Abstract
Topological semimetals can support onedimensional Fermi lines or zerodimensional Weyl points in momentum space, where the valence and conduction bands touch. While the degeneracy points in Weyl semimetals are robust against any perturbation that preserves translational symmetry, nodal lines require protection by additional crystalline symmetries such as mirror reflection. Here we report, based on a systematic theoretical study and a detailed experimental characterization, the existence of topological nodalline states in the noncentrosymmetric compound PbTaSe_{2} with strong spinorbit coupling. Remarkably, the spinorbit nodal lines in PbTaSe_{2} are not only protected by the reflection symmetry but also characterized by an integer topological invariant. Our detailed angleresolved photoemission measurements, firstprinciples simulations and theoretical topological analysis illustrate the physical mechanism underlying the formation of the topological nodalline states and associated surface states for the first time, thus paving the way towards exploring the exotic properties of the topological nodalline fermions in condensed matter systems.
Introduction
The discovery of the timereversal invariant topological insulator has stimulated an enormous research interest in novel topological states protected by different symmetries^{1,2,3}. One of the key properties of topological materials is the existence of symmetryprotected metallic edge or surface modes in bulkinsulating ground states, which is due to a topologically nontrivial ordering of bulk wave functions. Recently, because of the experimental observations of Weyl semimetals^{4,5,6,7,8,9,10,11,12,13,14}, the research interest in topological phenomena in condensed matter systems has partially shifted from insulating materials to semimetals and metals. A Weyl semimetal is a topological state of matter whose lowenergy bulk electrons are linearly dispersing Weyl fermions. The twofold degenerate Weyl nodes, carrying nonzero chiral charge, are connected on the boundary by Fermi arc surface states, which are predicted to exhibit unusual transport behaviours such as negative magnetoresistance and nonlocal transport current^{15,16,17,18}. In contrast to Weyl semimetals whose bulk Fermi surface has dimension zero, nodalline semimetals have extended band touching along onedimensional curves in k space, presenting a significant expansion of topological materials beyond topological insulators and Weyl semimetals, and new opportunities to explore exotic topological nodal physics. Linelike touchings of a conduction and valence band need extra symmetries besides translation, such as mirror reflection, to be topologically protected. Kinematically, this protection involves a finite fraction of Brillouin zone. For this reason, this leads potentially to many anomalies in electromagnetic and transport response^{19,20,21,22}. Similar to the case of Weyl nodes, one can define an integer topological invariant for the line node along which two nondegenerate bands touch^{19}. Despite the many theoretical discussions of nodalline semimetals, a material realization of topological nodalline fermions has been lacking for many years, just like Weyl semimetals.
In this work we performed systematic theoretical study and experimental characterization of the electronic structure of a spinorbit metal PbTaSe_{2}, indicating the existence of topological nodalline phase in this compound for the first time. The crystal lattice of this material lacks space inversion symmetry, which lifts the spin degeneracy of its electronic bulk bands. Our angleresolved photoemission (ARPES) measurements together with density functional theory (DFT) calculations show that the conduction band originated from Pb6p orbitals and the valence band from Ta5d orbitals cross each other, forming three nodalline states close to the Fermi energy. The nodal lines are protected by a reflection symmetry of the space group. The topological nodalline state in PbTaSe_{2} belongs to the symmetry class A+R (p=2) of symmetryprotected semimetals^{19}. We also demonstrate through effective Hamiltonian modelling and DFT simulations that the nodal lines are accompanied by surface bands. These surface states are due to the π Berry phase agglomerated around the nodal line in analogy to the states on the graphene zigzag edge. Our detailed theoretical modelling and calculation, aided by a systematic experimental characterization, establish the existence of topological nodalline fermions in the superconducting compound PbTaSe_{2}, opening the door for exploring the exotic properties of nodalline states in condensed matter.
Results
Crystal and electronic structure
Our PbTaSe_{2} single crystals were prepared by the chemical vapour transport method, see Fig. 1a. The samples were of high structural quality, which was confirmed by our Xray diffraction and scanning tunnelling microscopy (STM) measurements. The Xray diffraction peaks shown in Fig. 1b are consistent with the space group of PbTaSe_{2}, (187), therefore demonstrating the lack of inversion symmetry of our PbTaSe_{2} single crystals. This is crucially important for lifting the spin degeneracy, a necessary condition for the formation of topological nodal lines. To further check the chemical composition of our samples, we performed a photoemission corelevel scan. Clear Pb5d, Ta4f and Se3d corelevel peaks were observed in the photoemission spectrum, which confirms the correct chemical composition in our PbTaSe_{2} singlecrystal samples, shown in Fig. 1c. To verify the superconducting property of our samples, a transport measurement was carried out. The measured resistivity curve (Fig. 1d) shows a clear superconducting transition temperature at 3.8 K, consistent with the value reported in ref. 23. Figure 1e,f shows STM images of the cleaved (001) surface. The topography image clearly reveals a hexagonal lattice with few defects, demonstrating the high quality of our samples. Furthermore, no surface reconstruction was observed on the cleaved surface. The highresolution STM topography yields a lattice constant of 3.2 Å.
PbTaSe_{2} crystalizes in a hexagonal lattice system in which the unit cell consists of one Pb, one Ta and two Se atoms, and each atom resides on a hexagonal flat layer, shown in Fig. 2a. The stacking sequence of these atomic planes within the unit cell is PbSeTaSe: AABA (A, B and C, here refer to the three highsymmetry spots on a hexagonal lattice). The lattice can also be viewed as a Pb layer intercalating two adjacent TaSe_{2} layers with Pb atoms sitting above Se atoms. The Pb intercalation suppresses the softening of phonon modes associated with the charge density wave in TaSe_{2} and stabilizes the hexagonal lattice on the surface^{24}. This particular stacking does not preserve the space inversion symmetry; however, the lattice is reflectionsymmetric with respect to the Ta atomic plane. In other words, the Ta atomic planes are a mirror plane of the crystal lattice under the mirror operation R_{z} that sends z to −z. This reflection symmetry of the lattice provides a protection for the topological nodal lines, as discussed later on. The bulk and (001)projected surface Brillouin zones are shown in Fig. 2b. The A, H and L points are highsymmetry points on the k_{z}=π plane, which is a mirror plane of the bulk Brillouin zone. Figure 2c presents an overview of the band structure calculation for PbTaSe_{2}, which was performed by the method of generalized gradient approximation. Close to the Fermi level, two prominent features in the band structure are observed. One is a giant hole pocket around Γ, whose states are mainly derived from the Ta orbitals that are oriented out of the Ta atomic plane, taking the Ta plane as the x–y plane. The second major contribution to the density of states at the Fermi level comes from the four bands that cross each other near H. The two electronlike conduction bands originate from Pb6p_{x}/p_{y} orbitals and the two holelike valence bands from orbitals. We note that all these orbitals are invariant under R_{z}. A zoomin view of the bands around H without/with spinorbit coupling (SOC) is shown in Fig. 2d,e, respectively. Without the inclusion of SOC, the conduction and valence bands become spindegenerate. The two bands belong to different representations of the space group (the representation of the electronlike band is A′ and that of the holelike band is A′′); therefore, the intersection of the two bands is protected by the crystalline symmetry, forming a spinless nodal ring. Once SOC is turned on, each band split into two spin branches with opposite spin orientations and mirror reflection eigenvalues as indicated in Fig. 2e. Only the crossings of branches with opposite mirror reflection eigenvalues remain gapless as a result of symmetry protection, forming a pair of nodal rings. Interestingly, SOC also gives rise to a third nodal ring on k_{z}=0 plane. The detailed band dispersion and the rise of three nodal rings are very well captured by our effective k·p Hamiltonian (Supplementary Note 1 and Supplementary Fig. 1). Before proceeding to a detailed discussion of the nodalline states, we will present the results of our ARPES measurement, verifying the overall band dispersion of Pb conduction bands and Ta valence bands obtained from our DFT calculation.
ARPES and DFT results of PbTaSe_{2}
Figure 3a shows a brief overview of our ARPES band mapping, and the corresponding numerical calculation of the PbTaSe_{2} band structure is presented in Fig. 3b. The projected bulk bands and surface bands (as highlighted by white lines) were calculated for the Pbterminated (001) surface. The DFT band structure reproduces the ARPES spectrum very well. Specifically, in the ARPES spectral cut, a band marked as SS_{1} with high intensity poke the Fermi level between and . This is the surface band associated with the Pbterminated (001) surface and it corresponds to Dirac surface mode arising from a continuous bulk band gap opened by SOC^{23,24}. Around there are three concave bands whose binding energies at are 0.21, 0.75 and 0.80 eV. The top and bottom bands correspond to the electronlike bands derived from Pb6p orbitals. The middle band, marked as SS_{2}, is consistent with the surface band as plotted in Fig. 3b. The two bands at are tails of the two Ta5d bands that cross the two Pb6p bands forming the nodal rings in the vicinity of . Two Ta5d bands have to degenerate in energy at according to the Kramers theorem. The ARPES measured (001) Fermi surface with the incident photon energy of 64 eV, and the theoretical simulations are shown in Fig. 3c,d. At the Fermi level, our data show that the Fermi surface consists of three parts: a hexagonshaped pocket centred at with smeared intensity inside, a dogboneshaped contour centred at the point and several circles surrounding the point. Our ARPES data and calculation show an agreement on those features. Furthermore, the hexagon centred at and the intensity inside are the surface band and the bulk hole pocket at , respectively. The dogboneshaped contour corresponds to the one branch of the Ta valence band and the circles around are from the other branch of the Ta valence band; the surface states and the spinsplit conduction band are derived from Pb orbitals. As the binding energy decreases, we find that the Pb pockets at shrink while the Ta pockets expand outwards, which is in good accordance with the characteristics of the electronlike Pb bands and holelike Ta bands.
TaSe_{2} can be regarded as a building block of PbTaSe_{2}, and therefore its electronic structure can be traced from that of PbTaSe_{2}. To highlight the difference between electronic structures of TaSe_{2} and PbTaSe_{2}, we mapped out the Fermi surface and band structure along −− of the two compounds, shown in Fig. 4a,b. In the Fermi surface mapping of TaSe_{2}, there are one dogboneshaped contour centred at and only one circleshaped contour centred at . Those contours are from Ta valence bands, and are consistent with previous work^{25}. By contrast, the Fermi surface of PbTaSe_{2} has more ringshaped contours centred at , signifying the contribution from the Pb layers. It is easier to view this difference from the −− cut. TaSe_{2} does not show any electronlike bands at that exist in PbTaSe_{2}. Figure 4c shows the ARPES mapping of the Pb and Ta bands of PbTaSe_{2} with photon energies from 54 to 70 eV. The middle band at does not show any photonenergy dependence, which is consistent with the surface nature of this band. However, the other bands at and do not exhibit obvious changes with different photon energies either. This seems to contradict to the assignment of those bands as bulk bands according to our DFT calculations. The inconsistency can be understood by considering the fact that the Pb6p_{x}/p_{y} and orbitals that constitute those bands are primarily confined within the Pb and Ta atomic planes (which are parallel to the x–y plane), and, thus, the interlayer couplings (say, the coupling of one Pb6p_{x}/p_{y} orbital with another orbital on the adjacent Pb plane) is largely suppressed, which results in little k_{z}/photonenergy dependence. In addition, we mapped out the ARPES spectrum of PbTaSe_{2} along − with photon energy from 105 to 135 eV, shown in Fig. 4d. We observed that the intensity from the holelike bulk pocket around Γ varied prominently with photon energy. It can be attributed to the fact that the states on the hole pocket are mainly derived from the Ta orbitals that are oriented out of the Ta atomic plane and, thus, exhibit strong dispersion along the k_{z} direction. By contrast, the band marked as SS_{1} did not show any photonenergy dependence, indicative of the surface nature of this band. We note that both our theoretical calculation and our ARPES measurement unambiguously indicate the existence of the electronlike Pb6p bands around , which inevitably cross the holelike Ta5d bands with a similar energy, leading to the formation of the nodal rings. The remarkable consistency between ARPES result and the our firstprinciples calculation lay a solid foundation for our theoretical investigation of the topological nodal lines in PbTaSe_{2}.
Nodal lines and drumhead surface states of PbTaSe_{2}
From the discussion before, we know that the electronlike bands from the intercalated Pb layers are the essential components for forming the topologicalnodalring band structure. By comparing with the TaSe_{2} spectrum, our ARPES established unambiguously the existence of the Pb bands. To further examine the topological nodalline states and associated surface states, we calculated the band structure for Pb and Seterminated surfaces as shown in Fig. 5a–d. The projected bulk band on each cut from shows three nodal points at 0.05, 0.15 and 0.03 eV above the Fermi level. The first two closer to lie on the k_{z}=π plane, while the third one is on the k_{z}=0 plane. Let us refer to these three nodal lines as NL1, NL2 and NL3. Corresponding nodal points can be found on a cut of arbitrary orientation that includes . For example, the band structure along a generic direction − is shown in Fig. 5b,d. Unlike the projected bulk band, which is independent of surface termination, the dispersion of surface bands is found to be sensitive to the surface condition. However, in both cases we do find a surface band connecting to each nodal line, indicative of the topological nature of the bulk nodal lines. In the Pbtermination case, the surface bands disperse outwards with respect to , from NL1. The surface band connecting to NL2 grazes inwards at the edge of the lower bulk Dirac cone and merge into the bulk band. The surface band from NL3 disperses inwards with respect to , consistent with the SS_{2} band in our ARPES spectrum in Fig. 3, which forms a drumhead surface state contour. By contrast, on the Seterminated surface the surface band connecting to NL2 first moves into the bulk band gap and then fall into the bulk band region. The surface band from NL1 disperses outwards and connects to NL3. Please refer to Supplementary Fig. 2 and Supplementary Note 2 for a detailed visualization of connection of the surface bands to the bulk nodal lines. To get an overall view of the nodal ring and surface band, we plot in Fig. 5e,f the isoenergy contour in the vicinity of the NL1 nodal ring of the Pbterminated surface and the NL2 nodal ring of the Seterminated surface, as indicated by the red dashed lines in Fig. 5a–d. Indeed, gapless nodal points and surface states can be found at every inplane angle departing from . These nodal rings are protected against gap opening by the crystalline symmetry. Specifically, the states in the two Pb branches belong to two different representations of the space group, namely S_{3} and S_{4} as shown in Fig. 5g. The same is true for the two Se branches. In particular, with respect to the Ta atomic plane, the two representations have opposite mirror eigenvalues under the reflection operation. Therefore, gap opening is forbidden at the crossing point between two branches of different mirror eigenvalues, which results in the nodal rings discussed in this work. In this sense, the nodal rings are under the protection of the reflection symmetry. If we shift the Pb atom slightly in the vertical direction, thus breaking the reflection symmetry, all of the four branches are found to belong to the same S_{2} representation of the reduced space group and, in this case, a gap opening is allowed at every crossing point of these branches as illustrated in Fig. 5h. A similar gap opening is also found in NL3 on k_{z}=0 plane on breaking the reflection symmetry, please see Supplementary Fig. 3 and Supplementary Note 3.
Discussion
Let us briefly discuss the topological characterization of the nodal lines and the origin of the surface bands. The material PbTaSe_{2} is timereversalsymmetric, with timereversal symmetry represented by , where denotes complex conjugation and σ_{2} is the second Pauli matrix acting on the electron spin. The mirror symmetry R_{z} acts in spin space as iσ_{3} and therefore commutes with . This would place PbTaSe_{2} in class AII in the classification of ref. 19. However, since the nodal lines are centred around momenta H/H′ and K/K′, which are not invariant under timereversal, but pairwise map into each other, the timereversal symmetry imposes no constraints on the nodal lines individually. The material has, therefore, to be classified according to the timereversal breaking class AR, which admits an integer topological classification for Fermi surfaces of codimension 2, that is, lines (p=2 in ref. 19). The nodal lines carry a topological quantum number n^{+}, which is given by the difference in the number of occupied bands with R_{z} eigenvalue +i inside and outside the line. In the case at hand, n^{+}=−1 for the nodal line (NL3) in the k_{z}=0 plane, while n^{+}=+1 (NL1) and n^{+}=−1 (NL2) for the two nodal lines in the k_{z}=π plane. We have also computed numerically under the DFT framework the winding number , where A(k)=i∑_{a}〈u_{a,k}∇u_{a,k}〉 is the Berry connection of the occupied Bloch bands u_{a,k}〉. For a closed loop encircling any of the nodal lines, we find that γ=±π, indicating the topological protection of the nodal line as shown in Fig. 5i.
The topological origin of the observed surface states is rather subtle. Surface states associated with the topological invariant n^{+} via the bulkboundary correspondence should only appear on surfaces that preserve R_{z}. The (001) surface, however, breaks R_{z}. The reason why we still observe surface states can be understood from the Berry phase of π around the nodal line and the analogy to the edge states on the zigzag edge of graphene. Consider the bulk Hamiltonian on a plane in momentum space that contains both K and H. At low energies, each nodal ring pierces this plane twice, giving rise to two Dirac cones in this Hamiltonian. These two cones have a Berry phase +π and −π with respect to the orientation of the plane, precisely as in (spinless) graphene, as schematically depicted in Fig. 5j. We know that for any termination of a graphene sample, an edge state emanates from the projection of each of the Dirac points in the edge Brillouin zone (except for the pathological case where both Dirac points project on the same spot in the edge Brillouin zone). By this analogy, we also expect these edge states to emanate from the surface projections of the nodal lines in any direction away from the point, thereby forming a surface band. We note that the dispersion of this band and even whether it appears inside or outside the projection of the nodal line is not universal and depends on the details of the surface termination.
Recently, some preprint theoretical lines of work have reported on Ca_{3}P_{2} and Cu_{3}PdN, proposing that there may be nodalline states^{26,27,28}. We note that our work is distinct from those lines of work in many ways. Both Ca_{3}P_{2} and Cu_{3}PdN are centrosymmetric and, therefore, because of the coexistence of timereversal and inversion symmetry, have fourfold degeneracy at the nodal ring. By contrast, the degeneracy of nodalring states is two in PbTaSe_{2} because of the lack of inversion symmetry. In Ca_{3}P_{2} and Cu_{3}PdN, the nodalline states exist only in the absence of SOC. In real materials SOC, however, always exists. In PbTaSe_{2}, SOC is an essential ingredient for the formation of nodalring states. Our ARPES measurement established an experimental characterization of the topological nodalline material PbTaSe_{2}.
In summary, our direct experimental observation by ARPES of the coexistence of Pb concave bands and Ta convex bands centred at in the noncentrosymmetric superconductor PbTaSe_{2} is in good agreement with our firstprinciples band structure calculations, establishing the realization of the unusual ringshaped topological nodalline states in this compound. The topological nodal rings are protected by the reflection symmetry of the system. Meanwhile, the nodal rings are uniquely associated with drumheadlike surface states in a manner that resembles the connections of edge states and the nodal points in graphene. Considering the onedimensional nodal characteristics of the bulk band and the twodimensional topological ‘drumhead’ surface states, topological nodalline semimetals stand as a distinct class of topological materials beyond Weyl semimetals and topological insulators. For example, nodalline states possess an extra degree of freedom for manipulating novel properties of Weyl materials, which is the finite size of the nodal line. Furthermore, interactioninduced instabilities that have been broadly discussed for Weyl semimetals should be more likely occurring in nodalline states because of the higher density of states at the Fermi energy. In addition, superconductivity is induced by intercalating Pb layers to TaSe_{2}, which also offers the essential ingredient, the Pbconducting orbitals, for the formation of the topological nodalline states. Considering the intrinsic superconductivity, the spinsplit bulk nodalline band structure and the nontrivial surface states close to the Fermi level, it is possible that helical superconductivity and pwave Cooper pairing may exist in this compound without the aid of the proximity effect^{29,30}. Therefore, novel physics may arise from the interplay of the nodalline states and the emergent superconductivity of PbTaSe_{2}, which calls for further experimental investigation on PbTaSe_{2}. Our ARPES measurements, detailed DFT simulation and theoretical analysis demonstrate the fundamental mechanism for realizing topological nodalline fermions in PbTaSe_{2}, and pave the way for exploring the exotic properties of topological nodalline states in condensed matter systems.
Methods
Sample growth method
Single crystals of PbTaSe_{2} were grown with the chemical vapor transport (CVT) method using chlorine in the form of PbCl_{2} as a transport agent. For the pure synthesis of PbTaSe_{2}, stoichiometric amounts of the elements (purity of Pb and Ta: 6N, of Se: 5N) were loaded into a quartz ampoule, which was evacuated, sealed and fed into a furnace (850 °C) for 5 days. About 10 g of the prereacted PbTaSe_{2} were placed together with a variable amount of PbCl_{2} (purity 5 N) at one end of another silica ampoule (length 30–35 cm, inner diameter 2.0 cm and outer diameter 2.5 mm). All treatments were carried out in an Argon box, with continuous purification of the Argon atmosphere resulting in an oxygen and water content of less than 1 p.p.m. Again, the ampoule was evacuated, sealed and fed into a furnace. The end of the ampoule containing the prereacted material was held at 850 °C, while the crystals grew at the other end of the ampoule at a temperature of 800 °C (corresponding to a temperature gradient of 2.5 K cm^{−1}) during a time of typically 1 week. Compact single crystals of sizes of up to 8 × 5 × 5 mm^{3} were obtained.
ARPES and STM methods
ARPES measurements were performed at the liquid nitrogen temperature in the beamline I4 at the MAXlab in Lund, Sweden. The energy and momentum resolution was better than 20 meV and 1% of the surface Brillouin zone for ARPES measurements at the beamline I4 at the MAXlab. Samples were cleaved in situ under a vacuum condition better than 1 × 10^{−10} torr. Samples were found to be stable and without degradation for a typical measurement period of 24 h. STM experiments were conducted with a commercial system (Unisoku). Samples were cleaved at room temperature in a vucuum better than 2 × 10^{−10} mbar and were transferred to a STM head precooled to 77 K. Constantcurrent mode STM images were taken with chemicaletched Pt/Ir tips. Bias voltages were applied to the samples.
Computational method
We computed electronic structures using the normconserving pseudopotentials as implemented in the OpenMX package within the generalized gradient approximation schemes^{31,32}. Experimental lattice constants were used^{33}. A 12 × 12 × 4 MonkhorstPack kpoint mesh was used in the computations. The SOC effects are included selfconsistently^{34}. For each Pb atom, three, three, three and two optimized radial functions were allocated for the s, p, d and f orbitals (s3p3d3f2), respectively, with a cutoff radius of 8 Bohr. For each Ta atom, d3p2d2f1 was adopted with a cutoff radius of 7 Bohr. For each Se atom, d3p2d2f1 was adopted with a cutoff radius of 7 Bohr. A regular mesh of 300 Ry in real space was used for the numerical integrations and for the solution of the Poisson equation. To calculate the surface electronic structures, we constructed firstprinciples tightbinding model Hamilton. The tightbinding model matrix elements are calculated by projecting onto the Wannier orbitals^{35}. We use Pb p, Ta s and d, and Se p orbitals were constructed without performing the procedure for maximizing localization.
Additional information
How to cite this article: Bian, G. et al. Topological nodalline fermions in spinorbit metal PbTaSe_{2}. Nat. Commun. 7:10556 doi: 10.1038/ncomms10556 (2016).
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Acknowledgements
Work at Princeton University and Princetonled synchrotronbased ARPES measurements were supported by the US Department of Energy (DOE)/Basic Energy Sciences under DEFG0205ER46200. We gratefully acknowledge C.M. Polley, J. Adell, M. Leandersson and T. Balasubramanian for their beamline assistance at the Maxlab. We also thank A.P. Schnyder, C. Fang and M. Franz for discussions. The work at the Northeastern University was supported by the USDOE, Office of Science BES grant number DEFG0207ER46352, and was benefited from Northeastern University’s Advanced Scientific Computation Center (ASCC) and the NERSC supercomputing centre through DOE grant number DEAC0205CH11231. S.J. acknowledges National Basic Research Program of China under Grant Nos 2013CB921901 and 2014CB239302. H.L. acknowledges the Singapore National Research Foundation for the support under NRF Award No. NRFNRFF201303. F.C. acknowledges the support provided by the Ministry of Science and Technology in Taiwan under project number MOST1022119M002004. R.S. and F.C. acknowledge the support provided by the Academia Sinica research programme on nanoscience and nanotechnology project number NM004. T.R.C. and H.T.J. were supported by the National Science Council, Taiwan. H.T.J. also thanks the National Center for HighPerformance Computing, Computer and Information Network Center National Taiwan University and National Center for Theoretical Sciences, Taiwan, for technical support. Sample characterization was supported by the Gordon and Betty Moore FoundationsEmergent Phenomena in Quantum Systems Initiative through grant GBMF4547 (M.Z.H.).
Author information
Author notes
 Guang Bian
 , TayRong Chang
 & Raman Sankar
These authors contributed equally to this work.
Affiliations
Department of Physics, Laboratory for Topological Quantum Matter and Spectroscopy (B7), Princeton University, Princeton, New Jersey 08544, USA
 Guang Bian
 , TayRong Chang
 , SuYang Xu
 , Hao Zheng
 , Titus Neupert
 , Ilya Belopolski
 , Daniel S. Sanchez
 , Madhab Neupane
 , Nasser Alidoust
 , Chang Liu
 & M. Zahid Hasan
Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan
 TayRong Chang
 & HorngTay Jeng
Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan
 Raman Sankar
 & Fangcheng Chou
Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA
 Titus Neupert
Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
 ChingKai Chiu
Centre for Advanced 2D Materials and Graphene Research Centre National University of Singapore, 6 Science Drive 2, Singapore 117546, Singapore
 ShinMing Huang
 , Guoqing Chang
 , BaoKai Wang
 , ChiCheng Lee
 & Hsin Lin
Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore
 ShinMing Huang
 , Guoqing Chang
 , BaoKai Wang
 , ChiCheng Lee
 & Hsin Lin
Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA
 BaoKai Wang
 & Arun Bansil
Institute of Physics, Academia Sinica, Taipei 11529, Taiwan
 HorngTay Jeng
ICQM, School of Physics, Peking University, Beijing 100871, China
 Chenglong Zhang
 , Zhujun Yuan
 & Shuang Jia
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Contributions
G.B., S.Y.X., H.Z. and I.B. performed the experiments with assistance from D.S.S., C.L., N.A., M.N. and M.Z.H.; R.S., F.C., Z.Y., C.Z. and S.J. provided samples; T.R.C., G.B., H.T.J., S.M.H., G.C., B.W., C.C.L., H.L. and A.B. carried out the firstprinciples calculations; T.N., C.K.C., G.B. and M.Z.H. performed the theoretical modelling and analysis. M.Z.H. was responsible for the overall direction, planning and integration among different research units.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to M. Zahid Hasan.
Supplementary information
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Supplementary Information
Supplementary Figures 13 and Supplementary Notes 13
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