Generalized one-band model based on Zhang-Rice singlets for Tetragonal CuO
aa r X i v : . [ c ond - m a t . s up r- c on ] A p r Generalized one-band model based on Zhang-Rice singlets for Tetragonal CuO
I. J. Hamad and L. O. Manuel
Instituto de F´ısica Rosario (CONICET) and Universidad Nacional de Rosario,Bv. 27 de Febrero 210 bis, 2000 Rosario, Argentina
A. A. Aligia
Centro At´omico Bariloche and Instituto Balseiro, CNEA, CONICET, 8400 Bariloche, Argentina ∗ Tetragonal CuO (T-CuO) has attracted attention because of its structure similar to that of thecuprates. It has been recently proposed as a compound whose study can give an end to the longdebate about the proper microscopic modeling for cuprates. In this work, we rigorously derive aneffective one-band generalized t − J model for T-CuO, based on orthogonalized Zhang-Rice singlets,and make an estimative calculation of its parameters, based on previous ab initio calculations. Bymeans of the self-consistent Born approximation, we then evaluate the spectral function and thequasiparticle dispersion for a single hole doped in antiferromagnetically ordered half-filled T-CuO.Our predictions show very good agreement with angle-resolved photoemission spectra and withtheoretical multiband results. We conclude that a generalized t − J model remains the minimalHamiltonian for a correct description of single-hole dynamics in cuprates. PACS numbers: 75.20.Hr, 71.27.+a, 72.15.Qm, 73.63.Kv
More than three decades after their discovery, hightemperature superconductors still give rise to many de-bates. On the theoretical side, one of the most long-standing and important discussions is about the propermicroscopic model for describing superconductivity. Inthis respect and from the outset, attention was focusedon the spectral function of a single-hole doped on theparent half-filled compounds, whose quasiparticle (QP)dispersion relation is directly measured in angle-resolvedphotoemission (ARPES) experiments. Experimental ev-idence shows that this doped hole resides on the O 2p σ orbitals [5–7]. For the CuO planes that build up thecuprates, Zhang and Rice [4] proposed that a singlet,called Zhang-Rice (ZR) singlet, is formed between thespin of a cooper atom and the spin of the hole residing in alinear combination of four ligand oxygen orbitals aroundthe cooper atom. Integrating out the oxygen orbitals, aone-band effective model was proposed in which the effec-tive holes (representing ZR singlets) reside on the cooperatoms and propagate emitting spin excitations, magnons.In this model, adding two holes as nearest-neighbors inan antiferromagnetic background costs less energy thanif they are added far apart. This is a simplified view ofthe pairing glue of magnetic origin [22].Since the proposal of Zhang and Rice, an unclosed de-bate about the validity of one-band effective models hastaken place [6, 8–16, 23]. Several authors sustain thatonly the three-band model [1, 2] is valid for describingthe physics of the cuprates correctly, where the threebands come from two O 2p σ orbitals and one Cu 3d x − y orbital, not only for the insulating parent compound athalf-filling, but also for many other phases of the richphase diagram of the cuprates and related compounds.This issue is of central importance since many investiga-tions have been done in one-band models and hence their validity is, at least partially, questioned.Recently, tetragonal CuO (T-CuO) has been synthe-sized, by growing epitaxially CuO planes on a substrate[(001) SrTiO ] [19]. T-CuO can be considered as twointerpenetrating CuO sublattices sharing one oxygenatom and hence has two degenerate antiferromagneticground states, as shown in Fig. 3. ARPES experimentswere performed on this compound [20], showing substan-tial intralayer coupling between these two sublattices anda similar dispersion (with some differences) to that of thecuprate Sr CuO Cl . This material was addressed in arecent work [15] as a good candidate to discern whetherone-band models, based on ZR singlets, are valid for de-scribing the physics of CuO planes or if, instead, three-band models should be used.In this Letter, we rigorously derive an effective one-band model for T-CuO and compare its QP disper-sion with experimental ARPES results and theoreticalpredictions for the three-band model. Using a proce-dure based on previous derivations of generalized one-band effective Hamiltonians [10], we start from a spin-fermion model for T-CuO and we obtain then its effec-tive one-band model for the ZR singlets. The parametersof the model were calculated starting from parametersdetermined by constrained-density-functional computa-tions for La CuO [15], and estimating their variationsfor the T-CuO case. We find an effective hopping to firstnearest neighbors (NN) between CuO sublattices, andeffective hoppings to first, second, and third NN in thesame sublattice, together with superexchange parameters J (the usual NN antiferromagnetic one for CuO planes)and a ferromagnetic J ′ (NN in T-CuO, belonging to dif-ferent CuO sublattices).Using this model, we calculate the QP dispersionby means of the self-consistent Born approximation d c FIG. 1: (Color online) The two possible magnetic groundstates for T-CuO: Q = (0 , π ) (left) and Q = ( π,
0) (right).The coordinate versors point in the directions of c and d .Arrows indicate spins at Cu sites and circles correspond tothe O sites. (SCBA), a reliable and widely used many-body method.We compare our results with ARPES experiments in T-CuO, obtaining good qualitative and quantitative agree-ments. Our results also recover previous ones from athree-band calculation, including particular aspects thatwere claimed absent in a ZR picture. We then concludethat our method is correct for obtaining rigorous one-band effective models, and that the one-band model thatwe have derived describes correctly the physics of a singledoped hole in T-CuO.We start from a spin-fermion model (Cu spins and Oholes), obtained integrating out valence fluctuations atthe Cu sites [6, 9, 10, 15, 23]. With the adequate choiceof phases (Fig. S1 of Ref. 23) the Hamiltonian reads H sf = X iδδ ′ σ p † i + δ ′ σ p i + δσ (cid:20) ( t sf + t sf )( 12 + 2 S i · s i + δ ) − t sf (cid:21) − J d X iδ S i · s i + δ + J X iδ S i · S i +2 δ − t pp X jγσ p † j + γσ p jσ + t ′ pp X jγσ s γ (cid:16) p † j + γσ p jσ + H . c . (cid:17) − J ′ X iγ S i · S i + γ , (1)where i ( j ) labels the Cu (O) sites and i + δ ( j + γ )label the four O atoms nearest to Cu atom i (O atom j ).The spin at the Cu site i (O orbital 2p σ at site i + δ ) isdenoted as S i ( s i + δ ). The signs s γ = − γ k ˆx + ˆy and s γ = 1 in the perpendicular direction, being ˆx and ˆy the unit vectors along the directions of NN Cu atomsin the CuO planes (which are second NN in the T-CuOstructure). The parameter t ′ pp ≃ . t pp (Ref. 15). Thisis essentially the same Hamiltonian as that considered byAdolphs et al. [15] (we include virtual fluctuations viaCu +3 ) and its low-energy physics reproduces that of thethree-band model [10].Projecting the Hamiltonian over the subspace of or-thogonal ZR singlets, we have derived a one-band gener-alized t − J model for T-CuO. All the steps can be foundin Ref. 23. The one-band effective generalized t − J Hamiltonian is: H stJ = − X κ =0 t κ X iv κ σ (cid:16) c † iσ c i + v κ σ + H . c . (cid:17) ++ J X iv S i · S i + v − J ′ X iv S i · S i + v , (2)where the subscript κ = 0 refers to intersublattice hop-ping of NN Cu atoms in the T-CuO structure, while κ = 1 , ,
3, refer to first, second, and third NN withineach CuO sublattice, respectively. Instead of using ar-bitrary values for the parameters, we have calculatedthem, keeping the states corresponding to orthogonalizedZR singlets and using results from constrained-density-functional calculations [15]. These values are very simi-lar to those corresponding to the model used by Adolphs et al. [15], as shown in Table 3 of Ref. 23. We havechecked that the results for both sets are quite similar.To simplify the discussion we present here only the resultsfor the latter. The parameters in meV are t = − t = 369, t = − t = 65, J = 150, and J ′ = 0. Thiseffective model was proposed previously by Moser et al. [20]. Here we provide its justification and determine itsparameters.The spectral functions were calculated by means ofthe SCBA [36–39], a semianalytic method that has beenproven to compare very well with exact diagonaliza-tion (ED) results on finite clusters in different systems[32, 36, 37, 39, 41]. It is one of the more reliable andchecked methods up to date to calculate the hole Green’sfunction, and in particular its QP dispersion relation.However, some care is needed to map the QP weight be-tween different models [37]. In order to do such calcula-tion, we follow standard procedures [36]. On one handthe magnetic dispersion relation is obtained treating themagnetic part of the Hamiltonian at the linear spin-wavelevel, since the system we study has long-range order, andhence its magnetic excitations are semiclassical magnons.On the other hand, the electron creation and annihilationoperators in the hopping terms are mapped into holonsof a slave-fermion representation (details in Ref. 23).Within SCBA, we arrive to an effective Hamiltonian: H eff = X k ǫ k h † k h k + X k ω k θ † k θ k ++ 1 √ N X kq (cid:16) M kq h † k h k − q θ q + H . c . (cid:17) , (3) ǫ k = 2 t cos( k · c ) + 4 t cos( ak x ) cos( ak y ) +2 t [cos(2 ak x ) + cos(2 ak y )] ,ω k = q A k − B k ,M kq = 2 t { cos [( k − q ) · c ] u q − cos( k · c ) v q } +2 t [ u q ζ ( k − q ) − v q ζ ( k )] , (4)where ǫ k is the bare hole dispersion (with no coupling tomagnons), ω k is the magnon dispersion relation, with A k = 2 J − J ′ cos( c · k ), B k = J P v cos( v · k ) − J ′ cos( d · k ), and M kq is the vertex that couples thehole with magnons. Here ζ ( k ) = cos( ak x ) + cos( ak y ),and c = b ( ˆx + ˆy ), d = b ( − ˆx + ˆy ), being a = 2 b thedistance between Cu atoms in the CuO planes. Thevectors c and d are indicated in Fig. 3. We now com- Γ B X’ B’ M A’ X/M’ A Γ k E Q P ( k )- E ( Γ ) ( e V ) (0.π)(π,0) k y k x Γ BX’ B’M A’ X/M’ A (−π,π) (π,−π)(π,π)
FIG. 2: (Color online) Quasiparticle dispersion relation (rel-ative to Γ) along the path marked in the inset, the same asthe one measured in the ARPES experiment in Ref. 20. Abroadening equivalent to 20 meV was applied to the spectralfunctions (see text). pare our results with ARPES experiments performed onT-CuO, specifically with the those in Figs. 2 and 3 ofRef. 20. For that purpose, we adopt in Figs. 2 and 3, anelectron picture. In Fig. 2 we show the QP dispersionderived from our SCBA calculation. This should be com-pared with the blue points in Fig. 2 of Ref. 20, and alsowith the white points in the same figure, correspondingto exact diagonalization of a one-band Hubbard modelin 20 sites. In our calculation, a broadening equivalentto 20 meV (controlled by means of the parameter δ inEq. 4), similar to the experimental resolution (30 meV[20]), was applied to the spectral functions. Taking intoaccount the two possible magnetic ground states for T-CuO, we obtain the two QP dispersions shown in Fig.2. It can be observed that the dispersion correspond-ing to Q = ( π,
0) recovers all the main features of theexperimental dispersion, and hence our results can dis-tinguish between the possible degenerate magnetic ordersin the experiment. In particular, we recover the asym-metry between the points Γ and X ′ , B and B ′ , and A and A ′ . Moreover, we obtain, E ( A ) − E ( A ′ ) = 128 meV, E ( B ) − E ( B ′ ) = 64 meV, and E (Γ) − E ( M ) = 10 meV,while the experimentally measured energy differences are140 meV, 60 meV, and 180 meV, respectively [20]. The agreement is very good, except in the last case. Thisdiscrepancy is quite likely due to missing quasiparticlepeaks with small weight in the experiment (see also Fig.S4 of Ref. [23]). In that sense, we note that the Γ point(and points located in its vicinity) shows a very broadspectrum (see Figs. 2 and 3 in Ref. [20] ), and hencethere may be some uncertainty in the determination ofthe QP energy which could explain this discrepancy. Thebandwidth of the QP dispersion, along this path, takenfrom our SCBA calculation is 0 . eV , very similar to thebandwidth of the experimental dispersion, approximately0 . eV . We have also calculated an intensity curve along FIG. 3: (Color online) SCBA intensity map along the samepath as in Fig. 2. The assumed magnetic order is ( π, the same path as in the experiment, to compare withthe ARPES intensities (Fig. 2 of Ref. 20). We showonly the intensity corresponding to Q = ( π, X/M ′ point, where there is no in-tensity at all in the ARPES data, and on the other handa β band seems to merge with the QP band, specially atthe X/M ′ point but also possibly around the M point.So at these two points, in particular around the X/M ′ points, the comparison of our calculation with the exper-iment is obscured by these experimental facts. Finally,it is worth to mention that in the case that the illumi-nated area in the ARPES experiments contains domainswith both magnetic Q = ( π,
0) and Q = (0 , π ) vectors(as mentioned above, they are degenerate), the QP dis-persion should be a superposition of both curves shownin Fig. 2, which does not seem to be what is observed in Γ (π,π) (π,0) Γ (0,π) (π,0) k -9-8-7-6-5-4-3 E Q P ( k ) / J A A’ ky kx ππ FIG. 4: (Color online) SCBA hole’s dispersion relation in units of J (0.15 eV) along the path marked in the inset. Black dashedline: result corresponding to t = 0 (decoupled sublattices). Blue full line: full result with t = −
184 meV. the experiment [42]. The intensity curve Fig. 3 shouldalso change accordingly, but in our case we have checkedthat the only noticeable changes occur around the
X/M ′ point, at which nevertheless there is no intensity in theARPES data corresponding to the band ascribed to ZRsinglets [20].In general, the spectral function corresponding to adefinite momentum contains, in the hole picture, a lowenergy pole, whose energy defines the QP energy, and ahigh energy part which is related to the incoherent move-ment of the hole, having its origin in multimagnon pro-cesses [32]. When the quasiparticle weight is significant,the brighter areas in Fig. 3 will coincide with the energyof the QP in Fig. 2. On the contrary when the incoherentpart of the spectral function takes most of the spectralweight, this will not happen. Points like Γ and M havelow QP weight, while on the contrary for the lines B − B ′ , A ′ − A the QP weight is relatively high (some spectralfunctions can be seen in Fig S4 of ref. 23).It was claimed previously that the one-hole dispersionin T-CuO requires a three-band model to be describedcorrectly [15]. The evidence presented came from a vari-ational calculation on the spin-fermion model Eq. (2),whose results a one-band model supposedly cannot cap-ture. In particular, it was shown that the minimum thatthe QP dispersion has at ( π/ , π/
2) for CuO (or, equiv-alently in T-CuO, if the two CuO sublattices are dis-connected), shifts along the diagonal Γ ≡ (0 , − ( π, π ),towards the Γ point, when the two CuO sublattices areconnected to form T-CuO. This happens for Q = (0 , π ).Alternatively, the shift is along the antidiagonal towards X/M for Q = ( π, planes [16], where it was claimed that aone-band t − t ′ − t ′′ − J model has a minimum at ( π/ , π/ π/ , π/ t − J model [Eq. (37)] derived from H sf [Eq. (2)] we now calculate the QP dispersionalong the same path as in Ref. 15 and with the corre-sponding parameters (set B of Table III of Ref. 23), and Q = (0 , π ). Results are shown in Fig. 4, plotted adopt-ing the hole’s picture. As before, a broadening equivalentto 20 meV was applied to the spectral functions, but theresults do not depend significantly on this (unless broad-enings an order of magnitude larger are applied). It isclear that when both sublattices are connected throughthe t term, the QP dispersion relation derived from H sf is recovered. In particular, we obtain a shift of the QPminimum along the diagonal towards the Γ point, al-though this shift is lower (about half) in magnitude thanthe one obtained with the three-band model. This differ-ence might be due to the different theoretical treatmentsused by Adolphs et al. to solve H sf [Eq. (2)] and byus to solve H tJ [Eq. (37)] In this respect, we remarkit is very difficult to decide which theoretical treatmentgives more accurate results from quantitative differencesof this kind, since on one hand both compare very wellwith ED results in finite clusters, while on the other handno experiment so far could even measure this shift in theQP dispersion relation. We also remark that varying t ,the QP dispersion relation is not changed apart from aconstant shift (in agreement with previous results [16]).This is important since t is the parameter obtained withless accuracy.The shift in our model is not caused by the cou-pling of the hole with spin fluctuations, which in factconspires against it. This can be seen from the effec-tive Hamiltonian Eq. (4), since the bare-hole dispersion( i.e. with no coupling to magnons) ǫ k = 2 t cos( k · c ) +4 t cos( ak x ) cos( ak y ) + 2 t [cos(2 ak x ) + cos(2 ak y )] has aminimum, along the diagonal k x = k y , that shifts from( π/ , π/
2) towards the Γ point when the intersublatticehopping t is turned on. For example, the bare hole min-imum is at (0 . π, . π ) for the parameter set we used.However, when the interaction of the bare hole with spinfluctuations (magnons) is taken into account through thevertex M kq , the minimum shifts back towards ( π/ , π/ A and Γ points. Note that the SCBA contains aninfinite number of spin fluctuations while only a few areincluded in the treatment of Ref. 15. In any case, we haveshown that a ZR one-band model can explain a shift inthe QP minimum at ( π/ , π/ J , slightlyless than the result from the variational method in thethree-band model Eq. (2) [15].Overall, we conclude that our effective generalizedone-band model, rigorously derived from orthogonalizedZhang-Rice singlets, and without free parameters, notonly does recover characteristics of the three-band model,but also its predictions agree qualitatively and quantita-tively with ARPES experiments in tetragonal CuO.We thank A. Greco for fruitful discussions. IJH waspartially supported by PICT-2014-3290. IJH and LOMare partially supported by PIP 0364 of CONICET. AAAis sponsored by PIP 112-201101-00832 of CONICET andPICT 2013-1045 of the ANPCyT. ∗ Electronic address: [email protected][1] N. N¨ucker, H. Romberg, X. X. Xi, J. Fink, B. Gegen-heimer, and Z. X. Zhao, Phys. Rev. B , 6619 (1989).[2] M. Takigawa, P. C. Hammel, R. H. Heffner, Z. Fisk, K.C. Ott, and J. D. Thompson, Phys. Rev. Lett. , 1865(1989).[3] M. Oda, C. Manabe, and M. Ido, Phys. Rev. B , 2253(1996).[4] F. C. Zhang and T. M. Rice, Phys. Rev. B , 3759(1988).[5] L. F. Feiner, J. H. Jefferson, and R. Raimondi, Phys. Rev.Lett. , 4939 (1996); references therein; A. A. Aligia, F.Lema, M. E. Simon, and C. D. Batista Phys. Rev. Lett. , 3793 (1997); L. F. Feiner, J. H. Jefferson, and R.Raimondi Phys. Rev. Lett. , 3794 (1997).[6] V. J. Emery and G. Reiter, Phys. Rev. B , 11938(1988).[7] F. C. Zhang, Phys. Rev. B , 7375 (1989)[8] H. Q. Ding, G. H. 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B , 14092 (1997).[38] F. Lema and A. A. Aligia, Physica C , 307 (1998).[39] A. E. Trumper, C. J. Gazza, and L. O. Manuel, Phys.Rev. B , 184407 (2004).[40] I. J. Hamad, A. E. Trumper, A. E. Feiguin, and L. O.Manuel, Phys. Rev. B , 014410 (2008).[41] I. J. Hamad, L. O. Manuel, and A. E. Trumper Phys.Rev. B , 024402 (2012).[42] Note that in the exact diagonalization of the QP dis-persion, unless some small anisotropy is put by hand,the two magnetic degenerate ground states coexist in thecalculation and, hence, the results should reflect this fact[32]. Supplemental Material: Generalized one-band model based on Zhang-Rice singlets forTetragonal CuO
I. J. Hamad, L. O. Manuel, A. A. Aligia
THE STARTING MODEL t’ p −t’ px’y’ FIG. 1: Scheme of the 3d x − y (squares) and 2p σ (number 8) orbitals of the T-CuO planes. Blue and red orbitals belong todifferent CuO sublattices. Empty and filled parts of the orbitals have opposite signs. It is known that for energies below 1 eV, the physics of the superconducting cuprates is described by the three-band Hubbard model H b , which contains the 3d x − y orbitals of Cu and the 2p σ orbitals of O [1–3]. We denoteby ˆx and ˆy the unit vectors along the directions of nearest-neighbor (NN) Cu atoms in the CuO planes (which aresecond NN in the T-CuO structure) and a their distance. Experimental evidence about the symmetry of holes incuprate superconductors [5–7] shows that the undoped system has one hole in each Cu 3d x − y orbital, so that all Cuatoms are in the oxidation state 2+, while added holes enter the O 2p σ orbitals. Therefore, it is natural to eliminatethe Cu-O hopping t pd and the states with Cu + and Cu +3 (keeping them as virtual states) by means of a canonicaltransformation [8, 9]. The resulting effective Hamiltonian, which consists of Cu 1/2 spins and O holes is usually calledspin-fermion model. As usual, we change the phases of half the Cu and O orbitals so that the Cu-O hopping has thesame sign independent of direction (see Fig. 1) d iσ −→ e i Q · ( R i − R i ) d iσ , p jσ −→ e i Q · ( R j − R j ) p jσ , Q = πa ( ˆx − ˆy ) , (1)where R i ( R j ) is a fixed Cu (O) position. After this transformation, for one hole added to the undoped system, themodel can be written as [9, 10] H psf = X iδδ ′ σ p † i + δ ′ σ p i + δσ (cid:20) ( t sf + t sf )( 12 + 2 S i · s i + δ ) − t sf (cid:21) − J d X iδ S i · s i + δ − t pp X jγσ p † j + γσ p jσ + J X iδ S i · S i +2 δ . (2)Here i ( j ) labels the Cu (O) sites and i + δ ( j + γ ) label the four O atoms nearest to Cu atom i (O atom j ). The spinat the Cu site i (O orbital 2p σ at site i + δ ) is denoted as S i ( s i + δ ). The first term corresponds to an effective O-Ohopping with possible spin flip with a Cu spin, t sf ( t sf ) correspond to virtual processes through Cu + (Cu +3 ). Whenboth NN vectors coincide ( δ = δ ′ ), the second term contains a term of the form of the second one. The total Cu-ONN exchange is J K = 2( t sf + t sf ) − J d . In second-order perturbation theory, J d vanishes if the on-site O repulsionis neglected [8, 9], but in general J d >
0. The third term is the direct O-O hopping and the last one is the exchangebetween nearest Cu atoms.It has been shown that H psf with parameters slightly renormalized to fit the energy levels of a CuO cluster in somesymmetry sectors (solving small matrices) reproduces Cu and O photoemission and inverse photoemission spectraand spin-spin correlations functions of the three band model H b in a Cu O cluster [10]. This was later extended toangle-resolved Cu and O photoemission intensities [11]. Therefore we assume that H psf is an accurate representationof the low-energy physics of H b .The system of tetragonal CuO (T-CuO) consists of two interpenetrating CuO sublattices, one displaced withrespect to the other in a vector γ connecting two NN O ions (see Fig. 1). One of the sublattices can be described byEq. (2). The O orbitals of the other sublattice lie on the same site as the previous ones but are orthogonal to them.We label i ′ the Cu orbitals of the second sublattice and q i ′ + δσ the annihilation operators of the four O 2p σ orbitalsnearest to Cu site i ′ . The Hamiltonian that describes the second sublattice H qsf , has the same form as H psf with i replaced by i ′ and the O p operators by the q ones. Including the NN O-O hopping and the NN Cu-Cu exchangebetween both sublattices, the Hamiltonian reads H sf = H psf + H qsf + t ′ pp X jγσ s γ (cid:16) p † j + γσ q jσ + H . c . (cid:17) − J ′ X iγ S i · S i + γ , (3)where s γ = − γ k ˆx + ˆy and s γ = 1 in the perpendicular direction (see Fig. 1) and t ′ pp ≃ . t pp [12]. This isessentially the same Hamiltonian as that considered by Adolphs et al [12]. The last term is originated by perturbationtheory in fourth order in the Cu-O hopping t pd involving two O atoms, each one forming a Cu-O-Cu angle of 90 degrees,and virtual states with an O occupied by two holes in perpendicular orbitals (one p jσ and one q jσ ′ ). It is ferromagneticdue to the Hund rules at the O atoms. Estimating the difference between singlet and triplet two-hole states fromthat between D and P states in atomic O (1.97 eV [14]) and taking the rest of the parameters from constrained-density-functional calculations for La CuO (Ref. 15) we obtain J ′ = 2 . U pd between Cu and O. For example changing U pd from1.2 eV to 0, J ′ increases to 34 meV.In Table I we show an estimation of the parameters of H sf based on previous results [11] of the low-energyreduction procedure from H b with parameters derived from constrained-density-functional calculations (set A) andthe parameters used by Adolphs et al . [12] (set B).Since the structure of T-CuO is different from that of the cuprates, the estimation of the parameters is veryapproximate. It would be desirable to have estimations for the parameters of H b for T-CuO, in particular the charge-transfer energy ∆. In absence of them one can estimate the hopping terms taking into account that the CuO distanceis increased from the value b = a/ .
895 ˚A used in Ref. 15 to b = 1 . t pd ∝ d − / , t pp ∝ d − for the dependence on the distance d of the hopping parameters [18]. This leads to a reductionof t pp by a factor 0.94 and using that for small t pd , t sfi ∝ t pd a reduction of these effective hoppings by a factor 0.81might be expected, neglecting the influence of the change in on-site energies and repulsions. TABLE I: Parameters of the spin-fermion model for T-CuO in eV.set t sf t sf J d t pp t ′ pp J J ′ A 0.37 0.08 0.28 0.56 0.336 0.13 0.0027B 0.45 0 0.48 0.615 0.369 0.15 0
THE GENERALIZED t − J MODEL FOR CUO PLANES.
Zhang and Rice proposed that the low-energy physics of the cuprates is dominated by the now called Zhang-Ricesinglets (ZRS) [19]. In the language of H psf , for which fluctuations via Cu + and Cu +3 are included virtually, for eachCu site i these singlets have the form | i ˜ s i = 1 √ (cid:16) ˜ π † i ↑ d † i ↓ − ˜ π † i ↓ d † i ↑ (cid:17) | i , (4)˜ π iσ = 12 X δ p i + δσ , (5)where d † iσ creates a hole at the 3d x − y orbital of site i . Retaining only ZRS and neglecting the rest of the states(or including them perturbatively) and mapping these states | i ˜ s i ↔ | i i to the vacuum at site i (which correspondsto a full 3d shell) leads to a one-band generalized t − J model. Several systematic studies of this mapping weremade starting for either H b or H psf , which include more terms than just the NN hopping t and the exchange J . Seefor example Refs. [20–22]. A difficulty with the states | i ˜ s i is that they have a finite overlap for NN Cu sites i and i + 2 δ . Using these non-orthogonal singlets Zhang proved that the mapping from H psf to the t − J model is exact for t sf = t pp = 0 . [23] This procedure was generalized to include the other terms of H psf , leading to additional terms inthe generalized t − J model [20].However, orthogonalizing the states leads to a simpler mapping procedure which is in general preferred and is moreaccurate when t sf > t sf (fluctuations via Cu + dominate) [20], which is in general the case. The trick to obtainorthonormal states is to transform Fourier the ˜ π iσ operators, normalize in wave-vector space, and transform back [19],leading to π iσ = 1 N X k e − i k · R i β k X m e i k · R m ˜ π mσ , β k = (cid:2) cos ( k x b ) + cos ( k y b ) (cid:3) − / , (6)where R i is the two-dimensional position of the Cu site i and b = a/
2, where a is the lattice parameter of the CuO planes. The new operators π iσ satisfy canonical anticommutation rules. The mapping is now different: | i i ↔ | is i = 1 √ (cid:16) π † i ↑ d † i ↓ − π † i ↓ d † i ↑ (cid:17) | i . (7)Inverting Eq. (6), one has for the two 2p σ O orbitals per unit cell p i + b ˆx σ = 1 N X k β k e − i k · R i e − ik x b X m e i k · R m [cos( k x b ) π mσ + cos( k y b ) γ mσ ] ,p i + b ˆy σ = 1 N X k β k e − i k · R i e − ik y b X m e i k · R m [cos( k y b ) π mσ − cos( k x b ) γ mσ ] , (8)where the γ mσ correspond the so called non-bonding O orbitals which do not mix with the Cu 3d x − y orbitals bysymmetry. They are defined asking that the Fourier transforms π † kσ and γ kσ anticommute. In any case we neglectthese non-bonding orbitals in what follows.Using Eqs. (5) and (8) one can write˜ π iσ = X m λ ( R m ) π i + mσ , (9) λ ( R m ) = 1 N X k (cid:2) cos ( k x b ) + cos ( k y b ) (cid:3) / cos( k · R m ) == 1 N X k [1 + (cos( k x a ) + cos( k y a )) / / cos( k x x m ) cos( k y y m ) . (10)As expected, the sum in Eq. (9) is dominated by λ ( R m ) ≃ .
96 and the other terms decrease rapidly with distance(see Table II).The part independent of spin of the first term in Eq. (2) is12 ( t sf − t sf ) X iδδ ′ σ p † i + δ ′ σ p i + δσ = 2( t sf − t sf ) X iσ ˜ π † iσ ˜ π iσ = 2( t sf − t sf ) X ilσ ν ( R l ) π † i + lσ π iσ , (11)where using Eqs. (9), (10) and symmetry ν ( R l ) = X m λ ( R l + R m ) λ ( − R m ) = 1 N X kq m ( β k β q ) − e − i k · ( R l + R m ) e i q · R m == 1 N X k ( β k ) − e − i k · R l = 1 N X k [1 + (cos( k x a ) + cos( k y a )) /
2] cos( k x x l ) cos( k y y l ) . (12)It is easy to see that ν ( ) = 1 (contributing to a constant energy of the π orbitals which we drop), ν ( a ˆx ) = ν ( a ˆy ) = 1 / ν ( R l ) = 0. Calculating the matrix element h B | π † j ↑ π i ↑ | A i = − /
2, where | A i = d † j ↓ | is i and | B i = d † i ↓ | js i ,one realizes that the mapping Eq. (7) leads to P π † j ↑ π i ↑ P ←→ − d † i ↓ d j ↓ / , (13)for the corresponding operators, and the same interchanging spin up and down, where P is the projector on thelow-energy subspace of Zhang-Rice singlets (LESZRS). Thus, the spin independent part of the first term in Eq. (2)provides a contribution −
14 ( t sf − t sf ) X iδσ d † i +2 δσ d iσ (14)to the NN hopping of the one-band model.The spin dependent part of first term in Eq. (2) is( t sf + t sf ) X iδδ ′ ss ′ p † i + δ ′ s ′ p i + δs σ s ′ s · S i = 4 X iss ′ ˜ π † is ′ ˜ π is σ s ′ s · S i , (15)where σ s ′ s are the matrix elements between spins s ′ and s of a vector constructed from the three Pauli matrices.Replacing Eq. (9) in Eq. (15) one obtains several terms. Note that for at most one added hole in the system P ss ′ π † ls ′ π ms σ s ′ s · S i = 2 P s π † ls π ms s m · S i = 2 s l · S i P s π † ls π ms , where s l = P ss ′ π † ls ′ π ls σ s ′ s / π at site l . Then if either i = l or i = m , projection of this term in the LESZRS Eq. (7) leads to s i · S i = − /
4, and this term reduces to a hopping. Using λ ( − R m ) = λ ( R m ) and neglecting as before the on-siteenergy correction one obtains for the sum of all terms of this form6( t sf + t sf ) X imσ λ ( R m ) λ ( ) d † i + mσ d iσ . (16)For the rest of the terms one can use s zm = − S zm in the LESZRS, Eq. (13) and the mappings P π † j ↑ π i ↓ P ←→ d † i ↑ d j ↓ / d † i ↑ d j ↑ S + j / , P π † j ↓ π i ↑ P ←→ d † i ↓ d j ↑ / d † i ↓ d j ↓ S − j / , (17)leading to the following three-site terms4( t sf + t sf ) X l = i = mσ λ ( R l − R i ) λ ( R m − R i ) d † lσ d mσ S i · S m . (18)Using Eqs. (8) and neglecting non-bonding states absent in the LESZRS, the second term of Eq. (2) becomes − J d X lmiss ′ η ( R l − R i , R m − R i ) π † ls ′ π ms σ s ′ s · S i , (19)where η ( R l , R m ) = X α = x,y [ A α ( R l ) A α ( R m ) + B α ( R l ) B α ( R m )] ,A α ( R l ) = 1 N X k β k cos( k x x l ) cos( k y y l ) cos ( k α b ) ,B x ( R l ) = − N X k β k sin( k x x l ) cos( k y y l ) sin( k x a ) ,B x ( R l ) = − N X k β k cos( k x x l ) sin( k y y l ) sin( k y a ) . (20)As before we can separate from the sum the terms with either i = l or i = m , for which we can use s i · S i = − / A α ( − R l ) = A α ( R l ), B α ( − R l ) = − B α ( R l ), one obtains − J d X ilσ η ( R l , ) d † i + lσ d iσ − J d X l = i = mσ η ( R l − R i , R m − R i ) d † lσ d mσ S i · S m . (21)Finally, using Eqs (8) and neglecting non-bonding states the term in t pp of Eq. (2) becomes t pp X ilσ µ ( R l ) d † i + lσ d iσ , (22)where µ ( R l ) = 8 N X k β k cos ( k x b ) cos ( k y b ) . (23)Including the Cu-Cu exchange term and adding Eqs. (14), (16), (18), (21), and (22), one realizes that P H psf P canbe mapped into the following generalized t − J model: H ptJ = t sf − t sf X iδσ d † i +2 δσ d iσ + X imσ (cid:20) t sf + t sf ) λ ( R m ) λ ( ) − J d η ( R m , ) + t pp µ ( R m ) (cid:21) d † i + mσ d iσ ++ X l = i = mσ h t sf + t sf ) λ ( R l − R i ) λ ( R m − R i ) − J d η ( R l − R i , R m − R i ) i d † lσ d mσ S i · S m ++ J X iδ S i · S i +2 δ . (24)The main two-dimensional integrals that enter this expression are displayed in Table II. Note that λ ( R l ) and µ ( R l )are symmetric under the operations of the point group C v , while A α ( − R l ) = A α ( R l ) and B α ( − R l ) = − B α ( R l ).Some of these integrals were given previously [24]. There are small differences in some µ ( R l ). We believe that ourresults are more accurate. TABLE II: Two-dimensional integrals that enter H tJ . See Eqs. (24) and (20). R /a λ A x A y B x B y µ (0,0) 0.9581 0.4791 0.4791 0 0 1.4535(1,0) 0.1401 0.1989 -0.05877 0.2802 0 0.5465(1,1) -0.02351 -0.01753 -0.01753 0.2441(2,0) -0.01373 -0.02643 0.01270 -0.1277 THE GENERALIZED t − J MODEL FOR T-CUO
Naturally, the one-band model for the other CuO sublattice H qtJ (the mapping of P H qsf P to a generalized t − J model) has the same form as H ptJ above, with the only difference that i refers to Cu sites of the other sublattice. Inaddition, the exchange term proportional to J ′ in Eq. (3) retains the same form in the one-band model. Therefore,the remaining task is to map the term proportional to t ′ pp . Mapping using non-orthogonal singlets
We define the nonorthogonal ZRS for the second CuO sublattice in analogy to Eqs. (4) and (5): FIG. 2: Orbitals involved in the hopping between non-orthogonal ZRS belonging to different CuO sublattices. | i ˜ s i = 1 √ (cid:16) ˜ ρ † i ↑ d † i ↓ − ˜ ρ † i ↓ d † i ↑ (cid:17) | i , (25)˜ ρ iσ = 12 X δ q i + δσ , (26)Adolphs et al . argue that the orbitals ˜ π iσ and ˜ ρ nσ at NN sites do not mix [12]. However, in spite of a partialcancellation, the result is nonzero. An example is shown in Fig. 2 for R n = R i + b ( ˆx − ˆy ). In terms of the numbersof the figure ˜ π iσ = 12 X i =1 p iσ , ˜ ρ nσ = X i =3 q iσ . (27)Then t ′ pp X ijγσ s γ (cid:16) p † j + γσ q jσ + H . c . (cid:17) ˜ π † iσ | i = t ′ pp q † σ + q † σ ) | i + ... = t ′ pp ρ † nσ | i + ... (28)It is easy to see that the same value t ′ pp / R n − R i = − b ( ˆx − ˆy ), while the result is − t ′ pp / R n − R i = ± b ( ˆx + ˆy ). There are also contributions ± t ′ pp / | i ˜ s i ↔ | i i leads to a factor-1/2 [similar to Eq. (13)] plus some corrections due to non-orthogonality of the ZRS [20]. The details are beyond thescope of this work. In the following subsection, we derive the rigorous result using orthogonal ZRS. In any case, thesimpler results presented here show that the effective hopping is not zero. Mapping using orthogonal singlets
The term in t ′ pp of Eq. (3) can be written in the form H ′ = t ′ pp X ijγσ s γ (cid:16) p † j + γσ q jσ + H . c . (cid:17) == t ′ pp X iσ [ p † i + b ˆx σ ( q i + b ˆy σ + q i + a ˆx − b ˆy σ − q i − b ˆy σ − q i + a ˆx − b ˆy σ ) ++ p † i + b ˆy σ ( q i + b ˆx σ + q i − b ˆx + a ˆy σ − q i − b ˆx σ + q i + b ˆx + a ˆy σ ) + H . c . ] , (29)where the sum runs over all sites of the first CuO sublattice.Using Eqs. (8) and the corresponding ones for the second CuO sublattice: q n + b ˆx σ = 1 N X k β k e − i k · R n e − ik x b X m e i k · R m [cos( k x b ) ρ mσ + ... ] ,q n + b ˆy σ = 1 N X k β k e − i k · R n e − ik y b X m e i k · R m [cos( k y b ) ρ mσ − ... ] , (30)one obtains after some algebra H ′ = t ′ pp ξ ( R τ ) X iτσ (cid:16) π † iσ ρ i + τσ + H . c . (cid:17) , (31)where τ denotes the vectors connecting both CuO sublattices ( x τ and y τ below are both odd multiples of b ) and ξ ( R τ ) = − N X k sin( k x b ) sin( k y b ) sin( k x x τ ) sin( k y y τ ) . (32)It is easy to see that ξ ( R τ ) = − R τ = ± b ( ˆx + ˆy ), ξ ( R τ ) = 1 if R τ = ± b ( ˆx − ˆy ), and ξ ( R τ ) = 0 for other R τ .Therefore H ′ = t ′ pp X iγσ s γ (cid:16) π † iσ ρ i + γσ + H . c . (cid:17) . (33)using the mapping Eq. (13) and adding the other terms, the complete generalized t − J model for T-CuO takes theform H tJ = H ptJ + H qtJ − t ′ pp X iγσ s γ (cid:16) d † iσ d i + γσ + H . c . (cid:17) − J ′ X iγ S i · S i + γ . (34)To compare with experiment it is convenient to write the Hamiltonian in terms of the following operators c iσ = e i Q · ( R i − R i ) d iσ , (35)which restores the original phases of the Cu orbitals [changed before in Eq. (1)]. If the phases are not restored,the problem is of course equivalent, but the wave vectors are displaced by Q complicating the comparison withexperiment. This transformation within each CuO sublattice changes the sign of the NN hopping (at distances ± a ˆx , ± a ˆy ) leaving second and third NN hopping unchanged. In addition also the sign of the intersublattice hopping atdistances ± b ( ˆx − ˆy ) is changed, keeping the sign in the perpendicular direction, so that the corresponding term inEq. (34) becomes H NN = t ′ pp X iγσ (cid:16) c † iσ c i + γσ + H . c . (cid:17) . (36) SIMPLIFIED GENERALIZED t − J MODEL
The state of the art technique for studying the dynamics of one hole in an antiferromagnet is the self-consistent Bornapproximation (SCBA) [25–28]. It compares very well with exact diagonalization of small clusters [25, 27, 28, 32],while permitting an extensions to larger clusters. From previous studies for the antiferromagnetic order of CuO planes, one knows that the propagation of the hole is easier through each sublattice with spins pointing in the samedirection, in particular for hopping involving second and third NN, while it is inhibited for first NN in spite of thefact the corresponding hopping is larger, because the hopping distorts the antiferromagnetic alignement.The generalized t − J model for CuO planes, as described above, contains three-site terms which combine secondand third NN with spin-flip processes. These so called correlated hopping processes are argued to play an importantrole for superconductivity [29, 30]. However, the above argument indicates that for the propagation of the hole, onlythe spin-conserving part is important. Therefore, to simplify the model and bring it amenable to the SCBA treatmentwe retain only hopping up to third NN in the CuO planes and approximate S i · S m ≃ h S zi S zm i in Eq. (24). Thisleads to a simplified effective model for T-CuO similar to that considered by Moser et al [17]. H stJ = − X κ =0 t κ X iv κ σ (cid:16) c † iσ c i + v κ σ + H . c . (cid:17) + J X iv S i · S i + v − J ′ X iv S i · S i + v , (37)where the subscript κ = 0 refers to intersublattice hopping of NN Cu atoms in the T-CuO structure (connected bythe vectors v = ± b ( ˆx ± ˆy )), while κ = 1, 2, 3, refer to first, second, and third NN within each CuO sublattice.Comparison with Eqs. (24), (36) and using Eq. (35) leads to t = − t ′ pp ,t = t sf − t sf t sf + t sf ) λ ( ) λ ( a ˆx ) − J d η ( a ˆx , ) + t pp µ ( a ˆx ) , − t ≃ t sf + t sf ) λ ( ) λ ( a ( ˆx + ˆy )) − J d η ( a ( ˆx + ˆy ) , ) + t pp µ ( a ( ˆx + ˆy )) ++2 h S zi S zi + v i h t sf + t sf ) λ ( a ˆx ) − J d η ( a ˆx , a ˆy ) i , − t ≃ t sf + t sf ) λ ( ) λ (2 a ˆx ) − J d η (2 a ˆx , ) + t pp µ (2 a ˆx ) ++ h S zi S zi + v i h t sf + t sf ) λ ( a ˆx ) − J d η ( a ˆx , − a ˆx ) i . (38)Using Eqs. (20), Table II, and h S zi S zi + v i = − .
186 for the NN expectation value for the Heisenberg model in thesquare lattice (see next section ), one obtains t ≃ . t sf + 1 . t sf + 0 . t pp − . J d ,t ≃ . t sf + t sf ) − . t pp − . J d ,t ≃ . t sf + t sf ) + 0 . t pp − . J d . The fact that t sf + t sf and t pp enter with different sign in t leads to a large relative error in this parameter.Fortunately, the results seem to be rather insensitive to t .Using the estimated parameters for the spin-fermion model based on previous constrained-density-functional cal-culations (set A) or given by Adolphs et al [12] (set B), tabulated in Table I, we obtain the results presented in TableIII. TABLE III: Parameters of the effective model for T-CuO in meV.set t t t t J J ′ A -168 417 -2 69 130 3B -184 369 -11 65 150 0
THE SELF-CONSISTENT BORN APPROXIMATION
As Adolphs et al . [12], we assume the antiferromagnetic order of T-CuO given in the left of Fig. 3. The NN Cuatoms connected by the vectors ± c ( ± d ), have parallel (antiparallel) spins, where c = b ( ˆx + ˆy ) and d = b ( − ˆx + ˆy ). d c FIG. 3: (Color online) The two possible magnetic ground states for T-CuO Q = (0 , π ) (left) and Q = ( π,
0) (right). Thevectors c and d are indicated in the left figure. The primitive translation vectors in the plane, which also define the unit cell, are c and 2 d . The unit cell has thesame size as that of the CuO planes but it is different.Following Mart´ınez and Horsch [25], we perform the transformation c iσ −→ c i − σ in the sublattice in which thespins are pointing down [31], in such a way that all spins are pointing up after the transformation. Then we definethe spin excitations a † i = c † i ↓ c i ↑ , and the holon operators h i such that c i ↑ = h † i , c i ↓ = h † i a i , (39)in the Hilbert subspace we are considering. Spin waves
We first diagonalize the exchange part of the Hamiltonian Eq. (37) for the undoped system. In terms of the spinexcitations, it takes the form H e = J X iv S i · S i + v − J ′ X iv S i · S i + v = − JN ++ J X iv (cid:16) a † i a i + a † i + v a i + v + a i a i + v + a † i a † i + v (cid:17) −− J ′ X i (cid:16) a † i a i + a † i + d a i + d + a i a i + d + a † i a † i + d (cid:17) ++ J ′ X i (cid:16) a † i a i + a † i + c a i + c − a † i a i + c − a † i + c a i (cid:17) . (40)Using the Fourier transform a i = N − / P k e − i k · R i a k , one obtains H e + 2 JN = X k h A k a † k a k + B k ( a k a − k + H . c . ) i ,A k = 2 J − J ′ cos( c · k ), B k = J X v cos( v · k ) − J ′ d · k ) . (41)Dropping the constant 2 JN , H e is set into diagonal form introducing new bosonic operators:0 H e = X k ω k θ † k θ k , θ k = u k a k + v k a †− k ,ω k = q A k − B k , u k = 12 + A k ω k , v k = u k − ,u k > , sgn( v k ) = sgn( B k ) . (42) Spin-spin correlations
In the spin-wave formalism, the correlation function entering Eq. (38) becomes h S zi S zi + v i = h (cid:18) − a † i a i (cid:19) (cid:18) −
12 + a † i + v a i + v (cid:19) i = −
14 + h a † i a i i − h a † i a i a † i + v a i + v i , (43)where we have taken into account that the spins of sites i and i + v point in opposite directions. Decoupling the lastcorrelation function h a † i a i a † i + v a i + v i = h a † i a i ih a † i + v a i + v i + |h a † i a † i + v i| + |h a † i a i + v i| , (44)we obtain h S zi S zi + v i = − m − |h a † i a † i + v i| − |h a † i a i + v i| , (45)where m = 12 − h a † i a i i (46)is the sublattice magnetization.Transforming Fourier and suing the inverse of the second Eq. (42) a k = u k θ k − v k θ †− k , (47)the different correlation functions become at zero temperature h a † i a i i = 1 N X kq h (cid:16) u k θ † k − v k θ − k (cid:17) (cid:16) u q θ q − v q θ †− q (cid:17) i = 1 N X k v k , h a † i a † i + v i = 1 N X kq h (cid:16) u k θ † k − v k θ − k (cid:17) e iq · v (cid:0) u q θ † q − v q θ − q (cid:1) i = 1 N X k cos( k · v ) u q v k , h a † i a i + v i = 1 N X kq h (cid:16) u k θ † k − v k θ − k (cid:17) e iq · v (cid:16) u q θ q − v q θ †− q (cid:17) i = 1 N X k cos( k · v ) v k , (48)We have evaluated the two-dimensional integrals above for J ′ = 0. The result is h a † i a i i = 0 . h a † i a † i + v i = 0 . h a † i a i + v i = 0, leading to m = − . h S zi S zi + v i = − . . The hopping terms
The hopping terms of the Hamiltonian Eq. (37) can be separated in two: those involving two sites of the samesublattice (spin up or down), like the terms in t and t , and those connecting sites of different sublattices ( t and1 -2 -1 0 1 2 ω (eV) A ( k , ω ) Γ MA FIG. 4: (Color online) Spectral functions corresponding to the Γ, M and A points, with the hole picture adopted (for comparisonwith ARPES experiments the electron picture should be adopted). A broadening equivalent to ∼
20 meV was applied (seemain text). Dashed line: M spectral function with a broadening of 300 meV. With such a broadening the QP peak is whashedout and only the broad peak near 0 . half of the terms in t ). The latter give rise to a holon-magnon interaction. We neglect the terms creating two spinexcitations. Using the transformations introduced at the beginning of this section we obtain H t = − X κ =0 t κ X iv κ σ (cid:16) c † iσ c i + v κ σ + H . c . (cid:17) = t X i h h † i h i + c + h † i h i + d ( a i + a i + d ) + H . c . i ++ t X i h † i a i X v h i + v + H . c . ! + X κ =2 t κ X iv κ σ (cid:16) h † iσ h i + v κ σ + H . c . (cid:17) . (49)Using h i = N − / P k e − i k · R i h k , Eq. (47), and adding H e = P k ω k θ † k θ k we obtain, after some algebra H stJ = X k ǫ k h † k h k + X k ω k θ † k θ k + 1 √ N X kq M kq h † k h k − q θ q + H . c . ,ǫ k = 2 t cos( k · c ) + 4 t cos( ak x ) cos( ak y ) + 2 t [cos(2 ak x ) + cos(2 ak y )] ,M kq = 2 t { cos [( k − q ) · c ] u q − cos( k · c ) v q } + 2 t [ u q ζ ( k − q ) − v q ζ ( k )] ,ζ ( k ) = cos( ak x ) + cos( ak y ) . (50)The holon Green function G h ( k , ω ) is obtained from the self-consistent solution of the following equations: G − h ( k , ω ) = ω − ǫ k − Σ( k , ω ) + iǫ, Σ( k , ω ) = 1 N X q M kq G h ( k − q , ω − ω q ) . (51)In practice, the calculations are done in a large but finite system and the selfconsistency can be avoided calculatingsequentially Σ( k , ω ) for increasing values of ω , beginning with values (near − J ) such that Σ( k , ω − ω q ) = 0 for all k and q [33].An example of the hole spectral function calculated with the SCBA can be seen in Fig. 4 for the Γ, M and A points.A low broadening, equivalent to ∼
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