman diagram is thearchetypal example for all subsequent anomaly calcu-lations. Anomalies appear whenever a diagram has afermion loop (quarks) coupled to vector currents (pho-tons) and an odd number of axial currents ( π ).In 1983, Nielsen and Ninomiya predicted [9] that thechiral anomaly should be observable as an unusual neg-ative longitudinal magnetoresistance in semimetals thatfeature 3D Dirac states. C. Anomaly Cancellation
In the ABJ calculation of the decay rate of π , thetriangle diagram linked an axial current (quarks) with2 vector gauge bosons (photons). In electroweak the-ory and QCD, we can also have triangle diagrams withvertices linked only to gauge bosons. These diagramsare fatal because they render the theory unrenormaliz-able. Hence they must all cancel to zero. In the GlashowWeinberg Salam theory, each of the 3 vertices can becoupled to one of the 3 gauge bosons belonging to U (1), SU (2) or SU (3). Peskin and Schroeder [14] describe thecancellation of all the dangerous diagrams within eachgeneration of quarks and leptons. The anomaly cancel-lation, dubbed magical [14], imposes constraints on thechirality of fermions “running around” the triangle loop. D. The U A ( ) problem Anomalies have played crucial roles in resolving deeppuzzles at several energy hierarchies. By the early 70’squantum chromodynamics (QCD) based on quarks andleptons had gained universal acceptance, but a majorhurdle remained – the U A (1) problem [14, 43, 44]. Thesmall bare masses of the up and down quarks u and d (4 and 7 MeV, respectively) invite a massless fermiondescription. The invariance of L under the global axialunitary transformation U A (1) then suggests that the ax-ial vector current is conserved, i.e. ∂ µ J µ = 0. However,the assumed chiral symmetry immediately led to an un-welcome prediction: every hadronic state should have aparity pardner, in striking conflict with observation (thisis known as the parity-doublet problem). Hence the chi-ral symmetry must be spontaneously broken, but doingso raises a further problem – too many Nambu-Goldstonebosons.In a model incorporating just the lightest quarks u and d , breaking of the chiral symmetry generates altogether4 Nambu-Goldstone bosons – the isotriplet which corre-sponds to the pions come from breaking of SU L × SU R symmetry plus an extra isoscalar meson. The last is con-spicuously absent in the particle spectrum. When oneincludes the heavier strange quark s (mass 130 MeV),a similar problem arises. There should be 2 isoscalarmesons, i.e. 9 low-mass mesons altogether but only 8are observed (3 pions, 4 kaons and the η particle). Thepseudoscalar meson η (cid:48) is too massive to be part of thespectrum. This impasse constitutes the U A (1) prob-lem [14, 43, 45].To remove the unwanted U A (1) symmetry, one couldinvoke the anomaly mechanism, with A generalizedto [43] A = 4 g π tr( G µν ˜ G µν ) , (17)where g is the strong-interaction coupling parameter andtr means trace. The electromagnetic field tensor F µν inEq. 16 is replaced by the gluon tensor matrix G µν (with˜ G µν its dual). However, A is now a total derivative, ex-pressed as A = ∂ µ K µ , where K µ acts as a current [43, 45].If we define a new axial vector current ˜ J µ that includes K µ , viz. ˜ J µ ≡ J µ − K µ , we find that the correspondingaxial charge ˜ Q is conserved, i.e. ∂ µ ˜ J µ = 0 . (18)Hence the U A (1) problem is apparently unresolved. The solution discovered by ‘tHooft is that, in per-forming the path integrals, contributions from instantonsmust be included (see Sec. VIII E). E. Topological Origin
The ABJ anomaly was a bit of topology that fell intothe physics of the 1960’s. It soon emerged that A plays arole in gauge theory far more important than suggestedby the resolution of the π problem. An early hint toits topological origin was the finding that the one-looptriangle diagram receives no radiative corrections to allorders in perturbation calculations (the Adler-Bardeentheorem [48]). Moreover, A is identical whether calcu-lated from the triangle diagram (Eq. 16) or using Landaulevels (Eq. 5).The central role of A may be seen from Fujikawa’spath-integral approach to the anomaly [49]. Under atransformation of the field variables in the Dirac fermionLagrangian (hereafter, Euclidean metric is assumed), theJacobian J acquires a phase determined by A [16, 47, 50].In particular, the axial rotation ˆΨ → e iβγ ˆΨ induces thechange J → J e − β (cid:82) d x A ( x ) .The phase in J involves the integral (cid:82) d x A ( x ) (theaction S ) over a sphere in 4-space bounded by the 3-sphere S of radius R . For S to be finite, the gaugepotential A µ must vanish rapidly as R → ∞ , to leave thegauge-only form ˆ g (ˆ x ) − d ˆ g (ˆ x ), where ˆ g (ˆ x ) is a transforma-tion of the vacuum state in the direction ˆ x (ˆ g ∈ SU (2)).Because ˆ g is specified by 3 parameters, just like the 3-sphere S , the mapping S → SU (2) partitions into ho-motopic sectors each characterized by an integer windingnumber n ∈ Z ( n is also called the topological charge orPontryagin index) [15, 43, 45]. In Yang Mills theory, thewinding number of the mapping arises from instantons(see below).A lengthy calculation reveals that the integral (cid:82) d x A ( x ) equals an integer that measures the num-ber of chiral zero modes (a zero mode φ ± satisfies γ µ D µ φ ± = 0, where D µ = ∂ µ + A µ is the Dirac op-erator). We have (cid:90) d x A ( x ) = 2 i ( n + − n − ) , (19)where n ± is the number of zero-modes of chirality ± .The difference ( n + − n − ) is the index of the Weyl oper-ator D + [15, 16] (the index may be regarded as a gen-eralization of the familiar Euler characteristic χ ( M ) = V − E + F , an invariant that tracks the number of ver-tices V , edges E and faces F of a polyhedron [15]).Returning to the U A (1) problem, the apparent im-passe, Eq. 18, is equivalent to assuming that the action (cid:82) d x A vanishes, i.e. the Jacobian J is unchanged. Asshown by ‘tHooft, inclusion of instanton(s) makes theaction finite instead (Eq. 19). The instanton is a fluctu-ation of the vacuum in which G µν is finite within a local0region of spacetime but dies away rapidly as R → ∞ .According to Weinberg [45], the non-vanishing of the in-tegral in Eq. 19 suffices to solve the U A (1) problem. Thesolution of the U A (1) problem provided an early hint ofthe rich structure of the QCD vacuum.The calculation establishing Eq. 19 parallels that donefor the Atiyah-Singer (AS) index theorem [15, 16, 49].The AS index theorem is a far-reaching and deep gener-alization of the famous Gauss-Bonnet theorem, which re-lates the Gaussian curvature K of a closed, compact and orientable surface M to its Euler characteristic χ ( M ) bythe integral 12 π (cid:90) M KdA = χ ( M ) = 2 − g M , (20)where the genus g M counts the number of holes in M [15,51]. In Eq. 19, A plays the role of K while the indexreplaces χ ( M ). 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We are indebted to Stephen Adlerfor valuable comments and suggestions. N.P.O. acknowl-edges the support of the U.S. Army Research Office(ARO contract W911NF-16-1-0116), the U.S. NationalScience Foundation (Grant DMR 1420541) and the Gor-don and Betty Moore Foundation’s EPiQS Initiativethrough Grant GBMF4539.3 E k || B E, B
LN RN a J xy c xz B Iy b L w t d B k z E + +- -k y k x k z FIG. 1. Panel (a): Sketch of the Landau levels of Weylnodes. In the node with chirality χ = +1, the chiral level LL0disperses to the right (velocity v (cid:107) B ) while the node with χ = − E “pumps” electronsto the left at a rate given by the anomaly A . Occupancy ofthe levels is represented by thick curves. Panel (b) shows the6 contacts on a plate-like crystal mounted for the squeeze test.The local E -fields measured along the spine (blue dots) andalong the edge (yellow) define R spine and R edge , respectively.Panel (c) shows the heat-map of the current density J ( x, y )with B (cid:107) E (cid:107) ˆx . Current jetting concentrates J ( x, y ) alongthe spine (dark region) while depleting it at the edges. Theprofile of J x vs. y along the dashed line is sketched on theright. Panel (d): Schematic of the chiral zero sound (CZS) atan instant t in the oscillation cycle. The square shows 2 pairsof Weyl nodes in the k x - k y plane with B (cid:107) ˆz (blue and pinknodes are of chirality χ = +1 and -1, respectively). For eachnode, the accompanying sketch depicts the chiral dispersion E vs. k z as a solid line. At instant t , the occupancy of thechiral mode (thick lines) is higher in the 2 nodes drawn withlarger FS radii. All local charge deviations from equilibriumsum to zero. The CME charge currents also cancel pair-wise.Panels (a), (b) and (c) are from Liang et al. [17]. Panel (d) isbased on Ref. [23]. a b dc FIG. 2. The chiral anomaly observed in the LMR ofNa Bi. Panel (a) displays magnetoresistance curves measuredin Na Bi in parallel fields ( B (cid:107) (cid:104) J (cid:105) (cid:107) ˆx ) at selected T from 4.5to 300 K. Below ∼
100 K, the steep decrease of the longitu-dinal resistivity ρ xx in increasing B is direct evidence for thechiral anomaly. Panel (b) shows the variation of the inferredchange in conductivity ∆ σ xx versus φ in an in-plane B mak-ing an angle φ with ˆx , with magnitude B fixed at selectedvalues 0.5 to 2 T. In the polar plot (inset) displaying ∆ σ xx (radius) vs. φ , the conductivity enhancement appears as aplume directed along B . Application of the squeeze test todistinguish chiral anomaly LMR from current jetting effects.Panel (c) displays the field profiles of R spine ( B ) measured inNa Bi with B (cid:107) (cid:104) J (cid:105) in the squeeze test. If current jettingeffects dominate, concentration of J along the spine shouldlead to an increase in R spine with B (as observed in pure Bi).Here, R spine is observed to decrease instead. Panel (d): Themeasured profiles of R spine ( B ) (black curve) and R edge ( B )(red) allow the intrinsic curve R int vs. B (blue) to be ex-tracted numerically. In the quantum limit ( B > R int is10 × smaller than its value in zero B . Panels (a) and (b) areadapted from Xiong et al. [24]; (c) and (d) are from Liang etal. [17]. a bc d FIG. 3. The chiral anomaly in the half-Heusler semimetalGdPtBi. Panel (a) displays the field profiles ρ xx ( B ) measuredwith B (cid:107) (cid:104) J (cid:105) (cid:107) hatx (cid:107) ( ) with T fixed at the valuesindicated. Below 150 K, a large negative contribution to ρ xx appears. At 6 K, ρ xx displays a bell-shaped profile with astrongly negative LMR closely similar to the profile seen inNa Bi. Panel (b) shows the effect of tilting B out of the x - y plane by angle θ . The negative LMR remains prominent until θ exceeds 48 ◦ . Panels (a) and (b) are from Hirschberger etal. [25]. Panel (c): Application of the squeeze test to GdPtBi.As in Na Bi, the resistance on the spine, R spine ( B ), decreaseswith increasing B , consistent with current jetting effects beingsubdominant. In combination, the profiles of R spine (blackcurve) and R edge (red) may be used to obtain numerically theintrinsic resistance R int ( B ) (blue). In GdPtBi, the quantumlimit is reached above 24 T. Panels (c) and (d) are from Liang et al. [17]. a bc d FIG. 4. Unusual electronic properties of ZrTe . Panel (a)shows the initial observation of negative LMR measured atselected T from 5 to 150 K (Li et al. [27]). Panel (b) displaysthe temperature dependent evolution of the band structurefrom 2 to 255 K, measured by angle-resolved photoemissionalong the direction Γ- X (Zhang et al. [36]). Panel (c) displaysthe extreme sensitivity of the negative MR to slight tilting of B out of the layer. The negative component of the MR van-ishes for tilt angles | θ | > ◦ (Liang et al. [30]). In additionto the negative MR, ZrTe5 displays a striking array of Berrycurvature effects observable in its Hall resistivity ρ yx . Panel(d) shows the anomalous Hall effect (AHE) in a tilted mag-netic field B . Curves of the AHE contribution to the Hallresistivity ρ AHE are plotted vs. B at selected tilt angles θ between B and the a -axis (Liang et al. [30]). a b FIG. 5. Panel (a): The observed fractional change in re-sistivity, MR ≡ ( ρ ( B ) − ρ (0)) /ρ (0), observed in TaAs at 1.8K [32]. The negative MR is highly sensitive to slight tilts of B away from E . The negative MR vanishes when θ deviatesfrom 90 ◦ by 2 ◦ . Panel (b) shows the results of applying thesqueeze test to TaAs at 4 K [17]. The edge and spine resis-tances, R edge and R spine , respectively, strongly diverge once B deviates from zero (as shown by the arrows, both curvesare displayed on two scales). This implies that the observedLMR is dominated by current jetting artifacts because of thevery high carrier mobility. The inset shows the index plot ofLLs derived from the weak oscillations in R edge which yields B Q = 7.04 T. FIG. 6. Detection of non-local signal in the Dirac semimetalCd As . Panel (a): Schematic view of the valley diffusionprocess. Parallel (antiparallel) E and B fields generate thecharge imbalance between two Weyl nodes due to the chi-ral anomaly. The charge imbalance of different valleys candiffuse across the sample and be converted into a nonlocalvoltage along the direction of B . Panel (b): Schematic viewof the nonlocal resistance measurement with different diffu-sion channel width. Current is applied through terminal 1–2,while terminals 3–4 and 5–6 are used to measure the nonlocalresistance. The diffusion length L is 2 mm. Panel (c): Theelectron micrograph of the device (white scale bar is 2 mm).The contact regime in terminals 3–4 is slightly larger thanthat of 5–6. Panel (d): The two- terminal local resistance(R ) at 20 K. Panel (e): The nonlocal resistance (R andR ) at 20 K. Panel (f): The pure nonlocal resistance (R − NL and R − NL ) after subtracting the Ohmic diffusion at 20 K.Panel (g): Resistance ratio of R /R and R − NL /R − NL versus B at different T . Dashed lines labelled as Valley andOhmic correspond to ratios 0.50 and 0.17, respectively. FromZhang et al. [36]. a bGdPtBi Cd As FIG. 7. Curves of the thermopower S xx vs. B observed inGdPtBi (Panel a) and in Cd As (Panel b). Panel (a) showscurves of S xx ( B ) measured in GdPtBi at selected tilt angles θ of B relative to the x - y plane at T = 6.45 K. The ther-mal current density J Q is applied along the [110] direction.The pronounce decrease of S xx with increasing B at θ = 0 isconsistent with the increased dominance of the lowest (chiral)Landau level in which S xx is strongly suppressed (see text).Weak SdH oscillations are observed for the curve at θ = 0.Panel (b) plots S xx ( B ) in Cd As measured at selected val-ues of T . As the temperature decreases below 100 K, thecurves become increasingly negative. Panel (a) is adaptedfrom Hirschberger et al. [25]. Panel (b) is from Jia et al. [40]. a b FIG. 8. Panel (a): Raw experimental trace of the tem-perature gradient dT ( B ) in the semimetal TaAs as the field B is swept from -9 to 9 T at constant heater power with B (cid:107) (cid:104) J Q (cid:105) (cid:107) c and T fixed at 2 K. Giant quantum oscilla-tions are observed in the longitudinal thermal conductivity κ (cid:107) . Panel (b): The amplitudes of the quantum oscillationsin κ (cid:107) at low T (red curve) are nearly two orders of magni-tude larger than the oscillations in the thermal conductiv-ity κ e,WF inferred from the longitudinal conductivity σ (cid:107) us-ing the Wiedemann-Franz law (black curve). From Xiang etal. [41]. ab c FIG. 9. Optical detection of long-lived current inducedby charge pumping in the Weyl semimetal TaAs. Panel (a)shows the pump-induced fractional increase in probe reflec-tion ∆
R/R at zero magnetic field (black curve) and at B = 7T (yellow) as a function of the time delay between the pumpand probe pulses, with E pump (cid:107) E probe (cid:107) B . The prominentpeak close to zero time delay, common to both traces, tracksthe fast relaxation of hot carriers. However, when B is setat 7 T, the pump-probe trace exhibits a component (shadedarea) that relaxes on a time scale far longer than 400 ns. Thelong-lived component is associated with the chiral anomaly.Panel (b): Simulated pump-probe traces for fast hot carri-ers effects (red), metastable chiral pumping (blue), and thenet result (green). ∆ is defined as the maximum pump-induced change in probe reflection near zero time delay, and∆ characterizes the long-lived component. Panel (c) shows8 pump-probe traces as B is set at successively larger values.The inset shows ∆ (red) and ∆ (blue) vs. B . From Jadidi et al.et al.