Downloaded by guest on September 27, 2021 eemn h e intrso hspae hc r elzdin realized are which Ce phase, We semimetal this arcs. heavy- the the Fermi of feature signatures of that key out states the develop surface determine the nodes do Weyl as effect, the Kondo near semimetal lat- noncentrosymmetric Weyl–Kondo a The on a model tice. advance Anderson periodic setting we a natural in Here, phase a progress. present and, make coupling, systems to spin-orbit correlated strong strongly their of given prototype with a Heavy-fermion systems identified. are be In semimetals to yet recently. have not they until correlations, have strong evidenced metals and experimentally effect topological Hall well-estab- contrast, been quantum By a the insulators. by nontrivial, 2017) exemplified topological 8, is topologically September review that for be notion (received lished 2017 can 15, November states approved and Insulating CA, Irvine, California, of University Fisk, Zachary by Edited oivledgeso reo uha pnmmns which moments, and spin storage information as as purposes such such retrieval. for freedom harnessed expected of be are may degrees excitations Weyl involve correlated electronic to strongly low-energy any the in gener- semimetal, thus, in physics; freedom of low-energy degrees ating different mix typically elec- nonperturba- regimes in of tive correlations influence strong the Moreover, Weyl correlations. the to the tron that susceptible both expect particularly can Because one are (21–24). gapless, semimetals arcs are Fermi states surface of sur- and as form bulk well the as space, in momentum states the face in dispersions points nodal relativistic of massless in pairs with near excitations , chiral bulk of evi- possesses form recently the It was (18–20). (3D) experimentally dimensions three denced in (14–17). semimetal matter Weyl electronic The of phases topological essential obtaining an in plays role which coupling, spin-orbit with limit interacting have CeRu these including of , Ce Several inversion (6–13). broken hybridized cases a semimetals the two heavy-fermion the between heavy- between falls are intermediate a or Electronically the (3–5). within whether be bands on lies could depending heavy-fermion potential insulator, state the Kondo chemical a of resulting or hallmark metal The fermion effec- the renormalized is classification. strongly which large, system a mass, with carrier metal electrons. tive a conduction into to the leads moments with This local hybridize the can converts that entanglement quasiparticles electrons ensuing conduction the the state; generates with ground interaction spin-singlet The Kondo electrons. conduction the the of local spins the of the terms the in from considered originating be moments can systems antiferromag- these sim- phases, features the cases, In plest paramagnetic 2). that (1, other transitions diagram quantum-phase beyond-Landau and phase and Kondo-screened, rich order, a and netic tunable to highly rise states ground give their make correlations strong S semimetal Weyl an freedom. natu- of up that degrees phases open multiple quantum entangle such and rally of correlations studies systematic strong for with avenue metals search topological experimental of the for foundation theoretical much-needed and a www.pnas.org/cgi/doi/10.1073/pnas.1715851115 Lai systems Hsin-Hua heavy-fermion in semimetal Weyl–Kondo eateto hsc n srnm,Rc nvriy oso,T 77005; TX Houston, University, Rice Astronomy, and Physics of Department eiea ytm r en hoeial tde ntenon- the in studied theoretically being are systems Semimetal 3 c otne oepn t oio.I ev-emo systems, heavy-fermion that In field horizon. vibrant its a expand to represent continues electrons correlated trongly Bi nttt fSldSaePyis inaUiest fTcnlg,14 ina Austria Vienna, 1040 Technology, of University Vienna Physics, State Solid of Institute 4 Pd 3 (9). a,b,1,2 | od effect Kondo aa .Grefe E. Sarah , | ev-emo systems heavy-fermion f 3 Bi lcrn htKnocul to couple Kondo that electrons 4 Pd a,b,1 3 u nig rvd the provide findings Our . ik Paschen Silke , 4 Sn 6 6 )and 7) (6, c n iioSi Qimiao and , b ieCne o unu aeil,Rc nvriy oso,T 77005; TX Houston, University, Rice Materials, Quantum for Center Rice level term, first The tively. by physical the enforced to contains corresponding is model trons, The touching the (25). in symmetry band space-group and, electrons, its nonsymmorphic, noninteracting is it of 1A). because case (Fig. lattice B this and two read- choose A comprises We be lattices which can lattice, cubic diamond symmetry face-centered a interpenetrating inversion is of This breaking sys- incorporated. cubic ily the a which consider in we tem demonstration, proof-of-concept a For heavy-fermion is studied several in WKSM realized the be of to out signatures compounds. model. turn key microscopic which any the advances phase, well-defined work whether demonstrate Our this unclear we way. in priori robust Moreover, answer a a in affirmative is renormal- realized an it be are can ways, couplings state different Weyl electronic very in other the spin- ized and term, band, coupling, breaking conduction-electron inversion-symmetry orbit In the the large. because very much of be addition, is can width factor repulsion renormalization the electron–electron interaction-induced way. the than tunable where larger a regime in the inversion- coupling the In spin-orbital captures and it that breaking in symmetry realistic is It respectively. trons, strongly the correlated contains a model on This model lattice. microscopic noncentrosymmetric concrete 3D a in phase (WKSM) semimetal 1073/pnas.1715851115/-/DCSupplemental at online information supporting contains article This under 2 distributed is 1 article BY-NC-ND). (CC 4.0 access License NonCommercial-NoDerivatives open This Submission. Direct PNAS a is article This interest. of conflict no declare authors The paper. the wrote and data, analyzed research, uhrcnrbtos .HL,SEG,SP,adQS eindrsac,performed research, designed Q.S. and S.P., S.E.G., H.-H.L., contributions: Author owo orsodnemyb drse.Eal mirc.d [email protected]. or [email protected] Email: addressed. be may correspondence whom work. To this to equally contributed S.E.G. and H.-H.L. hc sraie narcnl icvrdheavy-fermion discovered recently a in realized compound. semimetal, effect. is Weyl–Kondo this Kondo associated of which the signature are key bulk called which a determine of the phenomenon We both many-body in surface, the the fermions with on Weyl arcs identify Fermi Weyl– and We a of semimetal. emergence corre- the Kondo strongly demonstrate a to on model lattice work lated theoretical report we corre- whether Here, strong with is lations. metals question in happen outstanding can An protection surface. topological the on the and in show both bulk semimetals electricity Weyl conducting Such while years. protection evidenced 2 topological past been the have tradi- during they metals have insulators, in topology in known nontrivial been with tionally states electronic While Significance h aitna o h eidcAdro oe obe to model Anderson periodic the for Hamiltonian The Weyl–Kondo a of discovery the report we work, this In E a,b,2 d n olm repulsion Coulomb a and 4f lcrn n ado conduction of band a and electrons PNAS H H d ecie the describes , = | aur ,2018 2, January H d + . H cd U 4f + h opigbtenthe between coupling The . www.pnas.org/lookup/suppl/doi:10. d and H raieCmosAttribution- Commons Creative | lcrn iha energy an with electrons c . o.115 vol. spd lcrn,respec- electrons, | d o 1 no. and spd | c elec- elec- 93–97 [1]

PHYSICS empty, forming a band insulator. All these imply that the nodal excitations develop out of the . To demonstrate the monopole flux structure of the Weyl nodes, we calculate the Berry curvature in the strong cou- pling regime. We show the results at the kz = 2π boundary of the 3D BZ, in the gray plane of Fig. 1C, whose dispersion is shown in Fig. 3B. In Fig. 3B, the arrows represent the field’s unit-length 2D projection onto the kx ky -plane, Ω(ˆ kx , ky , 2π) = −1  |Ω(~ kx , ky , 2π)| Ωyz (kx , ky , 2π), Ωzx (kx , ky , 2π) . The Weyl Fig. 1. The 3D noncentrosymmetric lattice and associated Brillouin zone node locations (blue/red circles) are clearly indicated by the (BZ). (A) Diamond lattice with hopping t and onsite energy ±m differenti- arrows flowing in or out, representing negative or positive ating A, B sublattices. The solid lines connect nearest neighbors. (B) Inter- monopole “charge.” locking tetrahedral sublattice cells illustrating how the distinction between We next analyze the surface states. Focusing on the (001) the A and B sublattices (zincblende structure) invalidates the inversion cen- surface, we find the following energy dispersion for the surface ter lying on the point marked “X.” (C) The BZ of the diamond lattice, with states: Weyl nodes shown in blue/red and high symmetry contour used for Fig. 2     2 2 in green. kx ky Vs + (Es ) E(kx , ky ) = −2 sin sin + 4 4 2Es

s 2      2 2  two species of electrons is described by a bare hybridization of kx ky Vs − (Es ) 2 − 2 sin sin − + Vs , strength V . The conduction-electron Hamiltonian Hc realizes 4 4 2Es a modified Fu–Kane–Mele model (26). Each unit cell has four species of conduction electrons, denoted by sublattices A and B [3] T  and physical spins ↑ and ↓: Ψk = ck↑,A ck↑,B ck↓,A ck↓,B . where we define the parameters (Vs , Es , µs ) = (rV , Ed + `, 2 There is a nearest-neighbor hopping t (chosen as our energy −(rV ) /(Ed + `)). (For the derivation, see Supporting Infor- unit) and a Dresselhaus-type spin-orbit coupling of strength λ. mation.) In Fig. 4, we show the energy dispersion along a high- The broken inversion symmetry, Fig. 1B, is captured by an onsite symmetry path in the k space. The solid lines represent the potential m that staggers between the A and B sublattices (17, surface states, and the dashed lines show where they merge with 27). The band basis is arrived at by applying a canonical trans- the bulk states and can no longer be sharply distinguished. The formation on Hc written in the sublattice and spin basis. It cor- surface electron spectrum has a width that is similarly narrow as responds to a pseudospin basis (27), defined by the eigenstates the bulk electron band (compare Fig. 4B with Fig. 3A), imply- | ± Di. We fix the d-electron level to below the conduction- ing that the surface states also come from the Kondo effect. electron band and consider the case of a quarter filling, corre- The surface Fermi arcs (where E(kx , ky ) = 0) connect the Weyl sponding to one electron per site. Further details of the model nodes along kx = 0 and ky = 0, separating the positive and neg- are given in Materials and Methods. ative energy surface patches, marked by the solid black lines in We consider the regime with the onsite interaction U being Fig. 4A. large compared with the bare c-electron bandwidth (U /t → ∞). As is typical for strongly correlated systems, the most dom- We approach the prohibition of d fermion double occupancy by inant interactions in heavy-fermion systems are onsite, making † † † an auxiliary-particle method (28): diσ = fiσbi . Here, the fiσ (bi ) it important to study them in lattice models (as opposed to the are fermionic (bosonic) operators, which satisfy a constraint that continuum limit). Our explicit calculations have been possible is enforced by a Lagrange multiplier `. This approach leads to a using a well-defined model on a diamond lattice that permits set of saddle-point equations, where bi condenses to a value r, inversion-symmetry breaking. Nonetheless, we expect our con- which yields an effective hybridization between the f quasipar- clusion to qualitatively apply to other noncentrosymmetric 3D ticles and the conduction c electrons. The details of the method systems. Finally, the WKSM is expected to survive the effect of a are described in Materials and Methods. time-reversal symmetry breaking term, such as a magnetic field; The corresponding band structure is shown in this is illustrated in Supporting Information. Fig. 2. Nodal points exist at the Fermi energy, in the bands for We now turn to the implications of our results for heavy- which the pseudospin (defined earlier) has an eigenvalue −D. fermion semimetals. The entropy from the bulk Weyl nodes ∗ They occur at the wave vectors kW , determined in terms of the will be dictated by the velocity v , and the corresponding hybridized bands, (+,+) (−,+) E−D (kW ) = E−D (kW ), (+,−) (−,−) E−D (kW ) = E−D (kW ), [2] for the upper and lower branches, respectively. The Weyl nodes appear along the Z lines (lines connecting the X and W points) in the three planes of the BZ, as illustrated in Fig. 1C. This is spe- cific to the zincblende lattice. For other types of lattices, the Weyl nodes may occur away from the high-symmetry parts of the BZ. We note that the bands near the Fermi energy have a width much reduced from the noninteracting value. This width is given by the Kondo energy. Compared with the bare width of the conduction-electron band, the reduction factor corresponds to r 2 (which is ∼0.067 in the specific case shown in Fig. 2). We Fig. 2. Energy dispersion of the bulk electronic states. Shown here is the remark that, in the absence of the hybridization between the f energy vs. wave vector k along a high-symmetry path in the BZ, defined and conduction electrons, the ground state would be an insulator in Fig. 1C. The bottom four bands near EF show a strong reduction in the instead of a semimetal: The f electrons would be half-filled and bandwidth. The bare parameters are (t, λ, m, Ed, V) = (1, 0.5, 1, −6, 6.6). In form a Mott insulator, while the conduction electrons would be the self-consistent solution, r ' 0.259 and ` ' 6.334.

94 | www.pnas.org/cgi/doi/10.1073/pnas.1715851115 Lai et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 rpris()adhsbe icse saptniltopolog- potential a as discussed been has and (6) properties the spe- CeRu noncentrosymmetric of Fermi propose the Eq. tion, role the we our the toward and and doping, taining tuned measurements chemical leads be heat scales energy, or can cific low-energy Fermi pressure they with by the system that energy from Kondo expectation a away the is likely to it are that fact system any the though this Even in 31). nodes (30, measurements magnetotransport the on in instance, For systems. 4f heavy-fermion WKSM other in candidate semimetals another be heavy-fermion another it of (29). heat suggesting system Eq. specific recently, the YbPtBi, fit More system, to system. used been pho- this has quantities in angle-resolved such spectroscopy high-resolution on toemission and studies further magnetotransport and (9). nodes for as Weyl magnitude motivation correlated the of strongly provides order with Ce system similar that WKSM evidence a didate strong by provides analysis the latter This in the reduction of Ce the bandwidth for reflecting temperature Kondo systems, scale—the energy correlated weakly for in described v well is heat Eq. specific Eq. trans- of its on terms based and behavior measurements, semimetal port display to discovered non- been Ce A system (4). heavy-fermion compounds centrosymmetric heavy-fermion semimetallic of number contributions large have also will entropy the thermopower. excitations the of over nodal to the form dominate that corresponding would implies The entropy component. the the phonon WKSM, to a sys- in contributions Therefore, correlated nodal temperature. weakly much Debye is the the electrons conduction than with the larger of contrasted bandwidth the be heavy- which in to tems, temper- typical is Debye the of This than ature. smaller temperature considerably is correlated Kondo systems weakly fermion The from WKSM semimetals. reflects the Weyl signature of key The distinction a important Information ). as an (Supporting quantities thermodynamical excitations of nodal inter- utility residual the the against of robust is actions expression this that stress We (Supporting form following the has Information): volume unit per heat specific 2. Fig. in as curva- same Berry the the are of a parameters distribution bare and The The points field. (B) node ture bandwidth. Weyl the the of at reduction degeneracies strong band the at showing nodes dispersion, Weyl four the of plane i.3. Fig. a tal. et Lai ∗ u hoeia eut rvd udnei h erhfrWeyl for search the in guidance provide results theoretical Our Eq. bsdsse eb elpyishsbe ugse based suggested been has physics Weyl CeSb, system -based hti he reso antd mle hnta expected that than smaller magnitude of orders three is that , S29 4 hrceiaino h elnds h lt r nte(k the in are plots The nodes. Weyl the of Characterization a eraiytse,gvnta hr saconsiderable a is there that given tested, readily be can eel nefcievelocity, effective an reveals Information) (Supporting 4 9.Afi ntrso u hoeia expression theoretical our of terms in fit A (9). c v ∼ 4f (k k lcrn nti ytm naddi- In system. this in electrons B z = T 2π /~v ga ln nFg 1C Fig. in plane (gray 4 ∗ Sn ) 3 6 k 4 B lodsly semimetal displays also . samaso ascer- of means a as 3 Bi 3 Bi 4 3 Pd Bi 4 Pd 4 3 Pd a recently has 3 fo the —from Energy (A) ). 3 sacan- a is x , [4] k y 4 ) ilb ntutv nacrann h oeta KMntr of nature WKSM studies potential the thermoelectrical CeRu ascertaining in and instructive be thermodynamic Eq. will further of that form the gest have to appear the of tance eemtvt ute tde hticroaeteraitcelec- realistic Ce the of findings incorporate structure Our that tronic studies by filling. further enforced electron motivate be and here well symmetry very may crystallographic nodes its Weyl Kondo-driven its WKSM and candidate a The Ce symmetries. such material with lattices crystallographic that lattice Kondo other likely 3D Kondo with inversion-symmetry-breaking it a in makes in arises This phase phase WKSM breaking. a symmetry of inversion emergence the for the for studies. search future systems; for Kondo to matter in this instructive transitions leave phase we be topological therefore at also state will insu- can WKSM distinct It semimetal topologically states. between Dirac transition lating a phase hand, the 3D, other at the in On arise way. systems robust break- noninteracting a inversion-symmetry in for of arises phase presence WKSM the the in on ing, defined and systems lattice Kondo 25). a for (21, such that state demonstrates semimetal group here study filling-enforced space Our a crystallographic for allows the which, symmetry electrons, in lattice, noninteracting diamond nonsymmorphic for an on defined model aist rb h elnds u okas estestage Weyl the the sets for also properties work signature Our additional thermody- nodes. of using calculations Weyl of for the possibility probe the to for renormaliza- interacting namics responsible large of is The factor types of previously. tion other order discussed from the semimetals WKSM Weyl on the typically distinguishes veloc- large, This nodal extremely the are (for ity) factors renormalization localized the the respondingly, that given the of effect moments zeroth-order a produce relations ttsit h uksae.Teprmtr r h aea nFg 2. Fig. in as same the are parameters The states. bulk the into states in specified path at lines boundary BZ black represent lines solid the dashed around The black BZ and (red). the arcs, Fermi blue represent with nodes the marked connecting nodes (anti-)Weyl the shows 4. Fig. ensgetdi h atrsuy h o-eprtr spe- low-temperature The CeRu have crystalline study. nature single latter in Weyl the heat the their cific capture in and exis- to nodes suggested the dispersing appear system, been linearly Kondo dis- not a of low-energy does in tence While the study expected (33). on renormalizations latter method disagree strong the projection calculations and by of Gutzwiller persion types studied the the- two been or field the 32) has mean dynamical (6, structure with ory combined electronic calculations Its initio ab (7). system ical AB eod u okpoie ro-fpicpedemonstration proof-of-principle a provides work our Second, a on focused have we First, observations. several with close We hr,i h KMsaeavne ee h lcrncor- electron the here, advanced state WKSM the in Third, 4 Sn nrydseso ftesraeeetoi tts h spectrum The states. electronic surface the of dispersion Energy 6 . 3 4f Bi k z h rydte iedntstedcyo h surface the of decay the denotes line dotted gray the A; 4 = lcrn otelweeg hsc u osnot does but physics low-energy the to electrons X 4f Pd High-symmetry (A) point. 2π 3 lcrn neleteWy xiain.Cor- excitations. Weyl the underlie electrons nrydseso ftesraesaeaogthe along state surface the of dispersion Energy (B) . PNAS a osmopi pc ru (220), group space nonsymmorphic a has 3 Bi 4 Pd | 3 aur ,2018 2, January . u hoeia eut sug- results theoretical Our 4. 4 Sn 6 saecnortkno the on taken contour k-space 3)ipisteimpor- the implies (34) | o.115 vol. | 10 o 1 no. 2 − 10 | 95 3 .

PHYSICS physics, such as the optical conductivity and related dynamical term specifies a Dresselhaus-type spin-orbit coupling between the second- quantities. nearest-neighbor sites (hhijii), which is of strength λ and involves vector  Fourth, the context of heavy-fermion systems links the WKSM D(k) = Dx (k), Dy (k), Dz(k) . Specifically, phase advanced here with the topological Kondo insulators (5) 3 ! X and the physics of their surface states (35). Nonetheless, the u1(k) = t 1 + cos(k · an) , [8] WKSM as a state of matter is distinct: In the bulk, it fea- n=1 tures strongly renormalized Weyl nodes instead of being fully 3 X gapped; on the surface, it hosts Fermi arcs from a band with a u2(k) = t sin(k · an), [9] width of the Kondo energy, which are to be contrasted with the n=1 surface Dirac nodes of the topological Kondo insulators. This also dictates that the WKSM will be realized in heavy-fermion Dx (k) = sin(k · a2) − sin(k · a3) − sin(k · (a2 − a1)) materials that are quite different from those for the topolog- + sin(k · (a3 − a1)), [10] ical Kondo insulators, as exemplified by the aforementioned and Dy , Dz are obtained by permuting the fcc primitive lattice vectors an. 3 4 3 ˘ † Ce Bi Pd . The canonical (unitary) transformation, Ψk = Sσ Ψk, leads to Finally, the emergence of a WKSM makes it natural for   X ˘ † hk+ 0 ˘ quantum-phase transitions in heavy-fermion systems from such Hc = Ψk Ψk, [11] 0 hk− a topological semimetal to magnetically ordered and other cor- k related paramagnetic states. Moreover, the existence of Weyl h = u (k)τ + u (k)τ + (m ± λD(k))τ . [12] nodes also enhances the effect of long-range Coulomb inter- k± 1 x 2 y z actions. While the density fluctuations of the f electrons are We have used a pseudospin basis (27), defined by the eigenstates |±Di with strongly suppressed in the local-moment regime, it still will be eigenvalues instructive to explore additional nearby phases such as charge- D · σ | ± Di = ± | ± Di [13] density-wave order (36); other types of long-range interactions D could produce topologically nontrivial Mott insulators (37). As where D(k) ≡ |D(k)|. The eigen energies of the | ± Di sectors are simply such, the study of WKSM and related semimetals promises to obtained to be shed new light on the global phase diagram of quantum critical q ετ = τ u (k)2 + u (k)2 + (m ± λD(k))2 [14] heavy-fermion systems (4) and other strongly correlated materi- ±D 1 2 als (38). where τ = (+, −). We use this transformation on the full Anderson model in In summary, we have demonstrated an emergent WKSM the strong coupling limit, at the saddle-point level where the Lagrange mul- tiplier ` , which enforces the local constraint b†b + P f † f = 1, takes a phase in a model of heavy-fermion systems with broken inver- i i i σ iσ iσ ` Ξ˘ = † Ξ sion symmetry and have determined the surface electronic spec- uniform value, . This corresponds to k Sσ k. Anticipating the separabil- ˘ T  ˘T ˘T  ˘T ˘T ˘T  tra which reveal Fermi arcs. The nodal excitations of the WKSM ity of the | ± Di sectors by specifying Ψk = ψk+, ψk− , Ξk = ξk+, ξk− ,     phase develop out of the Kondo effect. This leads to unique ˘T ˘ ˘ ˘T ˘ ˘ where ψk± = ψk±,A, ψk±,B and ξk± = ξk±,A, ξk±,B , we obtain the experimental signatures for such a phase, which are realized strong coupling Hamiltonian in a noncentrosymmetric heavy-fermion system. Our results are ! X   h − µ1 rV1  ψ˘ expected to guide the experimental search for f -electron-based Hs = ψ˘† ξ˘† ka 2 2 ka , [15] ka ka ˘ rV12 (Ed + `)12 ξ Weyl semimetals. In general, they open the door for studying k,a=± ka topological semimetals in the overall context of quantum phases s s s which separates as H = H++H−. We obtain the full spectra of the eight and their transitions in strongly correlated electron systems and, hybridized bands, conversely, broaden the reach of strongly correlated gapless and " r # 1  2 (τ,α) τ τ 2 quantum critical states of matter. E (k) = Es +ε ˜ + α Es − ε˜ + 4V , [16] ±D 2 ±D ±D s

τ τ Materials and Methods where α = (+, −) indexes the upper/lower quartet of bands, ε˜±D = ε±D −µ, The 3D Periodic Anderson Model. The three terms in the model of Eq. 1 are and (Es, Vs) = (Ed + `, rV). In Supporting Information, we prove that the presented here in more detail. The strongly correlated d electrons are spec- | + Di sector is always gapped, whereas the | − Di sector allows Weyl m 2 ified by nodes when 0 < 4|λ| < 1, and determine µ = −Vs /Es, which fixes the Fermi X † X energy at the Weyl nodes. H = E d d + U nd nd . [5] d d iσ iσ i↑ i↓ To determine r, `, Hs must be solved self-consistently from the saddle- i,σ i point equations The hybridization term is as follows: 1 X D˘† ˘ E 2 X  †  ξkaξka + r = 1, 2Nu Hcd = V diσ ciσ + H.c. . [6] k,a=± i,σ V X hD † E i ψ˘ ξ˘ + H.c. + ` = 0, In the above two equations, the site labeling i means i = (r, a), where r runs ka ka r [17] 4Nu over the Bravais lattice of unit cells and a runs over the two sites, a = A, B, k,a=±

in the unit cell. where Nu is the number of the unit cell. The equations are solved on a 64 × The conduction electron Hamiltonian Hc realizes a modified Fu– 64 × 64 cell of the diamond lattice, with error  ≤ O(10−5). = P Ψ† Ψ Kane–Mele model (26) and is expressed as Hc k k hk k, where T  ACKNOWLEDGMENTS. We acknowledge useful discussions with J. Analytis, Ψk = ck↑,A ck↑,B ck↓,A ck↓,B , and S. Dzsaber, M. Foster, D. Natelson, A. Prokofiev, B. Roy, F. Steglich, S. Wirth, hk = σ0 (u1(k)τx + u2(k)τy + mτz) + λ (D(k) · σ) τz. [7] and H. Q. Yuan. This work was supported in part by NSF Grants DMR- 1611392 (to H.-H.L. and Q.S.) and DMR-1350237 (to H.-H.L.); Army Research Here, σ = (σ , σ , σ ) and τ = (τ , τ , τ ) are the Pauli matrices acting on x y z x y z Office Grants W911NF-14-1-0525 (to S.E.G.) and W911NF-14-1-0496 (to S.P.); the spin and sublattice spaces, respectively, and σ0 is the identity matrix. Robert A. Welch Foundation Grant C-1411 (to S.E.G. and Q.S.); a Smalley In the first term, u1(k) and u2(k) are determined by the conduction elec- Postdoctoral Fellowship of Rice Center for Quantum Materials (to H.-H.L.); tron hopping, thiji = t between nearest-neighbor sites (hiji). The second and Austrian Science Funds Grant FWF I2535-N27 (to S.P.).

1. Coleman P, Schofield AJ (2005) Quantum criticality. Nature 433:226–229. 3. Aeppli G, Fisk Z (1992) Kondo insulators. Comm Cond Matter Phys 16:155–165. 2. Si Q, Steglich F (2010) Heavy fermions and quantum phase transitions. Science 4. Si Q, Paschen S (2013) Quantum phase transitions in heavy fermion metals and Kondo 329:1161–1166. insulators. Phys Status Solidi B 250:425–438.

96 | www.pnas.org/cgi/doi/10.1073/pnas.1715851115 Lai et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 3 ilyJ,LeS,BadmB aaewrnS 21)Fligefre nonsymmor- Filling-enforced (2017) SA Parameswaran Kondo B, Brandom anisotropic SB, Kondo Lee an topological JH, and Pixley in semimetals Dirac-Kondo 13. (2016) effect Q Si Nernst J, Dai isotropic H, Zhong X-Y, Giant Feng (2016) Kondo 12. the al. in et correlations U, antiferromagnetic and Stockert gap 11. Spin (1992) al. et TE, Mason 10. 2 ou ,Q 21)Rcn eeomnsi rnpr hnmn nWy semimet- three Weyl in in phenomena transport semimetals in Dirac developments Recent and (2013) X Weyl Qi P, (2017) Hosur A 22. Vishwanath EJ, points. Weyl Mele of NP, observation Experimental Armitage (2015) al. 21. et L, Lu 20. (2013) T Hughes insulators. B, topological Bernevig of Andrei birth The 17. (2010) J superconductors. Moore and 16. insulators Topological (2011) S-C Zhang insulators. X-L, Topological Qi Colloquium: 15. (2010) CL Kane MZ, Hasan 14. 9 vB,e l 21)Eprmna icvr fWy eiea TaAs. semimetal Weyl of discovery Experimental Fermi (2015) topological and al. semimetal et fermion BQ, Weyl Lv a of 19. Discovery (2015) al. et S-Y, Xu 18. a tal. et Lai .DsbrS ta.(07 od nuao osmmtltasomto ue yspin- by tuned transformation semimetal to insulator Kondo (2017) al. et S, Dzsaber Kondo 9. antiferromagnetic the in criticality quantum Pressure-tuned (2015) al. et Y, Luo 8. CeRu (2015) insu- al. et Kondo M, putative Sundermann the of 7. conductivity optical Anisotropic (2013) al. et insulators. V, Kondo Guritanu Topological (2010) 6. P Coleman V, Galitski K, Sun M, Dzero 5. hcKnosmmtl ntodimensions. two in semimetals Kondo phic arXiv:160502380. limit. carrier dilute the in insulators semimetal. cenisn. insulator coupling. orbit CeNi semimetal topology. CeRu lator Lett als. arXiv:170501111. solids. dimensional 5:031013. ductors 83:1057–1110. 3045–3067. arcs. ope edsPhysique Rendus Comptes 104:106408. Science PictnUi rs,Princeton). Press, Univ (Princeton c Rep Sci 4 hsRvLett Rev Phys Sn 349:613–617. 6 . hsRvLett Rev Phys 2−δ hsRvB Rev Phys hsRvLett Rev Phys 5:17937. As 2 . rcNt cdSiUSA Sci Acad Natl Proc 117:216401. 87:115129. 118:246601. 14:857–870. 69:490–493. 4 Sn oooia nuaosadTplgclSupercon- Topological and Insulators Topological 6 togycreae aeilwt nontrivial with material correlated strongly A : hsRvB Rev Phys 112:13520–13524. Nature 96:081105. 464:194–198. Science e o Phys Mod Rev 349:622–624. e o Phys Mod Rev hsRvX Rev Phys hsRev Phys 82: 4 ace ,e l 21)Aiorp fteKnoisltrCeRu Nikoli insulator Kondo 35. the of Anisotropy CeRu (2010) al. inCeRu et state of S, fermion Paschen Weyl Heavy 34. structure (2017) X Dai H, Electronic Weng C, Yue Y, Xu (2016) 33. K Held strongly- the in P, fermions Dirac (unexpected) Wissgott of form 32. new A CeSb. (2016) system al. Kondo et magnetic N, the Alidoust in fermions 31. Weyl canonical Possible a (2017) in al. fermions et Weyl C, (2017) Guo HQ Yuan 30. F, Steglich M, Smidman F, Wu CY, Guo 29. Weyl invariant (1993) time-reversal AC in Hewson states surface 28. and arcs dimensions. Fermi three Helical in (2013) insulators T Topological Ojanen (2007) 27. EJ Mele CL, Kane L, Fu 26. multilayer. insulator topological a in semimetal Weyl (2011) L and Balents semimetal AA, Burkov Topological (2011) 24. SY Savrasov A, Vishwanath AM, Turner X, Wan 23. 8 ica-rmaW hnG akKmY aet 21)Creae quantum Correlated (2014) L Balents Y, Kim Baek G, Chen W, insulator. Mott Witczak-Krempa Weyl (2016) 38. N Nagaosa T, semimet- Morimoto Weyl in phases 37. induced interaction Long-range (2014) V Aji S-P, Chao H, Wei 36. dimensions. three in semimetal Dirac (2012) al. et SM, Young 25. re fKnotplgclinsulators. topological Kondo of aries 200:012156. study. theory field 7:011027. mean dynamical 89:5. plus functional arXiv:160408571. monopnictides. cerium correlated Mater Quan arXiv:171005522. YbPtBi. semimetal heavy-fermion UK). Cambridge, semimetals. Lett 108:140405. Lett Rev iridates. pyrochlore of structure electronic 83:205101. the in states surface Fermi-arc hnmn ntesrn pnobtregime. spin-orbit strong the 5:57–82. in phenomena als. hsRvB Rev Phys 98:106803. 21)Todmninlhayfrin ntesrnl orltdbound- correlated strongly the on fermions heavy Two-dimensional (2014) P c ´ 107:127205. hsRvB Rev Phys 2:39. 89:235109. h od rbe oHayFermions Heavy to Problem Kondo The 87:245112. PNAS | aur ,2018 2, January hsRvB Rev Phys 90:235107. nuRvCnesMte Phys Matter Condens Rev Annu c Rep Sci | o.115 vol. 6:19853. CmrdeUi Press, Univ (Cambridge h u hsJB J Phys Eur The 4 Sn 4 6 Sn 4 . | Sn hsCn Ser Conf Phys J 6 hsRvLett Rev Phys density A : 6 o 1 no. . hsRvB Rev Phys hsRvX Rev Phys hsRev Phys | Phys NPJ 97

PHYSICS