Distinct pairing symmetries of superconductivity in infinite-layer nickelates
DDistinct pairing symmetries of superconductivity in infinite-layer nickelates
Zhan Wang, Guang-Ming Zhang,
2, 3, ∗ Yi-feng Yang,
4, 5, 6, † and Fu-Chun Zhang ‡ Kavli Institute for Theoretical Sciences and CAS Center for Topological Quantum Computation,University of Chinese Academy of Sciences, Beijing 100190, China State Key Laboratory of Low-Dimensional Quantum Physics andDepartment of Physics, Tsinghua University, Beijing 100084, China Frontier Science Center for Quantum Information, Beijing 100084, China Beijing National Lab for Condensed Matter Physics and Institute of Physics,Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China (Dated: June 30, 2020)We report theoretical predictions on the pairing symmetry of the newly discovered superconduct-ing nickelate Nd − x Sr x NiO based on the renormalized mean-field theory for a generalized modelHamiltonian proposed in [Phys. Rev. B , 020501(R)]. For practical values of the key parameters,we find a transition between a gapped ( d + is )-wave pairing state in the small doping region to agapless d -wave pairing state in the large doping region, accompanied by an abrupt Fermi surfacechange at the critical doping. Our overall phase diagram also shows the possibility of a ( d + is )-to s -wave transition if the electron hybridization is relatively small. In either case, the low-doping( d + is )-wave state is a gapped superconducting state with broken time-reversal symmetry. Ourresults are in qualitative agreement with recent experimental observations and predict several keyfeatures to be examined in future measurements. Introduction. -The recent discovery of superconduc-tivity (SC) in the single crystal thin films of infinite-layernickelates Nd − x Sr x NiO [1] has stimulated intensive de-bates on its underlying electronic structural propertiesand superconducting pairing symmetries [2–10]. Despitethe similarities in the crystal structure and 3 d configura-tion of the nickelate and cuprate superconductors, thereare increasing evidences suggesting that these two sys-tems might belong to different classes of unconventionalSC. Earlier first-principles calculations have revealed sub-tle differences in their band structures [11–21]. In exper-iments, the parent compound NdNiO displays param-agnetic metallic behavior at high temperatures with aresistivity upturn below about 70 K, showing no signof any magnetic long-range order [22]. This is in starkcontrast with the cuprates whose parent compound isa charge-transfer insulator with antiferromagnetic (AF)long-range order. As a consequence, the nickelates maybe modelled as a self-doped Mott insulator with two typesof charge carriers [8], with the low-temperature upturn[1, 23] arising from the Kondo coupling between low-density conduction electrons and localized Ni-3 d x − y moments [24]. This produces both Kondo singlets (dou-blons) and holes moving through the lattice of otherwisenickel spin-1/2 background, suppressing the AF long-range order, and causing a phase transition to a param-agnetic metal [8]. Latest measurements [5, 25] and first-principle calculations[9, 13] confirm this scenario and re-veal a special interstitial s orbital for the hybridization[9], which is missing in previous calculations.We expect these differences to have an immediate im-pact on the candidate pairing mechanism. In cuprates,additional holes are doped on the oxygen sites in the CuO planes [26–29] and combine with the 3 d x − y spinsof Cu-ions to form the Zhang-Rice singlets [30]. Hightemperature SC with robust d -wave pairing can be de-rived from an effective one-band t - J model [31–33]. Innickelates, Sr doping may not only introduce additionalholes on the oxygen sites to form the Ni-O spin singletsor holons (a spin zero state) [3, 34], but also reduce thenumber of conduction electrons, thus tilting the balancebetween the electron and hole carriers of distinct charac-ters. The Hall coefficient is then expected to vary grad-ually and change sign with doping or temperature. Onethus anticipates more rich physics in the nickelate super-conductors, whose pairing symmetry may be altered bythe hybridization. Indeed, latest experiment has revealeda non-monotonic change of T c in exact accordance withthe sign change of the Hall coefficient [35, 36].To further elucidate the pairing symmetry of the nick-elate superconductors, we employ here the renormalizedmean-field theory (RMFT) [37] and study the supercon-ducting pairing symmetry based on a generalized K - t - J model [8] in Eq. (1) and Eq. (2). Our calculations leadto a global phase diagram depending on the hole con-centration p and the conduction electron hopping ( t c /K )which controls the effective strength of the Kondo hy-bridization. At small doping and with reasonable choicesof parameters, our calculations reveal an unusual gapped( d + is )-wave SC with the time-reversal symmetry break-ing, which is distinctly different from the familiar cupratesuperconductivity. For large doping, we find either ex-tended s -wave pairing or pure d -wave pairing. The latteris quite robust and occupies a large region in the phasediagram. Comparison with experiment tends to favor atransition from the gapped ( d + is )-wave to gapless d -wave a r X i v : . [ c ond - m a t . s up r- c on ] J un pairing states with increasing hole doping. We furtherpredict that the SC transition is accompanied with anabrupt Fermi surface change associated with the break-down of the Kondo hybridization, causing potentially acrossover line in the temperature-doping phase diagramas observed in recent Hall measurements [35, 36]. Model Hamiltonian and RMFT. - We start by firstintroducing the generalized K - t - J model for the nickelatesuperconductors, given by H = H t − J + H K . The t - J partdescribes the hole doped lattice of Ni 3 d x − y spins withthe nearest-neighbor AF superexchange interactions, H t − J = − (cid:88) ijσ (cid:16) t ij P G d † iσ d jσ P G + h.c. (cid:17) + J (cid:88) (cid:104) ij (cid:105) S i · S j , (1)where d iσ and d † iσ are the annihilation and creation op-erators of the Ni 3 d x − y electrons, respectively, t ij isthe hopping integral between site i and j , and P G isthe Gutzwiller operator to project out doubly occupiedelectron states on the Ni sites. For simplicity, we con-sider only the nearest neighbor hopping (NN) t and next-nearest neighbor (NNN) hopping t (cid:48) . The AF superex-change J is induced by the O-2 p orbitals but greatlyreduced compared to that in cuprates. The Kondo hy-bridization part is given by, H K = − t c (cid:88) (cid:104) ij (cid:105) ,σ (cid:16) c † iσ c jσ + h.c. (cid:17) + K (cid:88) jα ; σσ (cid:48) S αj c † jσ τ ασσ (cid:48) c jσ (cid:48) , (2)where c iσ ( c † iσ ) are the annihilation (creation) operatorsof the conduction electrons from Nd 5 d , interstitial s , orother extended orbitals, t c describes the effective hopingamplitude of the conduction electrons projected on thesquare lattice sites of the Ni ions, τ α ( α = x, y, z ) arethe spin-1/2 Pauli matrices, and K is the effective Kondoexchange coupling.In the parent compounds LnNiO (Ln=Nd, La, Pr),the total electron density ( n c + n d ) is one per unit cell,hence the total holon density n h = n c . For Sr dopedcompounds, the hole doping p = n h − n c >
0. Analysesof the Hall coefficients at high temperatures suggest thatthe average number of the conduction electrons is alwayssmall, i.e., n c = N − (cid:80) jσ (cid:104) c † jσ c jσ (cid:105) (cid:28)
1, where N is thetotal number of the lattice sites.For the RMFT calculations, the Gutzwiller renormal-ization factor should be included to approximate the pro-jection operator that projects out the doubly occupiedstates. We have g t = n h / (1 + n h ) for the constraintelectron hopping t and t (cid:48) , g J = 4 / (1 + n h ) for the theAF Heisenberg exchange J , and g K = 2 / (1 + n h ) forthe Kondo exchange coupling K . Four different mean-field order parameters are then introduced to decouplethe quartic AF Heisenberg spin exchange and the Kondo exchange interactions: χ ij = (cid:104) d † i ↑ d j ↑ + d † i ↓ d j ↓ (cid:105) , B = 1 √ (cid:104) d † j ↑ c † j ↓ − d † j ↓ c † j ↑ (cid:105) , ∆ ij = (cid:104) d † i ↑ d † j ↓ − d † i ↓ d † j ↑ (cid:105) , D = 1 √ (cid:104) c † j ↑ d j ↑ + c † j ↓ d j ↓ (cid:105) . The resulting mean-field Hamiltonian has a bilinear formand can be expressed in the momentum space, H mf = (cid:88) k Ψ † k χ ( k ) K D ∆ ∗ ( k ) K ∗ B K ∗ D (cid:15) c ( k ) K ∗ B − k ) K B − χ ( − k ) − K ∗ D K B − K D − (cid:15) c ( − k ) Ψ k , where the Nambu spinors are defined as Ψ † k =( d † k ↑ , c † k ↑ , d − k ↓ , c − k ↓ ), and the matrix elements are χ ( k ) = − (cid:88) α (cid:18) tg t + 38 Jg J χ α (cid:19) cos( k · α ) − t (cid:48) g t (cid:88) δ cos( k · δ ) + µ ,(cid:15) c ( k ) = − t c (cid:88) α cos( k · α ) + µ , ∆( k ) = − Jg J (cid:88) α ∆ α cos( k · α ) ,K D = − g K K D √ , K B = − g K K B √ . (3)Here α denotes the vectors of the NN lattice sites and δ stands for those of the NNN sites. µ and µ are thechemical potentials fixing the numbers of the constraintelectrons d iσ and conduction electrons c iσ , respectively.The above mean-field Hamiltonian can be di-agonalized using the Bogoliubov transformation, (cid:16) d k ↑ , c k ↑ , d †− k ↓ , c †− k ↓ (cid:17) T = U k (cid:16) α k ↑ , β k ↑ , α †− k ↓ , β †− k ↓ (cid:17) T .The ground state is given by the vacuum of the Bogoli-ubov quasiparticles { α † k σ , β † k σ } , which in turn yields theself-consistent mean-field equations: B = 1 √ N (cid:88) k ( u ∗ k u k + u ∗ k u k − u ∗ k u k − u ∗ k u k ) ,D = 1 √ N (cid:88) k ( u ∗ k u k + u ∗ k u k + u ∗ k u k + u ∗ k u k ) ,χ α = 2 N (cid:88) k exp[ i k · α ]( u ∗ k u k + u ∗ k u k ) , ∆ α = 2 N (cid:88) k exp[ i k · α ]( u ∗ k u k + u ∗ k u k ) ,n c = 1 N (cid:88) k ( u k u ∗ k + u k u ∗ k + u k u ∗ k + u k u ∗ k ) , − n h = 1 N (cid:88) k ( u k u ∗ k + u k u ∗ k + u k u ∗ k + u k u ∗ k ) , (4) d+is d-waves-wave FIG. 1: Theoretical phase diagram of the superconductivitywith varying hopping t c /K and hole concentration p . At smalldoping, the pairing symmetry is primarily ( d + is )-wave SC.At large doping, the pairing is either s -wave SC for small t c /K or d -wave SC for large t c /K . where u k ij are given by the matrix elements of U k , andthe last two equations fix the chemical potentials µ and µ , respectively. Numerical results. -For clarity, we define ∆ s = | ∆ x + ∆ y | / d = | ∆ x − ∆ y | / s and d -wave pairing amplitudes, respectively. To nu-merically solve these self-consistent equations, we firstfix the practical parameters based roughly on the exper-imental analyses and first-principle results. The Kondocoupling K is considered to be the largest energy scaleand thus chosen as the energy unit ( K = 1). To sim-plify the discussions, only the numerical results for theNN hopping t = 0 .
2, the NNN hopping t (cid:48) = − .
05, andthe AF Heisenberg spin exchange J = 0 . n c = 0 . t c /K and the hole concentration p . We find a dominant d -wave pairing symmetry in thephase diagram, which, for small t c /K and large doping,turns into an extended s -wave state. Most intriguingly,we find a large region of the ( d + is )-wave pairing forsmall hole doping. This exotic pairing state breaks thetime-reversal symmetry and its presence reflects a uniquefeature of the nickelate superconductivity due to the in-terplay of the Kondo and Mott physics in comparisonwith the cuprates.Details on the transition from the mixed ( d + is )-waveSC to the pure d -wave SC can be found in Fig. 2(a) foran intermediate t c /K = 0 .
25. The critical hole doping is p ∗ ≈ .
13, which is comparable with the experiment butmay vary with t c and other controlling parameters. The (a)(b) (c)(d) (e) FIG. 2: RMFT results for t c /K = 0 .
25. (a) The mean-fieldparameters as a function of doping for g t ∆ (upper panel) and B and D (lower panel). (b) and (c) show the quasiparticleexcitation energy (background) and the Fermi surface (whitesolid line) defined as the minimal excitation energy at p = 0 . d + is )-wave and p = 0 . d -wave) as marked by the arrows in(a). (d) and (e) show the respective quasiparticle excitationgap along the Fermi surface. transition is accompanied with vanishing Kondo mean-field parameters B and D , implying a breakdown of theKondo hybridization in the large doping side. It also im-plies that the s -wave component is primarily associatedwith the Kondo hybridization effect and the d -wave com-ponent is from the usual t - J model. The correspondingFermi surface structures in these two different doping re-gions can be extracted from the minimal energy contourof the SC quasiparticle excitation energy. Two typicaldopings for p = 0 .
05 and p = 0 . p < p ∗ , the normal state has a large hole-like Fermi sur-face around four Brillouin zone corners, while for largedoping, two types of charge carriers are effectively de-coupled and give rise to two separate electron-like Fermisurfaces around the Brillouin zone center. The physics ofthe pure d -wave pairing region is similar to that of heavilyhole-doped cuprates for this particular doping. We have (a)(b) (c)(d) (e) FIG. 3: RMFT results for t c /K = 0 .
2. (a) The mean-fieldparameters as a function of doping for g t ∆ (upper panel) and B and D (lower panel). (b) and (c) show the quasiparticleexcitation energy (background) and the Fermi surface (whitesolid line) at p = 0 . d + is )-wave and p = 0 . s -wave) asmarked by the arrows in (a). (d) and (e) show the respectivequasiparticle excitation gap along the Fermi surface. thus a concurrent Lifshitz transition at the critical holedoping p ∗ , accompanying the transition between differentpairing states of the nickelate superconductivity.For comparison, the RMFT results for a smaller t c /K = 0 . d + is )-wave to thepure s -wave SC. In both phases, the SC are gapped andthe Kondo mean-field parameters D and B remain fi-nite. Hence the hybridization effect is not affected acrossthe transition. Again, the Fermi surfaces for p = 0 . p = 0 . Discussions and Conclusion. -The generalized K - t - J model contains several key energy scales that need to befixed for better experimental comparison in each individ-ual compound. While the conduction electron hopping t c may be roughly estimated from band calculations, theconstraint electron hoppings t and t (cid:48) are strongly renor-malized due to the background AF correlations. Follow-ing our previous analyses, the Heisenberg superexchange J is expected to be roughly the order of 10-100 meV,which is smaller than that of cuprates due to the largercharge transfer energy between O-2 p and Ni-3 d x − y or-bitals. The Kondo exchange interaction K is estimatedto be the order of 100-1000 meV [8, 9]. This justifiesour choice of J/K in current numerical calculations. Inany case, our results may serve as a qualitative guide forfuture studies on nickelate superconductors.It is worthwhile comparing our results with the avail-able experiment. Recent systematic measurements onthe resistivity and Hall coefficients in Nd − x Sr x NiO have revealed a non-monotonic doping dependence of thesuperconducting T c , whose local minimum coincides withthe sign change of the Hall coefficient [35, 36]. The latterfurther gives rise to a crossover line in the temperature-doping phase diagram of the nickelate superconductors.A straightforward comparison suggests that the experi-mental observation may correspond to our derived tran-sition from the ( d + is )-wave pairing to the d -wave or s -wave paring. The concurrent change in the Hall coef-ficient therefore marks a potential Fermi surface change,in resemblance of that observed in some heavy fermionsystems owing to the breakdown of the Kondo hybridiza-tion [38]. The latter also leads to a delocalization line inthe temperature-pressure or temperature-doping phasediagram [39]. It is thus attempted to link the experi-ment with our theoretical proposals, predicting the SCtransition from a gapped ( d + is )-wave state to a gap-less d -wave pairing state, with the crossover line in thetemperature-doping plane potentially associated with theFermi surface change due to the Kondo hybridization.If this is the case, one may further expect several keyfeatures to be examined in future experiment: 1) a super-conducting transition between gapped and gapless pair-ings with increasing doping to be best revealed by thescanning tunneling spectroscopy or the penetration depthmeasurement; 2) time-reversal symmetry breaking in thelow-doping gapped SC phase to be detected in the µ SRor Kerr experiments; 3) Fermi surface reconstruction ac-companying the superconducting transition to be mea-sured by the quantum oscillation experiments or angle-resolved photoemission spectroscopy. Additionally, theremay also exist other exotic properties associated with thequantum critical point, besides the non-Fermi liquid be-havior which has been observed in superconducting nick-elate thin films with ρ ∼ T α and α = 1 . − . − x Sr x NiO based on the RMFT for a generalized K - t - J model. Our calculations reveal an interesting inter-play between the Kondo and Mott physics. For practi-cal choices of the parameters, we find a transition froma gapped ( d + is )-wave state to a gapless d -wave statewith increasing doping. An extended s -wave pairing hasalso been predicted but requires sufficiently small hop-ping and large doping. For the former transition, ourcalculations suggest a concurrent Fermi surface changeand a corresponding crossover line in the temperature-doping phase diagram due to the breakdown of the Kondohybridization. Our proposal is in good agreement withavailable experiments and gives several key predictionsfor further verification. Note Added . As we are finishing this manuscript,single particle tunneling measurements [40] were reportedon superconducting nickelate thin films with T c ≈ . K ,and two distinct types of tunneling spectra were revealed:a V-shape feature with a gap maximum 3.9 meV, a U-shape feature with a gap about 2.35 meV, and somespectra with mixed contributions of the two components.These spectra were ascribed to different Fermi surfacesfrom the conduction and Ni 3 d x − y orbitals. However,according to our present calculations, these distinct tun-neling spectra observed at different locations on the thinfilms may be caused by different hole doping concen-trations due to surface effects, so the different spectralshapes may correspond to the different pairing states inour theory. In this sense, the tunneling experiment issupportive of our theoretical prediction of multiple su-perconducting phases. Acknowledgment .- This work was supported by theNational Key Research and Development Program ofMOST of China (2016YFYA0300300, 2017YFA0302902,2017YFA0303103), the National Natural Science Foun-dation of China (11774401, 11674278), the State KeyDevelopment Program for Basic Research of China(2014CB921203 and 2015CB921303), and the Strate-gic Priority Research Program of CAS (Grand No.XDB28000000). ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected][1] D. Li, K. Lee, B. Y. Wang, M. Osada, S. Crossley, H. R.Lee, Y. Cui, Y. Hikita, and H. Y. Hwang, Nature ,624 (2019).[2] A. S. Botana and M. R. Norman, Phys. Rev. X ,011024 (2020).[3] M. Jiang, M. Berciu, and G. Sawatzky, Phys. Rev. Lett. , 207004 (2020)..[4] H. Sakakibara, H. Usui, K. Suzuki, T. Kotani, H. Aoki,and K. Kuroki, arXiv:1909.00060 (2019).[5] M. Hepting, D. Li, C. J. Jia, H. Lu, E. Paris, Y. Tseng,X. Feng, M. Osada, E. Been, Y. Hikita, Y.-D. Chuang, Z.Hussain, K. J. Zhou, A. Nag, M. Garcia-Fernandez, M. Rossi, H. Y. Huang, D. J. Huang, Z. X. Shen, T. Schmitt,H. Y. Hwang, B. Moritz, J. Zaanen, T. P. Devereaux, andW. S. Lee, Nature Materials , 381 (2020).[6] Y. Normura, M. Hirayama, T. Tadano, Y. Yoshimoto,K. Nakamura, and R. Arita, Phys. Rev. B , 205138(2019).[7] X. Wu, D. D. Sante, T. Schwemmer, W. Hanke, H. Y.Hwang, S. Raghu, and Ronny Thomale, Phys. Rev. B , 060504 (2020).[8] G. M. Zhang, Y. F. Yang and F. C. Zhang, Phys. Rev.B , 020501(R) (2020).[9] Y. Gu, S. Zhu, X. Wang, J. Hu, H. Chen, Communica-tions Physics , 84 (2020).[10] J. Karp, A. S. Botana, M. R. Norman, H. Park, M. Zingl,and A. Millis, Phys. Rev. X , 021061 (2020).[11] V. I. Anisimov, D. Bukhvalov, and T. M. Rice, Phys.Rev. B , 7901-7906 (1999).[12] M. A. Hayward, M. A. Green, M. J. Rosseinsky, and J.Sloan, J. Am. Chem. Soc. , 8843 (1999).[13] K.-W. Lee and W. E. Pickett, Phys. Rev. B , 165109(2004).[14] A. S. Botana, V. Pardo, and M. R. Norman, Phys. Rev.Materials , 021801(R) (2017).[15] J. Chaloupka and G. Khaliullin, Phys. Rev. Lett. ,016404 (2008).[16] P. Hansmann, X. Yang, A. Toschi, G. Khaliullin, O. K.Andersen, and K. Held, Phys. Rev. Lett. , 016401(2009).[17] S. Middey, J. Chakhalian, P. Mahadevan, J. W. Freeland,A. J. Millis, and D. D. Sarma, Annu. Rev. Mater. Res. , 305 (2016).[18] A. V. Boris, Y. Matiks, E. Benckiser, A. Frano, P.Popovich, V. Hinkov, P. Wochner, M. Castro-Colin, E.Detemple, V. K. Malik, C. Bernhard, T. Prokscha, A.Suter, Z. Salman, E. Morenzoni, G. Cristiani, H.-U.Habermeier, and B. Keimer, Science , 937 (2011).[19] E. Benckiser, et. al., E. Benckiser, M. W. Haverkort, S.Br¨uck, E. Goering, S. Macke, A. Fra˜n´o, X. Yang, O. K.Andersen, G. Cristiani, H.-U. Habermeier, A. V. Boris,I. Zegkinoglou, P. Wochner, H. Kim, V. Hinkov, and B.Keimer, Nat. Mater. , 189 (2011).[20] A. S. Disa, D. P. Kumah, A. Malashevich, H. Chen, D.A. Arena, E. D. Specht, S. Ismail-Beigi, F. J. Walker,and C. H. Ahn, Phys. Rev. Lett. , 026801 (2015).[21] J. Zhang, A. S. Botana, J. W. Freeland, D. Phelan, H.Zheng, V. Pardo, M. R. Norman, and J. F. Mitchell, Nat.Phys. , 864 (2017).[22] M. A. Hayward and M. J. Rosseinsky, Solid State Sci-ences , 839 (2003).[23] A. Ikeda, Y. Krockenberger, H. Irie, M. Naito, and H.Yamamoto, Applied Physics Express , 061101 (2016).[24] Actually an alternative explanation for the resistivity up-turn at low temperatures is weak localization due to thepresence of disorder holes in the NiO plane. It can alsogive rise to a logarithmic temperature dependent correc-tion. However, the corresponding correction to the Hallcoefficient is independent of temperature in the same re-gion, which does not support this explanation.[25] B. H. Goodge, D. Li, M. Osada, B. Y. Wang, K.Lee, G. A. Sawatzky, H. Y. Hwang, L. F. Kourkoutis,arXiv:2005.02847.[26] J. P. Bednorz and K. A. Muller, Z. Phys. B , 189(1986).[27] P. W. Anderson, Science , 1196 (1987). [28] P. W. Anderson, P. A. Lee, M. Randeria, T. M. Rice, N.Trivedi, and F. C Zhang, J. Phys.: Condens. Matter ,R755 (2004).[29] P. A. Lee, N. Nagaosa, and X. G. Wen, Rev. Mod. Phys. , 17 (2006).[30] F. C. Zhang, and T. M. Rice, Phys. Rev. B , 3759(1988).[31] Z. X. Shen, D. S. Dessau, B. O. Wells, D. M. King, W.E. Spicer, A. J. Arko, D. Marshall, L. W. Lombardo,A. Kapitulnik, P. Dickinson, S. Doniach, J. DiCarlo, T.Loeser, and C. H. Park, Phys. Rev. Lett. , 1553 (1993).[32] D. A. Wollman, D. J. Van Harlingen, W. C. Lee, D. M.Ginsberg, and A. J. Leggett, Phys. Rev. Lett. , 2134(1993).[33] C. C. Tsuei, J. R. Kirtley, C. C. Chi, L. S. Yujahnes, A.Gutpa, T. Shaw, J. Z. Sun, and M. B. Ketchen, Phys.Rev. Lett. , 593 (1994). [34] Z. J. Lang, R. Jiang, and W. Ku, arXiv:2005.00022.[35] D. Li, B. Y. Wang, K. Lee, S. P. Harvey, M. Osada,B. H. Goodge, L. F. Kourkoutis and H. Y. Hwang,arXiv:2003.08506.[36] S. Zeng, C. S. Tang, X. Yin, C. Li, Z. Huang, J. Hu, W.Liu, G. J. Omar, H. Jani, Z. S. Lim, K. Han, D. Wan, P.Yang, A. T. S. Wee, A. Ariando, arXiv:2004.11281.[37] F. C. Zhang, C. Gros, T. M. Rice, and H. Shiba, Super-cond. Sci. Technol. , 36 (1988).[38] Q. Si, J. H. Pixley, E. Nica, S. J. Yamamoto, P. Goswami,R. Yu, and S. Kirchner, J. Phys. Soc. Jpn. , 061005(2014)[39] Y.-F. Yang, D. Pines, and G. Lonzarich, Proc. Natl.Acad. Sci. USA114