Evolution of possible Weyl semimetal states across the hole-doping induced Mott transition in pyrochlore iridates
Kentaro Ueda, Hikaru Fukuda, Ryoma Kaneko, Jun Fujioka, Yoshinori Tokura
aa r X i v : . [ c ond - m a t . s t r- e l ] D ec Evolution of possible Weyl semimetal states across the hole-doping induced Motttransition in pyrochlore iridates
K. Ueda, H. Fukuda, R. Kaneko,
1, 2
J. Fujioka, and Y. Tokura
1, 2, 4 Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan Faculty of Material Sciences, University of Tsukuba, Tsukuba 305-8577, Japan Tokyo Colledge, University of Tokyo, Tokyo 113-8656, Japan (Dated: December 24, 2020)We study possible Weyl semimetals of strongly-correlated electrons by investigating magneto-transport properties in pyrochlore R Ir O ( R =rare-earth ions), choosing three types of R ions todesign the exchange coupling scheme between R f and Ir 5 d moments; non-magnetic Eu (4 f ),isotropic Gd (4 f ), and anisotropic Tb (4 f ). In the doping-induced semimetallic state, distinctivefeatures of magnetoresistance and Hall effect are observed in R =Gd and Tb compounds due to theeffects of the exchange-enhanced isotropic and anisotropic Zeeman fields, respectively, exemplifyingthe double Weyl semimetal and the 2-in 2-out line-node semimetal as predicted by theories. In par-ticular, a Hall angle of R =Gd compound is strongly enhanced to 1.5 % near above the critical dopingfor the Mott transition. Furthermore, an unconventional Hall contribution is discerned for a lowerdoping regime of R =Gd compound, which can be ascribed to the emergence of Weyl points withthe field-distorted all-in all-out order state. These findings indicate that the hole-doping inducedMott transition as well as the characteristic f - d exchange interaction stabilizes versatile topologicalsemimetal states in a wide range of material parameter space. INTRODUCTION
The correlation between magnetism and topologicalelectronic states has been one of the most importantthemes of the modern condensed matter physics [1, 2].A magnetic Weyl semimetal (WSM) possesses crossingsof non-degenerate bands with linear dispersion under thebreaking of time-reversal symmetry with magnetic order[3]. Notably, the crossing (Weyl point) can be viewed as amagnetic monopole of Berry curvature in the momentumspace, proven to yield salient magnetotransport proper-ties [4]. Prominent example is the intrinsic anomalousHall effect (AHE) [5]. Recent enormous efforts on thissubject reveal that the position of Weyl points relativeto the Fermi energy E F is intimately related to the sizeof AHE, revealing astoundingly large AHE in a varietyof magnets [6–9]. Despite such remarkable findings, therole of electron-correlation as a source of magnetism intopological electronic states remains elusive. In particu-lar, the emergence of the magnetic topological states overthe electron-correlation induced metal-insulator transi-tion (MIT) [10] has seldom been explored.Pyrochlore oxide A B O is known to host exoticquantum magnetic ground states [11], and recently pro-posed to host topological electronic states for iridium ox-ides R Ir O ( R being rare-earth or Y ions) [3]. Themerit of pyrochlore oxides is that the effective electroncorrelation U/W and the band filling n (=1- δ , δ be-ing hole-doping) can be tuned by varying A -site ionsin a similar manner to perovskites [10]. For instance, R Ir O with a smaller ionic-size R shows an effectivelylarger bending of Ir-O-Ir bond angle and hence leads to asmaller one-electron band-width W ; thus the R -ion size is a good indicator of the inverse of the effective elec-tron correlation U/W , as shown in Fig.1(a). Among avariety of R Ir O , R =Pr compound, which is charac-terized by the largest R ionic radius, is found to pos-sess a quadratic band touching (QBT) of spin degener-ate valence and conduction bands near E F in the para-magnetic metal (PM) phase (see Fig. 1(b)) [12]. Sincethe QBT is protected by the cubic crystalline symme-try and the time-reversal symmetry, it is expected tobe robust against the perturbation as far as the sym-metry is kept intact. Such an electronic state is termedLuttinger SM or QBT SM that is theoretically antici-pated to be converted to versatile topological states bysymmetry-breaking procedures [13, 14]. Figure 1(b) dis-plays a schematic picture of the electronic band mod-ulation across Mott transitions. The strong Hubbardrepulsion U brings about the antiferromagnetic-like all-in all-out (AIAO) type magnetic order (depicted in theinset in Fig. 1(a)), and lifts the band degeneracy ofQBT, leading to the emergence of WPs along the high-symmetry axes [1, 15], termed AIAO WSM hereafter.Although WSM is generally robust against the perturba-tion, WPs in this system emigrate towards the Brillouinzone boundary with increasing U/W or the magnetic or-der parameter m . Consequently, with a tiny increase of U or m , WSM undergoes the pair annihilation of WPsat the zone boundary and turns into a gapped insulat-ing state (Fig. 1(b)), hampering the observation in themomentum space. In spite of such difficulty, intensivestudies on R =Nd and Pr compounds have unveiled anumber of emergent transport phenomena such as un-usual Hall effect [16, 17], magnetic field-induced MITs[18–20], and anomalous metallic states on the magneticdomain walls in the AIAO state[21, 22], which are all un-derstood in the context of underling or incipient WSMsttaes induced by the time-reversal symmetry breaking(Fig. 7 in Appendix).On the other hand, R Ir O with smaller R ionic ra-dius are not QBT-SM but fully-gapped insulators be-cause of larger U/W (Fig. 1(a)), and thus have been con-sidered to be no longer related to topological semimetallicstate [23]. However, recent studies demonstrate that thehole-doping by the chemical substitution of trivalent R ions with divalent A ions turns them into PM state[24–27] in which, remarkably, QBT-SM can ubiquitously sub-sist even if the chemical substitution introduces somedisorders [25] In this sense, the hole-doped R Ir O of-fers a fertile playground to study the correlation betweenmagnetism and topological electronic states, since car-rier doping can tune the position of the QBT node withrespect to E F , and moreover, a variety of local R -4 f mag-netic spins, which strongly affect itinerant Ir-5 d electronsthrough the f - d coupling, are available in this pyrochloresystem. RESULTSHole-doping induced insulator-metal transition
In this work, we choose three R ions to study therole of R magnetic moment in the magnetic-field inducedmodification of QBT state; nonmagnetic R =Eu (4 f ),isotropic R =Gd (4 f ), and Ising-type R =Tb (4 f ). Wesynthesized high-quality polycrystals by utilizing high-pressure apparatus which promotes the pyrochlore-latticeformation while keeping the right stoichiometry of com-pounds. Employing the growth condition described else-where [28], we obtained the hard and dense samples suit-able for the systematic transport measurements. As forthe chemical substitution of trivalent R ions with divalent A ions, we found that A =Ca (Cd) is suitable for R =Euand Tb (Gd) in the light of the ion-size matching. It al-lows us to examine the hole doping effect while suppress-ing other contributions such as the A -site doping-inducedchange of bandwidth or disorder. Zhu et al. demonstratethat the Ca-doping to Y Ir O enhances both metallic-ity and ferromagnetism [26]. They found that the tem-perature dependence of resistivity shows a minimum ataround 100 K for the high-doped metallic sample, whichis somewhat different from our result as shown below. Wespeculate that it is due to the mismatch of A -site ionicradius. Since the Ca ionic radius is much larger than thatof Y ions, the Ca doping modifies the effective bandwidthvia the change of Ir-O-Ir bond angles, and thereby bringsabout the resistivity minimum at around 100 K, which isalso observed in Eu Ir O under the hydrostatic pressureabove 7.88 GPa [29].As the A concentration x increases in ! " $ % & " ’ ( ) * % ! $ $+ , - " ) % + " --$ . , ( " ) % " $ !" ! ! !"! ! ! !"! ! ! (’)*+,-./0)1234.567)86.9:;2< ,=>: ?@ &1 A: (/ ! ! " ) B : @ C ) ; ! < " " / ) DEDFDEGDDEGFDEHD , . I . / @ / . ) ; J < (’(’!"! ! $% !" " ! < "/ & !"! $%& " FIG. 1. (color online). (a) Phase diagram of R Ir O as func-tions of rare-earth ionic radius, divalent-ion concentration,and temperature. PI stands for paramagnetic insulator, PMstands for paramagnetic metal, AIAO stands for all-in all-outstate, and WSM stands for Weyl semimetal. (b) Schematicpicture of modulation of electronic band structures as a func-tion of electron correlation U and hole doping δ . QBT standsfor quadratic band touching. (Gd − x Cd x ) Ir O and (Tb − x Ca x ) Ir O , whichcorresponds to the nominal hole-doping δ , both exhibitthe systematic reduction in resistivity ρ xx as well asin T N that shows up as a kink of ρ xx in accord withthe anomaly in M . They turn into PM at sufficientlylarge x , although the resistivity slightly increases below20 K as shown in Fig. 2. The results including T N are summarized in the phase diagram, Fig. 1(a).Incidentally, the upturn of ρ xx at low temperaturesin (Tb − x Ca x ) Ir O and (Gd − x Cd x ) Ir O , whichis absent in (Eu − x Ca x ) Ir O , is due perhaps tothe strong magnetic coupling between itinerant Irelectrons and localized R moments which producesincoherent carrier-electron scattering [30, 31]. Since ρ xx value of (Eu . Ca . ) Ir O is close to those of(Gd . Cd . ) Ir O and (Tb . Ca . ) Ir O , thesimilar electronic states that host QBT at Γ pointare anticipated to be realized in hole-doped analogs ofboth R =Gd and Tb compounds, which is corroboratedby magnetotranport measurements described in thefollowing. Magnetotransport of metallic R compounds Having thus realized the QBT state in various mag-netic ( R − x A x ) Ir O , we study the magnetic field de-pendence of ρ xx and Hall resistivity ρ yx in Fig. 3.(Eu . Ca . ) Ir O , whose R -site ion is non-magnetic,shows little magnetic field dependence of ρ xx (Fig. 3(a)) FIG. 2. (color online). Temperature dependence of resistivityfor (Eu − x Ca x ) Ir O ( x =0, 0.10), (Gd − x Cd x ) Ir O ( x =0,0.12), and (Tb − x Ca x ) Ir O ( x =0, 0.15). and ρ yx apart from the normal Hall effect (Fig. 3(b)).On the other hand, negative magnetoresistance is clearlyobserved for (Tb − x Ca x ) Ir O with x =0.15 in Fig. 3(c),indicating the coupling between the conduction elec-tron and R -ion moment. More importantly, ρ yx showsa complicated magnetic-field profile especially at 2 K(Fig. 3(d)); as the field increases, ρ yx abruptly increaseswith a maximum around 1 T, and subsequently plungesto the negative value, changing the sign at 7 T. Toidentify the origin of non-monotonic Hall responses for(Tb − x Ca x ) Ir O , we examine those of well-orientedsingle crystal Pr Ir O which is well characterized byQBT SM state at zero field and zero doping ( δ = 0) [12],and an Ising anisotropy of R moment similar to the Tbone. Figure 3(e) and 3(f) display ρ xx and ρ yx of Pr Ir O (single crystal) respectively, as a function of the mag-netic field along [001] axis which favors 2-in 2-out stateas depicted on top of Figs.3 (c), (e). Both ρ xx and ρ yx are qualitatively similar to those of (Tb − x Ca x ) Ir O ( x =0.15). A former study combining transport measure-ments on (Nd,Pr) Ir O and theoretical calculation sug-gests that the magnetic field can modulate the electronicstate into the line-node SM (LSM) depicted in the middlepanel of Fig.2 via the magnetic stabilization at the 2-in2-out state [18]. Thus, the close similarity of ρ xx and ρ yx in the two compounds with similar Ising anisotropy of R moments implies that the LSM can be also realized in(Tb − x Ca x ) Ir O polycrystals at high fields where the2-in 2-out configuration is dominant.On the other hand, a very different trend of ρ xx and ρ yx is observed in (Gd − x Cd x ) Ir O ( x =0.12). Especially,at the lowest temperature 2 K, ρ xx markedly decreasesby half at 14 T (Fig. 3(g)) while the absolute valueof negative ρ yx increases rapidly. This anomalous Hall-like behavior in the originally paramagnetic but field- magnetized state can show the large ρ yx value, 20 timeslarger than that of R =Eu compound (Fig. 3(h)), andthe observed value of the Hall angle is as large as 1.5 %.For instance, the Hall angle of ferromagnetic oxides suchas SrRuO [32] and (La,Sr)CoO [33] is approximately0.9 %, and that of isostrural ferromagnet Nd Mo O with the 2-in 2-out scalar spin chirality is about 1.3 %[34]. Such a large Hall effect can point to a substantialBerry-curvature contribution near E F in the momentumspace. Oh et al. theoretically show that the uniform mag-netic field along [001] crystalline direction lifts the banddegeneracy into two Weyl points hosting the monopolecharge ± R =Eu compound would showthe similar effect, but not in reality (Fig. 3(b)). The suf-ficiently large exchange splitting of QBT is only drivenby the f - d exchange coupling from six nearest-neighborGd spins which are easily aligned by external fields (Fig.4(a)). Magnetotransport across the filling-control-inducedmetal-insulator transition in (Gd − x Cd x ) Ir O In order to gain deeper insight into the possibletopological-state change in (Gd − x Cd x ) Ir O , we imple-ment a precise investigation on magnetotransport prop-erties across doping-induced Mott transition. Figure 4(b)shows the temperature dependence of ρ xx for several(Gd − x Cd x ) Ir O compounds with varying x . Accom-panied by the decline of ρ xx , T N , which is indicated byarrows in Fig. 4(b), is gradually suppressed with increas-ing x and no longer discernible for x > .
10. The temper-ature dependence of M for several x is displayed in Fig.8, Appendix. At x =0.07, M in the field-cooling processclearly deviates from that in the zero-field-cooling pro-cess below T N at which ρ xx shows a kink. The deviationdisappears down to 2 K for x =0.12, although the upturnof ρ xx subsist. The upturn of ρ xx is also observed in Ca-doped Nd Ir O below 20 K [27], and attributed to theordering or freezing of the Nd magnetic sublattice whichis observed by muon spin resonance [36]. Figures 4(c-j) show the magnetic field dependence of M and ρ yx at2 K for various compositions near the Mott transition,ranging from the x =0.07 compound, which undergoeslong-range AIAO order below T N , to the paramagneticsemimetal x =0.12 compound. As shown in Figs. 4(c)-(f), M increases monotonically with increasing field andfinally saturates at high fields for all compositions. Thesaturated M values are consistent with those expectedfrom fully-aligned Gd spins as depicted by dashed lines.The critical magnetic field, which is required to reach thesaturated M , gradually becomes smaller as x increases; !" !" ./0) ! ! ! !!" " ! " " " " ! " " " &’(’)*+),+ )-+ ).+)/+)0+)1+)2+ FIG. 3. (color online). Magnetic field dependence of longitudinal resistivity ((a),(c),(e),(g)) and Hall resistivity ((b),(d),(f),(h)).From left to right, the data are for (a,b) (Eu − x Ca x ) Ir O ( x =0.10), (c,d) (Tb − x Ca x ) Ir O ( x =0.15), (e,f) Pr Ir O . (c,d)(Gd − x Cd x ) Ir O ( x =0.12), respectively. Schematic magnetic configurations of R spins and expected electronic structuresare displayed on top of respective windows. it is roughly 11 T for x =0.07 while 8 T for x =0.12. Itis presumably because the AIAO-type antiferromagneticcorrelation of Ir moments in the small x region competeswith the collinear alignment of Gd moments via the f - d coupling. Figures 4 (g-j) display ρ yx at 2 K as a func-tion of the magnetic field. Interestingly, ρ yx for x =0.07shows a nonmonotonic field dependence that is explic-itly distinguished from the usual AHE (Fig. 4(g)). Asthe field increases, ρ yx increases with a broad maximumaround 3 T, and then markedly decreases with the signreversal. Finally ρ yx decreases modestly above 11 T atwhich M is saturated. This characteristic field depen-dency at 2 K fades out into the monotonic change as thetemperature is elevated (see Fig. 9 in Appendix). Onthe other hand, ρ yx for x =0.12 monotonically decreasesand saturates at around 8 T, seemingly in proportion to M (Fig. 4(j)).To understand this behavior, we employ the fittingfunction which is conventionally used for ferromagnets,expressed as ρ fit yx = R B + µ R s M , where R is the nor-mal Hall coefficient and R s is the anomalous coefficient.The Hall conductivity σ xy values for 0.07 ≤ x ≤ σ xy ∝ σ . xx (Fig. 5), which isempirically known for the intrinsic anomalous Hall effectin the large-scattering regions, e.g. σ xy < S/cm [37]. Conversely, this indicates the common origin, i.e. theanomalous Hall effect characteristic of DWSM, for thehigh-field σ xy irrespective of x or σ xx value. Thus we use R s = S H ρ . xx with the scaling coefficient S H . The solidlines in Figs. 4 are fitting curves ρ fit yx which reproduce ρ exp yx in the field-induced Gd-spin collinear region, i.e. athigh magnetic fields. While ρ exp yx of x =0.12 is well fit-ted with the above relation over the entire magnetic fieldrange, those of other compounds in Figs. 4, which takeAIAO states at zero field, show clear deviations from thefitting curves at low fields. The deviation becomes largeras temperature decreases (Fig. 10, Appendix). The ob-served ρ yx should reflect the variation of the electronicstates through the change of the magnetic configurations.At low fields, Ir spins form the AIAO type magnetic or-dering configuration. As the field increases, Gd spinsgradually point to the field direction and force Ir spinsto align in a parallel manner through the f - d exchangeinteraction, resulting in the breaking of the AIAO pat-tern accompanied with the remarkable change of ρ xx and ρ yx at high fields. In this regard, the origin of the devia-tion can be ascribed to the emergence of WPs with ”dis-torted” AIAO magnetic state. Note here that the AIAOmaintains the cubic crystalline symmetry, and hence thenet Berry curvature integrated over the Brillouin zone iscanceled out in the ideal AIAO WSM. However, when !" ! %&’%(’ %)’ %$’ %*’ %+’%,’ %-’ %.’ %/’ FIG. 4. (color online). (a) Schematic magnetic structures of Ir moments (green) and Gd moments (blue). (b) Temperaturedependence of resistivity for (Gd − x Cd x ) Ir O with various x . Arrows indicate the magnetic transition temperature. Magneticfield dependence H of magnetization M ((c),(d),(e),(f)) and Hall resistivity ((g),(h),(i),(j)) for various x in (Gd − x Cd x ) Ir O .From left to right, the data are for (c,g) x =0.07, (d,h) x =0.08, (e,i) x =0.09, (f,j) x =0.12, respectively. Thick black lines arefitting curves with the H -linear normal Hall plus the M -linear anomalous Hall terms (see text). ! !" !" !" )( ! * +, - $ FIG. 5. (color online). Log-log plot of Hall conductivity ver-sus longitudinal conductivity for (Gd − x Cd x ) Ir O with var-ious x . The dashed line indicates the relation σ xy ∝ σ . xx . the Weyl vectors connecting a pair of WPs are deformedby the Zeeman field, the Hall response can show up to aconsiderable value, as demonstrated in Nd Ir O underpressure [17] or epitaxially-strained film [38]. Hereafterwe define the deviating part ∆ ρ yx = ρ exp yx − ρ fit yx as a con-tribution of Berry curvature in AIAO WSM. DISCUSSION
To clarify the nature of the observed Hall effect in(Gd − x Cd x ) Ir O , we summarize the comprehensive re- sults of ρ xx and ρ yx in contour mappings along with thephase diagram in Fig. 6. We firstly show the phase dia-gram with the contour plot of ρ xx in the plane of tempera-ture and Cd concentration x in Fig. 6(a). As x increases, T N systematically decreases and disappears at around x =0.10. Concomitantly, ρ xx significantly decreases byseveral orders of magnitude down to ∼ ρ yx /ρ xx at 14 T and the ∆ ρ yx scaled with ρ . xx at 4 T, whichare anticipated to represent the Hall responses from theDWSM and the AIAO WSM state, respectively. Bothsignals are enhanced at low temperatures, suggestingthat the magnetic coupling between Gd-4 f and Ir-5 d mo-ments plays a vital role in the magnetotransport. As x increases, the Hall angle ρ yx /ρ xx at 2 K and 14 T in-creases steeply, reaches a maximum of 1.5 % at x =0.12,and mildly decreases yet sustains 0.7 % at x =0.20 (Fig.6(b)), which is far beyond the value of (Tb − x Ca x ) Ir O which exhibits 2-in 2-out LSM. Such a large Hall effectis attributable to efficient Berry-curvature generation inDWSM phase that host a pair of WPs with monopolecharges of ±
2. In addition, it can be also augmentedby the strong magnetic interaction between conductingelectrons and local moments that can manifest itself asthe upturn of ρ xx (Fig. 2). On the other hand, a fi-nite value of ∆ ρ yx /ρ . xx at 4 T (Fig. 6(c)) extends overthe large area of AIAO phase ( x < Ir O , which is constricted inthe narrow temperature window, i.e., within ∼ T N [17]. According to theoretical calculation [35],the WPs in AIAO WSM move away from [111] or equiva- !" !" ! ! ! " ! %&%’" ! ! ! " ! ! ! " ! " ! "" FIG. 6. (color online). (a) Phase diagram of(Gd − x Cd x ) Ir O with contour mapping of resistivity in theplane of temperature and Cd concentration x . Black marksdenote the magnetic transition temperature T N . Contourplots of (b) Hall angle ρ yx /ρ xx at 14 T and (c) ∆ ρ yx / ρ . xx at 4T, which represent the Hall responses from DWSM andAIAO WSM, respectively. lent high-symmetry axes under the uniform Zeeman fieldwhich breaks threefold rotation symmetry. Consequentlythe pair annihilation is prevented in the broad parame-ter region, leading to the expansion of WSM phase asdiscussed for the case of (Nd,Pr) Ir O [20]. In short,as seen in Fig. 6, the compounds undergo the criticaltransformation from the AIAO WSM to the QBT SMat zero field as well as to the DWSM under the fullyaligned Gd moments, as the hole doping proceeds acrossthe Mott transition point ( x ∼ SUMMARY
We investigate the magnetotransport properties in thecourse of hole-doping induced Mott transitions for py-rochlore iridates R =Eu, Gd, and Tb. We establish theversatile phase diagrams of the topological states, includ-ing the all-in all-out Weyl semimetal, the 2-in 2-out line-node semimetal, the quadratic-band-touching semimetaland the double-Weyl semimetal, as functions of R ionicradius, temperature, magnetic field, and the band fill-ing (hole doping). Among them, (Gd − x Cd x ) Ir O ex-hibit characteristic Hall effects which point to the differ-ent Berry-curvature generations in the two distinct topo-logical semimetals, i.e. the field-distorted all-in all-outWeyl semimetal and the double Weyl semimetal. Thepresent work shows that the control of electron correla-tion by tuning not only the bandwidth but also the bandfilling can unravel the hidden topological semimetal stateor dramatically expand its stable region in pyrocholoreiridates where the field selection of the specific Weylsemimetals (WSMs) is also possible via the exchange cou-pling between the rare-earth 4 f and Ir 5 d moments. ACKNOWLEDGEMENT
We thank Hiroaki Ishizuka for enlightening discussions.This work was supported by JSPS Grant-in-Aid for Sci-entific Research (No. 19K14647), and CREST (No. JP-MJCR16F1 and JPMJCR1874), Japan Science and Tech-nology Japan.
APPENDIXPossible topological semimetal states with differentmagnetic configurations
The electronic state of pyrochlore iridates in the para-magnetic metal phase is predicted to host a quadraticband touching (QBT) at Γ point which is actually ob-served by angle-resolved photoemission spectroscopy[12].Such an electronic state is anticipated to be convertedto versatile topological states by symmetry breaking, aspredicted in some theories [3, 14, 20, 35, 39, 40]. Figure 7displays a representative example for the electronic bandmodulation with magnetic ordering patterns. In the caseof 2-in 2-out type magnetic configuration, the line-nodesemimetal (LSM), which possesses a pair of Weyl points(WPs) along [001] direction and a line node in k z plane,can be stabilized. On the other hand, three pairs of WPsshow up when the 3-in 1-out type magnetic pattern isrealized. Furthermore, the exchange field of collinearly-aligned R spins (or extremely large Zeeman field) cangive rise to the double Weyl semimetal that host a pairof WPs with monopole charge of ± !" !" !""" ! ! " ! " ! " (’%&! ! " ($%& ! ! ! ! " ! " ! " (’%&! ! " ($%& ! $$!%%" ! $$! "" ! $$!%%" " ! " " " " ! " " " " ! " " " ! ! ! ! " ! " ! " (’%&! ! " ($%& !" FIG. 7. (color online). Schematic picture of electronic bandstructures with different magnetic ordering patterns stem-ming from the quadratic band touching semimetal at zeromagnetic field in Fig. 1(b). From top to bottom, line-node semimetal (LSM) with 2-in 2-out magnetic state, Weylsemimetal (WSM) with 3-in 1-out magnetic state, and dou-ble Weyl semimetal (DWSM) with collinear magnetic state.On the left side, respective magnetic ordering configuration isshown. The middle column display the distribution of Weylpoints and a line node in the Brillouin zone. The red andblue circles denote Weyl points with different chiralities andthe purple line denotes a line node. On the right side, weshow the schematic band energy levels for respective elec-tronic states. Red and blue squares (circles) denote Weylpoints with monopole charges of ± ± TRANSPORT AND MAGNETIZATIONPROPERTIES IN (Gd − x Cd x ) Ir O Figure 8 shows the temperature dependence of resis-tivity and magnetization for various (Gd − x Cd x ) Ir O .For x =0.07, the magnetization in field-cooling process(red lines) clearly deviates from that in zero-field-coolingprocess (blue line) at the magnetic transition tempera-ture, at which the resistivity shows slight upturn. As x increases, the transition temperature systematically de-creases while the resistivity decreases. The anomaly isno longer observed for x =0.12. ! ! !" FIG. 8. (color online). Temperature dependence of (a,c,e,g)resistivity and (b,d,f,h) magnetization for (Gd − x Cd x ) Ir O .From left to right, the Cd concentration is x =0.07, 0.08, 0.09,and 0.12. The vertical broken line indicates the magnetictransition temperature. Magnetotransport properties for hole-doped R =Gdcompounds Figure 9 shows the magnetic field dependence of re-sistivity, Hall resistivity, and magnetization for various(Gd − x Cd x ) Ir O . For all compounds, the resistiv-ity remarkably decreases with increasing field at 2 K.The reduction of resistivity becomes smaller as the tem-perature is elevated. Hall resistivity for x =0.07 showsnon-monotonic behavior whereas the magnetization sim-ply increases as discussed in the main text. Figure 10shows the magnetic field dependence of experimentally-obtained Hall resistivity (circles) and the fitting curve(solid lines). The deviation of the data from the fittingcurve is explicit at 2 K. This behavior fades out as tem-perature and x increases. For x =0.12, which is param-agnetic metal down to 2 K, the Hall resistivity seems tobe proportional to the magnetization in the whole tem-perature and magnetic field region, as shown in Figs. 9 [1] Armitage, N. P. and Mele, E. J. and Vishwanath, Ashvin,Rev. Mod. Phys. , 015001 (2018).[2] Y. Tokura, K. Yasuda, and A. Tsukazaki, Nature ReviewPhysics, , 126 (2019).[3] X. Wan, A. M. Turner, A. Vishwanath, S. Y. Savrasov,Phys. Rev. B , 205101 (2011).[4] Z. Zyuzin and A. A. Burkov, Phys. Rev. B , 115133(2012).[5] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, andN. P. Ong, Rev. Mod. Phys. , 1539 (2010).[6] Satoru Nakatsuji, Naoki Kiyohara, and Tomoya Higo,Nature , 212 (2015).[7] Suzuki, T., Chisnell, R., Devarakonda, A. Liu, Y. -T.,Feng, W., Xiao, D., Lynn, J. W. and Checkelsky, J. G.,Nature Phys. , 1119 (2016).[8] Enke Liu et al. , Nature Phys. , 1125 (2018). !" FIG. 9. (color online). The magnetic field dependence of(a-d) Hall resistivity, (e-h) resistivity, (i-l) magnetization for(Gd − x Cd x ) Ir O . From left to right, the data are x =0.07,0.08, 0.09, and 0.12. !" FIG. 10. (color online). Magnetic field dependence of Hallresistivity for (Gd − x Cd x ) Ir O ( x =0.07) at several temper-atures. Thick black lines are fitting curves (see text).[9] Ghimire, Nirmal J., Botana, A. S., Jiang, J. S., Zhang,Junjie, Chen, Y. -S., and Mitchell, J. F., Nature Com-mun. , 3280 (2018).[10] M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. , 1039 (1998).[11] J. S. Gardner, M. J. P. Gingras, and J. E. Greedan, Rev.Mod. Phys. , 53 (2010).[12] T. Kondo et al. , Nat. Commun. , 10042 (2015).[13] J. Cano, B. Bradlyn, Z. Wang, M. Hirschberger, N. P.Ong, and B. A. Bernevig, Phys. Rev. B , 161306(2017).[14] E.-G. Moon, C. Xu, Y.-B. Kim, and L. Balents, Phys.Rev. Lett. , 206401 (2013).[15] W. Witczak-Krempa, and Y.B. Kim, Phys. Rev. B ,045124 (2012).[16] Y. Machida, S. Nakatsuji, S. Onoda, T. Tayama, and T.Sakakibara, nature , 2120 (2010).[17] K. Ueda, R. Kaneko, H. Ishizuka, J. Fujioka, N. Nagaosa,and Y. Tokura, Nat. Commun. , 3032 (2018). [18] K. Ueda, J. Fujioka, B.-J. Yang, J. Shiogai, A. Tsukazaki,S. Nakamura, S. Awaji, N. Nagaosa, and Y. Tokura,Phys. Rev. Lett. , 056402 (2015).[19] Z. Tian et al. , Nat. Phys. , 134 (2016).[20] K. Ueda, T. Oh, B.-J. Yang, R. Kaneko, J. Fujioka, N.Nagaosa, and Y. Tokura, Nat. Commun. , 15515 (2017).[21] K. Ueda, J. Fujioka, Y. Takahashi, T. Suzuki, S. Ishiwata,Y. Taguchi, M. Kawasaki, and Y. Tokura, Phys. Rev. B , 075127 (2014).[22] Eric Yue Ma, Yong-Tao Cui, Kentaro Ueda, Shujie Tang,Kai Chen, Nobumichi Tamura, Phillip M. Wu, J. Fujioka,Y. Tokura, and Zhi-Xun Shen, Science , 538 (2015).[23] K. Ueda, J. Fujioka, and Y. Tokura, Phys. Rev. B ,245120 (2016).[24] H. Fukazawa and Y. Maeno, J. Phys. Soc. Jpn. , 2578(2002).[25] R. Kaneko, M.-T. Huebsch, S. Sakai, R. Arita, H. Shi-naoka, K. Ueda, Y. Tokura, and J. Fujioka, Phys. Rev.B , 161104 (2019).[26] W. K. Zhu, M. Wang, B. Seradjeh, F. Yang, and S. X.Zhang, Phys. Rev. B , 054419 (2014).[27] Z. Porter, E. Zoghlin, S. Britner, S. Husremovic, J. P. C.Ruff, Y. Choi, D. Haskel, G. Laurita, and S. D. Wilson,Phys. Rev. B , 054409 (2019).[28] K. Ueda, J. Fujioka, Y. Takahashi, T. Suzuki, S. Ishiwata,Y. Taguchi, and Y. Tokura Phys. Rev. Lett. , 136402(2012).[29] F. F. Tafti, J. J. Ishikawa, A. McCollam, S. Nakatsuji,and S. R. Julian, Phys. Rev. B , 205104 (2012).[30] M. Udagawa, H. Ishizuka, and Y. Motome, Phys. Rev.Lett , 066406 (2012).[31] Z. Wang, K. Barros, G.-W. Chern, D. L. Maslov, and C.D. Batista, Phys. Rev. Lett , 206601 (2016).[32] Fang, Zhong and Nagaosa, Naoto and Takahashi, Kei S.and Asamitsu, Atsushi and Mathieu, Roland and Oga-sawara, Takeshi and Yamada, Hiroyuki and Kawasaki,Masashi and Tokura, Yoshinori and Terakura, Kiyoyuki,Science , 92 (2003).[33] T. Miyasato, N. Abe, T. Fujii, A. Asamitsu, S. Onoda,Y. Onose, N. Nagaosa, and Y. Tokura, Phys. Rev. Lett. , 086602 (2007).[34] S. Iguchi, N. Hanasaki, and Y. Tokura, Phys. Rev. Lett. , 077202 (2007).[35] T. Oh, H. Ishizuka, and B.-J. Yang, Phys. Rev. B ,144409 (2018).[36] H. Guo, K. Matsuhira, I. Kawasaki, M. Wakeshima, Y.Hinatsu, I. Watanabe, and Z.-A. Xu, Phys. Rev. B ,060411(R) (2013).[37] S. Onoda, N. Sugimoto,and N. Nagaosa, Phys. Rev. B , 165103 (2008).[38] W. J. Kim, T. Oh, J. Song, E. K. Ko, Y. Li, J. Mun, B.Kim, J. Son, Z. Yang, Y. Kohama, M. Kim, B.-J. Yang,and T. W. Noh, Sci. Adv. , 1539 (2020).[39] Pallab Goswami, Bitan Roy, and Sankar Das Sarma,Phys. Rev. B95