Electronic structure and two-band superconductivity in unconventional high- T c cuprates Ba 2 CuO 3+δ
Kun Jiang, Congcong Le, Yinxiang Li, Shengshan Qin, Ziqiang Wang, Fuchun Zhang, Jiangping Hu
aa r X i v : . [ c ond - m a t . s up r- c on ] S e p Electronic structure and superconductivity in unconventional cuprates Ba CuO δ Congcong Le,
1, 2, 3
Kun Jiang,
4, 3
Yinxiang Li, Shengshan Qin,
1, 3
Ziqiang Wang, ∗ Fuchun Zhang,
1, 2 and Jiangping Hu
3, 5, 1, † Kavli Institute of Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China Chinese Academy of Sciences Center for Excellence in Topological Quantum Computation,University of Chinese Academy of Sciences, Beijing 100190, China Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,Chinese Academy of Sciences, Beijing 100190, China Department of Physics, Boston College, Chestnut Hill, MA 02467, USA Collaborative Innovation Center of Quantum Matter, Beijing, China (Dated: September 30, 2019)We study the recently discovered 73K high- T c superconductor Ba CuO δ at δ ≃ . CuO , but with dramatically different lattice parameters due to the CuO octahedron compres-sion. The resulting crystal field leads to an inverted Cu 3 d e g complex with the d x − y orbital sittingbelow the d z − r orbital and an electronic structure highly unusual compared to the conventionalcuprates. We conjecture that the material realizes a new path of in-plane positional oxygen doping,where the doped oxygens create matrices of compressed Ba CuO embedded in Ba CuO . Con-structing a strongly correlated two-orbital model at hole doping x = 2 δ of the Cu d state, we showthat the spin-orbital exchange interactions lead to a multiband antiphase d -wave superconductingstate, i.e. a nodal d ± pairing state. These findings suggest that the class of unconventional cuprateswith liberated orbitals as doped two-band Mott insulators can be a direction for realizing high-T c superconductivity with enhanced transition temperature T c . The current understanding of high- T c cuprate super-conductors [1] crucially relies on the crystal field due tothe Jahn-Teller distortion of the elongated CuO octahe-dra. The topmost Cu 3 d -electron e g states split accord-ingly into well separated lower d z − r ( d z ) and upper d x − y ( d x ) orbitals (Fig. 1(a)). In the parent com-pound, such as the prototypical single-layer La CuO (La214), the Cu is in the 3 d configuration with a fullyoccupied d z orbital and an active d x orbital partially oc-cupied by one electron. The strong correlation producesa spin- antiferromagnetic (AF) Mott insulator. Holedoping leads to an effective one-band model of Zhang-Rice singlets formed by the hole in the d x orbital and adoped hole in the planar oxygen 2 p orbitals [2]. The AFexchange interactions give rise to nodal d -wave high- T c superconductivity in the CuO planes [3]. Such a picturedescribes the vast majority of the conventional cuprates,where the dormant d z orbital plays only a minor role[4–7].The recent discovery of high- T c superconductorBa CuO δ at δ ≃ . d z orbital [9]. The polycrystal samples have been synthe-sized under high pressure in a strongly oxidizing envi-ronment and at high temperatures. Extraordinary prop-erties were observed by a combination of magnetization,specific heat, neutron scattering, x-ray absorption spec-troscopy (XAS), and µ spin-rotation ( µ SR) experiments: ∗ [email protected] † [email protected] (i) The polycrystalline BCO δ has an atomic structuresimilar to La214, but with dramatically different latticeparameters due to the octahedral compression, leading to inverted d z and d x orbitals as shown in Fig. 1(b); (ii)The extra O δ occupy the oxygen sites in the CuO δ planes with no evidence of ordering; (iii) Despite thelarge hole doping concentration reflected in the Cu L spectrum, the O K-edge XAS shows spectral the weighttransfer similar to optimally doped La214, indicative ofthe presence of strong Mott-Hubbard local correlations[10, 11]; and remarkably, (iv) The superconducting (SC)transition temperature T c ≃ d z orbitaland the two-band doped Mott insulator with strongly or-bital dependent correlation effects may provide a mech-anism for the dramatic T c enhancement.Based the neutron and µ SR measurements, it is in-structive to consider Ba CuO δ (Ba213 δ ) at δ ≃ . O δ oxygenated into the copper-oxygenchain plane of the stoichiometric Ba CuO (Ba213)[12, 13], which would have two domains with Cu-O chainsalong either the x or the y directions. Intriguingly,Ba CuO (Ba214), which is isostructural to La214, ma-terializes locally around the domain boundary as illus-trated in Fig. 1(c) for a diagonal domain wall. It is thusconceivable that the Ba214 would be clustered aroundthe randomly distributed domain walls, such that Ba213 δ can be viewed as δ · Ba214 ⊕ (1 − δ ) · Ba213, i.e. a matrixof Ba214 with CuO planes embedded in the backgroundof Ba213 with Cu-O chain planes. This inline with µ SRexperiments [8] that find no evidence for in-plane orderedoxygen superstructures. Since the Cu (3 d ) in Ba214 isnot a stable ionization state, self-doping by charge trans-fer takes place with the Cu in Ba213. Thus, Ba213 δ represents a new path of in-plane positional δ -oxygendoping, in sharp contrast to conventional cuprates wheredoping is achieved by either substitutional or interstitialdopants, all residing outside the CuO plane.Measuring the doping concentration referenced to the3 d configuration in both Ba214 and Ba213 by x and x ′ , we have the relation: δx + (1 − δ ) x ′ = 2 δ . Sincethe µ SR experiments did not observe apparent chargedisproportionation [8], we take the Cu atoms to be ofequal valence in the Ba213 and Ba214 regions, result-ing in a common heavily overdoped hole concentration x = x ′ = 2 δ ≃ CuO δ were viewed as Ba214 with randomly distributed oxygenvacancies in the plane, one would arrive at the same valueof x and the electronic structure to be discussed belowprovided that the vacancy potential is ignored. However,the high density of such vacancies ( ∼ ab initio simulation package(VASP) [14–16]. The details are described in the sup-plemental material (SM) [13]. The corresponding bandstructures are calculated with the generalized gradientapproximation (GGA) [17]. For Ba213, there is a single1D band at low energy near the Fermi level, resultingin two 1D Fermi surface sections in the Brillioun zone(Fig. S2). The primary d -electron content is d x − z forthe Cu-O chains along the x -direction, which is a lin-ear combination of the d x and d z orbitals. We ex-pect 40% hole-doping of such a single-band to give acorrelated metallic state. In the rest of the paper, wefocus on the compressed Ba214 and show that despitethe heavy hole-doping, a new two-band antiphase d -waveSC state emerges due to the multiorbital correlated elec-tronic structure and the spin-orbital exchange interac-tions.For compressed Ba214, the obtained DFT band struc-ture is shown in Fig. 1(d). There are two bands crossingthe Fermi level, originating predominately from the Cu3 d x and 3 d z orbitals. Since the distorted CuO octa-hedron has a shorter distance connecting the Cu to theapical oxygens than to the planar oxygen (Table S1), the d z orbital couples strongly with the apical oxygen p z or-bital. The inverted crystal field pushes the atomic energyof the d z orbital to be ∼ . d x orbital.Thus the atomic limit of the 3 d configuration has one-electron in the d z orbital while the d x orbital is occu-pied by two electrons (Fig. 1(b)), in contrast to conven-tional cuprates. Upon crystallization, although the much FIG. 1. Elongated octahedron in La214 (a) and compressedoctahedron in Ba214 (b) with 6 oxygens (red balls) and onecopper (blue balls). Note the inverted d x and d z orbitalsdue to different crystal fields. (c) Matrix of Ba214 with CuO clusters accumulated near a diagonal domain wall of the back-ground Ba213 with Cu-O chains. (d) DFT band structure ofcompressed Ba214 with structural parameters measured byneutron scattering. (e) Band dispersion of two-orbital TBmodel with color coded orbital contents in each band: d x orbital (red and 1) and d z orbital (blue and 0). smaller hopping integrals of the out of plane d z orbitalproduce a narrow band, the d x orbital generates a muchwider band through the larger hopping integrals via theplanar oxygen such that the two bands overlap near theFermi level (Fig. 1(d)). Thus, we propose that the com-pressed Ba214 is an unconventional cuprates where boththe d x and d z orbitals of the e g sector contribute tosuperconductivity. Although multi-orbital superconduc-tivity has been studied extensively for the t g electronsin the iron pnictides and chalcogenides superconductors[18–22], it is new for the e g electrons in the cuprates,Remarkably, in the presence of strong on-site Coulombrepulsion, we find that a strong-coupling theory for theelectronic structure predicts a nodal d -wave superconduc-tor with antiphase pairing gaps on the two Fermi surfacesat the high doping x = 0 . H = H t + H I , where H t describesthe DFT band structure near the Fermi level and H I theelectron correlations. The model is equivalent to a gen-eralization of the Zhang-Rice singlet construction wherethe charge degrees of freedom on the oxygen sites havebeen integrated out, and the Cu in the 3d configura-tion includes the spin singlets of a hole (3d ) on Cu-siteand a hole (2p ) on its neighbouring O-sites with com-patible symmetries to d x − y and d z − r [2]. Denoting aspin- σ electron in the effective d x - and d z -like orbitalsby d ασ , α = x, z respectively, we have H t = X kαβσ ε αβk d † kασ d kβσ + X kσ e α d † kασ d kασ , (1)where e α is the crystal field energy of each orbital.The lattice structure of Ba214 belongs to the D h pointgroup. The intra and interorbital hoppings can be ex-pressed in terms of the lattice harmonics of A g symme-try: γ k = cos k x + cos k y , α k = cos k x cos k y , and γ ′ k =cos 2 k x +cos 2 k y , and B g symmetry: β k = cos k x − cos k y and β ′ k = cos 2 k x − cos 2 k y , for up to third nearest neigh-bor hopping. The corresponding expression for the hop-ping energies in Eq. (1) is given by ε ααk = − t α γ k − t ′ α α k − t ′′ α γ ′ k and ε xzk = 2 t xz β k +2 t ′′ xz β ′ k , where the valuesof the hopping parameters are given in Table S2 togetherwith e α in the SM [13]. The band structure of the two-orbital TB model is shown in Fig. 1(e). It faithfully rep-resents the low energy DFT band structure in Fig. 1(d),including the orbital contents, as a two-band system witha bandwidth W ≃ ε xzk carries the B g harmonics, the band structure along thenodal direction (Γ → M ) is orbital diagonal. Moreover,the interband mixing is also weak along the Γ → X direc-tion because of the large band energy separation. How-ever, the two bands hybridize strongly when the bandenergies are close, as seen along X → M . Reproducedin Fig. 2(a), the Fermi levels are indicated by the dashedblack line for the d configuration with n e = 3 electrons,i.e. the undoped case at x = 0 with orbital occupations n z ≃ . n x ≃ .
7; and by the blue dashed line forthe experimentally realized doping level x = 0 .
4, wherethe bare occupations change to n z = 1 .
04 and n x = 1 . d z character around Γ and a much smallerhole-like d x FS around M . We next turn to the correla-tion effects beyond the band description.The correlation part of the Hamiltonian H I followsfrom the standard two-orbital Hubbard model [23–25] forthe e g complex H I = U X i,α ˆ n iα ↑ ˆ n iα ↓ + (cid:18) U ′ − J H (cid:19) X i,α<β ˆ n iα ˆ n iβ (2) − J H X i,α = β S iα · S iβ + J H X i,α = β d † iα ↑ d † iα ↓ d iβ ↓ d iβ ↑ , where the intra and interorbital repulsion U and U ′ arerelated to Hund’s rule coupling J H by U = U ′ +2 J H . Wewill set U = 7eV and hence U/W ≃ .
8, and J H = 0 . U in the rest of the paper. The strong correlated effects canbe studied using the multiorbital Gutzwiller projectionmethod [26–28]: H t + H I → H G = P G H t P G , where P G isthe finite- U projection operator that reduces the statisti-cal weight of the Fock states with multiple occupations.The projection can be conveniently implemented usingthe Gutzwiller approximation (GA) [26, 27] developed tostudy the multiorbital cobaltates [29], Fe-pnictides [28], FIG. 2. (a) Noninteracting band dispersion with the Fermilevels at x = 0 (black dashed line) and x = 0 . x = 0 and U = 4 . eV :half-filled flat d z band (red line) and completely filled d x band (black line). (c) Correlated band structure at x = 0 . U = 7eV. (d) Renormalized Fermi surfaces at x = 0 . and the monolayer CuO /Bi2212 [9], H G = X kαβσ g σαβ ε αβk d † kασ d kβσ + X kσ ( e α + λ σα ) d † kασ d kασ , (3)where the orbital dependent hopping and crystal fieldrenormalizations g σαβ and λ σα are determined self-consistently. We first study the undoped case at x = 0and n e = 3. The Gutzwiller solution shows that theground state is an antiferromagnetic (AF) insulator. Itis remarkable that the AF order is orbital selective, i.e.the electron correlation induces the interorbital carriertransfer [29] and drives the d z band to half-filling with n z = 1. The paramagnetic interacting band dispersionsare shown in Fig. 2(b) at U = 4 . eV , exhibiting a half-filled flat d z band, while the d x band is pushed belowthe Fermi level and completely filled. An insulating gapopens up with AF ordered moments coming from the lo-calized d z electrons.For large hole doping at x = 0 . n e = 2 . n z = 1 .
02 and n x = 1 .
58 closeto their bare values, the band narrowing effect is strongand orbital-dependent. The strongly correlated d z bandremains close to half-filling and narrows significantly bya factor g zz = 0 .
33 while the d x band only narrows bya factor g xx = 0 .
87. It is thus rather natural to considera picture of two-orbital doped Mott insulator where theSC state arises from the spin-orbital superexchange in-teractions of the Kugel-Khomskii type [9, 23, 25]. In thespin-orbital basis ( d x ↑ , d x ↓ , d z ↑ , d z ↓ ) T , the fermion bilin-ears at each site can be represented as a tensor product T µi S νi , where S µi and T µi , µ = 0 , x, y, z are one half of theidentity and Pauli matrices acting in the spin and orbitalsectors, respectively. For example, d † ix ↓ d iz ↑ = T + i S − i ,where T ± i = T xi ± iT yi and S ± i = S xi ± iS yi . The spin-orbital superexchange interactions can thus be derivedand written as H J − K = X h ij i (cid:20) J S i · S j + X µν I µν T µi T νj (4)+ X µν K µν ( S i · S j )( T µi T νj ) (cid:21) where the J -term is the SU(2) invariant Heisenberg spinexchange coupling, while the terms proportional I µν and K µν describe the anisotropic orbital and spin-orbital en-tangled superexchange interactions, respectively, sincethe orbital rotation symmetry is broken by the lattice in H t . In Ref. [9], it was shown that despite the orbital T zi order induced by the crystal field splitting, the transverseorbital T ± i fluctuations contribute to SC pairing. Thus,including all contributions to spin-singlet pairing repre-sented by ∆ αβ † ij = d † iα ↑ d † jβ ↓ − d † iα ↓ d † jβ ↑ in the spin andspin-orbit entangled quadruple exchange interactions inEq. (4), we arrive at the effective Hamiltonian describingthe possible SC ground states in the strongly correlatedtwo-orbital model H = P G H t P G − X h ij i (cid:20) J s X αβ ∆ αβ † ij ∆ αβij + K X α = β (∆ αα † ij ∆ ββij + ∆ αβ † ij ∆ βαij ) (cid:21) (5)The couplings ( J s , K ) are explicit functions of t αβ , U ,and J H [9], but will be considered as phenomenologicalparameters in our effective theory.In the following, we set J s = 200meV and K = 80meV,and study the emergent SC states in such a two-banddoped Mott insulator at high doping x = 0 . h ∆ αβij i = 1 N s X k ,αβ ∆ αβ b αβ ( k ) e i k · ( r i − r j ) , (6)self-consistently in the Gutzwiller approximation, where N s is the number of lattice sites and b αβ ( k ) the form fac-tors of different symmetries in the D h point group ofthe crystal: b αα ( k ) = γ k and b xz ( k ) = β k in the A g and b αα ( k ) = β k and b xz ( k ) = γ k in the B g channel. Re-markably, we find that the ground state is a prominentsuperconductor with B g symmetry. In Fig. 3(a) and3(b), the gap functions are plotted along the FSs, whichexhibit a two-band d ± -wave structure with 8 nodes (4 on θ/π -40-2002040 ∆ ( m e V ) -40 -20 0 20 40 E (meV) d I / d V ( a . u . ) Totald z Γ θ
M(a)(b) (c)
FIG. 3. (a) Normal state FSs at x = 0 .
4. (b) Anisotropic pair-ing energy gaps as a function of angle θ depicted in (a) aroundtwo FSs (blue for FS around Γ and red around M ), showingtwo antiphase d -wave gap functions. (c) Total (red line) and d z orbital contribution (black dashed line) to local tunnel-ing density of states, showing two d -wave gaps with coherentpeaks around ± ± each FS) and an overall sign-change between the two FSs.This is a strong coupling d -wave analogy of the proposed s ± pairing gap function in Fe-based superconductors.We stress that the new mechanism of doping an orbital-selective Mott insulator in the two-band unconventionalcuprate is crucial for the emergent high- T c SC state in thehighly overdoped region. The close to half-filled, stronglycorrelated d z band is responsible for the large pairinggap ( ∼ d x banddevelops a smaller ( ∼ c su-perconductivity in doped Mott insulators [3]. It is con-ceivable that this mechanism provides an explanation forthe nearly doubled T c compared the isostructural single-band La214. The calculated orbital resolved local densityof states, N α ( ω ) = P kσ Im R β e iωτ h T τ d kασ ( τ ) d † kασ (0) i ,is shown in Fig. 3(c), exhibiting the mixing of two d -wavegap functions and the dominate spectral weight from the d z band. We also find that increasing the δ -oxygen den-sity to δ = 0 .
45 leads to a different SC state of A g sym-metry near the d configuration with a two-band s ± gapfunction [9], in qualitatively agreement with the recentweak-coupling RPA calculations [30].In summary, we presented a theoretical description ofthe atomic and electronic structure, and the emergenceof superconductivity of the newly discovered 73K high- T c Ba CuO δ at δ = 0 . d z and d x orbitals highly overdoped ( x = 0 .
4) from the d config-uration. This is complimentary to the CuO monolayergrown on Bi Sr CaCu O δ [31], where the liberation ofthe d z orbital and nodeless two-band superconductivityhave been argued to arise from the crystal field of the un-balanced octahedron and the heavy hole-doping throughinterface carrier transfer [9]. Constructing a minimaltwo-orbital Hubbard model using the DFT band struc-ture, we found that the correlated electronic states canbe described by a novel doped two-orbital Mott insulatorand materialize a new two-band high-T c SC state with d ± -wave gap functions through the spin and spin-orbitalexchange interactions. The multiband and the orbital se-lectivity are crucial for the T c enhancement at such highdopings via a combination of strong pairing interaction and the large superfluid density. The basic predictionof d z orbital libration and two-band superconductivityshould be amenable to experimental tests for two pair-ing gaps in the heat capacity, NMR, and tunneling mea-surements in the current polycrystal samples. While fur-ther experimental and theoretical studies are necessary,the present theory offers a conjecture that Ba CuO δ highlights a class of unconventional cuprates, includingthe 95K Sr CuO δ [32–34], where in-plane positional δ -oxygen doping under high-pressure can serve as a routetoward higher T c by utilizing the electrons partially oc-cupying both the 3 d e g orbitals.We thank C.Q. Jin and S. Uchida for useful discus-sions. The work is supported in part by the Ministryof Science and Technology of China 973 program (No.2017YFA0303100, No. 2015CB921300), National Sci-ence Foundation of China (Grant No. NSFC-1190020,11534014, 11334012), and the Strategic Priority ResearchProgram of CAS (Grant No. XDB07000000); and theU.S. Department of Energy, Basic Energy Sciences GrantNo. DE-FG02-99ER45747 (K.J and Z.W). Z.W. thanksthe Institute of Physics, CAS for hospitality. The firsttwo authors (C. Le and K. Jiang) contributed equally. [1] J.G. Bednorz, K.A. Muller, Z. Phys. B , 189 (1986).[2] F. C. Zhang and T. M. Rice, Phys. Rev. B , 3759(R)(1988).[3] Patrick A. Lee, Naoto Nagaosa, and Xiao-Gang Wen, Rev.Mod. Phys. , 17 (2006).[4] C.-C. Chen, et. al. Phys. Rev. Lett. , 177401 (2010).[5] X. Wang, H.T. Dan, and A.J. Millis, Phys. Rev. B ,014530 (2011).[6] H. Sakakibara, H. Usui, K. Kuroki, R. Arita, and H. Aoki,Phys. Rev. Lett. , 057003 (2010).[7] H. Sakakibara, H. Usui, K. Kuroki, R. Arita, and H. Aoki,Phys. Rev. B , 064501 (2012).[8] W. M. Li, et al., PNAS , 12156 (2019).[9] K. Jiang, X. Wu, J. Hu and Z. Wang, Phys. Rev. Lett. , 227002 (2018).[10] D. C. Peets, D. G. Hawthorn, K. M. Shen, Y.-J. Kim, D.S. Ellis, H. Zhang, S. Komiya, Y. Ando, G. A. Sawatzky,R. Liang, D. A. Bonn, and W. N. Hardy, Phys. Rev. Lett. , 087402 (2009).[11] Xin Wang, Luca de Medici, and A. J. Millis, Phys. Rev.B , 094522 (2010).[12] Kai Liu, Zhong-Yi Lu, Tao Xiang, Phys. Rev. Materials , 044802 (2019).[13] See Supplemental Material for more detailed discussions.[14] G. Kresse and J. Hafner, Phys. Rev. B , 558 (1993).[15] G. Kresse and J. Furthmuller, Comput. Mater. Sci. , 15(1996).[16] G. Kresse and J. Furthmuller, Phys. Rev. B ,11169(1996).[17] J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev.Lett. , 3865 (1996).[18] I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du,Phys. Rev. Lett. , 057003 (2008).[19] S. Graser, T. A. Maier, P. J. Hirschfeld, and D. J. Scalapino, New J. Phys. , 025016 (2009).[20] F. Wang, H. Zhai, Y. Ran, A. Vishwanath, and D. Lee,Phys. Rev. Lett. , 047005 (2009).[21] A. Chubukov, D. Efremov, and I. Eremin, Phys. Rev. B ,206404 (2008).[23] C. Castellani, C.R. Natoli, and J. Ranninger, Phys. Rev.B , 4945 (1978).[24] A. Georges, L. de Medici, and J. Mravlje, Annu. Rev.Condens. Matter Phys. , 137 (2013).[25] K. I. Kugel and D. I. Khomskii, Sov. Phys. JETP , 725(1973); K. I. Kugel and D. I. Khomskii, Sov. Phys. Usp. , 231 (1982).[26] J. Bunemann, W. Weber, and F. Gebhard, Phys. Rev. B , 6898 (1998).[27] F. Lechermann, A. Georges, G. Kotliar, and O. Parcollet,Phys. Rev. B , 155102 (2007).[28] S. Zhou, Z. Wang, Phys. Rev. Lett. , 096401 (2010).[29] Sen Zhou, Meng Gao, Hong Ding, Patrick A. Lee, andZiqiang Wang, Phys. Rev. Lett. , 206401 (2005).[30] T. A. Maier, T. Berlijn, and D. J. Scalapino, Phys. Rev.B , 224515 (2019).[31] Y. Zhong, Y. Wang, S. Han, Y. Lv, W. Wang, D. Zhang,H. Ding, Y. Zhang, L. Wang, K. He, R. Zhong, J. A.Schneeloch, G. Gu, C. Song, X. Ma, Q. K. Xue, ScienceBulletin , 1239 (2016).[32] T. H. Geballe and M. Marezio, Physica C , 680(2009).[33] Q. Q. Lin, H. Yang, X.M. Qin, Y.Yu, L.X. Yang, F.Y.Li,R.C.Yu, C.Q.Jin, S.Uchida, Phys. Rev. B , 100506(2006).[34] Z. Hiroi, M. Takano, M. Azuma, Y. Takeda, Nature ,315 (1993). TABLE S1. Experimental determined crystal Structure ofBa CuO δ in space group I /mmm with lattice constant a = 4 . A and b = 12 . A .Atom site x y z occupancyBa 4e 0 0 0.35627 1Cu 2a 0 0 0 1O
4e 0 0 0.1438 1O
4c 0 0.5 0 0.592
SUPPLEMENTAL MATERIALA. Density functional calculations
Our calculations are performed using density func-tional theory (DFT) employing the projector augmentedwave (PAW) method encoded in the Vienna ab initiosimulation package (VASP) [1–3]. Generalized-gradientapproximation (GGA) [4] for the exchange correlationfunctional is used. Throughout the work, the cutoff en-ergy is set to be 500 eV for expanding the wave functionsinto plane-wave basis. In the calculation, the Brillouinzone (BZ) is sampled in the k space within Monkhorst-Pack scheme[5]. On the basis of the equilibrium struc-ture, the k mesh used is 10 × ×
4. In our calculation,we adopt the experimental parameters listed in Table.S1[6] and use the stoichiometric formula Ba CuO (Ba214)and Ba CuO (B213), whose crystal structures are shownin Fig. S1(a) and Fig. S2(a). The calculated band struc-tures are shown in Fig. S1(c) and Fig. S2(c) for Ba214and Ba213 respectively. Note that the convention of BZusing the BCO primitive unit cell is slightly differentthan the normal cuprate convention. To be consistentwith other cuprates, we use the common convention ofcuprates. As can be seen from Fig. S2, Ba213 has 1DCu-O chain planes. We choose the chain to be along the x direction. As a result, the hopping in the y direction isgreatly reduced due to the lack of oxygen on the bond,leading to an 1D band with two 1D Fermi surface sec-tions shown in Fig. S2(b) and (c), which are consistentwith the calculations in Ref. [7]. B. Orbital construction and Tight-binding modelparameters
To construct an effective model describing the bandstructures near Fermi level, we can analyze orbital char-acters of the electronic structure. The d x − y orbitalmixed with the anti-symmetric combination of the in-plane oxygen p x and p y orbitals is similar to the Zhang-Rice singlet of common cuprate contributing a holepocket around the M point. Due to the compressed oc-tahedron, the d z orbital strongly hybridizes with the p z orbital from the top apical oxygen (O A ) and bottom api-cal oxygen (O B ), as shown in the left side of Fig. S1(b). FIG. S1. (a) Crystal structure of Ba CuO . (b) Effectivemolecular orbital through hybridization between d z and api-cal oxygen p z orbitals. Right side is the schematic molecularorbital φ . (c) Calculated DFT band structure.FIG. S2. (a) Crystal structure of Ba CuO with Cu-O chainin x direction. (b) The Fermi surface of Ba CuO . (c) Cal-culated DFT band structure. One can consider a local molecular model describingthis hybridization. Taking p Az ,d z ,p Bz as the basis, theeffective Hamiltonian of the molecular model can be writ- TABLE S2. Hopping parameters of the TB model in eV.hopping integral 1rd ( t ) 2nd ( t ′ ) 3rd ( t ′′ )intra-orbital t x -0.4968 0.0503 -0.0652intra-orbital t z -0.2135 -0.0190 -0.0219inter-orbital t xz ten as H local = ǫ t t ǫ − t − t ǫ (S1)where ǫ and ǫ are the on-site energies of the p z and d z orbitals. t is the hopping parameter between p z and d z orbitals. The eigenvalues of Eq. (S1) can be found as E = 12 ( ǫ + ǫ + p t + ( ǫ − ǫ ) ) E = ǫ E = 12 ( ǫ + ǫ − p t + ( ǫ − ǫ ) ) . (S2)The corresponding eigenvectors are φ = − p Az + p Bz − t ( − ǫ + ǫ + p t + ( ǫ − ǫ ) ) d z φ = p Az + p Bz φ = − p Az + p Bz − t ( − ǫ + ǫ − p t + ( ǫ − ǫ ) ) d z . The schematic molecular orbital φ is plotted in right sideof Fig. S1(b). From Fig. S1(c), we can also find that φ and φ are located around Fermi level and -4.5eV respec-tively. φ is entirely attributed to O A/B -p z orbitals anddistributes around -2.6eV. Hence, φ with the d z likebonding orbital and Zhang-Rice singlet with the d x likebonding orbital dominate the electronic structure aroundthe BCO Fermi level. Based on the DFT results andorbital fields, we construct a two-orbital tight-binding(TB) model of Cu e g complex for the BCO. The Hamil-tonian is given in Eq. (1) in the main text. Denoting d ασ , α = x ( d x ) , z ( d z ) H t = X kσ ε xxk d † kxσ d kxσ + X kσ ε xzk ( d † kxσ d kzσ + h.c. )+ X kσ ε zzk d † kzσ d kzσ + X kσ e α d † ασ d ασ (S3)where ε αβk is the kinetic energy due to intra and interor-bital hopping, and e α is the on-site energy of d z and d x orbitals. Up to third nearest neighbor hopping, we have ε xxk = − t x γ k − t ′ x α k − t ′′ x γ ′ k ε zzk = − t z γ k − t ′ z α k − t ′′ z γ ′ k ε xzk = 2 t xz β k + 2 t ′′ xz β ′ k (S4)where the intraorbital hopping involves lattice harmonicsof A symmetry γ k = cos k x + cos k y , α k = cos k x cos k y ,and γ ′ k = cos 2 k x + cos 2 k y , and the interorbital hoppinginvolves B harmonics β k = cos k x − cos k y and β ′ k =cos 2 k x − cos 2 k y . The hopping parameters for the 1st ( t ),2nd ( t ′ ), and 3rd ( t ′′ ) nearest neighbors are listed in TableS2. The on-site energy of d x and d z are e x = 2 . eV and e z = 3 . eV . [1] G. Kresse and J. Hafner, Phys. Rev. B , 558 (1993).[2] G. Kresse and J. Furthmuller, Comput. Mater. Sci. , 15(1996).[3] G. Kresse and J. Furthmuller, Phys. Rev. B ,11169(1996).[4] J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. , 3865 (1996).p[5] H. J. Monkhorst and J. Pack, Phys. Rev. B , 5188(1976).[6] W. M. Li, et al., PNAS116