パイロクロア型イリジウム酸化物で直接観測する{フェルミノード状態

Observation of a quadratic Fermi node in the pyrochlore iridate

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[1] T. Kondo et al., Nat. commn. 6, 10042 (2015). [2] M. Nakayama, T. Kondo et al., Phys. Rev. Lett. 117, 056403 (2016).

(a) Pr2Ir2O7

(b,c) Γ L

ARPES (d)

ARPES (e) ARPESΓ

(T=70K)

the ARPES intensities at EF becomes strongest at the zone centre,and the band touching at EF is confirmed in the dispersion mapsalong kx and ky (Fig. 3d) and the corresponding symmetrizedEDCs (Fig. 4f, also see the raw EDCs in Supplementary Fig. 7). Incontrast, these features are missing in the kx–ky sheet across Lpoint (hn¼ 39 eV) (see Fig. 4c), where a rather flat, gappeddispersion is observed (Fig. 4e,g).

Second, we demonstrate that our conclusion is insensitive tothe different analytic schemes. This is significant especiallybecause the symmetrization technique is relevant for the particle–

hole symmetric state, which is unknown in Pr2Ir2O7. Accord-ingly, we have tested another widely used measure of dividing theARPES spectra by the Fermi-Dirac function (FD). Figure 5c plotsthe EDCs along a momentum cut crossing G (light-blue arrow inthe inset of Fig. 5d) measured at various temperatures. Instead ofsymmetrization, the curves are divided by the energy-resolutionconvoluted Fermi function at the measurement temperatures toremove the effect of the Fermi cutoff. It is clearly seen that thequadratic dispersion touches EF, in agreement with the earlieranalysis. In Fig. 5e, we compare the dispersions determined by the

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0.2

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10 eV

18 eV

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Z

L

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18 eV

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10 eV

kx(Å–1)

Fitting (m* = 6.3me)

k(111)(Å–1)

kx (Å –1)

Γ

Γ

Γ

Figure 3 | ARPES spectra revealing a quadratic Fermi node in the 3D band of Pr2Ir2O7. (a) Brillouin zone, showing a momentum sheet along whichARPES data were measured. The momentum cuts at hn¼ 10 and 18 eV crosses the G and L points in the 1st Brillouin zone. (b) Energy dispersions along thekx direction measured at hn¼ 7, 8, 9 and 10 eV. The corresponding momentum cuts are indicated in a by dashed coloured lines. The band dispersionobtained by the first-principles band calculation is superimposed (grey curve). The data close to EF is fitted by a parabolic function, e(k)pk2 (light-bluedotted curve). The estimated effective mass at G, meff¼ 6.3m0 (m0: free electron mass), is in agreement with the band calculation. (c) The calculated banddispersion in the kx" k(111) sheet, painted with orange in a. On it, the ARPES data in b are plotted. (d) The ARPES data (symmetrized energy distributioncurves (EDCs)) along the kx direction measured at several photon energies. All the spectra shown were accumulated at T¼ 15 K. The energy positions ofspectral peaks are marked by bars and dashed curves. (e) EDCs and the corresponding symmetrized EDCs along the G–L direction, measured at k pointsmarked with green circles in a. The energy positions of dip and hump, suggesting mode coupling, are marked by arrows and circles, respectively, on EDCs.Error bars in b represent uncertainty in estimating the spectral peak positions.

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10042

4 NATURE COMMUNICATIONS | 6:10042 |DOI: 10.1038/ncomms10042 | www.nature.com/naturecommunications

fuller identification of the non-Fermi liquid state andexplication of its physics in Pr2Ir2O7 requires higherresolution data and more elaborate analysis, beyond the scopeof this paper.

Quadratic band touching. In passing, we point out that broadspectral weight emerges beyond EF as seen in the Fermi-function-divided image for the 75-K data (arrow in Fig. 5f). The relatedspectral intensity is obtained off G, showing an upturn behaviourbeyond EF (arrow in Fig. 5b and Supplementary Figs 10 and 11,and see Supplementary Notes 5 and 6). While the strongsuppression of quasiparticle peaks at elevated temperatures pre-vents us from a definitive determination of the conduction banddispersion, the observation of spectral weight above EF iscompatible with the predicted existence of a quadratic bandtouching on the unoccupied side (Fig. 5d).

DiscussionPrior measurements6,16,31 showing ferromagnetic spin-ice-typecorrelations among the Pr moments below 2K may be explaineddue to unconventional ferromagnetic RKKY interactions arisingfrom the point-like Fermi surface. The Fermi node also leads tostrong sensitivity to small time-reversal breaking perturbations,producing Weyl points close to the Fermi energy12 and agigantic anomalous Hall effect. This is also in accord with theexperimental fact that Pr2Ir2O7 was the first material found toexhibit a large spontaneous Hall effect in a spin liquid state atzero field6,31.

Our results suggest that tuning of a unique quantum criticalpoint between an antiferromagnetic Weyl semimetal and the non-Fermi liquid nodal phase may be possible by alloying orhydrostatic pressure. Correlated topological phases and deviceapplications with iridate films could be accessed by controllingthe strain-induced breaking of the cubic crystal symmetry and

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Figure 5 | Temperature evolution of ARPES spectra and evidence for the Fermi nodal state. (a,b) Temperature variation of FD-divided EDCs measured atG (hn¼ 10.5 eV) and off G (hn¼ 21.2 eV), respectively. In d, the observed k points for a and b are marked by purple and orange dashed lines, respectively.The unoccupied side is displayed up to the energy of 3kBT. The spectral peak positions are marked by coloured circles. The original EDCs used for theanalysis are plotted in the insets. (c) The FD-divided EDCs along a momentum cut indicated in d with a light-blue arrow measured at various temperatures.The spectral peak positions are indicated by red bars and dashed curves. The Fermi node is marked by a magenta circle. At T¼ 150K, the spectral peaks arestrongly suppressed, and thus the dispersion cannot be determined. (d) Band dispersions obtained by the first-principles calculation, magnified in theregion close to EF. In the inset, the Brillouin zone of Pr2Ir2O7 is shown. (e) The energy dispersion along a momentum cut crossing G (light-blue arrow in d),determined from the peak positions of the FD-divided spectra at T¼ 75K in c (red circles). The same plots, but determined from the peak positions ofsymmetrized spectra, are superimposed (blue circles). The quadratic dispersion (light-blue dashed curve) nicely fits the data. (f) ARPES dispersion mapcrossing G (light-blue arrow in d) measured at hn¼ 10.5 eV. The original ARPES intensities are divided by the energy-resolution convoluted Fermi functionat the measured temperature of T¼ 75K to remove the cutoff effect near EF. The thick black arrow points to the remarkable spectral weight above EF,consistent with the existence of conduction touching band in the unoccupied side. Error bars in e represent uncertainty in estimating the spectral peakpositions.

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10042

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symmetrization and the FD-division methods, and confirm theconsistency between the two (see Supplementary Fig. 8 andSupplementary Note 3 for more details).

Comparison between ARPES results and band calculations.In Fig. 3b, we compare our experimental results near G with theab initio dispersion (grey curve), which has a purely quadraticshape of the electronic structure given by e(k)pk2. Close to EF,we find an agreement between the two, demonstrating that thequadratic curve (light-blue dotted line) fits well to our data withalmost the same effective mass, meff¼ 6.3m0 (m0 is the mass of afree electron). On the other hand, for energy below " 0.012 eV,the measured dispersion deviates remarkably from the parabolicshape. This contrasts to the calculation, which matches withe(k)pk2 up to much higher binding energies. It is possible thatthe deviation is due to correlation effects beyond the bandcalculation. Intriguingly, the total bandwidth in the occupied sideis estimated to be B40meV (arrows in Fig. 4f,g), which isactually much narrower than that (4100meV) of the bandcalculation (see Fig. 2g and Supplementary Fig. 9e,h). Bandnarrowing relative to the density functional theory is indeed awell-known characteristic of correlated electrons. If that isthe case, however, the agreement of the effective mass around Gbetween the data and calculation would be a coincidence. Thisis understandable, considering that the band shape of Pr2Ir2O7with comparable energy scales between the spin–orbit interactionand the electron correlations is sensitive to different calcu-lation methods (see Supplementary Note 4; SupplementaryFigs 1 and 9).

Another possible cause of the discrepancy between the dataand calculations is the strong coupling of electrons to the bosonicmodes, which also could significantly renormalize the band shape.Indeed, the peak–dip–hump shape, which is a characteristicsignature of strong mode coupling, is seen in EDCs (Fig. 3e),being consistent with this scenario. One of candidates for thebosonic modes is the phonons, which are usually coupled to thecorrelated systems very strongly. Another candidate is suggestedby the similarity of the slightly distorted band shape tomeasurements in graphene, where it was attributed to electron–plasmon coupling24. Namely, the origin could be the same in thetwo cases: emission of collective modes through vertical interbandtransitions becomes possible above a threshold energy, modifyingthe spectrum.

Correlation-driven anomalous temperature evolution. Thestrongly correlated feature of 5d electrons of Pr2Ir2O7 should beobserved in the ARPES spectra. We find such a signature in thetemperature evolution of the spectral shape. Figure 5a,b plotsFermi-function-divided EDCs at G (hn¼ 10.5 eV) and off G(k(111)¼ 0.31 Å" 1 reached at hn¼ 21.2 eV). The sharp peakclearly seen in the low-temperature spectra is strongly sup-pressed at elevated temperatures. The behaviour is clearly moremarked than the thermal broadening effects observed in theother strongly collated systems such as the well-studied cup-rates, which have the ‘marginal’ Fermi liquid state25,26. We notethat the peak suppression in the present data is acceleratedacross B100 K, differently from the typical thermal broadeningwith a gradual increase over a wider temperature range. We findthat, associated with the suppression, a large portion of spectralweight above EC" 0.1 eV is transferred to higher bindingenergies in Pr2Ir2O7. A plausible origin for it is the polaroniceffects, which could become crucial in the purely screenedelectronic systems, as also proposed for the other stronglycorrelated systems such as the perovskite iridates, manganitesand lightly doped cuprates27–29. These features are compatiblewith the non-Fermi liquid behaviour predicted theoretically forthe Fermi node phase in refs 11,30, and may also berelated to recent observations of quantum critical behaviour inthermodynamic measurements of Pr2Ir2O7 (ref. 17). However, a

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Energy (eV) Energy (eV)

~ 40 meV ~ 40 meV ~ 40 meVg

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XWK

Z

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a

f

Figure 4 | ARPES intensity mapping at the Fermi level across C and Lpoints. (a) Brillouin zone, showing momentum sheets along which ARPESdata were measured. The momentum cuts at hn¼ 52 and 39 eV crosses theG and L points, respectively, in the 3rd Brillouin zone. (b,c) ARPESintensities at EF in the kx" ky sheet crossing G and L measured at hn¼ 52and 39 eV, respectively. For the data at hn¼ 39 eV, only the positive side ofky were measured, and it is reflected to the negative side. (d) ARPESdispersion maps for hn¼ 52 eV, measured along ky and kx crossing G (greenand orange arrows in b, respectively). (e) The same data as d, but forhn¼ 39 eV, which cross the L point. (f,g) EDCs of d and e, respectively,symmetrized about EF. The dimension arrows estimate the bandwidth ofB40meV in the occupied side.

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10042 ARTICLE

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symmetrization and the FD-division methods, and confirm theconsistency between the two (see Supplementary Fig. 8 andSupplementary Note 3 for more details).

Comparison between ARPES results and band calculations.In Fig. 3b, we compare our experimental results near G with theab initio dispersion (grey curve), which has a purely quadraticshape of the electronic structure given by e(k)pk2. Close to EF,we find an agreement between the two, demonstrating that thequadratic curve (light-blue dotted line) fits well to our data withalmost the same effective mass, meff¼ 6.3m0 (m0 is the mass of afree electron). On the other hand, for energy below " 0.012 eV,the measured dispersion deviates remarkably from the parabolicshape. This contrasts to the calculation, which matches withe(k)pk2 up to much higher binding energies. It is possible thatthe deviation is due to correlation effects beyond the bandcalculation. Intriguingly, the total bandwidth in the occupied sideis estimated to be B40meV (arrows in Fig. 4f,g), which isactually much narrower than that (4100meV) of the bandcalculation (see Fig. 2g and Supplementary Fig. 9e,h). Bandnarrowing relative to the density functional theory is indeed awell-known characteristic of correlated electrons. If that isthe case, however, the agreement of the effective mass around Gbetween the data and calculation would be a coincidence. Thisis understandable, considering that the band shape of Pr2Ir2O7with comparable energy scales between the spin–orbit interactionand the electron correlations is sensitive to different calcu-lation methods (see Supplementary Note 4; SupplementaryFigs 1 and 9).

Another possible cause of the discrepancy between the dataand calculations is the strong coupling of electrons to the bosonicmodes, which also could significantly renormalize the band shape.Indeed, the peak–dip–hump shape, which is a characteristicsignature of strong mode coupling, is seen in EDCs (Fig. 3e),being consistent with this scenario. One of candidates for thebosonic modes is the phonons, which are usually coupled to thecorrelated systems very strongly. Another candidate is suggestedby the similarity of the slightly distorted band shape tomeasurements in graphene, where it was attributed to electron–plasmon coupling24. Namely, the origin could be the same in thetwo cases: emission of collective modes through vertical interbandtransitions becomes possible above a threshold energy, modifyingthe spectrum.

Correlation-driven anomalous temperature evolution. Thestrongly correlated feature of 5d electrons of Pr2Ir2O7 should beobserved in the ARPES spectra. We find such a signature in thetemperature evolution of the spectral shape. Figure 5a,b plotsFermi-function-divided EDCs at G (hn¼ 10.5 eV) and off G(k(111)¼ 0.31 Å" 1 reached at hn¼ 21.2 eV). The sharp peakclearly seen in the low-temperature spectra is strongly sup-pressed at elevated temperatures. The behaviour is clearly moremarked than the thermal broadening effects observed in theother strongly collated systems such as the well-studied cup-rates, which have the ‘marginal’ Fermi liquid state25,26. We notethat the peak suppression in the present data is acceleratedacross B100 K, differently from the typical thermal broadeningwith a gradual increase over a wider temperature range. We findthat, associated with the suppression, a large portion of spectralweight above EC" 0.1 eV is transferred to higher bindingenergies in Pr2Ir2O7. A plausible origin for it is the polaroniceffects, which could become crucial in the purely screenedelectronic systems, as also proposed for the other stronglycorrelated systems such as the perovskite iridates, manganitesand lightly doped cuprates27–29. These features are compatiblewith the non-Fermi liquid behaviour predicted theoretically forthe Fermi node phase in refs 11,30, and may also berelated to recent observations of quantum critical behaviour inthermodynamic measurements of Pr2Ir2O7 (ref. 17). However, a

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Figure 4 | ARPES intensity mapping at the Fermi level across C and Lpoints. (a) Brillouin zone, showing momentum sheets along which ARPESdata were measured. The momentum cuts at hn¼ 52 and 39 eV crosses theG and L points, respectively, in the 3rd Brillouin zone. (b,c) ARPESintensities at EF in the kx" ky sheet crossing G and L measured at hn¼ 52and 39 eV, respectively. For the data at hn¼ 39 eV, only the positive side ofky were measured, and it is reflected to the negative side. (d) ARPESdispersion maps for hn¼ 52 eV, measured along ky and kx crossing G (greenand orange arrows in b, respectively). (e) The same data as d, but forhn¼ 39 eV, which cross the L point. (f,g) EDCs of d and e, respectively,symmetrized about EF. The dimension arrows estimate the bandwidth ofB40meV in the occupied side.

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10042 ARTICLE

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