Multiple topological Dirac cones in a mixed-valent Kondo semimetal: g-SmS
Chang-Jong Kang, Dong-Choon Ryu, Junwon Kim, Kyoo Kim, J.-S. Kang, J. D. Denlinger, G. Kotliar, B. I. Min
MMultiple topological Dirac cones in a mixed-valent Kondo semimetal: g -SmS Chang-Jong Kang , ∗ Dong-Choon Ryu , ∗ Junwon Kim , KyooKim , , J.-S. Kang , J. D. Denlinger , G. Kotliar , , and B. I. Min † Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea MPPHC CPM, Pohang University of Science and Technology, Pohang 37673, Korea Department of Physics, The Catholic University of Korea, Bucheon 14662, Korea Advanced Light Source, Lawrence Berkeley Laboratory, Berkeley, California 94720, USA Condensed Matter Physics and Materials Science Department,Brookhaven National Laboratory, Upton, New York 11973, USA (Dated: August 16, 2019)We demonstrate theoretically that the golden phase of SmS ( g -SmS), a correlated mixed-valentsystem, exhibits nontrivial surface states with diverse topology. It turns out that this materialis an ideal playground to investigate different band topologies in different surface terminations.We have explored surface states on three different (001), (111), and (110) surface terminations.Topological signature in the (001) surface is not apparent due to a hidden Dirac cone inside thebulk-projected bands. In contrast, the (111) surface shows a clear gapless Dirac cone in the gapregion, demonstrating the unambiguous topological Kondo nature of g -SmS. Most interestingly, the(110) surface exhibits both topological-insulator-type and topological-crystalline-insulator (TCI)-type surface states simultaneously. Two different types of double Dirac cones, Rashba-type and TCI-type, realized on the (001) and (110) surfaces, respectively, are analyzed with the mirror eigenvaluesand mirror Chern numbers obtained from the model-independent ab initio band calculations. PACS numbers:
Introduction . Topological insulators have been studiedintensively in recent years as a new phase of quantummatter and for possible applications to spintronics andquantum computing [1, 2]. The physical significance oftopological insulators is that they possess metallic sur-face states that are protected by time-reversal symme-try, which leads to the exhibition of robust surface statesunder any perturbations without breaking the symmetry[1, 2]. Since the theoretical study proposed that non-trivial topology emerges in strongly correlated mixed-valent or Kondo systems [3, 4], subsequent experimentshave been conducted to examine the topological proper-ties in a candidate material SmB [5–18]. Several angle-resolved photoemission spectroscopy (ARPES) measure-ments confirmed a topologically-driven metallic surfacestate [11–15] and its spin helicity in the (001) surface[16], where these topological properties are induced solelyfrom a single Dirac cone. But a few ARPES reports claimthat the observed surface states are just trivial [17, 18].So the controversy still remains.Another candidate for a topological Kondo materialwas proposed theoretically in a Sm mixed-valent/Kondosystem, golden phase of SmS ( g -SmS). But, unlike SmB that has a simple-cubic structure, g -SmS crystallizes inrock-salt-type face-centered cubic (fcc) structure and hasa semi-metallic electronic structure. The difference incrystal symmetry gives rise to a richer or more intricatetopological structure. Due to the odd number of bandinversions in the bulk band structure, g -SmS was sug-gested to be a topological compensated semimetal [19].However, (001) surface states exhibit double Dirac cones with a tiny gap instead of gapless Dirac cones [20]. Thisresult is in contrast with that of Kasinathan et al. [21],who reported the existence of gapless Dirac cones on the(001) surface of the isoelectronic compound SmO. Thuswhether a gap exists in the double Dirac cones on the(001) surface of g -SmS is still unclear, which is an impor-tant issue to be clarified in connection with its topologicalnature.The double Dirac cones appear when two single-Diraccones, which are induced from band inversion at non- FIG. 1: (Color Online) Bulk and surface BZs of fcc g -SmS.There are mirror-symmetry lines along ¯Γ − ¯ M and ¯Γ − ¯ X onthe (100) surface BZ, and along ¯Γ − ¯ X and ¯Γ − ¯ Y on the (110)surface BZ. a r X i v : . [ c ond - m a t . s t r- e l ] A ug FIG. 2: (Color Online) Semi-infinite TB slab calculations for the (001) surface of g -SmS with (a) normal SOC and (b) theenhanced SOC strength by ten times. While the gap in the double Dirac cones in (a) is too tiny to be identified, the gap isclearly shown in (b) with the enhanced SOC strength. Mirror eigenvalues along ¯Γ − ¯ M are denoted in (b). (c),(d) The evolutionsof the Wannier charge centers (WCCs), respectively, for k y = 0 and k x = k y mirror-symmetry planes. Two distinct MCNs,( C , C d ) = ( − , +1), are obtained from the Wilson-loop calculations. (e) Schematic diagram for the gap-opening mechanismin the double Dirac cones along ¯Γ − ¯ M − ¯Γ (see Fig. 1). The bands having the same mirror eigenvalue (M) hybridize with eachother to produce the energy gap. (f),(g) The FS, and its spin texture around ¯ M . The spin texture shows the Rashba-type spinpolarization. For the comparison to experiment, one needs to take into account the scale factor of Z ≈ y -axis DFTenergies in (a) and (b) due to the band renormalization effect. equivalent bulk k -points, are projected onto one k -pointin the surface Brillouin zone (BZ) (see Fig. 1). The dou-ble Dirac cones were detected in ARPES measurementsfor CeBi [22, 23] and also for topological-crystalline in-sulators (TCIs) [24] of SnTe [25–27] and SnSe [28, 29],having the same rock-salt structure as g -SmS. Note thatdouble Dirac cones observed in the Sn-chalcogenides havebasically different topology from those in CeBi. The Sn-chalcogenides have the gapless double Dirac cones, whileCeBi has the gapped double Dirac cones. Therefore, it isimperative to identify the topological nature of the dou-ble Dirac cones realized in g -SmS.In this work, we have investigated the surface states of g -SmS, which is one of the representative Sm compoundsexhibiting mixed-valent Kondo properties, employing thedensity functional theory (DFT) and the Wilson-loop cal-culations. The strong correlation effect of 4 f electrons inKondo systems can be captured by renormalizing DFT4 f bands with a reasonable scale factor. Indeed we haveshown before that the DFT band structures near theFermi level ( E F ) have similar shape to those obtained bythe dynamical mean-field theory (DMFT) at low tem-perature after rescaling the DFT band with the DMFTrenormalization factors [14, 20, 30, 31]. With this rescal-ing in the bulk, the slab calculations properly describethe surface states [32].Here we have demonstrated for g -SmS that (i) it has nontrivial mirror Chern numbers (MCNs), (ii) the (001)surface has the gapped double Dirac cones seemingly ofRashba-type, (iii) the (111) surface has a clear singleDirac cone in the gap region, confirming that g -SmS isindeed a topological Kondo system, and (iv) the (110)surface has the TCI-type double Dirac cones as wellas the intriguing topological-insulator (TI)-type singleDirac cone. Method . For the DFT calculations, we have usedthe projector-augmented wave band method [33], imple-mented in VASP [34]. We have employed the generalized-gradient approximation [35] for the exchange-correlationfunctional. A lattice constant of a = 5 . A was usedfor g -SmS. To investigate surface electronic structures,we have constructed the tight-binding (TB) Hamilto-nian from DFT results, using the Wannier interpolationscheme implemented in WANNIER90 code [36], and then TABLE I: Products of parity eigenvalues of the occupiedstates at the time-reversal invariant momentum (TRIM)points of the bulk BZ of fcc g -SmS. It indicates nontrivial Z topology. Γ 3 X L Z g -SmS + − + 1 performed semi-infinite TB slab calculations using theGreen function method [37] implemented in Wannier-Tools [38]. We have double-checked the surface bandstructures by performing the DFT slab calculations withboth the WIEN2K [39] and the VASP [34] codes. Thetopological nature of surface states is analyzed in termsof the mirror eigenvalues and MCNs [40–44], which areobtained from the Wilson-loop calculations [45, 46] basedon the model-independent ab initio calculations. (001) surface . Since g -SmS shows Sm 4 f -5 d band in-version at X of the bulk BZ [19, 20], it provides a non-trivial Z number, as shown in Table I. On the (001)surface of g -SmS, one X point is projected onto ¯Γ, whiletwo non-equivalent bulk X points ( X and X (cid:48) in Fig. 1)are projected onto the ¯ M point of the surface BZ, andso the single and double Dirac cones are expected to berealized, respectively, at ¯Γ and ¯ M .Figure 2 shows the (001) surface band structures ob-tained from semi-infinite TB slab calculations. The sin-gle Dirac cone at ¯Γ is hardly detectable in Fig. 2(a) dueto an overlap with the bulk band structures. On theother hand, the double Dirac cones are noticeable at ¯ M ,which seem to be gapless as claimed by Kasinathan etal. [21] for SmO. However, a tiny band gap actually ex-ists. The gap opening is clearly identified for the calcula-tion with ten-times enhanced spin-orbit coupling (SOC)strength in Fig. 2(b) [47], which manifests the gappeddouble Dirac cones of Rashba-type in agreement with aprevious report [20]. As shown in Fig. 1, on the (001) sur-face, there are two mirror-symmetry lines along ¯Γ − ¯ X and ¯Γ − ¯ M . Therefore, the gap opening along ¯ X − ¯ M isobvious because there is no mirror symmetry to protectthe band crossings. But, further analysis is required forthe surface states along ¯Γ − ¯ M .To explore the origin of the gap opening along ¯Γ − ¯ M ,we have calculated MCNs and the mirror eigenvalues ofthe surface states. In the bulk BZ, there are two indepen-dent mirror-symmetry planes: the k y = 0 and k x = k y planes [48]. Note that two mirror operators with re-spect to the k y = 0 and k y = π planes are symmetri-cally equivalent. Therefore g -SmS has two independentMCNs, C ( ≡ C + k y =0 ) and C d ( ≡ C + k x = k y ), where the ‘+’sign refers to the mirror eigenvalue of + i for the corre-sponding mirror-symmetry plane.Figures 2(c) and 2(d) show the evolutions of the Wan-nier charge centers (WCCs) for the k y = 0 and k x = k y mirror planes. The corresponding MCNs are obtained tobe C = − C d = 1, which implies the existence ofat least one gapless Dirac cone along both ¯ M − ¯Γ − ¯ M and ¯ X − ¯Γ − ¯ X lines on the (001) surface (see Fig. 1).Accordingly, the MCN of C = − − ¯ M besides the gapless single Dirac cone buried at ¯Γ.Indeed, mirror eigenvalue analysis provides a more directexplanation on the gapping. Depicted schematically inFig. 2(e) are the double Dirac cones along ¯Γ − ¯ M − ¯Γ, FIG. 3: (Color Online) Surface electronic structures from thesemi-infinite TB slab calculations for the (111) surface of g -SmS with (a) Sm- and (b) S-termination. Topological Dirac-cone surface states are clearly shown at ¯ M in both (a) and(b). (c),(d) The FS and the energy contour at E = 50 meV,respectively, for the Sm-terminated case. Their spin textureswith spin helicities of Rashba-type are also provided. which are composed of two single Dirac cones arising fromband inversions at two non-equivalent bulk X points, X (red) and X (cid:48) (blue). Two bands of each single Diraccone have opposite mirror eigenvalues: + i for the dottedand − i for the solid line in Fig. 2(e). Then the crossingDirac-cone bands with the same mirror eigenvalues hy-bridize with each other to produce the hybridization gaparound ¯ M , resulting in the gapped double Dirac cones.Namely, the double Dirac cones at ¯ M have neither theTI nor the TCI character [24]. As mentioned earlier,this kind of double Dirac-cone feature around ¯ M was de-tected in ARPES for CeBi [22, 23]. Note, however, thatthe double Dirac cones detected in CeBi appear due to a p - d (not f - d ) band inversion.Figure 2(f) presents the Fermi surface (FS) of the (001)surface band structure with normal SOC strength. Thecrossing oval-shaped FS is apparent around ¯ M , whicharises from the double Dirac cones at ¯ M in Fig. 2(a). Onthe other hand, the FS around ¯Γ is derived from bothbulk and surface band structures. The spin texture ofthe FS around ¯ M is provided in Fig. 2(g). The spin-helical structure around each ellipse is evident, reflectingthat the gapped double Dirac cones have the spin textureof Rashba-type (not Dresselhaus-type) [41–44]. (111) surface . In order to ascertain the topologicalnature of g -SmS more evidently, we have investigatedthe (111) surface states. As shown in Fig. 1, each non- FIG. 4: (Color Online) (a) Semi-infinite TB slab calculations for the (110) surface of g -SmS. The double Dirac cones of TCI-type are clearly manifested around ¯ X . A single Dirac-cone surface state is also seen at ¯Γ, even though it is buried inside thebulk-projected bands. (b) Mirror eigenvalues of the double Dirac cones along ¯Γ − ¯ X − ¯Γ. Mirror eigenvalues of + i and − i arepresented in aqua and navy-blue colors, respectively. (c)-(f) The FS and energy contours on the (110) surface. equivalent bulk X point is projected onto a different ¯ M point of surface BZ on the (111) surface. Hence, the gap-less single Dirac cone could be realized at each ¯ M point.Note that the (111) surface has two kinds of terminations:Sm- and S-terminations. As shown in Figs. 3(a) and 3(b),both terminations indeed possess the gapless single Diraccone at ¯ M in the gap region. Figures 3(c) and 3(d) showthe FS and the energy contour at E = 50 meV, respec-tively, for the Sm-terminated case. They clearly revealthe helical spin textures of Rashba-type, which originatefrom the single Dirac-cone surface states. These featuresprovide unambiguous evidence of the topological naturein semimetallic g -SmS. (110) surface . As in the case of the (001) surface,two non-equivalent bulk X points ( X and X (cid:48) ) are pro-jected onto ¯ X of the (110) surface BZ (see Fig. 1). Sothe (110) surface also has double Dirac cones at ¯ X . Itis thus worthwhile to check whether these double Diraccones at ¯ X would produce the TCI-type or the Rashba-type surface states as in the (001) surface. As shown inFig. 4(a), the double Dirac cones at ¯ X manifest a hall-mark of TCI-type surface states with a Dirac point off theTRIM points (marked by an arrow). They are gappedalong ¯ S − ¯ X because ¯ S − ¯ X is not a mirror-symmetry line,while, along the mirror-symmetry line ¯ X − ¯Γ, they showthe prominent Dirac point in the gap region. To ensurethe band crossing in-between ¯ X − ¯Γ, we have analyzedtheir mirror eigenvalues. It is shown in Fig. 4(b) that thecrossing surface bands have opposite mirror eigenvalues,+ i and − i , so that the band crossings are protected bythe mirror symmetry, which is distinct from the case ofthe (001) surface. Note that, besides the double Diraccones at ¯ X , the (110) surface exhibits a single Dirac cone of TI-type at ¯Γ in Fig. 4(a). Its Dirac point is clearlymanifested at ¯Γ, even though it is buried inside the bulk-projected bands at E ≈ − . X display the Lifshitz-like transition as a function ofbinding energies, which is another manifestation of theTCI-type double Dirac-cone surface states [25]. How-ever, the energy contours around ¯Γ look more compli-cated than a single Dirac cone. This intricate features areexpected to come from the TI-type single Dirac-cone sur-face state distorted along the other mirror-symmetry line,¯Γ − ¯ Y . As shown in Fig. 4(a), two neighboring surfacestates along ¯Γ − ¯ Y , one of which corresponds to the Dirac-cone state and the other to a trivial state, have the samemirror eigenvalues, + i , and so they are hybridized to begapped. Hence, most interestingly, the (110) surface hasboth the TCI and the intricate TI nature. It is notewor-thy that this kind of TI/TCI feature was also proposedin the (110) surface of SmB [40–44]. But one shouldkeep in mind that they have different crystal structures,fcc g -SmS vs simple-cubic SmB , which induce differentmirror symmetries in the two materials.Finally, we would like to comment on the experimentalverification of the topological nature of g -SmS. Note that g -SmS is a phase under pressure [20]. Therefore, it is noteasy to probe its topological fingerprints by employingconventional ARPES because it is difficult to apply ex-ternal pressure in ARPES. Instead of applying externalpressure, one may utilize chemical pressure or strain tosimulate the g -SmS phase with a reduced volume. Forexample, Y-substituted SmS (Sm − x Y x S) or SmS-filmgrown on an iso-structural sulphide having a smaller vol-ume can be used. In fact, ARPES measurements havebeen performed on metallic Sm − x Y x S [49, 50], but nosurface state has been observed yet.
Conclusion . We have demonstrated that g -SmS hasthe gapless single Dirac cone in the gap region of its(111) surface BZ, which provides unambiguous evidenceof the topological Kondo nature in mixed-valent g -SmS.The double Dirac cones realized in the (001) and (110)surfaces of g -SmS result in the Rashba-type and TCI-type surface states, respectively, which are elaborated bythe mirror eigenvalues and MCNs, obtained by ab initio band structure calculations. It is thus worth challengingexperimentalists to identify, via high-resolution ARPES,the gapped double Dirac cones of Rashba-type surfacestates, the gapless TI-type Dirac cone, and the TCI-typedouble Dirac cones, respectively, for the (001), (111), and(110) surfaces of g -SmS or Sm − x Y x S. Acknowledgments . Chang-Jong Kang and Dong-Choon Ryu contributed equally to this work. Thiswork was supported by the National Research Founda-tion (NRF) Korea (Grants No. 2016R1D1A1B02008461,No. 2017R1A2B4005175, and No. 2019R1A2C1004929),Max-Plank POSTECH/KOREA Research Initiative(Grant No. 2016K1A4A4A01922028), the POSTECHBSRI Grant, and KISTI supercomputing center (GrantNo. KSC-2017-C3-0057). The ALS is supported by U.S.DOE under Contract No. DE-AC02-05CH11231. C.-J.K. and G.K. were supported by the National ScienceFoundation Grant DMR-1733071. ∗ Co-first authors † Corresponding author: [email protected][1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010).[2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011).[3] M. Dzero, K. Sun, V. Galitski, and P. Coleman, Phys.Rev. Lett. , 106408 (2010).[4] M. Dzero, K. Sun, P. Coleman, and V. Galitski, Phys.Rev. B , 045130 (2012).[5] S. Wolgast, C¸ . Kurdak, K. Sun, J. W. Allen, D.-J. Kim,and Z. Fisk, Phys. Rev. B , 180405(R) (2013).[6] D. J. Kim, S. Thomas, T. Grant, J. Botimer, Z. Fisk,and J. Xia, Sci. Rep. 3, 3150 (2013).[7] D. J. Kim, J. Xia, and Z. Fisk, Nat. Mater. , 466(2014).[8] S. R¨oßer, T.-H. Jang, D.-J. Kim, L. H. Tjeng, Z. Fisk,F. Steglich, and S. Wirth, Proc. Natl. Acad. Sci. U.S.A. , 4798 (2014).[9] L. Jiao, S. R¨oßer, D. J. Kim, L. H. Tjeng, Z. Fisk, F.Steglich, and S. Wirth, Nat. Commun. , 13762 (2016).[10] G. Li, Z. Xiang, F. Yu, T. Asaba, B. Lawson, P. Cai, C.Tinsman, A. Berkley, S. Wolgast, Y. S. Eo, D.-J. Kim, C.Kurdak, J. W. Allen, K. Sun, X. H. Chen, Y. Y. WangmZ. Fisk, and Lu Li, Science , 1208 (2014).[11] N. Xu, X. Shi, P. K. Biswas, C. E. Matt, R. S. Dhaka,Y. Huang, N. C. Plumb, M. Radovi´c, J. H. Dil, E. Pom- jakushina, K. Conder, A. Amato, Z. Salman, D. McK.Paul, J. Mesot, H. Ding, and M. Shi, Phys. Rev. B ,121102(R) (2013).[12] J. Jiang, S. Li, T. Zhang, Z. Sun, F. Chen, Z. R. Ye, M.Xu, Q. Q. Ge, S. Y. Tan, X. H. Niu, M. Xia, B. P. Xie,Y. F. Li, X. H. Chen, H. H. Wen, and D. L. Feng, Nat.Commun. , 3010 (2013).[13] M. Neupane, N. Alidoust, S.-Y. Xu, T. Kondo, Y. Ishida,D. J. Kim, C. Liu, I. Belopolski, Y. J. Jo, T.-R. Chang,H.-T. Jeng, T. Durakiewicz, L. Balicas, H. Lin, A. Bansil,S. Shin, Z. Fisk, and M. Z. Hasan, Nat. Commun. 4, 2991(2013).[14] J. D. Denlinger, J. W. Allen, J.-S. Kang, K. Sun, J.-W.Kim, J. H. Shim, B. I. Min, D.-J. Kim, and Z. Fisk,arXiv:1312.6637 (2013).[15] C.-H. Min, P. Lutz, S. Fiedler, B. Y. Kang, B. K. Cho,H.-D. Kim, H. Bentmann, and F. Reinert, Phys. Rev.Lett. , 226402 (2014).[16] N. Xu, P. K. Biswas, J. H. Dil, R. S. Dhaka, G. Landolt,S. Muff, C. E. Matt, X. Shi, N. C. Plimb, M. Radovi´c, E.Pomjakushina, K. Conder, A. Amato, S. V. Borisenko,R. Yu, H.-M. Weng, Z. Fang, X. Dai, J. Mesot, H. Ding,and M. Shi, Nat. Commun. , 4566 (2014).[17] Z.-H. Zhu, A. Nicolaou, G. Levy, N. P. Butch, P. Syers,X. F. Wang, J. Paglione, G. A. Sawatzky, I. S. Elfimov,and A. Damascelli, Phys. Rev. Lett. , 216402 (2013).[18] P. Hlawenka, K. Siemensmeyer, E. Weschke, A.Varykhalov, J. S´anchez-Barriga, N. Y. Shitsevalova, A.V. Dukhnenko, V. B. Filipov, S. Gab´ani, K. Flachbart,O. Rader, and E. D. L. Rienks, Nat. Commun. , 517(2018).[19] Z. Li, J. Li, P. Blaha, and N. Kioussis, Phys. Rev. B ,121117(R) (2014).[20] C.-J. Kang, H. C. Choi, K. Kim, and B. I. Min, Phys.Rev. Lett. , 166404 (2015).[21] D. Kasinathan, K. Koepernik, L. H. Tjeng, and M. W.Haverkort, Phys. Rev. B , 195127 (2015).[22] K. Kuroda, M. Ochi, H. S. Suzuki, M. Hirayama, M.Nakayama, R. Noguchi, C. Bareille, S. Akebi, S. Ku-nisada, T. Muro, M. D. Watson, H. Kitazawa, Y. Haga,T. K. Kim, M. Hoesch, S. Shin, R. Arita, and T. Kondo,Phys. Rev. Lett. , 086402 (2018).[23] P. Li, Z. Wu, F. Wu, C. Cao, C. Guo, Y. Wu, Y. Liu, Z.Sun, C.-M. Cheng, D.-S. Lin, F. Steglich, H. Yuan, T.-C.Chiang, and Y. Liu, Phys. Rev. B , 085103 (2018).[24] L. Fu, Phys. Rev. Lett. , 106802 (2011).[25] T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, and L.Fu, Nat. Commun. , 982 (2012).[26] Y. Tanaka, Z. Ren, T. Sato, K. Nakayama, S. Souma, T.Takahashi, K. Segawa, and Y. Ando, Nat. Phys. , 800(2012).[27] S.-Y. Xu, C. Liu, N. Alidoust, M. Neupane, D. Qian, I.Belopolski, J. D. Denlinger, Y. J. Wang, H. Lin, L. A.Wray, G. Landolt, B. Slomski, J. H. Dil, A. Marcinkova,E. Morosan, Q. Gibson, R. Sankar, F. C. Chou, R. J.Cava, A. Bansil, and M. Z. Hasan, Nat. Commun. ,1192 (2012).[28] P. Dziawa, B. J. Kowalski, K. Dybko, R. Buczko, A.Szczerbakow, M. Szot, E. (cid:32)Lusakowska, T. Balasubrama-nian, B. M. Wojek, M. H. Berntsen, O. Tjernberg, andT. Story, Nat. Mater. , 1023 (2012).[29] Y. Okada, M. Serbyn, H. Lin, D. Walkup, W. Zhou, C.Dhital, M. Neupane, S. Xu, Y. J. Wang, R. Sankar, F.Chou, A. Bansil, M. Z. Hasan, S. D. Wilson, L. Fu, and V. Madhavan, Science , 1496 (2013).[30] J. Kim, K. Kim, C.-J. Kang, S. Kim, H. C. Choi, J.-S.Kang, J. D. Denlinger, and B. I. Min, Phys. Rev. B ,075131 (2014).[31] C.-J. Kang, J. Kim, K. Kim, J. Kang, J. D. Denlinger,and B. I. Min, J. Phys. Soc. Jpn. , 024722 (2015).[32] For g -SmS, a proper value of Z is known to be around0.1 (see Ref. [20]).[33] G. Kresse and D. Joubert, Phys. Rev. B , 1758 (1999).[34] G. Kresse and J. Furthm¨uller, Phys. Rev. B , 11169(1996); Comput. Mater. Sci. , 15 (1996).[35] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.Lett. , 3865 (1996).[36] A. A. Mostofi, J. R. Yates, G. Pizzi, Y. S. Lee, I. Souza,D. Vanderbilt, N. Marzari, Comput. Phys. Commun. , 2309 (2014).[37] M. P. Lopez Sancho, J. M. Lopez Sancho, and J. Rubio,J. Phys. F : Met. Phys. , 851 (1985).[38] Q. S. Wu, S. N. Zhang, H.-F. Song, M. Troyer, and A. A.Soluyanov, Comput. Phys. Commun. , 405 (2018).[39] P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka,and J. Luitz, WIEN2K (Karlheinz Schwarz, TechnischeUniversitat Wien, Austria, 2001).[40] M. Ye, J. W. Allen, and K. Sun, arXiv:1307.7191 (2013).[41] M. Legner, A. R¨uegg, and M. Sigrist, Phys. Rev. Lett. , 156405 (2015).[42] P. P. Baruselli and M. Vojta, Phys. Rev. Lett. ,156404 (2015).[43] P. P. Baruselli and M. Vojta, Phys. Rev. B , 195117(2016).[44] M. Legner, A. R¨uegg, and M. Sigrist, Phys. Rev. B ,085110 (2014).[45] R. Yu, X. L. Qi, A. Bernevig, Z. Fang and X. Dai, Phys.Rev. B , 075119 (2011).[46] A. A. Soluyanov and D. Vanderbilt, Phys. Rev. B ,235401 (2011).[47] We used ten-times larger SOC in Fig. 2(b) to enhance thevisibility of an existing tiny gap. It could, however, be abetter representation of reality by shifting up the SOC-split j = 7 / E F .[48] Here we conisdered the (001) surface instead of the (100)surface.[49] K. Imura, T. Hajiri, M. Matsunami, S. Kimura, M.Kaneko, T. Ito, Y. Nishi, N. K. Sato, and H. S. Suzuki,J. Korean Phys. Soc. , 2028 (2013).[50] M. Kaneko, M. Saito, T. Ito, K. Imura, T. Hajiri, M.Matsunami, S. Kimura, H. S. Suzuki, N. K. Sato, JPSConf. Proc.3