Magnon-assisted dynamics of a hole doped in a cuprate superconductor
MMagnon-assisted dynamics of a hole doped in a cuprate superconductor
I. J. Hamad and L. O. Manuel
Instituto de F´ısica Rosario (CONICET) and Facultad de Ciencias Exactas,Ingenier´ıa y Agrimensura, Universidad Nacional de Rosario,Bv. 27 de Febrero 210 bis, 2000 Rosario, Argentina
A. A. Aligia
Centro At´omico Bariloche and Instituto Balseiro,Comisi´on Nacional de Energ´ıa At´omica, CONICET, 8400 Bariloche, Argentina
We calculate the quasiparticle dispersion and spectral weight of the quasiparticle that resultswhen a hole is added to an antiferromagnetically ordered CuO plane of a cuprate superconductor.We also calculate the magnon contribution to the quasiparticle spectral function. We start from amultiband model for the cuprates considered previously [Nat. Phys. , 951 (2014)]. We map thismodel and the operator for creation of an O hole to an effective one-band generalized t − J model,without free parameters. The effective model is solved using the state of the art self-consistent Bornapproximation. Our results reproduce all the main features of experiments. They also reproducequalitatively the dispersion of the multiband model, giving better results for the intensity nearwave vector ( π, π ), in comparison with the experiments. In contrast to what was claimed in [Nat.Phys. , 951 (2014)], we find that spin fluctuations play an essential role in the dynamics of thequasiparticle, and hence in both its weight and dispersion. PACS numbers: 75.20.Hr, 71.27.+a, 72.15.Qm, 73.63.Kv
I. INTRODUCTION
More than three decades after the discovery of hightemperature superconductors, the issue of the appropri-ate microscopic minimal model that correctly describesthe low-energy physics is still debated. There is howevera consensus on the validity the three-band model H b fordescribing the physics of the cuprates at energies below ∼ σ orbitals (those pointing in the direction of the nearestCu sites) and one Cu 3d x − y orbital. At higher ener-gies other orbitals should be considered.
Other modelsused to describe the cuprates are the spin-fermion model H sf , obtained from H b after eliminating the Cu-O hop-ping by means of a canonical transformation (only the d configuration of Cu is retained, represented by a spin 1 / and the generalized t − J model H GtJ , which consistsof holes moving in a background of Cu 1 / J , nearest-neighbor hopping t , and additional terms of smaller magnitude. H GtJ is derived as a low-energy effective one for H b or H sf , assuming that the low-energy part of themultiband models is dominated by Zhang-Rice singlets(ZRSs), which in H sf consist of singlets formed betweenthe spin of a cooper atom and the spin of the hole re-siding in a linear combination L of four ligand oxygenorbitals around the cooper atom. In H b , in whichcharge fluctuations are allowed, the ZRS also includesstates with two holes in the Cu 3d x − y orbital and inthe O L orbital. The proposal of Zhang and Rice hasinitiated a debate about the validity of a one-band modelthat continues at present. ? –25 . This issue is of centralimportance since models similar to the t − J model were used to explain many properties of the cuprates, in-cluding superconductivity. An important probe for the models is the spectralfunction of a single-hole doped on the parent half-filledcompounds, whose quasiparticle (QP) dispersion rela-tion is directly measured in angle-resolved photoemis-sion (ARPES) experiments.
Experimental evidenceshows that this doped hole resides mainly on the O 2p σ orbitals. Naively one might expect that this fact isa serious problem for H GtJ , since O holes are absentin the model. However, mapping appropriately the cor-responding operators, Cu and O photoemission spectracan be calculated with both H sf , and H GtJ . Never-theless, while the experimental dispersion observed inSr CuO Cl , has been fit using H GtJ , an unsatisfac-tory aspect is that the “bare” t − J model with onlynearest-neighbor hopping t was unable to explain the ob-served dispersion, and ad hoc hopping to second and thirdnearest neighbors were included. In Ref. 20, the QP dispersion E QP ( k ) and its intensity Z QP ( k ) for adding an O hole in an undoped CuO planewere calculated, using H sf solved with an approximatevariational method using realistic parameters. The dis-persion obtained agrees with experiment. However, thereported intensity increases as k moves from ( π , π ) to( π, π ), in contrast to experiment. In the mentioned ref-erence, it was also claimed that spin fluctuations play aminor role in the dynamics of the hole.In this work we map the H sf used in Ref. 20 to an H GtJ without adjustable parameters, extending to Z QP the procedure we used before for T-CuO. The resulting H GtJ is solved using the state of the art self-consistentBorn approximation (SCBA). We obtain results in agree-ment with experiment for both E QP and Z QP . We also a r X i v : . [ c ond - m a t . s up r- c on ] F e b t pp t pp ’ JdJ t sf FIG. 1. Structure of the CuO planes and sketch of the pa-rameters of the three-band spin fermion model [Eq. (1)].Filled (empty) circles represent Cu (O) sites. calculate the hole’s spectral function, by taking into ac-count multimagnon contributions within the SCBA. Inthis way we argue that the spin fluctuations play anessential role in the hole’s dynamics. In particular thewidth of E QP is determined by the nearest-neighbor spinexchange J . II. SPIN-FERMION MODEL AND THEONE-BAND MODEL DERIVED FROM IT
We start from the spin-fermion model (Cu spins andO holes), obtained from H b integrating out valence fluc-tuations at the Cu sites. With the adequate choiceof phases the Hamiltonian reads H sf = (cid:88) iδδ (cid:48) σ p † i + δ (cid:48) σ p i + δσ (cid:20) ( t sf + t sf )( 12 + 2 S i · s i + δ ) − t sf (cid:21) − t pp (cid:88) jγσ p † j + γσ p jσ + t (cid:48) pp (cid:88) jδσ (cid:16) p † i + δσ p i − δσ + H . c . (cid:17) − (cid:88) iδ J d S i · s i + δ + J (cid:88) iδ S i · S i +2 δ , (1)where i ( j ) labels the Cu (O) sites, i + δ ( j + γ ) labelthe four O atoms nearest to Cu atom i (O atom j ), and p † jσ creates an O hole at the 2p σ orbital of site j withspin σ . The spin at the Cu site i (O orbital 2p σ at site i + δ ) is denoted as S i ( s i + δ ). As in Ref. 20 we includehopping t (cid:48) pp between second-neighbor O orbitals with aCu in between, and J d (which reduces part of the firstterm for δ (cid:48) = δ ) not included in earlier studies. Themodel is represented in Fig. 1. In units of the Cu-Cu spinexchange J = 1, the parameters chosen for the multibandmodel of Ref. 20 are: t sf = 2 . t sf = 0, t pp = 4 . ,t (cid:48) pp = 2 .
40, and J d = 3 . t − J model: H GtJ = − (cid:88) κ t κ (cid:88) iv κ σ (cid:16) c † iσ c i + v κ σ + H . c . (cid:17) + J (cid:88) iv S i · S i + v , (2)where c † iσ creates a hole at the Cu site i with spin σ ,and κ = 1 , ,
3, refer to first, second, and third nearest-neighbors v κ within the sublattice of Cu atoms. Ad-ditional terms are small and do not affect the hole dy-namics. The derivation of this one-band Hamiltonianand the calculation of its parameters follow the proce-dure detailed in the supplemental material of Ref. 25,here generalized to include the effect of second nearest-neighbor O hopping t (cid:48) pp . The contribution of this termfor a hopping τ R between sites at a distance R = ( x, y )becomes: τ R = 2 t (cid:48) pp N (cid:88) k cos( k x x ) cos( k y y ) × (cid:18) − cos ( k x b ) + cos ( k y b )cos ( k x b ) + cos ( k y b ) (cid:19) , (3)where b = a/ N isthe number of sites of the cluster. The contribution ofthe other terms of H sf to the different terms of H GtJ has been described in detail before. . The resulting pa-rameters of H GtJ are, taking J = 0 .
15 eV as the unit ofenergy, t = 1 . t = − . t = 0 . III. TREATMENT OF THE ONE-BAND MODEL
We calculate the QP spectral functions –from whichthe single hole’s dispersion and weight are directlyderived– and the magnon contributions to the hole’s wavefunction (WF) by means of the SCBA, a semiana-lytic method that compares very well with exact diago-nalization (ED) results on small clusters in different sys-tems.
It is one of the most reliable and checkedmethods up to date to calculate the hole Green’s func-tion, and in particular its QP dispersion relation. How-ever, some care is needed to map the QP weight betweendifferent models. In order to do such calculation, wefollow standard procedures. On one hand, the magnondispersion relation is obtained treating the magnetic partof the Hamiltonian at the linear spin-wave level, since thesystem we study has long-range antiferromagnetic order,and it is well known that its magnetic excitations aresemiclassical magnons. On the other hand, the elec-tron creation and annihilation operators in the hoppingterms are mapped into holons of a slave-fermion repre-sentation (details in Ref. 25). Within SCBA, we arriveto an effective Hamiltonian: H eff = (cid:88) k (cid:15) k h † k h k + (cid:88) k ω k α † k α k ++ 1 √ N (cid:88) kq (cid:16) M kq h † k h k − q α q + H . c . (cid:17) , (4)with (cid:15) k = 4 t cos( ak x ) cos( ak y ) + 2 t [cos(2 ak x ) + cos(2 ak y )] ,ω k = (cid:113) A k − B k ,M kq = 2 t [ u q ζ ( k − q ) − v q ζ ( k )] , (5)where (cid:15) k is the bare hole dispersion (with no couplingto magnons), ω k is the magnon dispersion relation, with A k = 2 J , B k = J (cid:80) R cos( R · k ), and M kq is the ver-tex that couples the hole with magnon excitations. Here ζ ( k ) = cos( ak x ) + cos( ak y ) being a the distance betweenCu atoms in the CuO planes and where u q and v q arethe usual Bogoliubov coefficients.The heart of the SCBA method lies in the self-consistent Dyson equation for the hole’s self energy Σ k ( ω ) = 1 N (cid:88) q | M kq | G k − q ( ω − ω q ) , being G k ( ω ) = ( ω − (cid:15) k − Σ k ( ω )) − the hole Green’sfunction. From the self-energy the QP energy canbe computed, by means of the self-consistent equation E QP ( k ) = Σ k ( E QP ( k )), and also the holon spectralweight, defined as Z h ( k ) = (cid:18) − ∂ Re Σ k ( ω ) ∂ω (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E QP ( k ) . (6)Although Eq. (6) in principle allows the calculation ofthe spectral weight directly, in practice within the SCBAit is impossible to apply it due to the strong irregularitiesin the derivative of ReΣ k . Instead, the spectral weightis calculated by integrating the QP peak in the spectralfunction. A. QP spectral function and magnon coefficients ofthe QP wave function
For the calculation of the magnon contributions to theQP’s WF we follow the steps taken in Refs. 48–50. TheQP WF with momentum k can be expressed as a sumof terms, each of which involves the contribution of agrowing number of magnons. Hence, within the SCBA,the QP WF results by taking the n → ∞ limit of: | Φ n k (cid:105) = Z h ( k ) (cid:34) h † k + 1 √ N (cid:88) q g k , q h † k − q α † q + ... + 1 √ N n (cid:88) q ,....., q n g k , q g k , q ....g k n − , q n h † k n α † q ...α † q n (cid:35) | AF (cid:105) , where k i = k − q − ... − q i , | AF (cid:105) is the undoped anti-ferromagnetic ground state, and g k n , q n +1 = M k n , q n +1 G k n +1 ( E QP ( k ) − ω q − .... − ω q n +1 ) . (7)It can be seen that each contributing term to the QP WFinvolves a growing number of magnons, starting from thefirst zero magnon term whose relative weight is given bythe holon spectral weight Z h ( k ).The QP WF satisfies the normalization condition S k = lim n →∞ (cid:104) Φ n k | Φ n k (cid:105) = ∞ (cid:88) m =0 A ( m ) k = 1 . (8)Each coefficient A ( m ) k is the m -magnon contribution tothe QP WF and is defined as A ( m ) k = z k N m (cid:88) q ,....., q n g k , q g k , q ......g k m − , q m , (9)while for the particular case m = 0, A (0) k ≡ Z h ( k ). Inthis way, within the SCBA the relative weight of each n -magnon term for the spin polaron can be evaluated fora specific moment of the Brillouin zone.In order to estimate the effective number of magnonsnecessary to have a reliable QP WF we can find the min-imum n such that S ( n ) k = (cid:104) Φ n k | Φ n k (cid:105) = (cid:80) nm =0 A ( m ) k (cid:39) , within certain precision. IV. RESULTS
In this Section we present the SCBA calculations for H G t − J , using the previously estimated parameters andthe experimental value J ≡ J = 0 .
15 eV.
A. Quasiparticle dispersion relation
Fig. 2 shows the SCBA QP dispersion relation corre-sponding to our one-band generalized t − J model alongwith the QP dispersion relation of the three-band model,obtained variationally (Ref. 21). We recall that in ourmodel there are no free parameters. All of them are rig-orously obtained from the three-band model and ex-periments. The agreement of the one-band model andthe multiband model dispersions is very good near theQP ground state momentum (cid:0) π , π (cid:1) and all along the di-agonal and antidiagonal lines. In the rest of the chosenpath, the agreement is semiquantitative. Compared withthe ARPES measurements, our results seem to bet-ter capture the quasi-degeneracy between the ( π, π ) and( π,
0) points, with the energy at ( π,
0) a little higher thanat ( π, π ). It should be stressed that, for simplicity, we aretaking a hole picture, so the dispersion relation should bereversed in order to be compared with ARPES. From thedispersion relation only, it is not possible to conclusivelydiscern whether the SCBA solution of the generalized ( π,0) (0,0) (π,π) (0,π) (π,0) -2,5-2-1,5-1-0,50 E Q P / J J=J J = 2J J = J /23 band model, 3 magnons FIG. 2. QP hole dispersion relation. The solid black (blue)curve corresponds to the QP dispersion relation for the one-band generalized t − J model (three-band model) calculatedwith SCBA (variational approach). The broken red (green)curve corresponds to the SCBA QP dispersion for a one-bandmodel with an exchange interaction twice (half) the value ofthe experimental one. t − J model or the variational solution of the three-bandmodel predictions agree better with ARPES.To analyze the role of the spin fluctuations for the holemotion within our theory, we also plot in Fig. 2 the SCBAQP dispersion relation for the same H G t − J parametersbut half and double exchange interaction J values. Thefirst point to be noticed is that as a first approxima-tion, the bandwidth is directly proportional to J . When J = 2 J , that is, the spin fluctuations are enhanced incomparison with the hole kinetic energy, the relative dis-persion bandwidth (in units of the corresponding J ) isdecreased, and now the energy of the k = (0 ,
0) and( π, π ) points is slightly higher than that at the ( π, J = J / , i.e. when spin fluctuations arelowered, has the same structure as for J = J but itsrelative bandwidth is larger than that of the three-bandmodel. Therefore, it is evident that the spin fluctuationshave a noticeable impact on the global dispersion formand its bandwidth. In particular, the increase of the ex-change interaction gives rise to more localized QP states. B. Quasiparticle spectral weight
In Fig. 3 we show the QP spectral weight for the one-band and the three-band models along the same Brillouinzone path as in Fig. 2. Care must be taken in order tocalculate the QP spectral weight within the one-bandmodel, since almost all the contribution to the photoe-mission spectra comes from the addition of an O hole.However in the one-band model the O degrees of free-dom have been integrated. In order to compute the Ocontribution to the ARPES QP intensity Z QP ( k ) withinthe SCBA, we follow the procedures of Ref. 36: we first calculate the holon spectral weight Z h ( k ) , we then cal-culate the spectral weight for emitting a physical electron(see Ref. 40), and finally from this we calculate the O in-tensity by means of a simple analytical relation betweenboth, detailed in Ref. 36. In general, the calculated Ointensity is higher than that of the variational calculationof the three-band model. Despite so, it can be seen thatalong the diagonal (0 , → ( π, π ), the intensity is largenear the ground state ( π/ , π/
2) momentum (note thatit’s not symmetric around ( π/ , π/ ,
0) and ( π, π ). Nev-ertheless, those momenta do not show degeneracy in theintensity, as it happens for the holon weight within theSCBA .The general trend of the intensity calculated with thegeneralized t − J model by means of the SCBA coincideswith experiments, in contrast with the results of thevariational three-band model calculations . In particu-lar, the experiments show an almost vanishing QP pho-toemission weight close to (0 ,
0) and ( π, π ) (see Fig. 1 ofRef. 31), that is correctly captured by our results, whilein the three-band model calculation the ( π, π ) point hasan appreciable QP weight. In Ref. 20 it was shown thatappealing to a five-band model a partial decrease of theQP weight is obtained at ( π, π ), while our more sophis-ticated SCBA calculation already captures this spectralfeature in the one-band generalized t − J model. Hence,we believe that the one-band model provides a quanti-tatively correct description of the photoemission spectrafor the undoped cuprates. ( π,0) (0,0) ( π,π) (0, π) ( π,0) Z Q P ( k ) SCBA3 band model, 2 magnons
FIG. 3. QP spectral weight: the blue curve corresponds to theO contribution to the photoemission intensity calculated withSCBA, while the red curve corresponds to the QP weight func-tion of the three-band model calculated variationally (takenfrom Ref. 20).
C. Magnon contributions to the QP wave function
In Fig. 4 we show the magnon coefficients A ( m ) k for m = 1 , , ( π,0) (0,0) ( π,π) (0, π) ( π,0) A i A A A A A ( π,0) (0,0) ( π,π) (0, π) ( π,0) S k S k FIG. 4. Solid curves: Magnon coefficients A ( m ) k of the SCBAQP wave function for N = 400. Broken curve: sum of thefirst 4 magnon coefficients. cluster of N = 400 sites, using 25000 frequencies. Wehave checked that the results are essentially the sameas for N = 1600 sites, which is an indication that the N = 400 cluster is a very good approximation for thethermodynamic limit. We have chosen this cluster sizebecause, for 1600 sites, the calculation of the fourth co-efficient A (4) k is computationally expensive. For compar-ison, we put in Table I the A ( m ) m = 1 , , , − ( π, π ). It is worth to mention that for acorrect computation of all the magnon coefficients is es-sential to get a very precise QP dispersion relation and itsspectral weight, as can be seen from Eqs. (7) and (9). Forthis purpose, it is necessary to use a very large numberof frequencies.What can be clearly seen in Fig. 4 and Table I isthat the one- and the two-magnon coefficients can be,for many momenta, greater or of the same order of mag-nitude that the zero magnon coefficient A (0) k , which werecall is the holon spectral weight Z h ( k ). The three-magnon coefficient A (3) is small for all momenta but isby no means negligible. On the other hand, A (4) k is alwaysvery small, even compared to A (3) k . From the magnon co-efficients it can be concluded that spin fluctuations corre-sponding to several magnons are essential to build up theQP wave function. Since our one-band generalized t − J model is rigorously derived from a multiband model and,as we have shown above, it reproduces the main featuresof the experimental QP dispersion relation and photoe-mission intensity, it can be stated that the spin polaron is the appropriate physical picture of the QP in cuprates.Fig. 4 also displays the partial sum of the norm S (4) k ,that is the sum of the first four magnon coefficients. Itis evident that for those momenta where the holon QPweight Z h ( k ) is not so small ( Z h (cid:38) . k = (0 ,
0) and ( π, π ), where theholon QP spectral weight is much smaller than 0.05, thenormalization condition is far from being satisfied. Sincethe four-magnon coefficients A (4) k are much smaller thanthe three-magnon ones A (3) k , it is plausible to assume thatthe following coefficients would be smaller, and so theremust be a “magnon proliferation”, that is, the QP wouldbe composed of a great number of magnons, and the sumrule can only be reasonably satisfied with a huge num-ber of magnon coefficients, corresponding to very slowlyconvergent series. k x /π k y /π A (0) k A (1) k A (2) k A (3) k S k A ( m ) k for m = 1 , , N = 1600 cluster, for selected momenta along the di-agonal of the Brillouin zone. By symmetry, A ( m )( π + k, π + k ) = A ( m )( π − k, π − k ) . In the pure t − J model ( t = t = 0), for a J/t ra-tio as our
J/t , Ramsak and Horsch have shown thatthe QP is also composed of several magnons, and thatfor some momenta the one-magnon coefficient is largerthan the zero-magnon one, and even that the two- andthree-magnon terms are important to fulfill the normal-ization condition. This behaviour is analogous to the onewe have found in this work. However it is known that forthe pure t − J model the hole can only propagate byemitting and absorbing spin fluctuations . Besides, itdoes not reproduce the experimentally measured disper-sion . With our generalized model, with second- andthird-nearest neighbor hoppings, we were able to repro-duce the experiments. It is usually argued that, since t or t allow free hopping processes, in which the hole canmove along a magnetic sublattice without disturbing theN´eel order, the correct dispersion obtained by includingfurther hoppings in the model implies that spin fluctua-tions do not play an important role in the QP formation.Our results indicate that this is not the case, and thatfor the generalized H Gt − J the multimagnon processes areequally important in the formation of the QP as in thepure t − J model. In previous works we have alreadyshown that even when there is a “free hopping” chan-nel that allows the hole to move without generating spinfluctuations of the magnetic background, the hole motionis promoted by emitting magnons, since this is all in allenergetically favourable. V. CONCLUSIONS
Recent variational calculations have suggestedthat one-band models cannot give a correct descrip-tion of cuprates superconductors, based on the argu-ment that these models, without ad-hoc terms, fail todescribe even the ARPES photoemission spectra for ahole doped into an antiferromagnetically ordered CuO layer. Also, these works have questioned a long-held be-lief about the spin polaron nature of a single hole dopedin undoped cuprates. To elucidate these claims, in thiswork we have performed a rigorous derivation of a one-band Zhang-Rice singlet based generalized t − J modelfor cuprate superconductors, with no free parameters,starting from a three-band model. Its hopping terms,appreciable up to third nearest neighbors, are obtainedfrom the three-band model parameters, while the ex-change interaction J between copper sites is taken fromexperimental measurements.With the well established SCBA, we have computedthe QP dispersion relation and the oxygen contributionto the photoemission intensity, obtaining a satisfactoryagreement with ARPES experiments, improving the above mentioned variational three-band model calcula-tions . Particularly, we have reproduced the experimen-tal abrupt drop of the QP spectral weight going awayfrom ( π , π ) to ( π, π ), that, within the variational calcu-lation can only be partially obtained appealing to a morecomplicated five-band model.In addition, we have analyzed the structure of theSCBA QP wave function computing its magnon coeffi-cients, and we have found that the spin fluctuations playan essential role in the building up of the QP. This hap-pens even for our generalized t − J model where secondand third NN hoppings would allow the hole motion with-out emitting magnon excitations of the antiferromagneticbackground.From our results we can conclude that rigorously de-rived one-band models are appropriate for the descrip-tion of (at least slightly doped) cuprate superconductors,while the physical nature of a single hole doped in CuO layer corresponds to a spin polaron quasiparticle withspin fluctuations as its main ingredient. VI. ACKNOWLEDGEMENTS.
I. J. H and L. O. M. are partially supported by PIP0364 of CONICET, Argentina. A. A. A. is sponsored byby PIP 112-201501-00506 of CONICET, and PICT 2017-2726 and PICT 2018-01546 of the ANPCyT, Argentina. V. J. Emery, “Theory of high-T c superconductivity in ox-ides,” Phys. Rev. Lett. , 2794–2797 (1987). C.M. Varma, S. Schmitt-Rink, and Elihu Abrahams,“Charge transfer excitations and superconductivity in“ionic” metals”,” Solid State Communications , 681 –685 (1987). R. Raimondi, J. H. Jefferson, and L. F. Feiner, “Effectivesingle-band models for the high- T c cuprates. II. Role ofapical oxygen,” Phys. Rev. B , 8774–8788 (1996). M. E. Sim´on, A. A. Aligia, C. D. Batista, E. R. Gagliano,and F. Lema, “Excitons in insulating cuprates,” Phys. Rev.B , R3780–R3783 (1996). Mi Jiang, Mirko Moeller, Mona Berciu, and George A.Sawatzky, “Relevance of Cu − d multiplet structure inmodels of high- T c cuprates,” Phys. Rev. B , 035151(2020). V. J. Emery and G. Reiter, “Quasiparticles in the copper-oxygen planes of high- T c superconductors: An exact so-lution for a ferromagnetic background,” Phys. Rev. B ,11938–11941 (1988). C. Batista and A.A. Aligia, “Validity of the t - j model:Quantum numbers for ( Cu O ) − ,” Solid State Commu-nications , 419 – 422 (1992). C. D. Batista and A. A. Aligia, “Effective hamiltonianfor cuprate superconductors,” Phys. Rev. B , 8929–8935(1993). F. C. Zhang and T. M. Rice, “Effective hamiltonian for thesuperconducting Cu oxides,” Phys. Rev. B , 3759–3761(1988). A. A. Aligia, M. E. Sim´on, and C. D. Batista, “Systematicderivation of a generalized t − J model,” Phys. Rev. B ,13061–13064 (1994). V. I. Belinicher, A. L. Chernyshev, and L. V. Popovich,“Range of the t − J model parameters for cuo planes:Experimental data constraints,” Phys. Rev. B , 13768–13777 (1994), and references therein. L. F. Feiner, J. H. Jefferson, and R. Raimondi, “Effectivesingle-band models for the high- T c cuprates. I. Coulombinteractions,” Phys. Rev. B , 8751–8773 (1996). A. A. Aligia, “Comment on “relevance of Cu-3 d multipletstructure in models of high- T c cuprates”,” Phys. Rev. B , 117101 (2020). F. C. Zhang, “Exact mapping from a two-band model forCu oxides to the single-band Hubbard model,” Phys. Rev.B , 7375–7377 (1989). Hong-Qiang Ding, Gladys H. Lang, and William A. God-dard, “Band structure, magnetic fluctuations, and quasi-particle nature of the two-dimensional three-band Hubbardmodel,” Phys. Rev. B , 14317–14320 (1992). C. D. Batista and A. A. Aligia, “Validity of the t − J model,” Phys. Rev. B , 4212–4215 (1993). L.-C. Duda, J. Downes, C. McGuinness, T. Schmitt,A. Augustsson, K. E. Smith, G. Dhalenne, andA. Revcolevschi, “Bandlike and excitonic states of oxygenin CuGeO : observation using polarized resonant soft-x-ray emission spectroscopy,” Phys. Rev. B , 4186–4189(2000). Bayo Lau, Mona Berciu, and George A. Sawatzky, “High-spin polaron in lightly doped CuO planes,” Phys. Rev.Lett. , 036401 (2011). W-C. Lee and T. K. Lee, “Comment on “High-spin polaronin lightly doped CuO planes”,” arXiv:1108.5413. Hadi Ebrahimnejad, George A. Sawatzky, and MonaBerciu, “The dynamics of a doped hole in a cuprate isnot controlled by spin fluctuations,” Nature Physics ,951–955 (2014). H Ebrahimnejad, G A Sawatzky, and M Berciu, “Differ-ences between the insulating limit quasiparticles of one-band and three-band cuprate models,” Journal of Physics:Condensed Matter , 105603 (2016). N. B. Brookes, G. Ghiringhelli, A.-M. Charvet, A. Fuji-mori, T. Kakeshita, H. Eisaki, S. Uchida, and T. Mi-zokawa, “Stability of the Zhang-Rice singlet with dopingin Lanthanum Strontium Copper Oxide across the super-conducting dome and above,” Phys. Rev. Lett. , 027002(2015). A. Chainani, M. Sicot, Y. Fagot-Revurat, G. Vasseur,J. Granet, B. Kierren, L. Moreau, M. Oura, A. Yamamoto,Y. Tokura, and D. Malterre, “Evidence for weakly cor-related oxygen holes in the highest- T c cuprate supercon-ductor HgBa Ca Cu O δ ,” Phys. Rev. Lett. , 057001(2017). Clemens P. J. Adolphs, Simon Moser, George A. Sawatzky,and Mona Berciu, “Non-Zhang-Rice singlet character ofthe first ionization state of T-CuO,” Phys. Rev. Lett. ,087002 (2016). I. J. Hamad, L. O. Manuel, and A. A. Aligia, “Gener-alized one-band model based on Zhang-Rice singlets fortetragonal CuO,” Phys. Rev. Lett. , 177001 (2018). Andr´es Greco, “Evidence for two competing order parame-ters in underdoped cuprate superconductors from a modelanalysis of fermi-arc effects,” Phys. Rev. Lett. , 217001(2009), and references therein. Andr´es Greco, Hiroyuki Yamase, and Mat´ıas Bejas, “Ori-gin of high-energy charge excitations observed by reso-nant inelastic X-ray scattering in cuprate superconduc-tors,” Communications Physics , 3 (2019). C. D Batista, L. O Manuel, H. A Ceccatto, and A. AAligia, “Superconductivity and incommensurate spin fluc-tuations in a generalized t − J model for the cuprates,”Europhysics Letters (EPL) , 147–152 (1997). Aabhaas V. Mallik, Gaurav K. Gupta, Vijay B. Shenoy,and H. R. Krishnamurthy, “Surprises in the t − J model:Implications for cuprates,” Phys. Rev. Lett. , 147002(2020). N. M. Plakida, “Superconductivity in the t − J model,”Condensed Matter Physics , 32 (2002). B. O. Wells, Z. X. Shen, A. Matsuura, D. M. King, M. A.Kastner, M. Greven, and R. J. Birgeneau, “ e versus krelations and many body effects in the model insulatingcopper oxide Sr CuO Cl ,” Phys. Rev. Lett. , 964–967(1995). S. Moser, L. Moreschini, H.-Y. Yang, D. Innocenti,F. Fuchs, N. H. Hansen, Y. J. Chang, K. S. Kim, A. L. Wal-ter, A. Bostwick, E. Rotenberg, F. Mila, and M. Grioni,“Angle-resolved photoemission spectroscopy of tetragonalCuO: Evidence for intralayer coupling between cupratelikesublattices,” Phys. Rev. Lett. , 187001 (2014). N. N¨ucker, H. Romberg, X. X. Xi, J. Fink, B. Gegen-heimer, and Z. X. Zhao, “Symmetry of holes in high- T c superconductors,” Phys. Rev. B , 6619–6629 (1989). M. Takigawa, P. C. Hammel, R. H. Heffner, Z. Fisk, K. C.Ott, and J. D. Thompson, “ O nmr study of local spinsusceptibility in aligned YBa Cu O powder,” Phys. Rev.Lett. , 1865–1868 (1989). M. Oda, C. Manabe, and M. Ido, “STM images of a super-conducting Cu-O plane and the corresponding tunnelingspectrum in Bi Sr CaCu O δ ,” Phys. Rev. B , 2253–2256 (1996). J. Eroles, C. D. Batista, and A. A. Aligia, “Angle-resolvedCu and O photoemission intensities in CuO planes,” Phys.Rev. B , 14092–14098 (1999). A. Nazarenko, K. J. E. Vos, S. Haas, E. Dagotto, andR. J. Gooding, “Photoemission spectra of Sr CuO Cl : Atheoretical analysis,” Phys. Rev. B , 8676–8679 (1995). T. Xiang and J. M. Wheatley, “Quasiparticle energy dis-persion in doped two-dimensional quantum antiferromag-nets,” Phys. Rev. B , R12653–R12656 (1996). V. I. Belinicher, A. L. Chernyshev, and V. A. Shubin,“Single-hole dispersion relation for the real CuO plane,”Phys. Rev. B , 14914–14917 (1996). F. Lema and A. A. Aligia, “Quasiparticle photoemissionintensity in doped two-dimensional quantum antiferromag-nets,” Phys. Rev. B , 14092–14095 (1997). Gerardo Martinez and Peter Horsch, “Spin polarons in the t − J model,” Phys. Rev. B , 317–331 (1991). F. Lema and A.A. Aligia, “Spectral function and quasi-particle weight in the generalized t–J model,” Physica C:Superconductivity , 307 – 317 (1998). A. E. Trumper, C. J. Gazza, and L. O. Manuel, “Quasipar-ticle vanishing driven by geometrical frustration,” Phys.Rev. B , 184407 (2004). I. J. Hamad, A. E. Trumper, A. E. Feiguin, and L. O.Manuel, “Spin polaron in the J − J Heisenberg model,”Phys. Rev. B , 014410 (2008). I. J. Hamad, L. O. Manuel, and A. E. Trumper, “Effects ofsemiclassical spiral fluctuations on hole dynamics,” Phys.Rev. B , 024402 (2012). R. Coldea, S. M. Hayden, G. Aeppli, T. G. Perring, C. D.Frost, T. E. Mason, S.-W. Cheong, and Z. Fisk, “Spinwaves and electronic interactions in La CuO ,” Phys. Rev.Lett. , 5377–5380 (2001). C. L. Kane, P. A. Lee, and N. Read, “Motion of a sin-gle hole in a quantum antiferromagnet,” Phys. Rev. B ,6880–6897 (1989). George F. Reiter, “Self-consistent wave function for mag-netic polarons in the t − J model,” Phys. Rev. B , 1536–1539 (1994). A. Ramˇsak and P. Horsch, “Spatial structure of spin po-larons in the t − J model,” Phys. Rev. B , 4308–4320(1998). Adolfo E. Trumper, Claudio J. Gazza, and Luis O.Manuel, “Quasiparticle excitations in frustrated antiferro-magnets,” Physica B: Condensed Matter , 252 – 256(2004), proceedings of the Workshop ”At the Frontiers ofCondensed Matter ”. Magnetism, Magnetic Materials, andtheir Applications. Andrea Damascelli, Zahid Hussain, and Zhi-Xun Shen,“Angle-resolved photoemission studies of the cuprate su-perconductors,” Rev. Mod. Phys.75