Failure of t-J models in describing doping evolution of spectral weight in x-ray scattering, optical and photoemission spectra of the cuprates
aa r X i v : . [ c ond - m a t . s up r- c on ] N ov Failure of t − J models in describing doping evolution of spectral weight in x-rayscattering, optical and photoemission spectra of the cuprates R. S. Markiewicz, Tanmoy Das, and A. Bansil
Physics Department, Northeastern University, Boston MA 02115, USA (Dated: June 14, 2018)We have analyzed experimental evidence for an anomalous transfer of spectral weight from highto low energy scales in both electron and hole doped cuprates as a function of doping. X-rayscattering, optical and photoemission spectra are all found to show that the high energy spectralweight decreases with increasing doping at a rate much faster than predictions of the large U − limitcalculations. The observed doping evolution is however well-described by an intermediate couplingscenario where the effective Hubbard U is comparable to the bandwidth. The experimental spectraacross various spectroscopies are inconsistent with fixed- U exact diagonalization or quantum MonteCarlo calculations, and indicate a significant doping dependence of the effective U in the cuprates. PACS numbers: 71.10.-w,71.30.+h,71.45.-d,71.35.-y
I. INTRODUCTION
The key to unraveling the mechanism of cuprate su-perconductivity is to ascertain the effective strength ofcorrelations since pairing is widely believed to arise fromelectron-electron interactions rather than from the tra-ditional electron-phonon coupling. Two sharply differ-ent scenarios have been proposed and remain subject ofconsiderable debate. One viewpoint holds that U ≫ W where U is the Hubbard U , and W ∼ t is the bandwidthwith hopping parameter t . In this case, a ‘pairing glue’ isnot necessary as the pairs are bound by a superexchangeinteraction J = 4 t /U , and the dynamics of the pairs in-volves virtual excitations above the Mott gap set by theenergy scale U . In the opposing view, U ∼ W , and pair-ing is mediated by a bosonic ‘glue’, which originates fromantiferromagnetic (AFM) spin fluctuations . It is clearthus that the determination of the size of the effective U and its variation with doping are essential ingredientsfor understanding the mechanism of superconductivity aswell as the magnetic phase diagram of the cuprates.Since the electronic dispersion at half filling has a gapof magnitude ∼ U which is clearly visible in x-ray ab-sorption (XAS), angle-resolved photoemission (ARPES),and optical spectra, one way to estimate the size of U isto follow the evolution of high-energy spectral weight asa function of doping. Whereas for a conventional bandinsulator the spectral weights of the bands above andbelow the gap are independent of doping, this is not thecase for a Mott insulator. In the latter case, for U → ∞ ,removing one electron creates two low energy holes – onefrom the lower Hubbard band (LHB), but a second onefrom the upper Hubbard band (UHB), since without anelectron on the atom there is no U -penalty in addingan electron. Paradoxically, as U decreases, the rate ofthis anomalous spectral weight transfer (ASWT) actu-ally increases . For infinite U double occupancy (DO) isalways forbidden, so no matter how few electrons are inthe LHB, there will be an equivalent number of holes inthe UHB. In contrast, for smaller U values DO is reducedcollectively, via long-range magnetic order. As the mag- netic order disappears at a quantum critical point , amuch higher degree of DO is restored, and the UHB cancompletely vanish.Hence, by measuring the high energy spectral weight(HESW) as a function of doping, we can estimate thedegree of correlation in cuprates. Here we quantify theseresults for XAS, ARPES, and optical measurements, anddemonstrate that the doping evolution of ASWT is sim-ilar across all these spectroscopies for both electron andhole doped cuprates. Moreover, the observed doping evo-lution is inconsistent with large U values, and also withfixed- U Hubbard model calculations, but it is consis-tent with a doping-dependent effective U of intermediatestrength U . W .This paper is organized as follows. In Section II weexplain how to quantify the rate of ASWT with doping,and show that similar rates are found for several differentspectroscopies. In Section III we show that these ratesare consistent with an intermediate coupling model of thecuprates. A discussion of the results is given in SectionIV, and conclusions are presented in Section V. II. QUANTIFYING ANOMALOUS SPECTRALWEIGHT TRANSFER
We motivate a definition of the rate of ASWT withdoping in Fig. 1(a). In a strongly correlated system, re-moving x electrons produces (1 + x ) states above theFermi level, which are distributed between p ≥ x lowenergy (in-gap) states and W UHB = 1 + x − p states inthe UHB. Then the ASWT can be quantified by the co-efficient β , defined such that in this process the weightof the UHB reduces to W UHB = 1 − βx and the low en-ergy holes gain weight by p = (1 + β ) x . The value of β is found theoretically to depend on U such that β = 1for a very strongly correlated ( U → ∞ ) Mott insulator,while reducing U leads to larger values of β . Figure 1(b)illustrates a variety of calculations of the HESW vs dop-ing. Exact diagonalization (ED) calculations on smallclusters (dashed lines) find β = 1 for the t − J model FIG. 1: (color online) (a) Schematic diagram of ASWT forhole doped cuprates. (b) Estimates of W UHB (for hole dop-ing) and W LHB (for electron doping) from various experi-mental results (see legend) are compared with our the-oretical results (open symbols of same color). Dashed linesof various colors show exact diagonalization calculations fordifferent values of U taken from Ref. 4. QMC results fromFig 4 are plotted as blue stars. All curves are normalized to W UHB → or for a U → ∞ Hubbard model, β ≃ . U = 10 t and β ≃ . U = 5 t . Shown also inFig. 1b are QMC results for U = 8 t , t ′ = 0 [where t and t ′ are hopping parameters] , which are consistent withthe ED results. Hence, β = 1 confirms strong correla-tions and the ‘no double occupancy’ (NDO) hypothesis,while a faster falloff ( β >
1) indicates otherwise, and sup-ports a real gap collapse model ( W UHB ∼ x = 1 /β ).For an electron doped system the HESW is associatedwith the LHB, and is described by the mirror image ofFig 1(a) with respect to E F .Shown also in Fig. 1(b) is our key result, the HESW ofa variety of cuprates as a function of doping, extractedfrom a number of spectroscopies. The results are strik-ingly similar over a variety of spectroscopies, as expected,but also over several families of cuprates for both electron and hole doping. Shown in Fig. 1(b) are XAS results onLa − x Sr x CuO (LSCO) , ARPES on Nd − x Ce x CuO ± δ (NCCO) , and optical absorption on both NCCO and LSCO , compared with additional XAS data forLSCO, YBCO and Tl-2201 from Ref. 15. All experimen-tal measures of HESW find a rapid falloff of the spec-tral weight with doping, and at low doping decrease al-most linearly with doping with approximately the sameslope of β ≃ .
7, consistent with U eff < t , suggest-ing that the cuprates are far from the strong correlationlimit. The observed falloff supports a real gap collapseat x UHB ∼ /β = 0 .
27. Notably, the value of U eff is in-compatible with the measured gap at half filling. For ex-ample, optical spectra find a gap consistent with U ∼ t ,but the HESW calculations for fixed U = 8 t are far fromthe experimental results. On the other hand, the exper-imental data can be explained by intermediate couplingmodel calculations with a doping dependent effective U . The calculated results are plotted in Fig 1(b) as opensymbols of same color as the corresponding experimentaldata.Since the HESW is an intrinsic property of the elec-tronic structure of cuprates, it should show up in allspectral probes, and Fig. 1(b) confirms this. However,it is important to realize that the ASWT will play outquite differently in different spectroscopies. First, asis clear from Fig. 1(a), there is a strong electron-holeasymmetry to the effect: the changes will be muchsmaller in the Hubbard band which lies at the Fermilevel. Hence, for maximum sensitivity to ASWT in ahole-doped cuprate, the probe should be sensitive toempty states, and to filled states for electron-dopedcuprates. Thus, ARPES or X-ray emission spectro-scopies are well-suited for studying ASWT in electron-doped cuprates, while XAS is appropriate for hole-doped cuprates. Optical and resonant inelastic x-rayscattering studies would work for both cases, asthey measure a joint density of states. On the otherhand, Compton scattering and positron annihilation will not be sensitive to ASWT because these spectro-scopies measure only the total spectral weight of occupiedstates, but not how this spectral weight gets rearrangedin energy with doping. In principle, scanning tunnel-ing microscopy could follow either sign of charge, butwould require a wide energy range, ∼ n ( ω ), in the lower panel of Fig. 2. Ineach spectrum, the UHB [LHB for electron doping] isdenoted by a gray shaded region, and the W UHB [or W LHB ], Fig. 1(b), is defined as the integrated den-sity over that region, starting from a cutoff frequency ω c , taken as independent of doping. A complicationis involved in comparing our one-band calculation withthe experimental data, in that the antibonding band in FIG. 2: (color online) (a) ARPES spectra along the nodal direction of NCCO for various dopings. (b-c) Optical conductivityof NCCO and LSCO . (d) K-edge XAS results are compared with our theorerical DOS (broadened with experimentalresolution of 0.4eV). All DOS curves are shifted by a doping independent x-ray edge energy value of 528.4 eV. (e-h) Integratedspectral weights corresponding to the spectra after subtracting a background associated with higher-energy bands , shown asblack dashed lines in frames (a-c). (e) Integrated ARPES spectral weight (integrated around a small momentum window tomimic the experiment and averaged over nodal and antinodal directions), normalized to (1 + x ) at E F . (f-g) Effective numberof electrons (Eq. 1) calculated from the optical spectra for NCCO and LSCO. (h): Integrated XAS intensity. In all frames,dashed lines are experimental data ; solid lines of same color are the present calculations, while the edge of the shadedregion marks the crossover energy ω c , discussed in the text. cuprates lies near to other bands, and the role of the lat-ter must be disentangled before the spectral weight canbe estimated. At high energies, we subtract off a back-ground from the experimental spectra associated withinterband transitions to higher-lying bands not includedin the present one band calculations. . We use adoping-independent background contribution shown asblack dashed lines in Figs. 1(a-c). In all cases we com-pare the data with calculations based on the QP-GWmodel (solid lines), discussed below.For electron doping, ARPES can detect the full LHBand hence determine W LHB to the extent that matrixelement effects are doping independent. The ARPESresults for NCCO are compared with our theoretical re-sults in Fig. 2(a). At half-filling the energy distribu-tion curve (EDC) along the nodal direction shows theso-called charge-transfer gap from the Fermi level to theLHB. A significant redistribution of spectral weight is ev-ident at x = 0 .
04 as the LHB approaches the Fermi leveland by x = 0 .
10, virtually all of the spectral weight of theLHB has shifted to the vicinity of the Fermi level. The top of the LHB crosses E F at x ≃ .
15, forming a holepocket, and the spectral weight near E F undergoes anabrupt increment. To extract the total spectral weightassociated with the LHB we have integrated the spectralweight from − . ω , Fig. 2(e). ARPES data areavailable along only two high-symmetry directions, so wetake their average as representative of the net spectralweight, and at each doping normalize n ( ω ) to (1 + x ) at E F .In Fig. 2(b,f), analysis of the HESW in NCCO based onthe optical absorption spectra proceeds similarly .There is a large Mott gap below 2 eV in the undopedmaterial, but with doping there is a strong transfer ofspectral weight from the gap to low energy features – theDrude peak and the midinfrared (MIR) peak – with anisosbetic (equal absorption) point around ω ∼ . N eff ( ω ) = 2 m Vπe ~ Z ω −∞ σ ( ω ′ ) dω ′ , (1)where m and e are the free electron mass and chargeand N is the number of Cu-atoms in a cell of volume V .The weight of the LHB is extracted as W LHB = 1 + x − N eff ( ω c ). A similar analysis of W UHB was carried out onLSCO spectra in Fig. 2(c,g), and the results included inFig. 1(b). These optical results are consistent with theanalysis of Comanac, et al. . It is interesting to notethat the ω c which separates the high-energy Hubbardbands and the low-energy in-gap states coincides with theisosbetic or equal absorption point in the optical spectrai.e., the residual charge-transfer gap.For hole doping, W UHB was determined by x-rayabsorption spectroscopy (XAS) , which detects theempty states above the E F . In this spirit, we comparethe measured XAS spectra with the calculated empty-state DOS in Figs. 2(d) and 2(h). The behavior of thespectral weight transfer is very similar to the ARPESresult for NCCO in Fig. 2(a).The overall similarity of the doping dependence of theexcess electron [or hole] count n ( ω ) between ARPES, op-tical and XAS experiments is striking, and is well cap-tured in the model calculations in Figs. 2(e)-(h). TheHESW plotted in Fig. 1(b) illustrates one importantcharacteristic of these curves to demonstrate the univer-sality of the doping dependence, but the detailed agree-ment is clearly much more extensive. This observa-tion motivates our choice of the cut-off frequencies inFig. 2. Since experimental and theoretical values areextracted in the same way, it is simplest to chose adoping-independent ω c for each spectroscopy. The natu-ral choice is the minimum spectral weight regions evidentin Fig. 2, separating low and high energy scales. Thesecorrespond to the waterfall region in single particle spec-tra of ARPES and XAS or the isosbetic point in opticalspectra which is also the manifestation of the waterfalleffect as discussed in Ref. . Our ω c values are chosen asaverage values which fall near this minimum. III. INTERMEDIATE COUPLING MODEL OFASWT
The theoretical calculations in Figs. 1 and 2 are basedon the QP-GW model , an extension of our earlierHartree-Fock (HF) model of AFM gap collapse tothe intermediate coupling regime by introducing a GW-like self-energy correction.
The self-energy in QP-GW model is dominated bya broad peak in Σ ′′ which produces the ‘waterfall’effect in the electronic dispersion by redistribut-ing spectral weight into the coherent in-gap states andan incoherent residue of the undressed UHB and LHB.With underdoping, the in-gap states develop a pseudogapwhich we model as a ( π, π )-ordered spin density wave.The doping evolution of both electron and hole-dopedcuprates is dominated by a magnetic gap collapse nearoptimal doping. The present calculations are obtainedwith the same parameter sets as in Ref. 16; in particu- ∫ Σ ′′ [ e V ] NCCO LSCO ∫ − ∞ Σ ′′ ∫ ∞ Σ ′′ FIG. 3: (color online) Integrated imaginary part of the calcu-lated self-energy as a function of doping. lar the doping dependence of U is shown in Fig. 5 of thatpublication. Our analysis identifies two main factors thatcause ASWT. Firstly, the pseudogap collapses with dop-ing, shifting the optical MIR peak to low energies whiletransferring weight to the Drude peak. Secondly, theresidual incoherent weight associated with the Hubbardbands decreases with doping due to decrease in magnonscattering. This is reflected in the doping dependence ofthe peak in Σ ′′ . The strength of this peak can be mea-sured by the area under the Σ ′′ curve, Fig. 3. This givesa direct measure of the tendency of the spectrum to splitinto coherent and incoherent parts, and hence a measureof the weight of the Hubbard bands. Fig. 3 shows thisquantity as a function of doping above and below theFermi level for both NCCO and LSCO. In both materi-als, R Σ ′′ dω below E F , seen in ARPES, shows a muchfaster fall-off with doping. This fast fall-off seems to ter-minate around x ∼ . − .
25 close to the point wherethe HESW extrapolates to zero, x UHB = 1 /β ∼ . The unusual doping dependence of the experimental W UHB in Fig. 1b can be understood within our modelas follows. The magnetic gap collapses near x ∼ and hole doped case , and beyond thisdoping there is at most only a weak dip in the densityof states, indicating a separation of the band into twocomponents – now coherent and incoherent parts. How-ever, since we work with fixed cutoff, we count all emptystates in the band above ω c as part of the UHB. Thesechange slowly with doping, decreasing linearly to zero at x = 0 .
2. Hence the break in slope indicates the magneticgap collapse.
IV. DISCUSSION
To better understand the failure of QMC calculationswith fixed U = 8 t to explain the observed ASWT, in Fig-ure 4(a-b) we plot the DOS and the associated electron FIG. 4: (color online) (a) DOS computed in QMC . All re-sults are normalized to their peak values. (b): Correspondingelectron number n ( ω ), the integral of the DOS, normalized to2 for a full band. The QMC results for some of the dopingsare not available at higher energies below E F , so we nor-malize n ( ω ) to (1 − x ) at E F . (c) and (d) QMC DOS withexperimental broadening and K-edge energy shift (solid lines)is compared with experimental data (dashed lines) as inFig. 2(d-h). count calculated in QMC . For ω c = 0 . , W UHB is in good agreement withthe exact diagonalization results for the corresponding U = 8 t , blue stars in Fig. 1(b), but has a considerablyweaker falloff than found in experiment. Note that thesame result would follow by choosing a doping dependent ω c pinned to the DOS minimum. Consistent with this, wecarried out a similar analysis of the XAS spectra basedon QMC-based DOS with doping-independent U = 8 t in Figs. 4(c-d). The QMC spectra (solid lines) are notconsistent with the experimental results, clearly overesti-mating the weight of the UHB for finite dopings. Similarconclusions were reached in Refs. 15,40. Note that themean-field result is similar: for U = 8 t , the gap collapsewould be shifted to much higher doping x ∼ . Since the bare U should be doping independent, theapparent doping dependence of the effective U in ourmodel arises to compensate for interactions not includedin the underlying calculation. We have been able explainthis doping dependence as due to long-range Coulombscreening . Alternatively, it should be noted that thedoping dependence of U can also be significantly reducedby going to a three-band model . Indeed, it is com-mon practice in the LDA+U literature to try to calcu-late a screened U by incorporating interactions involvingother bands or longer range Coulomb interactions. In thissense, our result is a natural extension to incorporate the doping dependence of this screening, which is particularlyimportant near the metal-insulator transition. However, there is an ongoing debate on this issue thatwe would like to address. Some screening is presentwithin the one band Hubbard model, and it is impor-tant to see whether the full doping dependence of Ucould be understood on the basis of a more exact treat-ment of the Hubbard model – i.e., whether the physicsof cuprates can be fully understood within a single-bandHubbard model. Clearly, as more correlations are addedthe doping dependence of the effective U systematicallydecreases from Hartree-Fock calculations to the presentQP-GW model, to recent DMFT calculations that cansuccessfully describe the doping evolution of the cuprateswith fixed- U models . However, neither exact di-agonalization nor QMC with fixed U = 8 t capture theASWT, Fig. 1(b), and the doping dependence of U wasnot found in recent Gutzwiller calculations . Indeed, bycomparing the experimental results with exact diagonal-ization calculations, Fig. 1(b), a value U < t at finite x is indicated, consistent with our results. One way toreconcile the DMFT with the QMC and exact diagonal-ization results might be to note that both of the lattercalculations are for a pure Hubbard model, neglectingband structure effects by restricting the overlap to near-est neighbor only. Hence, it will be necessary to includeat least a t ′ in the exact diagonalization and QMC calcu-lations to ensure that all three calculations converge ona common behavior for the one-band Hubbard model. V. CONCLUSIONS
In conclusion, we have shown that the spectral weightof the UHB [LHB for electron-doped cuprates] collapseswith doping at a rate much faster than can be explainedin a t − J or U = ∞ Hubbard model. Such a fast falloffwould seem to require a real Mott gap collapse consistentwith an intermediate coupling
U < W scenario. We findthat the rate of ASWT is universal – the same acrossseveral spectroscopies and many different cuprates. Theplot of HESW vs doping in Fig. 1(b) provides a uniquesignature of the effective Hubbard U in these materials. Acknowledgments
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