Unusual change in the Dirac-cone energy band upon two-step magnetic transition in CeBi
Hikaru Oinuma, Seigo Souma, Kosuke Nakayama, Koji Horiba, Hiroshi Kumigashira, Makoto Yoshida, Akira Ochiai, Takashi Takahashi, Takafumi Sato
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Unusual change in the Dirac-cone energy band upon two-step magnetic transition inCeBi
Hikaru Oinuma, Seigo Souma, , Kosuke Nakayama, Koji Horiba, Hiroshi Kumigashira, , Makoto Yoshida, Akira Ochiai, Takashi Takahashi, , , and Takafumi Sato , , Department of Physics, Tohoku University, Sendai 980-8578, Japan Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan WPI Research Center, Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan Institute of Materials Structure Science, High Energy AcceleratorResearch Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan Institute of Multidisciplinary Research for Advanced Materials (IMRAM), Tohoku University, Sendai 980-8577, Japan Max-Planck-Institute for Solid State Research, Heisenbergstrsse 1, 70569 Stuttgart, Germany (Dated: October 3, 2019)We have performed angle-resolved photoemission spectroscopy (ARPES) on CeBi which undergoesa two-step antiferromagnetic (AF) transition with temperature. Soft-x-ray ARPES has revealed theinverted band structure at the X point of bulk Brillouin zone for CeBi (and also for LaBi) as opposedto LaSb with non-inverted band structure. Low-energy ARPES on CeBi has revealed the Dirac-coneband at the ¯Γ point in the paramagnetic phase associated with the bulk band inversion. On theother hand, a double Dirac-cone band appears on entering the first AF phase at T = 25 K, whereasa single Dirac-cone band recovers below the second AF transition at T = 14 K. The present resultsuggests an intricate interplay between antiferromagnetism and topological surface states in CeBi. PACS numbers: 71.20.-b, 73.20.At, 79.60.-i
I. INTRODUCTION
One of intriguing challenges in heavy-fermion physicsis to understand the relationship between magnetism andelectronic structure associated with the interplay of con-duction and f electrons. Cerium monopnictides CeX p (X p = P, As, Sb, and Bi) are a typical example of heavy-fermion systems showing an exotic and complicated mag-netic phase diagram as a function of temperature, pres-sure, and magnetic field, known as a “devil’s staircase”[1–4]. CeX p is a Kondo semimetal with the rock-saltstructure having two hole pockets at the Γ point in thebulk fcc Brillouin zone (BZ) arising from the X p p or-bital, together with an electron pocket at the X pointoriginating from the Ce 5 d orbital [see Fig. 1(a)] [5].Ce 4 f electrons are nearly localized in the crystal lat-tice (Ce ) and thereby the overall electronic structureof CeX p is essentially the same as that of LaX p with no4 f electrons [5, 6].Unlike the case of RKKY(Ruderman-Kittel-Kasuya-Yosida)-type magnetic order in rare-earth metals, thecomplex magnetic properties of CeX p , in particularCeSb, is explained in terms of the p-f mixing [7] whichis based on the strong hybridization between the X p p state and the crystal-field-split level of the Ce 4 f state[8], as supported by the observation of an energy shift ofthe p band in angle-resolved photoemission spectroscopy(ARPES) of CeSb [9–12] and the change in the Fermi sur-face in the de Haas-van Alphen experiment [13] across themagnetic transition. As a result of strong hybridization,the Ce 4 f state shows ferromagnetic coupling within thelayer and leads to the complicated magnetic structurestabilized by a subtle balance between the entropy andthe interlayer exchange coupling [14–16]. Such p-f mixing and the electron correlation are thought to be responsi-ble for the observed anomalous behavior in the electricaltransport and lattice parameters [4, 17–19] as well as therich magnetic properties, making the physics of CeX p particularly fertile.Recently, Zeng et al . predicted from the first-principlesband-structure calculations [20] that LaX p becomes atopological insulator due to the band inversion at the X point of bulk BZ. This report has renewed the inter-est for RX p (R = La and Ce) systems in topological as-pects and triggered intensive theoretical and experimen-tal investigations [12, 21–48], resulting in the discoveryof extremely large magnetoresistance and unusual resis-tivity plateau in LaSb and LaBi [23–30], as well as theobservation of Dirac-cone-like energy band in some RX p compounds [38–48]. While RSb was turned out to betopologically trivial [37, 41, 43, and 46], ARPES studiesof LaBi [37, 39, 40, 42, and 44] have commonly revealedthe Dirac-cone-like energy band of topological origin atthe ¯Γ and ¯ M point of the surface BZ, associated with theband inversion at the X point [note that the ¯ M point isequivalent to the projected bulk X point; see Fig. 1(a)].Taking into account such magnetic and topological as-pects of RX p family, one would naturally expect thatCeBi is a unique candidate to study the interplay betweenmagnetism and topological properties, since it shows in-teresting magnetic phases characterized by a two-step an-tiferromagnetic (AF) transition at T = 25 and 14 K un-der zero-magnetic field [3, 4], in addition to the expectedtopological nature. In a broader perspective, it is of greatimportance to experimentally clarify the role of antifer-romagnetism to the topological properties, which is cur-rently a target of intensive debates in theories [49–61]while no concrete experimental data have been hithertoreported.In this article, we report the soft-x-ray (SX) andvacuum-ultraviolet (VUV) ARPES study of CeBi. Wefound that the paramagnetic phase of CeBi is character-ized by a single Dirac-cone energy band at the ¯Γ pointoriginating from the band inversion at the X point in bulkBZ, whereas the energy dispersion of the Dirac-cone bandis strongly reconstructed in the AF phase; surprisingly italso depends on the types of magnetic structures. Wediscuss the implications of our observation in relation tothe magnetic band folding and the symmetry of the AFphase. II. EXPERIMENTAL
CeBi single crystal and its reference materials LaBiand LaSb were grown by the Bridgman method witha tungsten heater furnace. High-purity starting mate-rials of La/Ce (4N) and Sb/Bi (6N) with the ratio of1:1.005 were sealed in a tungsten crucible using an elec-tron beam welder. The crucible was heated above theirmelting points and then slowly pulled down from theheater. The obtained crystals were characterized by thex-ray diffraction measurements. SX-ARPES measure-ments were performed with a Scienta-Omicron SES2002electron analyzer with energy-tunable synchrotron lightat BL2 in Photon Factory (PF), KEK. We used linearlypolarized light (horizontal polarization) of 400-630 eV.VUV-ARPES measurements were performed with a MBSA1 electron analyzer with the Xe discharge lamp at To-hoku University. We used the Xe-I line (8.437 eV). Theenergy resolutions for SX- and VUV-ARPES measure-ments were set to be 150 and 5 meV, respectively. Sam-ples were cleaved in situ in an ultrahigh vacuum of ∼ × − Torr along the (100) crystal plane. The Fermilevel ( E F ) of samples was referenced to that of a gold filmevaporated onto the sample holder. III. RESULTS AND DISCUSSION
First, we present the overall energy band structure ofthe bulk valence band. To discuss the electronic statesof CeBi in terms of the topology, it is essential to clarify(i) the role of spin-orbit coupling (SOC) and (ii) the in-fluence of Ce 4 f electrons to the energy bands. Sincethe bulk-band inversion at the X point is thought tobe directly linked to the topological nature of this sys-tem [20, 37, 39–44, 46, and 48], we experimentally de-termined the bulk-band structure passing through the X point (along the Γ X cut) of bulk 3D BZ [see Fig. 1(a)]with SX photons ( hν ∼
500 eV), and compa the resultsbetween LaSb and LaBi to address the issue (i) since thereplacement of Sb with Bi leads to a stronger SOC due tothe heavier atomic mass of Bi. In addition, we comparedLaBi ( f system) and CeBi ( f system) to address theissue (ii). The result for LaSb shown in Fig. 1(b) signi- FIG. 1. (color online). (a) Bulk and surface BZs of RX p together with the k cut (red solid line) where the ARPES datafor (a)-(c) were obtained. (b) Second-derivative-intensity plotof EDC for LaSb at T = 30 K measured along the Γ X cut inbulk BZ with SX photons. (c), (d) Same as (b) but for LaBiand CeBi, respectively. We estimated the photon energy thattraces the Γ X cut to be hν = 510, 500, and 505 eV, for LaSb,LaBi, and CeBi, respectively, by the normal-emission ARPESmeasurements. Red dashed curves in (a)-(c) are guides for theeyes to trace the band dispersion. White arrows in (c) and(d) highlight the intersection of Bi 6 p and La/Ce 5 d bands.Non-dispersive feature at E B ∼ f final state [62, 63]. fies an electronlike La 5 d band crossing E F around the X point and several holelike Sb 5 p bands at the Γ point. Itis noted that these bands are well separated from eachother around the X point due to the absence of band in-version [43]. On the other hand, in LaBi [Fig. 1(c)], theLa 5 d and Bi 6 p bands cross each other midway betweenthe X and Γ points (marked by white arrows), in sup-port of the inverted band structure. We found that sucha band inversion is also resolved in CeBi consistent withthe previous study [46], although there exist some quan-titative differences between CeBi and LaBi, e.g. , the Bi6 p band around E F is flatter in CeBi than in LaBi, lead-ing to the movement of the band-crossing point towardthe Γ point in CeBi.To clarify a possible link between the bulk-band in-version and the appearance of topological surface states(SS), we have performed high-resolution ARPES mea-surements with VUV photons (the Xe-I line; hν = 8.437eV) around the ¯Γ point where a distinct Dirac-cone-likeenergy band was resolved in previous ARPES studies onLaBi and CeBi [38, 40, 42, 44, 46–48]. As seen fromthe ARPES-intensity plot of LaSb [Fig. 2(a)], no bandsare observed inside the inner Sb 5 p band crossing E F .In a sharp contrast, in LaBi, there exists an X-shapedDirac-cone-like band besides the holelike Bi 6 p band [Fig.2(b)]. This is better visualized in the second-derivative-intensity plot of momentum distribution curves (MDCs) FIG. 2. (color online).(a), (b) ARPES intensity of LaSb andLaBi, respectively, measured around the ¯Γ point at T = 30 Kwith the Xe-I line ( hν = 8.437 eV). (c), (d) Second-derivative-intensity plot of MDCs for LaBi and CeBi, respectively. shown in Fig. 2(c). Such a crucial difference betweenLaSb and LaBi strongly suggests that the band inversionand the resultant change in the bulk-band topology areindeed related to the appearance of the Dirac-cone SS.This is also collaborated with the observation of a simi-lar Dirac-cone SS in the paramagnetic phase ( T = 30 K)in CeBi, as displayed in Fig. 2(d) (note that the spec-tral feature of CeBi is broader due to the poorer surfacequality).Now we proceed to our most important finding, astriking temperature dependence of the topological SSin CeBi. We systematically performed a temperature-dependent ARPES measurement across the two-step AFtransition at T AFI (= 25 K) and T AFII (= 14 K) [see Fig.3(c) for the corresponding magnetic structure [3, 4]]. Asshown in Fig. 3(a), the energy distribution curve (EDC)at the ¯Γ point in the paramagnetic phase ( T = 30 K)consists of a single peak around the Dirac-point energyat ∼ E B ∼ ∼ T = 24 K ( ∼ T AFI ), but they aretransformed again into a single peak below T = 13 K( ∼ T AFI ) at ∼ ∼ T = 30 K in the paramagneticphase [Fig. 4(a)], there exit two Dirac-cone-like bands at FIG. 3. (color online). (a) Temperature dependence of EDCat the ¯Γ point for CeBi measured with the Xe-I photons ( hν =8.437 eV) across the two-step magnetic transition at 25 and14 K. EDCs at around T AFI and T AFII are highlighted byred color. (b) Second-derivative-intensity plot of EDC atthe ¯Γ point as a function of temperature. Note that thetemperature-independent flat feature near E F is due to theFermi-edge cut-off. (c) Schematic view of magnetic structurein the (top) AF-I and (bottom) AF-II phases (the case fortype-F domain). T = 19 K in the AF-I phase [Fig. 4(b)] which are energet-ically separated from each other by ∼ T = 6K in the AF-II phase [Fig. 4(c)], the two Dirac-conebands again disappear and a single Dirac-cone band re-covers. In contrast, the holelike Bi-6 p bulk band shows nodiscernible band reconstruction even across T AFII (notethat this band is gradually pushed up on lowering tem-perature likely due to p-f mixing as reported in CeSb[9–12]; see Appendix A for detailed temperature depen-dence of the bulk bands). To the best of our knowledge,the present result is the first experimental observation ofthe antiferromagnetism-induced drastic reconstruction ofthe Dirac-cone SS.Now we discuss the origin of observed intriguing changein the Dirac-cone band dispersion. The appearance of asingle Dirac-cone band at the ¯Γ point in the paramagneticphase is reasonable since a single bulk X point is pro-jected onto the surface ¯Γ point as shown by a schematicband diagram in the k x - k z plane in Fig. 4(d). In thiscase, the band inversion takes place just once, resultingin a single Dirac-cone band at the ¯Γ point. In the AF-Iphase, the magnetic moment of Ce 4 f electron aligns fer-romagnetically in a single Ce layer as shown by red orblue shade in Fig. 4(h), but aligns antiferromagneticallybetween adjacent layers (compare blue and red shades).Under zero magnetic field, there exist three types of mag-netic domains, two type-A domains with a ferromagneticlayer perpendicular to the surface [one of two such casesis shown in Fig. 4(h)], and a type-F domain with a ferro-magnetic layer oriented parallel to the surface [Fig. 4(i)]. FIG. 4. (color online). (a)-(c) Plot of second-derivative in-tensity of MDCs around the ¯Γ point for CeBi measured at T = 30 K (paramagnetic phase), 19 K (AF-I phase), and 6 K(AF-II phase). (d)-(f) Schematic view of band folding for theDirac-cone SS in the k x - k z plane. (g)-(i) Crystal and mag-netic structures for paramagnetic, AF-I type-A domain, andAF-I type-F domain, respectively [3, 4]. Red and blue shadeshighlight the ferromagnetic Ce layer. Such AF domains give rise to a superstructure potentialof 2 × × × × × × × × M points in thesurface BZ, it is expected that the Dirac cone originallylocated at the ¯ M point is folded onto the ¯Γ point, and viceversa . Thus, one of double Dirac cones seen in Fig. 4(b)may be attributed to the Dirac cone originally located atthe ¯ M point in the paramagnetic phase. Such a Diraccone at ¯ M was identified in previous VUV-ARPES stud-ies on LaBi and CeBi [40, 42, 44, 46–48], although theshape is more complex than that at the ¯Γ point probablybecause two bulk X points are simultaneously projectedonto the surface ¯ M point [40] (note that the ¯ M pointis not accessible in our ARPES measurement due to theinsufficient k range of Xe-I photons). In this context,the observed double Dirac-cone feature is not fully ex-plained in terms of a simple overlap of the Dirac conesat ¯Γ and ¯ M in the paramagnetic phase, and one mayneed to invoke a complex mechanism beyond the simpleband-folding picture (see Appendix B for more detaileddiscussion of the band folding). Thus, the band-foldingpicture is just a likely possibility and its verification re-quires high-resolution domain-selective measurements.Here we point out some other possibilities to explainthe emergence of a double Dirac-cone feature in the AF- I phase. We think that this Dirac-cone-like feature ishardly explained in terms of the folded bulk band. Be-cause, the k z point in the present measurement ( k z =0 . π ) estimated from the inner-potential value ( V =13.5 eV) is far from the bulk X point ( k z = 0 or π ).Thus, in the present experimental setup, the electronlikebulk band should be located far above E F after foldingand as a result unable to produce the observed Dirac-cone feature near E F even after band-folding. Moreover,the bulk band in the paramagnetic phase does not showa Dirac-cone-like dispersion at the X point. This is ob-viously incompatible with the bulk origin of the doubleDirac cone. One may also think that one of the Dirac-cone bands in the AF-I phase could be a trivial SS whichis seen in the paramagnetic phase of LaSb [40]. However,such trivial SS is not well visible in the present studyof LaSb [Fig. 2(a)], probably because of the difference inthe experimental conditions (such as photon energy, lightpolarization, and sample geometry). Since we used thesame experimental conditions for LaSb and CeBi in ourARPES study, the trivial SS would have shown up evenin the paramagnetic phase in CeBi if some of observedbands in the AF phase are attributed to the trivial SS.However, this is not the case in our experiment since weresolve a single non-trivial SS (a Dirac cone) in the para-magnetic phase of CeBi, as shown in Figs. 2(d) and 4(a).It is noted that the observed double Dirac-cone featuremay originate from an overlap of SS in the type-F andtype-A domains which are energetically inequivalent toeach other. One cannot rule out this possibility becausethe ¯ M point is out of the measurable k range with aXe lamp. To resolve this issue, it is highly desirable toperform a high-resolution domain-selective ARPES mea-surements covering both the ¯Γ and ¯ M points to experi-mentally distinguish the SS originating from the type-Aand type-F domains.One may point out that the gapless topological SSshould not appear in the AF phase since the topologicalproperties can be destroyed by the antiferromagnetism via time-reversal-symmetry (TRS) breaking. However,we think that the topological SS can be maintained evenin the AF phase when we take into account a combinedsymmetry. It has been predicted that even antiferromag-nets, where both TRS (Θ) and primitive lattice transla-tional symmetry ( T / ) are broken, can still host a topo-logical phase if the crystal preserves the combined sym-metry S = Θ T / [49]. This is likely the case for thetype-A domain in the AF-I phase because the AF vectorsatisfies this translation condition. This may be the rea-son why the Dirac-cone SS is still observed in the AF-Iphase. It is noted that, since the combined symmetry isbroken at the surface of the type-F domain, the Dirac-cone SS is expected to exhibit a finite energy gap at theDirac point. However, this gap would be very small anddifficult to resolve with the present experimental resolu-tion.The appearance of a single Dirac cone in the AF-IIphase [Fig. 4(c)] as opposed to the double Dirac cone inthe AF-I phase is puzzling. The AF domain in the AF-IIphase creates the 4 × × M , resulting in the emergence of several Dirac conesbetween ¯Γ and ¯ M . However, this is not experimentallyobserved since such multiple Dirac cones at correspond-ing k regions are not resolved in the present ARPES ex-periment. We speculate that the surface electrons do notstrongly feel the periodic potential with such a long pe-riodicity so that the influence of the band folding wouldbe weakened compared in the AF-I phase with a shorterperiodic potential. But we point out here that the aboveinterpretation based on the in-plane band-folding is oneof possibilities. Moreover, there exist some unresolved is-sues as to (i) why the bulk bands do not show clear bandfolding in the AF phase unlike the SS, and (ii) whetherthe electronic states at ¯ M in the AF phase can be ex-plained by the band-folding picture. The verification re-quires the domain-selective high-resolution measurementby micro-beam-spot ARPES, combined with the sophis-ticated first-principles band-structure calculations thattake into account the magnetic structure. IV. SUMMARY
The present ARPES study of CeBi has revealed the ex-istence of topological Dirac-cone SS at the ¯Γ point asso-ciated with the band inversion at the X point of bulk BZ.We uncovered an unexpected change in the energy dis-persion of the Dirac-cone SS associated with the two-stepAF transition. Intriguingly, we found that a single Dirac-cone band observed in the paramagnetic phase abruptlybecomes a double Dirac cone on entering the AF-I phasebelow 25 K, whereas the doubling disappears in the AF-II phase below 14 K. The present result strongly suggestsa crucial role of antiferromagnetism to the appearance ofthe Dirac-cone SS, and opens a pathway toward under-standing the interplay between magnetism and topologyin exotic topological materials. ACKNOWLEDGMENTS
We thank K. Sugawara, M. Kitamura, R. Yukawa, S.Ideta, and K. Tanaka and for their assistance in theARPES experiments. This work was supported by JST-CREST (No: JPMJCR18T1), JST-PRESTO (No: JP-MJPR18L7), MEXT of Japan (Innovative Area “Topo-logical Materials Science” JP15H05853), JSPS (JSPSKAKENHI No: JP17H01139, JP26287071, JP25220708,JP18J20058, JP18H01160), KEK-PF (Proposal number:2018S2-001, 2018G653), and UVSOR (Proposal number:30-858, 30-846, 30-554, 30-568).
Appendix A: TEMPERATURE DEPENDENCE OFBULK-BAND DISPERSION
We have examined the influence of p-f mixing on theobserved electronic structure, by inspecting the changein the bulk valence-band dispersion with temperature.As shown in Figs. 5(a) and 5(b), one can recognize inthe paramagnetic phase ( T = 30 K) a couple of holelikeBi-6 p bands (inner and outer) which cross E F besidesthe surface band. Interestingly, the magnitude of theFermi wave vectors ( k F ) of these bands increases in theAF-II phase, as shown by white arrows. To investigatethe detailed temperature evolution of the k F position, weshow in Fig. 5(c) the second-derivative intensity of MDCsat E F as a function of temperature. One can clearlysee a systematic increase in the absolute value of k F onlowering temperature. A similar behavior in the valenceband has been observed in CeSb [9–12] and is interpretedin terms of the energy shift of valence bands due to the p-f mixing. Thus, our experimental results clearly show thatthe p-f mixing affects the band structure also in CeBi. FIG. 5. (a), (b) Plots of second-derivative intensity of MDCsaround the ¯Γ point for CeBi measured at T = 30 K (para-magnetic phase) and 6 K (AF-II phase) [same as Figs. 4(a)and 4(c)]. Green dashed curves are a guide for the eyes totrace the holelike Bi 6 p / bulk bands. Arrows indicate theposition of the k F points for the bulk bands. (c) Temperaturedependence of the second-derivative intensity of MDCs at E F . Appendix B: ANALYSIS OF EDCs and MDCs
We show in Figs. 6(a)-6(i) the EDCs, MDCs, and theirnumerical analyses, at representative temperatures in theparamagnetic phase ( T = 30 K), the AF-I phase ( T = 19K), and the AF-II phase ( T = 6 K). As shown in Figs.6(a) and 6(b), one can see in the paramagnetic phase( T = 30 K) a single X-shaped band in both EDCs andMDCs. We traced the peak position of the EDCs andMDCs by numerical fittings with Lorentzians around thepeak top, and found that the obtained band energies forEDCs and MDCs reasonably coincide with each other at T = 30 K near the Dirac point, as shown in Fig. 6(c).In the AF-I phase [Fig. 6(f)], on the other hand, theEDC- and MDC-peaks show a finite deviation aroundthe ¯Γ point, whereas they reasonably overlap with eachother away from the ¯Γ point. This suggests that the FIG. 6. (a), (b) Near- E F EDCs and MDCs of CeBi, respec-tively, measured along the Γ M cut at T = 30 K (param-agnetic phase). (c) Comparison of band energies estimatedfrom the peak position of the EDCs and MDCs in (a) and(b). (d)-(f) Same as (a)-(c) but for the AF-I phase ( T = 19K). (g)-(i) Same as (a)-(c) but for the AF-II phase ( T = 6 K).(j) Schematic band-folding picture in the AF-I phase whichtakes into account the folding of light and heavy Dirac conesaround the ¯ M point. Note that the heavy Dirac-cone band isnot clearly seen in the present study. lower Dirac-cone bands are substantially rounded aroundthe Dirac point. The rounded behavior of the Dirac-cone dispersion may be explained in terms of a finite gap-opening at the Dirac point and/or the inherent curvatureof the lower Dirac-cone band in the AF phase due tothe complicated hybridization between the original andfolded bands.Here we elaborate on our observation of the doubleDirac-cone feature based on the band-folding picture. Ithas been reported that the SS at the ¯ M point in the sur-face Brillouin zone (corresponding to the X point in bulk)is characterized by light and heavy Dirac-cone bands [46,48] with an energy gap at the Dirac point due to hy-bridization, as schematically shown in Fig. 6(j). In our experiment, we set the analyzer slit parallel to the Γ M direction. In this geometry, when the bands at the ¯ M point are folded onto the ¯Γ point due to antiferromag-netism, we expect to observe a narrow energy dispersionof the heavy Dirac-cone band besides the light one aroundthe ¯Γ point in Fig. 4(b). However, we have not resolvedsuch a heavy Dirac-cone band, presumably because of thematrix-element effect of photoelectron intensity. As illus-trated in Fig. 6(j), the band which is topped at E B ∼ ∼ E B ∼ E B on moving away fromthe ¯Γ point. This feature is also seen in the MDCs inFigs. R2(h) and R2(i) as a holelike band which has atop at E B ∼ E B on movingaway from the ¯Γ point while rapidly reducing its intensity.This holelike dispersion is hardly explained in terms of agapped upper Dirac-cone band because the upper Diraccone should have an electronlike dispersion. The reasonwhy the energy position of the Dirac-crossing point in theMDCs and the peaks in the EDCs does not well coincidewith each other [see Fig. 6(i)] may be due to the deviationof the Dirac-cone-like band from a linear dispersion, aswell as the unusually weak intensity of the 0.1-eV peakaway from the ¯Γ point (which makes it difficult to followits dispersion from the MDC plots). One may argue thatthe existence of a couple of holelike bands in the AF-II phase can be explained in terms of a double Dirac-cone band similar to that in the AF-I phase but shiftedtoward E F due to the p-f mixing. However, whereas thebulk holelike band certainly shows a gradual shift of thedispersion toward E F on lowering temperature due tothe p-f mixing [see Fig. 5(c)], the observed abrupt shiftof the band energy across T AFII cannot be well explainedin terms of the p-f mixing, since such an abrupt shift isabsent in the bulk bands. J. Rossat-Mignod, P. Burlet, J. Villain, H. Bartholin, W.Tcheng-Si, D. Florence, and O. Vogt, Phys. Rev. B , 440 (1977). J. Rossat-Mignod, P. Burlet, S. Quezel, J. M. Effantin, D.Delacˆote, H. Bartholin, O. Vogt, and D. Ravot, J. Magn.Magn. Mater. , 398 (1983). H. Bartholin, P. Burlet, S. Quezel, J. Rossat-Mignod, andO. Vogt, Le J. Phys. Colloq. , C5 (1979). M. Kohgi, K. Iwasa, and T. Osakabe, Physica B ,417 (2000). A. Hasegawa, J. Phys. Soc. Jpn. , 677 (1985). R. Settai, T. Goto, S. Sakatume, Y. S. Kwon, T. Suzuki, Y.Kaneta, and O. Sakai, J. Phys. Soc. Jpn. , 3026 (1994). H. Takahashi and T. Kasuya, J. Phys. C , 2697 (1985); , 2709 (1985); , 2721 (1985); , 2731 (1985); ,2745 (1985); , 2755 (1985). H. Heer, A. Furrer, W. Halg and O. Vogt, J. Phys. C ,5207 (1979). H. Kumigashira, H.-D. Kim, A. Ashihara, A. Chainani, T.Yokoya, T. Takahashi, A. Uesawa, and T. Suzuki, Phys.Rev. B , 13654 (1997). T. Ito, S. Kimura, and H. Kitazawa, Physica B , 268(2004). A. Takayama, S. Souma, T. Sato, T. Arakane, and T. Taka-hashi, J. Phys. Soc. Jpn. , 073702 (2009). S. Jang, R. Kealhofer, C. John, S. Doyle, J. Hong, J. H.Shim, Q. Si, O. Erten, J. D. Denlinger, and J. G. Analytis,Sci. Adv. , eaat7158 (2019). H. Kitazawa, Y. S. Kwon, A. Oyamada, N. Takeda, H.Suzuki, S. Sakatsume, T. Satoh, T. Suzuki, and T. Kasuya,J. Magn. Magn. Mater. , 40 (1988). J. von Boehm and P. Bak, Phys. Rev. Lett. , 122 (1979). K. Nakanishi, J. Phys. Soc. Jpn. , 1296 (1989). T. Kasuya, Y. S. Kwon, T. Suzuki, K. Nakanishi, F.Ishiyama, and K. Takegahara, J. Magn. Magn. Mater. , 389 (1990). Y. Okayama, H. Takahashi, N. Mori, Y. S. Kwon, Y. Haga,and T. Suzuki, J. Magn. Magn. Mater. , 113 (1992). T. Kasuya, Y. Haga, Y. S. Kwon, and T. Suzuki, PhysicaB , 9 (1993). K. Iwasa, A. Hannan, M. Kohgi, and T. Suzuki, Phys. Rev.Lett. , 207201 (2002). M. Zeng, C. Fang, G. Chang, Y.-A. Chen, T. Hsieh, A.Bansil, H. Lin, and L. Fu, arXiv:1504.03492 (2015). P.-J. Guo, H.-C. Yang, B.-J. Zhang, K. Liu, and Z.-Y. Lu,Phys. Rev. B , 235142 (2016). P.-J. Guo, H.-C. Yang, K. Liu, and Z. Y. Lu, Phys. Rev.B , 081112(R) (2017). N. N. Stepanov, N. V. Morozova, A. E. Kar’kin, A. V.Golubkov, and V. V. Kaminskii, Phys. Solid State , 2369(2015). F. F. Tafti, Q. D. Gibson, S. K. Kushwaha, N. Hal-dolaarachchige, and R. J. Cava, Nat. Phys. , 272 (2016). S. S. Sun, Q. Wang, P. J. Guo, K. Liu, and H. C. Lei, NewJ. Phys. , 082002 (2016). N. Kumar, C. Shekhar, S.-C. Wu, I. Leermakers, O. Young,U. Zeitler, B. H. Yan, and C. Felser, Phys. Rev. B ,241106(R) (2016). F. F. Tafti, Q. D. Gibson, S. K. Kushwaha, J. W. Krizan,N. Haldolaarachchige, and R. J. Cava, Proc. Natl. Acad.Sci. , E3475-E3481 (2016). N. Kumar, C. Shekhar, J. Klotz, J. Wosnitza, and C.Felser, Phys. Rev. B , 161103(R) (2017). F. F. Tafti, M. S. Torikachvili, R. L. Stillwell, B. Baer,E. Stavrou, S. T. Weir, Y. K. Vohra, H.-Y. Yang, E. F.McDonnell, S. K. Kushwaha, Q. D. Gibson, R. J. Cava,and J. R. Jeffries, Phys. Rev. B , 014507 (2017). R. Singha, B. Satpati, and P. Mandal, arXiv:1703.06100v1. F. Wu, C. Y. Guo, M. Smidman, J. L. Zhang, and H. Q.Yuan, Phys. Rev. B , 125122 (2017). C. Guo, C. Cao, M. Smidman, F. Wu, Y. Zhang, F.Steglich, F. C. Zhang, and H. Yuan, Npj Quantum Mater. , 39 (2017). L. Ye, T. Suzuki, C. R. Wicker, and J. G. Checkelsky,Phys. Rev. B , 081108(R) (2018). Y.-Y. Wang, L.-L. Sun, S. Xu, Y. Su, and T.-L. Xia, Phys.Rev. B , 045137 (2018). Y.-Y. Wang, H. Zhang, X.-Q. Lu, L.-L. Sun, S. Xu, Z.-Y. Lu, K. Liu, S. Zhou, and T.-L. Xia, Phys. Rev. B ,085137 (2018). Z. Li, D.-D. Xu, S.-Y. Ning, H. Su, T. Iitaka, T. Tohyama,and J.-X. Zhang, Int. J. Mod. Phys. B , 1750217 (2017). L.-K. Zeng, R. Lou, D.-S. Wu, Q. N. Xu, P.-J. Guo, L.-Y. Kong, Y.-G. Zhong, J.-Z. Ma, B.-B. Fu, P. Richard, P.Wang, G. T. Liu, L. Lu, Y.-B. Huang, C. Fang, S.-S. Sun,Q. Wang, L. Wang, Y.-G. Shi, H. M. Weng, H.-C. Lei, K.Liu, S.-C. Wang, T. Qian, J.-L. Luo, and H. Ding, Phys.Rev. Lett. , 127204 (2016). Y. Wu, T. Kong, L.-L. Wang, D. D. Johnson, D. Mou, L.Huang, B. Schrunk, S. L. Bud’ko, P. C. Canfield, and A.Kaminski, Phys. Rev. B , 081108(R) (2016). N. Alidoust, A. Alexandradinata, S.-Y. Xu, I. Belopolski,S. K. Kushwaha, M. Zeng, M. Neupane, G. Bian, C. Liu,D. S. Sanchez, P. P. Shibayev, H. Zheng, L. Fu, A. Bansil,H. Lin, R. J. Cava, and M. Z. Hasan, arXiv:1604.08571v1. X. H. Niu, D. F. Xu, Y. H. Bai, Q. Song, X. P. Shen, B.P. Xie, Z. Sun, Y. B. Huang, D. C. Peets, and D. L. Feng,Phys. Rev. B , 165163 (2016). J. He, C. Zhang, N. J. Ghimire, T. Liang, C. Jia, J. Jiang,S. Tang, S. Chen, Y. He, S.-K. Mo, C. C. Hwang, M.Hashimoto, D. H. Lu, B. Moritz, T. P. Devereaux, Y. L.Chen, J. F. Mitchell, and Z.-X. Shen, Phys. Rev. Lett. ,267201 (2016). J. Nayak, S.-C. Wu, N. Kumar, C. Shekhar, S. Singh, J.Fink, E. E. D. Rienks, G. H. Fecher, S. S. P. Parkin, B.Yan, and C. Felser, Nat. Commun. , 13942 (2017). H. Oinuma, S. Souma, D. Takane, T. Nakamura, K.Nakayama, T. Mitsuhashi, K. Horiba, H. Kumigashira, M.Yoshida, A. Ochiai, T. Takahashi, and T. Sato, Phys. Rev.B , 041120(R) (2017). R. Lou, B.-B. Fu, Q. N. Xu, P.-J. Guo, L.-Y. Kong, L.-K.Zeng, J.-Z. Ma, P. Richard, C. Fang, Y.-B. Huang, S.-S.Sun, Q. Wang, L. Wang, Y.-G. Shi, H. C. Lei, K. Liu, H.M. Weng, T. Qian, H. Ding, and S.-C. Wang, Phys. Rev.B , 115140 (2017). Y. Wu, Y. Lee, T. Kong, D. Mou, R. Jiang, L. Huang, S.L. Bud’ko, P. C. Canfield, and A. Kaminski, Phys. Rev. B , 035134 (2017). K. Kuroda, M. Ochi, H. S. Suzuki, M. Hirayama, M.Nakayama, R. Noguchi, C. Bareille, S. Akebi, S. Kunisada,T. Muro, M. D. Watson, H. Kitazawa, Y. Haga, T. K. Kim,M. Hoesch, S. Shin, R. Arita, and T. Kondo, Phys. Rev.Lett. , 086402 (2018). B. Feng, J. Cao, M. Yang, Y. Feng, S. Wu, B. Fu, M. Arita,K. Miyamoto, S. He, K. Shimada, Y. Shi, T. Okuda, andY. Yao, Phys. Rev. B , 155153 (2018). P. Li, Z. Wu, F. Wu, C. Cao, C. Guo, Y. Wu, Y. Liu, Z.Sun, C.-M. Cheng, D.-S. Lin, F. Steglich, H. Yuan, T.-C.Chiang, and Y. Liu, Phys. Rev. B , 085103 (2018). R. S. K. Mong, A. M. Essin, and J. E. Moore, Phys. Rev.B , 245209 (2010). H. Guo, S. Feng, and S.-Q. Shen, Phys. Rev. B , 045114(2011). C. Fang, M. J. Gilbert, and B. A. Bernevig, Phys. Rev. B , 085406(R) (2013). S. Miyakoshi and Y. Ohta, Phys. Rev. B , 195133 (2013). T. Yoshida, R. Peters, S. Fujimoto, and N. Kawakami,Phys. Rev. B , 085134 (2013). R. A. M¨uller, N. R. Lee-Hone, L. Lapointe, D. H. Ryan,T. Pereg-Barnea, A. D. Bianchi, Y. Mozharivskyj, and R.Flacau, Phys. Rev. B. , 041109(R) (2014). C.-X. Liu, R.-X. Zhang, and B. K. VanLeeuwen, Phys.Rev. B , 085304 (2014). R.-X. Zhang and C.-X. Liu, Phys. Rev. B , 115317(2015). C. Fang and L. Fu, Phys. Rev. B , 161105(R) (2015). J. Yu, B. Yan, and C.-X. Liu, Phys. Rev. B , 235158(2017). W. Brzezicki and M. Cuoco, Phys. Rev. B , 155108(2017). N. Hao, F. Zheng, P. Zhang, and S.-Q. Shen, Phys. Rev.B , 165102 (2017). K.-W. Chang and P.-J. Chen, Phys. Rev. B
7, 195145(2018). A. Franciosi, J. H. Weaver, N. M˚artensson, and M. Croft,Phys. Rev. B , 3651 (1981). J. W. Allen, S. J. Oh, I. Lindau, J. M. Lawrence, L. I.Johansson, and S. B. Hagstr¨om, Phys. Rev. Lett.46