Quadratic Fermi node in a 3D strongly correlated semimetal

Strong spin–orbit coupling fosters exotic electronic states such as topological insulators and superconductors, but the combination of strong spin–orbit and strong electron–electron interactions is just beginning to be understood. Central to this emerging area are the 5d transition metal iridium oxides. Here, in the pyrochlore iridate Pr2Ir2O7, we identify a non-trivial state with a single-point Fermi node protected by cubic and time-reversal symmetries, using a combination of angle-resolved photoemission spectroscopy and first-principles calculations. Owing to its quadratic dispersion, the unique coincidence of four degenerate states at the Fermi energy, and strong Coulomb interactions, non-Fermi liquid behaviour is predicted, for which we observe some evidence. Our discovery implies that Pr2Ir2O7 is a parent state that can be manipulated to produce other strongly correlated topological phases, such as topological Mott insulator, Weyl semimetal, and quantum spin and anomalous Hall states.

-2.    Figure 3: a, Brillouin zone, indicating the measured momentum plane across , L and K (painted by orange) and momentum cuts for various photon energies (dashed lines). b, Energy distribution curves (EDCs) measured by ARPES along kx (defined in a). The used photon energy is described on the top of each panel, and the corresponding k (111) location is indicated in a. In the right panel (green and magenta curves), the EDCs at kx = 0 are extracted. The magenta arrows mark the energy position of spectral peak. A blue dashed line is plotted in each panel to confirm an energy dispersion of the peaks. c, The symmetrized EDCs of b. The red bars and dashed curves indicate the spectral peaks and the energy dispersion, respectively. In the right panel (green and magenta curves), the curves at kx = 0 are extracted. d, ARPES images symmetrized about EF obtained at various photon energies (bottom panels), and the second derivative of each image (top panels). The measured momentum cuts are indicated in a with dashed lines.  (111) and perpendicular to it (kx)], and an estimation of the kz-broadening ( kz or k (111) ). a, Brillouin zone of Pr2Ir2O7. b, The energy dispersions along the kx and k (111) directions (red and blue circles, respectively), determined from the peak positions of spectra in c and d. The former is measured at h⌫=10eV (red dashed line in the right panel of a), and the latter at various photon energies (blue circles in the right panel of a). The experimental kz broadening ( kz or k (111) ) is estimated to be ⇠ 5% of Brillouin zone at h⌫=10eV ( point), which is indicated with a light blue belt and black dimension arrows. c, d The ARPES spectra (symmetrized EDCs) along kx and k (111) , respectively. Error bars in b represent uncertainty in estimating the spectral peak positions.  Figure 5: a, Brillouin zone, indicating the measured momentum plane across , L and W (painted by orange) and momentum cuts for various photon energies (dashed lines). b, Energy distribution curves (EDCs) measured by ARPES along kx (defined in a). The used photon energy is described on the top of each panel, and the corresponding k (111) location is indicated in a. In the right panel (green and magenta curves), the EDCs at kx = 0 are extracted. The magenta arrows mark the energy position of spectral peak. A blue dashed line is plotted in each panel to confirm an energy dispersion of the peaks. c, The symmetrized EDCs of b. The red bars and dashed curves indicate the spectral peaks and the energy dispersion, respectively. In the right panel (green and magenta curves), the curves at kx = 0 are extracted.   Figure 7: a, c, ARPES intensities at EF along momentum sheets perpendicular to the k (111) axis (see panel b), crossing (h⌫ = 52 eV) and L (h⌫ = 39 eV), respectively. For the data at h⌫ = 39 eV, only the positive side of ky were measured, and it is reflected to the negative side. b, Brillouin zone, indicating the momentum sheets for the data of a and c, which are painted by red and blue, respectively. d, e, Band dispersion maps along ky and kx marked by a green and orange arrows in a, respectively. f , g, Band dispersion maps along ky and kx marked by a green and orange arrows in c, respectively. h, i, The left panels plot EDCs extracted from d and e, respectively. The magenta arrows indicate energies of spectral peaks. A blue dashed line is added to confirm the energy dispersion. To the right, the symmetrized EDCs are shown. Green and orange bars and dashed curves indicate the spectral peaks and the energy dispersion of those. j, k, The same data as h and i, but extracted from f and g, respectively.  Figure 8: Consistency between the results obtained with the FD-division method and the symmetrization method. a, The ARPES spectra (EDCs) measured at T = 75K along a momentum cut crossing indicated in d with a light blue arrow. The green dashed line is a guide to eyes for a confirmation of dispersion in the peaks (green arrows). b, The same EDCs in a, but divided by the Fermi function at 75K convoluted with the experimental energy resolution. c, The same EDCs in a, but symmetrized about EF . The color bars and dashed curves in b and c indicate the spectral peaks and the energy dispersions, respectively. d, Brillouin zone of Pr2Ir2O7 with a measured momentum cut (a light blue arrow). e, The corresponding ARPES image after the treatment of FD-division. The black arrow points to the remarkable intensity above EF , implying the existence of conduction band touching on the unoccupied side. f , The band dispersion extracted from the peak positions of spectra in b and c. The light blue dashed curve is a quadratic dispersion fitted to the data. Error bars in f represent uncertainty in estimating the spectral peak positions.  Here we show the data up to the higher temperature of 300K. b, EDCs used for the analysis in a. c, Brillouin zone: a momentum sheet measured at h⌫ = 21.2 eV and the k point where EDCs of a were taken are marked by a blue sheet and a green circle, respectively. d, The following two energy distribution curves are compared; one taken just after the sample cleaving and the other taken after completing the temperature scan presented in b. These two curves almost perfectly overlap, signifying that there is no aging of sample surface during the entire measurement. The inset shows the calculated band dispersion along the high symmetry lines (the same figure as Fig. 5d).  Figure 11: Signature of the unoccupied conduction band of Pr2Ir2O7 observed by a laser ARPES with ultra-high energy resolution. a, The ARPES spectra of Au (electrically contacted to the samples) measured at various temperatures by a 7-eV laser ARPES (the upper panel). In the bottom panel, the same spectra, but divided by the Fermi function at the measured temperature convoluted with the experimental energy resolution, are plotted. The energy resolution is estimated to be " = 2.4 meV in the present experimental setting as demonstrated in the inset panel using the Au spectrum at the lowest temperature (T = 2.7K). b, Temperature evolution of ARPES spectra for the sample (the upper panel) and the same curves after the FD-division (the lower panel), measured at the midpoint between the and L points reached at h⌫ = 7eV. The experimental setting and the FD-division method applied to the data are exactly the same as for the Au in a. The arrows in a and b mark the thermally populated spectral weight above EF , exhibited up to 3kBT . c, The comparison of spectral shape before and after the temperature scan. d, Comparison between the spectra of Au and the sample (EDCs and FD-divided EDCs extracted from a and b). e, Band calculation close to EF . The kz (or k (111) ) value at h⌫ = 7eV is located at the midpoint between the and L points (a dashed green line and a green kx ky plane).

Cross midpoint
Cross L point

Cross midpoint
Cross L point

Cross midpoint
Cross L point     In this supplemental section, we discuss the electronic structure calculations in the paramagnetic state. Calculations were carried out using both the standard generalized gradient approximation (GGA) [1] and the Tran-Blaha modified Becke-Johnson (TB-mBJ) [2] exchange potential as implemented in Wien2k [3], for a series of di↵erent A site ions. An RKmax parameter 7.0 was chosen and the wave functions were expanded in spherical harmonics up to l wf max = 10 inside the atomic spheres and l pot max = 4 for non-mu n tins. Bulk A 2 Ir 2 O 7 (A=rare earth element) has a cubic crystal structure with the space group Fd3m. The experimental lattice parameters [4][5][6][7] were used for A= Pr, Nd, Eu and Y in both GGA and TB-mBJ paramagnetic calculations, and spin-orbit coupling was applied to both the heavy rare earth element and the Ir electrons. The paramagnetic GGA and TB-mBJ calculations put the 4f states of the rare earth element at the Fermi energy; to avoid this, since the 4f electrons are highly localized, their potential is shifted by a constant.
The TB-mBJ method is believed to produce improved results for small band gap systems [8], and has been widely used in studies of topological insulators [9]. Both methods yield almost indistinguishable results away from the Fermi energy, with a slight decrease of band-width in the TB-mBJ calculation, and small di↵erences near E F , as shown in Supplementary Fig. 1. We observed the symmetry-required nodal band touching at for all calculations, but the Fermi energy is shifted from the nodal point by an amount that decreases with increasing A (A= Y, Eu, Nd and Pr) site ionic radius in both methods [10]. This occurs because of accidental crossing of states near the L point, where the valence band rises and approaches the Fermi energy, especially in the smaller rare earths. For example, a small shift of the Fermi level of about 30 meV in the GGA approximation is observed in Pr 2 Ir 2 O 7 , while the shift is about 70 meV in Y 2 Ir 2 O 7 . In general, the GGA calculation underestimates the e↵ects of correlations, which tend to narrow the bands and to push occupied states deeper below the Fermi energy. Hence our expectation is that the accidental crossing near L, which is responsible for the Fermi level shift in GGA, will be suppressed by correlations. The TB-mBJ method may be regarded as a crude way to do this. Indeed, in the presumed more accurate TB-mBJ method, all the paramagnetic band structures for the A 2 Ir 2 O 7 series show smaller shifts of the Fermi level compared to their GGA counterparts. In the case of A=Pr, we find the shift vanishes and the nodal point occurs precisely at the Fermi energy, and a shift value of around 7 meV is observed in the case of A=Y. While there is a universal trend of smaller shift of the Fermi energy as rare earth ionic radius increases for the rare earth elements we have tested, the small di↵erences between GGA and TB-mBJ suggest that some theoretical uncertainty remains. We note, however, that we expect methods that include correlations more accurately, such as LDA+DMFT, are likely to further suppress the states near the Fermi energy at L even beyond TB-mBJ, favoring placing the Fermi level precisely at the node.
The above calculation treats Pr as non-magnetic by artificially shifting the levels away from the Fermi energy. More properly, these levels are highly localized and, due to large Coulomb repulsion, form local moments, which fluctuate thermally in the range of the ARPES measurements. A calculation which includes the localized but thermally fluctuating paramagnet state of the Pr spins requires Dynamical Mean Field Theory (DMFT), which is beyond the scope of this paper. However, we expect that the Pr moments are in any case so strongly localized that they couple only weakly to the Ir electrons. We can test this, and thereby gauge the magnitude of such e↵ects, by studying the system at low temperature using the GGA+U technique. This treats the localization of the 4f moments but does not allow them to fluctuate. It has indeed been inferred that in Pr 2 Ir 2 O 7 , the Pr moments becomes quasi-static and form some 2-in/2-out structure below 0.3K.
As the 2-in/2-out states are degenerate, we instead focus on the simpler all-in/all-out configuration of 4f moments. Although this is not the proper arrangement in Pr 2 Ir 2 O 7 , it still shows the magnitude of the e↵ects to be expected from Pr magnetic order, and specifically from time-reversal symmetry breaking. To include the non-collinear magnetism, the calculation is performed in WIENNCM (non-collinear magnetic version of the Wien2k package) [11]. We take U on the 4f electrons to be 6eV and U on the Ir 5d electrons to be 0eV. It is inevitable that the Ir 5d electrons acquire a small magnetic moment, through their interaction with the Pr ones. The largest Ir moment introduced in the GGA+U calculation is below 0.004µ B . Supplementary Fig. 2 shows the magnetic band structure of Pr 2 Ir 2 O 7 . Generally the structure of the Ir bands is very similar to that in the paramagnetic GGA calculation. This proves the weak coupling of Pr moments to the Ir electrons. However, there is indeed, as expected by symmetry, a small energy splitting of the bands in the GGA+U band structure, comparing to the originally degenerate bands in the GGA calculation. This is consistent with broken time reversal symmetry, which, as expected from low energy k · p theory, leads to the appearance of Weyl points along the L branch. The point appears near the point, along the -L line, as shown in Supplementary Fig. 2. Note that the band bending between the L and W points is larger in the GGA+U calculation than in the TB-mBJ one, so a Fermi pocket appears in this calculation in this region in the GGA+U result.

Supplementary Note 2: Energy distribution curves (EDCs) used for the symmetrization analysis
In Fig. 3 of the main paper, we show the energy distribution curves (EDCs) symmetrized about E F to remove the cut-o↵ e↵ect by Fermi function. Here we present the row EDCs used for the analysis (Supplementary Fig. 3). Moreover, we show some data which are discussed but not presented in the main paper ( Supplementary Fig. 5 and 6). Supplementary Fig. 3b shows the EDCs measured at various photon energies from h⌫ = 7 eV to 18 eV. The measured momentum cuts (dashed lines in Supplementary Fig. 3a) are located in the k x k (111) plane crossing , L, and K. As marked by magenta arrows, the quasiparticle peaks clearly disperse along k x , and that at k y = 0 (green curve in each panel) most approaches E F at h⌫ = 10 eV as examined in the right panel of Supplementary Fig. 3b. The symmetrized EDCs of Supplementary Fig. 3b are plotted in Supplementary Fig. 3c. (The same data at h⌫ = 7, 8, 9, and 10 eV are shown in Figs. 3d of the main paper.) It is clearly seen that the quadratic dispersion approaches E F with increasing the photon energy, touches E F at 10 eV, and move away from E F again with a further increase of h⌫. This behavior is more clearly demonstrated in the right panel of Supplementary Fig. 3c, where the curves at k x = 0 are extracted. The corresponding ARPES images symmetrized at E F are exhibited in the bottom panels of Supplementary Fig. 3d. The upper panels plot the second derivatives of those. The systematic variation of dispersions with photon energy, including the Fermi node state at h⌫ = 10 eV, are clearly seen.
We have further confirmed the existence of Fermi node by observing a di↵erent momentum sheet crossing , L, and W points using another piece of sample. The k x k (111) plane and measured momentum cuts corresponding to used photon energies (9.5 eV to 11.5 eV) are indicated in Supplementary Fig. 5a. Supplementary Fig. 3b shows the EDCs along k x obtained at di↵erent photon energies. The h⌫ value was swept by a fine step ( 0.5 eV, see dashed lines in Supplementary Fig. 5a). The quasiparticle peak disperses along k x , and most approaches E F at ⇠ 10 eV as examined in the right panel of Supplementary  Fig. 5b. The corresponding symmetrized EDCs are plotted in Supplementary Fig. 5c. We identify the band touching to E F in the data at h⌫ = 10.5 eV. This is more clearly presented in the right panel of Supplementary Fig. 5c, where the curves at k x = 0 are extracted. The broad shape of quasiparticle peak in Pr 2 Ir 2 O 7 does not allow us to determine the exact photon energy for the point. Nevertheless, our data signify that the Pr 2 Ir 2 O 7 has a 3D band structure with a Fermi node theoretically predicted.
In order to further validate the 3D nature of band structure in Pr 2 Ir 2 O 7 , we examined it with higher photon energies (39 6 h⌫ 6 60 eV), which reach the 3rd Brillouin Zone in the k (111) direction. Supplementary Fig. 6b plots the EDCs at various photon energies corresponding to momentum points marked by green circles in Supplementary Fig. 6a. The curves are symmetrized in Supplementary Fig. 6c to judge whether or not the dispersion touches E F . A gap clearly observed at 40 eV gets smaller with increasing h⌫, and it becomes negligible at 52 eV, where a broad single peak is obtained. With a further increase of h⌫, the gap opens again showing two peak structure in the spectra, being consistent with the Fermi nodal state. The energy dispersion along k (111) is consistent with that seen in the 1st Brillouin Zone observed with lower photon energies. The periodic trend over a wide range of photon energy ensures that our data provide the intrinsic 3D bulk band structure of Pr 2 Ir 2 O 7 .
In the right-and left-side panels of Supplementary Fig. 7, we show the band dispersion along two k x k y planes across (h⌫ = 52 eV) and L (h⌫ = 39 eV), respectively. The ARPES intensities at E F (top panels), the dispersion maps along k x and k y (middle panels), and the corresponding symmetrized EDCs (bottom panels) are also presented in Fig. 4 of the main paper. Here we plot the original EDCs before the symmetrization (left panels in Supplementary Figs. 7h-7k). As marked by magenta arrows in each panel, the quasiparticle peaks are observed. Importantly, while the dispersion is clearly seen across the point (Supplementary Fig. 7h and 7i), it is negligible across the L point ( Supplementary Fig. 7j  and 7k), which agrees to the Fermi node state.
Supplementary Note 3: Agreement between the results obtained with symmetrized spectra and FD-divided spectra In the main paper, we use two di↵erent techniques to remove the Fermi cut-o↵ e↵ect from the ARPES spectra (EDCs). One is the symmetrization method, which flips the EDC about E F and add it to the original curve. This technique is widely used to visualize the opening of a gap, and also to determine the Fermi crossing point (k F point). The method is easy to use and applicable even for the low temperature spectra with negligible intensity above E F , thus it has been commonly used for a gap estimation in the superconducting and density-wave materials. A weakness of this method is that it requires the particlehole symmetry, which is not for sure in the unknown materials like Pr 2 Ir 2 O 7 . The more straightforward way of eliminating the Fermi cut-o↵ e↵ect would be to divide the EDCs by the Fermi function (FD) at measured temperature. For the realistic analysis, the function actually used is convoluted with the experimental energy resolution. However, the FD-division method, though conceptually straightforward, also has a di culty; the high temperature data are required, since it utilizes the thermally populated intensity above E F . Moreover, the high quality data with a high signal/noise ratio are essential in order to reliably reproduce the spectral shape close to E F . Therefore, the best way to extract the band dispersion close to E F is to apply both the analyses of the symmetrization and FD-division methods to the same data, and confirm the consistency between the two results.
In Supplementary Fig.8, we demonstrate that the results obtained with these two methods agree with each other, pointing to the realization of the quadratic Fermi nodal state in Pr 2 Ir 2 O 7 . Supplementary Figure 8a plot the EDCs along a momentum cut across (light blue arrow in Supplementary  Fig.8d). Here we use the data at T = 75K, which fulfill the following two conditions: the surviving of quasiparticles and the su cient intensity of thermally populated spectral weight above E F . The EDCs in Supplementary Fig.8a divided by the Fermi function and symmetrized about E F are exhibited in Supplementary Figs. 8b and 8c, respectively. The e↵ect of Fermi cut-o↵ near E F is removed after these treatments, and importantly, the dispersion of the spectral peaks (color bars and dashed curves) touches E F at the point (magenta curves) in both the cases. The results are more clearly demonstrated in Supplementary Fig. 8f, where the peak positions in Supplementary Figs. 8b and 8c are extracted. We find that the plots obtained by the symmetrization and the FD-division techniques (red and blue circles, respectively) almost perfectly matches with each other, both showing the Fermi node state. The consistency between the two di↵erent analyses strongly support our conclusion. A further support is presented in Supplementary Fig. 8e, which plot the the ARPES image after the FD-division treatment. We find the remarkable intensities above E F (pointed by a tick black arrow), which implies the existence of the conduction band touching on the unoccupied side, as expected in the band calculation.
Supplementary Note 4: Comparison of ARPES data with band calculations: agreement with the TB-mBJ calculation, and disagreement with the GGA calculation The relevance of band calculation could be judged by directly comparing it with the experimental results. The ARPES is the most direct technique to determine the momentum-resolved band structure, thus it allows us to achieve this aim. Here we carefully compare the ARPES data and two di↵erent band calculations of GGA and TB-mBJ, and demonstrate that the TB-mBJ method more correctly reproduces the real band structure in Pr 2 Ir 2 O 7 .
There are mainly two di↵erences between the results of the GGA and TB-mBJ calculations as summarized in the left and middle panels in Supplementary Fig. 9. The first di↵erence is seen in the energy dispersion from toward L point: in GGA, it approaches E F (red arrow in Supplementary Fig. 9b), whereas goes toward higher binding energies in the TB-mBJ (red arrow in Supplementary Fig. 9e). Secondly, the two calculations are distinguished by whether or not the Fermi pocket exists around the L point: the GGA has the doughnut-shaped Fermi pockets surrounding the L point (see Supplementary  Fig. 9a and red dashed circle in 9b), whereas there is no Fermi crossings in the result of TB-mBJ with fully gapped along the L-W (Supplementary Fig. 9e).
We summarize our ARPES results in the right-side panels of Supplementary Fig. 9. We find that the band structure disperses toward higher binding energies from the towards L point in a monotonic fashion (red arrow in Supplementary Fig. 9h). Secondly, a gap keeps opening all the way from to W point with no indication of Fermi crossings (or Fermi pockets). These two features are reproduced by the TB-mBJ calculation, while disagree with the GGA calculation. The TB-mBJ is thus more relevant than the GGA for the calculation of electronic system in Pr 2 Ir 2 O 7 . Most importantly, the absence of pockets o↵ the point requires the touching of quadratic valence and conduction band just at E F to hold the charge neutrality, which justifies our conclusion.
Nonetheless, we also find a clear di↵erence between the ARPES data and the TB-mBJ calculation. The ARPES results exhibit a much narrower band width of ⇠ 40meV in the occupied side than the calculation (>100 meV). Intriguingly, the band width of Pr 2 Ir 2 O 7 is thus more than one order of magnitude narrower than the band structure observed in HgTe [12]. It is actually consistent with the view that the Pr 2 Ir 2 O 7 is a strongly correlated analog of HgTe with a mathematically identical quadratic node at the Fermi energy.
Supplementary Note 5: Dramatic peak suppression at elevated temperatures and aging check after the temperature sweeping In Fig. 5 of the main text, we investigate the temperature evolution of ARPES spectra measured, and demonstrate a strong suppression of the spectral peak at elevated temperatures. Here we confirm that this is an intrinsic property of Pr 2 Ir 2 O 7 rather than an aging e↵ect on the sample surface. Supplementary Fig. 10b and 10a show EDCs and the same curves symmetrized about E F over a wide range of temperature from 15K up to 300K. (The same data up to 150K are presented in the main paper.) We used 21.2 eV photons (He discharge lamp), which locate the k point in nearly the midpoint between and L (a green circle in Supplementary Fig. 10c and a green dashed line in the inset of Supplementary Fig. 10d). The upturn behavior toward higher energies beyond E F (arrow in Supplementary Fig. 10a) is visible regardless of temperature, which is consistent with the theoretical prediction that the quadratic band similar to that in the occupied side is stayed in the unoccupied side (see inset of Supplementary Fig. 10d). In order to check where or not there is an aging e↵ect on the spectra, we cooled the sample down to the lowest temperature after the temperature scan, and compare the spectral shapes before and after the measurement, as shown in Supplementary Fig. 10d. The spectral shape just after the sample cleaving is almost perfectly reproduced, which validates that the remarkable variation in spectral shape with temperature is an intrinsic feature of Pr 2 Ir 2 O 7 , mostly due to the strong electron correlation.
Supplementary Note 6: Signature of the unoccupied conduction band observed by a laser ARPES with an ultra-high energy resolution The ARPES is the technique to measure the occupied side of band dispersion. However, the energy states slightly above E F can be evaluated by examining the thermally populated spectral weight divided by the Fermi function (FD). The electronic system of Pr 2 Ir 2 O 7 is strongly correlated, and the quasiparticle peaks vanish at elevated temperatures. This feature prevents one from a definitive determination of the unoccupied band dispersion with this technique. Nevertheless, we have obtained the upturn behavior in the spectra above E F ( Fig. 5b and Supplementary Fig.10a), signifying the existence of the conduction band in the unoccupied side. To verify it further, here we apply the FD-division technique to the data measured by a 7-eV laser ARPES, which is capable of an ultra-high energy resolution (⇠2meV), bulk sensitive (a long photoelectron escape depth: ⇠ 100Å), and high signal/noise ratio measurements. These performances has huge advantages in reproducing the intrinsic spectral shape above E F by the FD-devision. The k z position corresponding to 7eV photons is located at the midpoint between and L, where the unoccupied conduction band is predicted to stay at almost the same energy distance from E F as the occupied valence band (Supplementary Fig.11e). Therefore, the signature for the conduction band is expected to be obtained from the spectral tail above E F at high temperatures. First, we have tested the FD-division technique to the spectra of the evaporated Au, which is electrically contacted to the samples. We confirmed, in Supplementary Fig.11a, that the Au spectra divided by the energy-resolution convoluted FD becomes flat at least up to 3k B T above E F , meaning that the analysis we use e↵ectively removes the Fermi cut-o↵ e↵ect near E F . We have accumulated the spectra of Pr 2 Ir 2 O 7 with exactly the same experimental setting as that for the Au measurements, and also applied the identical FD-division analysis to the data in the Supplementary Fig.11b. We have reproduced the previous results of spectral peak suppression at elevated temperatures, indicating the significance of strong correlation in this material. Furthermore, the upturn behavior toward higher energies beyond E F is also obtained in the present data (see the bottom panel of Supplementary Fig.11b). Here we emphasize that the thermally populated spectral weight above E F seen in the data is large enough to reliably extract the intrinsic electronic state in the unoccupied side by the FD-division technique (see Supplementary Fig.11d) at least up to 3k B T above E F . This feature is exactly expected when the conduction touching-band exists in the unoccupied side. The consistent results from the data sets at di↵erent photon energies justify our conclusion.
After the above measurement of temperature scan, we cooled the sample again down to the lowest temperature and compared the spectral shape with that taken just after the sample cleaving. As demonstrated in Supplementary Fig.11c, the two curves are almost perfectly overlapped with each other, meaning that the sample aging is negligible. This assures that the strong suppression of spectral peaks at elevated temperatures seen in Supplementary Fig.11b  The TB-mBJ paramagnetic band structure calculations show Pr 2 Ir 2 O 7 has a nodal Fermi point, at which the doubly degenerate, quadratically dispersing conduction and valence bands touch at the zone center, right at the Fermi level. This suggests that Pr 2 Ir 2 O 7 is very sensitive to perturbations, such as time reversal symmetry or cubic symmetry breaking terms, giving rise to the possibility of various novel phases. Indeed, the e↵ect of small uniaxial strain near the point can be deduced, up to a sign, by symmetry within the e↵ective mass method [13]. For the proper sign, uniaxial strain is predicted from this approach to open a gap and produce a topological insulator. However, on pure symmetry grounds, it is not clear whether uniaxial compression or expansion is appropriate.
In this supplemental section, we consider this problem microscopically. Uniaxial strain is applied along the h111i direction, which breaks the cubic symmetry and converts the system into a rhombohedral crystal structure with space group R3m. Since TB-mBJ has no exchange functional [2], the GGA functional is used for structural optimization, both for the lattice parameters and the atomic positions. In the hexagonal unit cell setting, the in-plane lattice parameter is fixed and the out-of-plane lattice parameter is varied to minimize the total energy. We consider the case of 1% (a =1.01a 0 , a 0 = 7.35Å, the bulk in-plane lattice parameter in the hexagonal cell) and 5% (a =1.05a 0 ) in-plane tensile strain, which is equivalent to uniaxial compressive pressure along h111i direction, in Supplementary Figs. 12c and 12d, respectively, and 5% (a =0.95a 0 ) in-plane compressive strain in Supplementary Fig. 12a. Using the relaxed structure, the TB-mBJ paramagnetic band structure calculation shows that a gap opens with in-plane tensile strain, and the gap increases as the strain intensifies, as shown in Supplementary Figs. 12c and 12d. The quadratic band touching state at the point is very sensitive to the cubic symmetry breaking perturbation, and immediately the degeneracy at the point is split as strain is applied. We indeed observe the direct gap at the point is substantial and grows proportionally with strain. However, there is a closing of di↵erent states away from , closer to the Z point (⇡,⇡,⇡), which results in a reduced total gap. A small gap, about 4 meV, exists in the case of 1 % tensile strain ( Supplementary Fig. 12c) and the gap is 60 meV in the case of 5 % strain ( Supplementary Fig. 12d). The GGA calculation shows metallic state in the 1 % case and a much reduced gap, 8 meV, with the 5% tensile strain, which is attributed to the well-known fact that GGA and LDA greatly underestimated the gap. On the other hand, by applying in-plane compressive strain ( Supplementary Fig. 12a), Pr 2 Ir 2 O 7 becomes metallic and shows a band crossing along the h111i direction. This band crossing is indeed expected from the e↵ective mass theory of the perturbed Fermi node, for the strain of sign opposite to that which produces the topological gap [13].
To confirm that the gapped state produced by uniaxial pressure along h111i direction is topological, we further examine the Z 2 topological invariant. Since the strained structure preserves inversion symmetry, the Z 2 invariant is expressed by [14][15][16] ( is the parity eigenvalue at the time-reversal invariant momentum i (TRIM), which are tabulated in Supplementary Table 1. The Z 2 class is (1;000), which verifies that Pr 2 Ir 2 O 7 under uniaxial pressure along h111i direction indeed turns into a strong topological insulator with strong correlelation. Note that the parity eigenvalues at all points except the zone center are essentially already defined in the cubic structure, so that the topological nature is already incipient without strain.
We also consider the surface spectrum of the strained material for further evidence for the topological insulating state. In the latter, an odd number of gapless surface Dirac cones are expected, and we indeed find them, as we now show. To locate the surface state, we first obtain the tight binding Hamiltonian for h111i strained Pr 2 Ir 2 O 7 using maximally localized Wannier functions [17,18], in the J e↵ = 1/2 basis. Both the orbital and spinor parts are rotated to the local axes of each inequivalent Ir atom. The rhombohedral unit cell is modified to the hexgonal unit cell, and we consider a thick slab of Pr 2 Ir 2 O 7 to accommodate with the surface state. The slab is fixed to include 73 Ir layers, with identical Ir kagome layers on the upper and lower surface of the slab. Supplementary Fig. 13a shows the band structure of the bulk and the surface state for Pr 2 Ir 2 O 7 under 5% uniaxial strain on the h111i surface, choosing the large strain to better distinguish bulk and surface states. The Dirac points are located at the M points. On the surface Brillouin zone there are four TRIM, which are located at and three M points, which are equivalent by 120 rotations. There are three symmetry operations of the crystal with space group R3m, including the inversion, 3-fold rotation along the h111i direction and two-fold rotation around x direction (orthorgonal to h111i). The 2D Brillouin zone also obeys these symmetry operations, making the three M points all equivalent. In total there are three Dirac cones around the TRIM on the 2D surface, which is consistent with the parity argument.