Supplementary Figure S1: Number of Fermi surfaces. Electronic dispersion around Γ a = 0 and Γ b = π/a. In (a) the number of Fermi surfaces is even,

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1 Supplementary Figure S1: Number of Fermi surfaces. Electronic dispersion around Γ a = 0 and Γ b = π/a. In (a) the number of Fermi surfaces is even, whereas in (b) it is odd. An odd number of non-degenerate Fermi surfaces leads to the topologically nontrivial TI phase in the bulk.

2 Supplementary Figure S2: Calculated bulk band structure of SmB 6. Top: Firstprinciples calculation of the bulk band structure of SmB 6, showing a 15 mev Kondo insulating gap, before and after 10 or 15 mev self-energy broadening. Due to the fact that the Kondo gap is relatively small, it is challenging to resolve the detailed dispersion of the Dirac surface states after self-energy broadening is considered. Bottom: Firstprinciples calculation on Bi 2 Se 3 surface electronic structure. The bulk band gap of Bi 2 Se 3 is as large as 300meV. Thus, after the same self-energy broadening (right panel), the surface states within the band-gap can still be very well resolved.

3 Supplementary Figure S3: ARPES dispersion map at the and pockets. (a) ARPES dispersion map for the pocket measured with photon energy of 26 ev at temperature of 7 K in SSRL. (b) Corresponding momentum distribution curves. (c) ARPES dispersion map for the pocket measured with laser- based ARPES at temperature of 6 K. (d) ARPES dispersion map for the band measured with laser-based ARPES with temperature of 6 K.

4 Supplementary Figure S4: ARPES dispersion map. These spectra are measured along the momentum space cut. These ARPES dispersion spectra are measured at SSRL beamline 5-4 with different photon energy at temperature of 10 K over the wide binding energy range.

5 Supplementary Figure S5: Surface states within the hybridization gap. (a) Schematic view of the hybridization above and below the hybridization temperature. (b) Calculation of surface electronic structure taken from ref. [12].

6 Supplementary Figure S6: Calculated bulk band structure of SmB 6. (a) The calculated band structure of SmB 6 by GGA. (b) Zoom in of (a) near Fermi level along the Γ X. The bandgap of about 15 mev was obtained in the calculation. (c) Orbital decomposed band structure near the Fermi level along Γ X. The size of the blue and yellow spheres is proportional to the weight of Sm 5d and 4f orbitals, respectively.

7 Supplementary Discussion: A diagrammatical highlight of the logical sequence for the experiment A diagrammatical highlight of the logical sequence for the experiments is sketched below: Step 1: Transport background of SmB 6, a Kondo insulator with low temperature anomaly (i) SmB 6, a heavy fermion compound, has long been under extensive studies in transport experiments. At high and intermediate temperatures (T>30 K), SmB 6 is found to behave metallic-like, but the resistivity goes up as decreasing temperature, which suggests the occurrence of Kondo hybridization leading to a Kondo gap opened at the Fermi level. However, at very low temperature below 7 K, the resistivity starts to saturate to a finite value, which cannot be explained by the physics picture of a simple Kondo insulator alone. (ii) As reported in a recent transport study [25], the Kondo hybridization gap is found to open at T 30 K. As further decreasing the temperature below 30 K, the Kondo gap increases and saturates at 15 mev at T 6 K. At the same temperature (T 6 K), the resistivity anomaly occurs, which suggests the existence of in-gap states. Step 2: The predicted topological phase in SmB 6 (i) Kondo hybridization: At temperatures below T<30 K, Kondo hybridization gap gradually opens at the Fermi level. In momentum space, the hybridization gap is found to locate in the vicinity of the three X points (X1, X2 and X3) along the momentum space cut-direction (see figure 3 main-text). (ii) Prediction of topological phase: Very recently, it has been predicted that the Kondo insulator SmB 6 can host topological insulator phase, and therefore realizes a topological Kondo insulator (TKI). The predicted topological surface states are consistent with the low temperature transport resistivity anomaly. Such new phase of topological matter is of great interest since it enables to test the interplay between topological order and strong electron- electron correlation. According to recent theoretical studies [12], the bulk bandinversion occurs at the Kondo hybridization gap, which locates near the X points. Since there are three X points in the first BZ, therefore, there are in total three band-inversions in SmB 6, which leads to a nontrivial Z2 topological insulator phase. This is further confirmed by the bulk band parity calculation, where the 3X points show negative (inverted) parity and other 2 high symmetry points in the BZ all show positive (noninverted) parity. (iii) Predicted topology of surface Fermi surface: At the (001) surface, these three X points, X1, X2 and, project onto the, and, respectively. Therefore, there exist three distinct topological surface state Fermi surfaces that enclose the, and points at the (001) surface. This is indeed confirmed by the surface electronic structure calculation in [12]. Step 3: Conditions required for the experimental demonstration of the TKI phase in SmB 6

8 General conditions: It has been widely accepted that ARPES and spin-resolved ARPES are the experimental tools to identify a three-dimensional topological insulator phase. Specifically, one needs to use ARPES to show that, (1.a) a surface Fermi surface with odd number of pockets must exist. (1.b) Fixing the chemical potential at an energy level within the bulk band-gap and by traversing the momentum space from the surface Brillouin zone (BZ) center to the BZ edge at a Kramers point, the system must feature an odd number of surface band crossings the Fermi level (EF), as shown in Fig. S1. An even number of surface state crossings will be a signature of topological triviality. (2) The surface states cannot be even number like the Rashba surface states. (3) These pockets can only be around the Kramers points of the lattice Brillouin zone not any high symmetry points of the BZ. (4) Surface states must have non-trivial Berry s phase. Berry s phase is a phase whose decisive proof has to come from an interfere experiment (not spin-arpes which is not directly phase sensitive since there is no interference involved). (5) There must be (a) bulk-boundary correspondence and (b) non-trivial entanglement entropy (and this is the most precise proof of topological-order: see, Ref. [30, 31]). Temperature: In the predicted TKI phase in SmB 6, the Kondo gap serves as the bulk band-gap in a regular Z2 TI such as in Bi 2 Se 3. As mentioned above, transport experiments found that the Kondo hybridization gap gradually increases as temperature is decreased from 30 K [25] and saturates at 15 mev when temperature reaches about 6 K. Therefore, it is critically important to work at a temperature sufficiently below 30 K so that the Kondo gap is pronounced and fully formed and the predicted topological surface states within the Kondo gap can be well-defined in energy and momentum space. Furthermore, the transport anomaly also occurs around 7 K. Therefore, to unambiguously observe the topological surface states, temperature T<=6 K is necessary. Energy resolution: Even at temperature T<=6 K, the full Kondo gap (occupied to unoccupied states) is only 15 mev. And the predicted surface states lie throughout within this Kondo gap. It is possible that Fermi level (chemical potential) lies somewhere inside the gap. Therefore, in order to resolve any dispersion of those surface states, it is critical to have an energy resolution sufficiently better than 7 mev. For example, E 15/2 7 mev would be the minimum requirement. Finite quasi-particle lifetime: Assuming ARPES has a near-ideal energy resolution ( E<1 mev) for this problem at hand, another very important factor has to be taken into account is that quasi-particles have finite lifetime (since the gap is small), which leads to a selfenergy broadening to the ARPES dispersion spectrum. For example, Fig. S2 shows the theoretically calculated SmB 6 bulk d f band structure (blue lines) with a Kondo gap of 15 mev before and after a 10 or 15 mev Lorentzian broadening. It can be seen that after self-energy broadening, it becomes challenging to clearly track the dispersion of the surface states (red lines) at a precise qualitative level, which are within the 15 mev Kondo gap, as shown in Fig. S2. In contrast, similar broadening process applied on Bi 2 Se 3 calculation gives a spectrum where the bands can be well-resolved, because the surface states disperse in a much larger bulk energy gap ( 300 mev).

9 Therefore, finite quasi-particle lifetime is a critical factor that makes it is challenging to resolve the dispersion of surface states in SmB 6. Ultra-high quality SmB 6 samples (for example, grown by molecular beam epitaxy) with much higher transport mobility is necessary for such purpose. The above demonstration of the self-energy broadening effect in a 15 mev Kondo gap in SmB 6 also justify our emphasis of using the partially angle (momentum) integrated ARPES spectral intensities in our present work. Since it is almost impossible to completely resolve the E-k dispersion of the surface states in the angle (momentum) resolved ARPES dispersion maps if a 10 mev self-energy is present. Step 4: Spin-resolved measurements Spin-resolved measurement is not feasible to measure in-gap state in SmB 6 due to the currently available energy/momentum resolution and temperature limit of spin-resolved ARPES system: typically the energy resolution of SR-ARPES at Synchrotrons is about 50 mev >> 5 mev Kondo scale [see [32]], and with laser 14 mev >> 5 mev (see [33]) and lowest accessible temperature with best spin-resolution is about 20 K >> 8K transport anomaly scale. With the latest technology at hand it is currently impossible to prove points 4-5 of general conditions on SmB 6 and point 5 of general conditions in any TI material. We have presented general points (1-3) for SmB 6 and it is more important than the Bi 2 Se 3 series because SmB 6 is not a band (topological) insulator of simple Bloch type but a Kondo-like or mixed-valence (correlated) insulator which is novel and interesting beyond the Bi-based compounds. Circular dichroism & Berry s Phase: Recently the observation of the finite circular dichroic (CD) ARPES signal on the low energy states of SmB 6 is argued to be an evidence of nontrivial spin texture. However, it has been shown that CD-ARPES does not uniquely represent the spin texture profile of the topological surface states and therefore the π Berry s phase [34 36]. For example in Ref. [34], CD-ARPES measurements on Bi 2 Te 3 shows dramatic photon energy dependence, where even the sign of the CD- ARPES signal flips as the photon energy is changed. Therefore, it is not possible to draw unique conclusion on the spin-texture of the surface states via CD-ARPES measurements. Moreover, in Ref. [36], even the bulk bands of TI Bi 2 Te 2 Se shows strong CD signal at certain photon energy. Therefore, CD-ARPES cannot be used to provide a unique measure of the surface spin texture or Berry s phase. Step 5: Summary of our key results (a) We have performed low-temperature ( 6 K) and high-resolution ( 4 mev and 0.1 deg) ARPES measurements on SmB 6. The low temperature and high-energy resolution, for the first time, provide access to study the low temperature resistivity anomaly (<=8K) as well as the predicted in-gap surface states (within 15 mev (theoretical) Kondo gap <= 6K). (b) We show that 1. At very low temperature T= 6 K corresponding to the transport anomaly, we observe a pronounced ARPES spectral intensity in-gap signal within a 4 mev energy window but finite everywhere inside the gap and lying inside the 17 mev (experimental, ARPES) Kondo insulating gap, which experimentally shows the existence of in-gap states.

10 2. The in-gap states are found to become suppressed with rising temperature and disappear before 30 K, which correspond to the Kondo hybridization onset temperature at which the Kondo gap closes. Such in-gap states are further observed to be robust against thermal cycling (see 6 K thermal recycling), consistent with their topologically protected nature predicted in theoretical calculations. 3. ARPES Fermi surface (k-space map) mapping at the energy of the in-gap states shows three distinct Fermi pockets that enclose the, and points, respectively, which, strikingly, is consistent with the predicted surface state Fermi surface topology for the topological surface states of the Kondo phase in SmB The observed Fermi surface topology of the in-gap states, as well as their temperature dependence across the transport anomaly and Kondo hybridization temperatures, collectively not only provide a unique insight illuminating the nature of the residual conductivity anomaly but also serve as a strong evidence in support of the predicted topological Kondo insulator phase in SmB 6. (c) We note that the studies of in-gap states in the 4 mev energy window within the Kondo gap using high resolution ARPES is mainly based on the angle (momentum)- integrated ARPES data. The detailed energy and momentum dispersion of the in-gap surface states cannot be obtained in angle (momentum)-resolved dispersion ARPES data due to intrinsic self-energy/resolution broadening (given the best mobility of SmB 6 samples) in a 15 mev Kondo gap issue as discussed above. In summary, we have performed systematic laser- and synchrotron-based ARPES measurements. For synchrotron-based ARPES measurements, we observe quasi-2d states with clear dispersion near the Fermi level. More importantly, for ultra-high resolution and ultra- low temperature laser-arpes data, we observe that 1. At very low temperature T 6 K corresponding to the transport anomaly, we observe a pronounced ARPES spectral intensity features within the Kondo gap ( 4meV features) energy window lying inside the 17 mev Kondo insulating gap, which experimentally shows the existence of in-gap states. 2. The in-gap states are found to become suppressed while raising temperature and completely disappear at T= 30 K, which correspond to the Kondo hybridization onset temperature at which the Kondo gap closes. Such in-gap states are further observed to be robust against thermal cycling, consistent with their protected/robust nature. 3. ARPES Fermi surface measurements at energy that correspond to the in-gap states show three (odd as opposed to even) distinct Fermi pockets that enclose the, and points, respectively. 4. The observed Fermi surface topology of the in-gap states, as well as their temperature dependence across the transport anomaly and Kondo hybridization temperatures, collectively not only provide a unique insight illuminating the nature of the residual conductivity anomaly but also serve as a strong supporting evidence for the existence of a topological Kondo insulator phase in SmB 6. In-gap states The temperature-dependent behavior of states near the Fermi level is observed at both the

11 and the points (see Fig. 2d main text). At 7 ev laser-based photoemission process, photoionization cross-section for the f orbitals is weak, so the measured states should significantly correspond to the partial d orbital character of the hybridized band if the hybridization exists there. Based on the temperature dependent electronic structure presented in Figure 2 of main text, we observe two relevant temperatures, which are around 15 K and 30 K. We note that these in-gap states are robust below 15 K, appear to be weak above 15 K and finally vanish above 30 K (Figs. 2c and d in the main text). The robustness of these states below 15 K is in correspondence with the 2D conductivity channels in SmB 6. It is important to note that the in-gap states at point can only be observed with high- resolution measurements ( 4meV and 0.1 deg) performed at low temperature (<8 K) which reveals the importance of high energy-momentum resolution of the ARPES measurement (Figure S3 versus Figure S4). Bulk band calculations of SmB 6 Figure S6 shows the calculated bulk band structure of SmB 6, which agrees with the previous calculations [2,10 11,37,38]. With fined tuned parameters, our GGA calculations give an insulating ground state with a small energy gap of 15 mev (Fig. S6b). From the orbital- decomposed band structure (Fig. S6c), we find that the flat Sm 4f bands are located around EF from -0.5 ev. An itinerant Sm 5d band with a larger dispersion hybridizes with the 4f bands, forming an anti-crossing band shape. Our GGA calculations qualitatively agree with the observed bulk band dispersion. Supplementary Methods: In order to further cross check the bulk band dispersion, the first-principles bulk band calculations were performed based on the generalized gradient approximation (GGA) [39] using the projector augmented-wave method [40,41] as implemented in the VASP package [42,43]. The experimental crystallographic structure was used [44] for the calculations. The spin- orbit coupling was included self-consistently in the electronic structure calculations with a Monkhorst-Pack k mesh. Supplementary References: [30] Zhu, W. et al. Identifying Non-Abelian Topological Order through Minimal Entangled States. arxiv: (2013). [31] Gu, Z.-C et al. Tensor-entanglement-filtering renormalization approach and symmetryprotected topological order. Phys. Rev. B 80, (2009). [32] Hoesch, M. et al. Spin-polarized Fermi surface mapping. Journal of Electron Spectro. Relate. Phenom 124, (2002). [33] Xue, Q.-K. et al. Topological insulators: Full spin ahead for photoelectrons. Nat. Phys. 9, (2013). [34] Scholz, M. R. et al. Reversal of the Circular Dichroism in Angle-Resolved Photoemission from Bi2Te3. Phys. Rev. Lett. 110, (2013). [35] Jozwiak, C. et al. Photoelectron spin-flipping and texture manipulation in a topological insulator. Nat. Phys. 9, (2013). [36] Neupane, M. et al. Oscillatory surface dichroism of an insulating topological insulator

12 Bi2Te2Se. Phys. Rev. B 88, (2013). [37] Dzero, M. et al. Theory of topological Kondo insulators. Phys. Rev. B 85, (2012). [38] Ye, M. et al. Topological crystalline Kondo insulators and universal topological surface states of SmB6. arxiv: (2013). [39] Perdew, P. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 77, 3865 (1996). [40] Blochl, P. E. Projector augmented-wave method. Phys. Rev. B 50, (1994). [41] Kresse, G., Joubert, D. From ultrasoft pseudopotentials to the projector augmentedwave method. Phys. Rev. B 59, 1758 (1999). [42] Kresse, G. & Hafner, J. Ab initio molecular dynamics for open-shell transition metals. Phys. Rev. B 48, (1993). [43] Kress, G. & Furthmuller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B. 54, (1996). [44] Malyshev, A. L. et al. Neutron powder diffraction studies of mixed Ce1 xlax11b6 and isotope substitured SmB6. Mater. Sci. Forum 62, (1990).