The origin of the apparent discrepancies between the one-particle spectra of
the Hubbard and t-J models is revealed: Wavefunction corrections, in addition
to the three-site terms, should supplement the bare t-J. In this way a
quantitative agreement between the two models is obtained, even for the
intermediate-$U$ values appropriate for the high-Tc cuprate superconductors.
Numerical results for clusters of up to 20 sites are presented. The momentum
dependence of the observed intensities in the photoemission spectra of
Sr2CuO2Cl2 are well described by this complete strong-coupling approach.Comment: 4 two-column RevTeX pages, including 4 Postscript figures. Uses epsf.
  Accepted for publication in Physical Review B, Rapid Communicatio

A. Nazarenko

A.B. Harris

A.R. Kampf

B.O. Wells

C.T. Chen

E. Dagotto

F.C. Zhang

H. Eskes

Henk Eskes

J. Bała

K.J. von Szczepanski

L.F. Feiner

M.S. Hybertsen

P. Horsch

P.W. Anderson

P.W. Leung

R. Eder

R. Preuss

Robert Eder

S. Uchida

S.A. Trugman

W. Stephan

X.G. Wen

English

arXiv

Crossref

Hubbard model versus<i>t</i>-<i>J</i>model: The one-particle spectrum

Eskes, H

Eder, R

University of Groningen

   University of GroningenHubbard model versus t-J modelEskes, H; Eder, RPublished in:Physical Review BDOI:10.1103/PhysRevB.54.R14226IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1996Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Eskes, H., & Eder, R. (1996). Hubbard model versus t-J model: The one-particle spectrum. PhysicalReview B, 54(20), 14226-14229. https://doi.org/10.1103/PhysRevB.54.R14226CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 25-04-2023arXiv:cond-mat/9609233 v1   24 Sep 1996Hubbard model versus t-J model: The one-particle spectrumHenk EskesInstitute Lorentz, Leiden University, P.O. Box 9506, 2300 RA Leiden, The NetherlandsRobert EderLaboratory of Applied and Solid-State Physics, Materials Science Centre,University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands(August 13, 2006)The origin of the apparent discrepancies between the one-particle spectra of the Hubbard and t-Jmodels is revealed: Wavefunction corrections, in addition to the three-site terms, should supplementthe bare t-J model. In this way a quantitative agreement between the two models is obtained,even for the intermediate-U values appropriate for the high-Tc cuprate superconductors. Numericalresults for clusters of up to 20 sites are presented. The momentum dependence of the observedintensities in the photoemission spectra of Sr2CuO2Cl2 are well described by this complete strong-coupling approach.71.27.+a,79.60.-i,71.10.+x,74.72.-hThe past ten years the interest in Hamiltonians mod-eling correlated electrons has greatly increased. This islargely due to the mysterious normal-state properties ofthe high-Tc cuprate superconductors and the realizationthat local correlations on the Cu sites are large in thesecompounds. The two most intensively studied Hamilto-nians are the Hubbard and t-J models. Although at firstsight such models may seem oversimplified, it has beenargued that the two dimensional square-lattice Hubbardmodel captures the essential dynamics of the cupratesin a window of a few eV around the chemical poten-tial [1–4]. The materials belong to the intermediate-Uregime, U/t ≈ 10, t ≈ 0.3−0.4 eV [3,4]. Further supportfor a single-band Hubbard approach comes from mea-sured one-particle and optical spectra of the cuprates.These show a dramatic reshuffling of intensity as a func-tion of doping on the scale of a few eV [5,6]. This char-acteristic behavior is a fingerprint of strong correlationsand is described well by the Hubbard model [5,7,8].Although the t-J model is frequently referred to as be-ing the large-U limit of the single-band Hubbard model,the (one-particle) spectra of the two models differ sig-nificantly in the intermediate-U regime of interest. (Fora recent review on numerical results, see Ref. [7]) Prob-ably because of this, the two models are often treatedas being “distinct”. One obvious difference concernsthe momentum occupation number nk at half filling,equal to the one-electron removal sum rule. For the t-Jmodel this quantity is trivially equal to 1/2, while for theintermediate-U Hubbard model it is a strongly varyingfunction of k. Even though in the infinite-U limit nktends to 1/2, as in the t-J model, the convergence to thisvalue by increasing U is very slow.In this Letter we show that these discrepancies can beattributed to the neglect of certain terms in the large-U perturbation theory. Specifically, the weight distribu-tions in the spectral functions show a high sensitivity tothe first-order renormalization of the wave functions. Thecomplete strong-coupling approach is found to quantita-tively reproduce the Hubbard spectra for U ≥ W , whereW is the U = 0 band width. For these U values the per-turbation series in t/U agrees well with the full HubbardHamiltonian.The derivation of the strong-coupling Hamiltonian bymeans of a canonical transformation, written as an ex-pansion in t/U , is well known [8,9]. Starting point is theHubbard model,H = V + T = U∑ini,↑ni,↓ − t∑i,δ,σa†i,σai+δ,σ. (1)Here ni,↑ is the occupation number of the original Hub-bard fermions ai,σ, and δ is a vector connecting nearestneighbors. After the transformation the states in thelowest Hubbard sector (no doubly occupied sites) are de-scribed to order t2/U by the strong-coupling Hamilto-nian,Hsc = −t∑i,δ,σc†i,σci+δ,σ −J2∑i,δ(S̃i · S̃i+δ −14ñiñi+δ)−J4∑i,δ 6=δ′,σ(c†i+δ,σñi,σ̄ci+δ′,σ − c†i+δ,σ̄c†i,σci,σ̄ci+δ′,σ),(2)where J = 4t2/U . The motion of the above transformedci,σ fermions will leave the number of doubly occupiedsites unchanged, and S̃i and ñi,σ are the spin and occu-pation number of these new fermions. Simply neglectingthe ‘three-site term’, i.e. the second line of (2), one ob-tains the t-J model.Why the full strong-coupling model has received farless attention than the t-J model is unclear. It might beexpected that the three-site term is very important in theone-hole limit (and low doping region) since, unlike the1nearest-neighbor hopping, a direct next-nearest-neighborhopping process does not frustrate the spin system. How-ever, it was shown by a numerical study [10], spin-polaronapproaches [11,12] and a linear-spin-wave self-consistentBorn (LSW-SCB) calculation [13] that the main effect ofthis term on the quasiparticle dispersion is an increase ofthe bandwidth of the dominant ‘quasiparticle peak’.By virtue of its construction via a canonical transfor-mation, the Hamiltonian (2) reproduces the eigenvaluespectrum of the Hubbard model for large U . To obtainfurther information, such as matrix elements or expec-tation values, operators acting in the Hilbert space ofthe Hubbard model have to be subjected to the samecanonical transformation. For the electron annihilationoperator this procedure leads to the expression [8,9],Pai,σP = Pci,σP+tU∑δP(ñi,σ̄ci+δ,σ − c†i,σ̄ci,σci+δ,σ̄)P. (3)P is defined as the projector on states with no dou-ble occupancy of the transformed fermions (states with〈Ṽ 〉 = 0). The first-order correction terms (second line)describe the dressing of the annihilation operator with‘virtual’ charge fluctuations involving doubly-occupiedsites. While their prefactor t/U seems to suggest thatthey represent only a minor correction, their impact onthe calculated spectra is dramatic – one reason beingthat the summation over neighbors introduces the co-ordination number z as an additional ‘hidden’ prefac-tor. When comparing strong-coupling calculations di-rectly with experiments it is therefore crucial to takethese corrections into account. For the optical conduc-tivity such corrections were shown to lead to qualita-tively different results [14]. Another consequence is arapid reshuffling of weight between the upper (UHB) andlower-Hubbard band (LHB) as a function of hole doping,observed in both one-particle and optical experiments.The first-order corrections to transformed operators suchas Eq. (3) lead to modified expressions for sum rules thatconsistently describe the rapid intensity changes observedin experiments [5,6,8,9].For the numerical calculations of the one-particle spec-tra we implemented the strong-coupling model Eq. (2).Electron removal is described by the operator Eq. (3).This operator does not change double-occupancy andonly the LHB part of the Hubbard spectrum is pro-duced. (There is, in fact, a similar independent proce-dure to calculate the UHB response as well.) The t/Uterm in Eq. (3) amounts to the removal of an electronfrom a neighbor of i. This leads to an extra phase factorexp(ik·δ). Because of this the total weight is enhanced atΓ and reduced at (π, π) and Eq. (3) leads to the correctbehavior of n(k) (see Fig. 2 of Ref. [8]). Note that theimplementation of the first-order correction is straight-forward and does not increase memory or CPU time inFIG. 1. The one-particle spectrum of the two-dimensional4x4 strong-coupling model (solid line) compared with theHubbard model (dotted line), at half filling (left) and for twoholes (right). U/t = 10. Only the LHB part of the Hubbardspectrum is shown. Dashed line is the chemical potential inthe two-hole case. Hubbard data reproduced from Ref. [15]a Lanczos-like diagonalization calculation. In a similarway such operator corrections can be taken into accountin analytic approaches, for instance in the LSW-SCB cal-culation of the one-hole spectrum.In Fig. 1 we compare the two-dimensional 16-site 4x4strong-coupling model with the LHB electron removaland addition spectra of the Hubbard model, reproducedfrom Ref. [15]. Note that U = 10 is comparable to thebandwidth W = 8. No shift or scaling has been appliedto the spectra, showing that the value of the gap betweenthe UHB (not shown, starting at energy ≈ 8t) and LHBis accurately described. (While there is no UHB in thestrong coupling model, the magnitude of the energy gapstill can be inferred from the relation ∆ = U − 2 · Emin,with Emin ≈ 2t the minimal ionization energy of themodel.) The resulting gap of ≈ 6t is consistent with themeasured insulating gap of ≈ 2eV .As can be seen, excitation energies and weights of thedominant features in the spectra are reproduced correctlyby the strong-coupling approach. The agreement is bestfor the physically most relevant low excitation energies,but the strong coupling approach also gives a completedescription of the higher excitations, with energies notmuch smaller than U . The total width is slightly overes-timated by the second-order t2/U terms in the Hamilto-nian, due especially to those peaks which obtain a largeenergy correction by the three-site terms, for instance at(π, π). There is no significant deterioration of the agree-ment upon doping, demonstrating that the discrepancies2between the doping dependences of low energy sum-rulesfor t− J and Hubbard originate from matrix element ef-fects, rather than an incompleteness of the Hilbert spaceof the strong coupling model. We explicitly checked thatsecond-order corrections to the Fermi operators lead toonly minor changes of the peak intensities. For U valuesbelow 10 the agreement between Hubbard and its strong-coupling limit gradually deteriorates, which we interpretas a breaking down of the perturbation approach.Stimulated by the good agreement in the 4x4 systemwe now present large-U Hubbard spectra for clusters upto 20 sites. Because the momenta of the 16 and 18 siteclusters nicely complement each other and are all on thehigh-symmetry axis, we show a mix of these two clus-ters in Figs. 2 and 3. The curves on the left show a scanfrom (0, 0) to (π, π). The plots on the right go from (0, 0)to (π, 0) and from (π, 0) to (π, π). We shifted all 4 × 4spectra by the same small energy such that the lowestpeak at k = (0, 0) is at the same energy for both clus-ters. For the two-hole case we shifted the t-J data by anadditional 0.3t to have the chemical potentials at roughlythe same position. The ground-state momentum is (0, 0)in all cases (there is an extra degeneracy for 16-sites, 2holes). In both figures the results are compared with the“bare” t-J model.As mentioned before there are substantial differ-ences between t-J and strong coupling. At half-fillingthe low energy ‘quasiparticle peaks’ are at higherenergies (implying a larger Hubbard gap) for theFIG. 2. The momentum dependence of the elec-tron-removal spectrum of the strong-coupling Hubbard model(solid line) compared with the t-J model (dotted line). Theplot is a mix of data from the 16 and 18 site clusters.U/t = 10. n = 1. The spectra for k = (π/2, π/2), (π/2, 0),(π, 0) and (π/2, π) are calculated using the 2D 16-site cluster;the others, using the 18-site cluster.FIG. 3. Combined electron addition and removal spectrafor the two-hole ground state. See caption of Fig. 2.t-J model and their dispersion has a smaller width.Moreover, the momentum dependence of the weights isqualitatively different. The t-J spectrum at (0, 0), andindeed through much of the Brillouin zone, is more com-pact than for large-U Hubbard. The strongest differenceis at (π, π). Here strong coupling shows a little peak athigh energy, while the t-J model has pronounced struc-tures at a more intermediate energy. This reflects thesmall value of nk for k = (π, π) in the Hubbard model,which is reproduced by the first-order corrections to theFermi operator Eq. (3).The spectra at half filling can be compared with re-cent angle-resolved photoemission data on insulatingSr2CuO2Cl2 [16]. A significant feature of these measure-ments is the strong momentum dependence of the weightsof the spectra. The region in momentum space with themost intense peaks is not centered around (π/2, π/2),but displaced towards (0, 0). Beyond (π/2, π/2) the in-tensity drops very sharply and seems to vanish around(π, π) [17]. As shown in Fig. 4 these features are wellreproduced by the strong-coupling Hubbard model. Incontrast, the intensities in t-J are much more symmetricaround (π/2, π/2) and change only little in that partof momentum space. On average the peaks in strong-coupling Hubbard are more intense than those in thet-J model, at the expense of the incoherent part of thespectrum. Note that also in the t-J model there aredifferences in weight between k and k + (π, π). This isdue to the coupling of the created hole to quantum-spinfluctuations in the ground state of the Heisenberg anti-ferromagnet [12]. Experimentally the strong changes atthe antiferromagnetic Brillouin-zone boundary look like3FIG. 4. Weights (proportional to the radius of the circles)of the lowest energy peak for the strong-coupling Hubbardmodel (left) and t-J model (right). Data collected from the16, 18 and 20 site clusters. U/t = 10, n = 1.a kind of ‘pseudo Fermi surface’: The spectral weightobserved drops significantly, but an actual Fermi-levelcrossing does not take place. We note that the largedifference observed between (π, 0) and (π/2, π/2) in thephotoemission measurements actually is in disagreementwith the near degeneracy of these momenta in both thestandard Hubbard and t-J model. As discussed by vari-ous authors this may be explained by adding extra next-neighbor hopping terms to the Hamiltonian [13,18].The increased quasiparticle weight and the strong dropof intensity when passing the half-filled Fermi surfaceare consistent with a continuous cross-over to a small-Uspin-density-wave (SDW) like shadow-band scenario [19].Surely Fig. 4 shows that such a strong drop in weight isnot necessarily a weak-coupling feature.The interpretation of the doping dependence of thespectral function is a subtle issue [20–23], and will not bediscussed here. For roughly 10% holes (Fig. 3) there areintense features appearing in the electron addition spec-trum for all momenta outside the magnetic zone. Thesefeatures are more intense than for t-J and because ofthe sum rule the extra weight must come from the up-per Hubbard band. This weight-transfer effect has beenclearly observed in O-1s X-ray absorption [5].In conclusion, we have shown that the strong-couplingapproach is a good alternative to Hubbard calculationseven in the intermediate U regime relevant for the high-Tc superconductors. Only the complete strong-couplingmodel gives a faithful representation of the propertiesof the Hubbard model. The corrections to t-J lead toa smooth crossover to small U . For comparisons withexperiments such as angle-resolved photoemission, O-1sX-ray absorption (and optical conductivity) it is impor-tant to use the renormalized wavefunctions (rotated op-erators), as these can lead to even qualitatively differentresults.We acknowledge stimulating discussions with GeorgeA. Sawatzky, Jan Zaanen and Andrzej M. Oleś. HE issupported by the Stichting voor Fundamenteel Onder-zoek der Materie (FOM), which is financially supportedby the Nederlandse Organisatie voor WetenschappelijkOnderzoek (NWO). RE is supported by a postdoctoralfellowship granted by the Commission of the EuropeanCommunities.[1] P.W. Anderson, Science 235, 1196 (1987).[2] F.C. Zhang and T.M. Rice, Phys. Rev. B 37, 3757 (1988).[3] M.S. Hybertsen et al., Phys. Rev. B 41, 11068 (1990).[4] L.F. Feiner, J.H. Jefferson, and R. Raimondi, Phys. Rev.B 53, 8751 (1996), and references therein.[5] C.T. Chen et al., Phys. Rev. Lett. 66, 104 (1991).[6] S. Uchida et al., Phys. Rev. B 43, 7942 (1991);[7] E. Dagotto, Rev. Mod. Phys. 66, 763 (1994), and refer-ences therein.[8] H. Eskes and A. M. Oleś, Phys. Rev. Lett. 73, 1279(1994); H. Eskes et al., Phys. Rev. B 50, 17980 (1994).[9] A.B. Harris and R.V. Lange, Phys. Rev. 157, 295 (1967).[10] K.J. von Szczepanski, et al., Phys. Rev. B 41, 2017(1990).[11] S.A. Trugman, Phys. Rev. B 37, 1597 (1988); ibid. Phys.Rev. B 41, 892 (1990).[12] R. Eder and K.W. Becker, Z. Phys. B 78, 219 (1990);ibid. Phys. Rev. B 44, 6982 (1991).[13] J. Ba la, A.M. Oleś and J. Zaanen, Phys. Rev. B 52, 4597(1995).[14] W. Stephan and P. Horsch, Int. J. Mod. Phys. B6, 141(1992). P. Horsch and W. Stephan, Phys. Rev. B 48,10595 (1993).[15] P.W. Leung et al., Phys. Rev.B 46, 11779 (1992).[16] B.O. Wells et al., Phys. Rev. Lett. 74, 964 (1995).[17] It is important to realize that other effects can influencethe momentum dependence of intensities as well, suchas the changing angle of the escaping electron and theactual underlying Cu-3d, O-2p orbital structure.[18] See, for instance, A. Nazarenko et al., Phys. Rev. B 51,8676 (1995).[19] A.R. Kampf and J.R. Schrieffer, Phys. Rev. B 42, 7967(1990).[20] W. Stephan and P. Horsch, Phys. Rev. Lett. 66, 2258(1991).[21] R. Preuss, W. Hanke, and W. von der Linden, Phys. Rev.Lett. 75, 1344 (1995).[22] R. Eder, Y. Ohta, and T. Shimozato, Phys. Rev. B 50,3350 (1994); R. Eder and Y. Ohta, to appear in Phys.Rev. B.[23] X.G. Wen and P.A. Lee, Phys. Rev. Lett. 76 503 (1996).4

Hubbard model versus t-J model:The one-particle spectrum

University of Groningen Research Database

  
 University of Groningen
Hubbard model versus t-J model
Eskes, H; Eder, R
Published in:
Physical Review B
DOI:
10.1103/PhysRevB.54.R14226
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from
it. Please check the document version below.
Document Version
Publisher's PDF, also known as Version of record
Publication date:
1996
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Eskes, H., & Eder, R. (1996). Hubbard model versus t-J model: The one-particle spectrum. Physical
Review B, 54(20), 14226-14229. DOI: 10.1103/PhysRevB.54.R14226
Copyright
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number of authors shown on this cover page is limited to 10 maximum.
Download date: 10-02-2018
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96
Hubbard model versus t-J model: The one-particle spectrum
Henk Eskes
Institute Lorentz, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands
Robert Eder
Laboratory of Applied and Solid-State Physics, Materials Science Centre,
University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
(August 13, 2006)
The origin of the apparent discrepancies between the one-particle spectra of the Hubbard and t-J
models is revealed: Wavefunction corrections, in addition to the three-site terms, should supplement
the bare t-J model. In this way a quantitative agreement between the two models is obtained,
even for the intermediate-U values appropriate for the high-Tc cuprate superconductors. Numerical
results for clusters of up to 20 sites are presented. The momentum dependence of the observed
intensities in the photoemission spectra of Sr2CuO2Cl2 are well described by this complete strong-
coupling approach.
71.27.+a,79.60.-i,71.10.+x,74.72.-h
The past ten years the interest in Hamiltonians mod-
eling correlated electrons has greatly increased. This is
largely due to the mysterious normal-state properties of
the high-Tc cuprate superconductors and the realization
that local correlations on the Cu sites are large in these
compounds. The two most intensively studied Hamilto-
nians are the Hubbard and t-J models. Although at first
sight such models may seem oversimplified, it has been
argued that the two dimensional square-lattice Hubbard
model captures the essential dynamics of the cuprates
in a window of a few eV around the chemical poten-
tial [1–4]. The materials belong to the intermediate-U
regime, U/t ≈ 10, t ≈ 0.3−0.4 eV [3,4]. Further support
for a single-band Hubbard approach comes from mea-
sured one-particle and optical spectra of the cuprates.
These show a dramatic reshuﬄing of intensity as a func-
tion of doping on the scale of a few eV [5,6]. This char-
acteristic behavior is a fingerprint of strong correlations
and is described well by the Hubbard model [5,7,8].
Although the t-J model is frequently referred to as be-
ing the large-U limit of the single-band Hubbard model,
the (one-particle) spectra of the two models differ sig-
nificantly in the intermediate-U regime of interest. (For
a recent review on numerical results, see Ref. [7]) Prob-
ably because of this, the two models are often treated
as being “distinct”. One obvious difference concerns
the momentum occupation number nk at half filling,
equal to the one-electron removal sum rule. For the t-J
model this quantity is trivially equal to 1/2, while for the
intermediate-U Hubbard model it is a strongly varying
function of k. Even though in the infinite-U limit nk
tends to 1/2, as in the t-J model, the convergence to this
value by increasing U is very slow.
In this Letter we show that these discrepancies can be
attributed to the neglect of certain terms in the large-
U perturbation theory. Specifically, the weight distribu-
tions in the spectral functions show a high sensitivity to
the first-order renormalization of the wave functions. The
complete strong-coupling approach is found to quantita-
tively reproduce the Hubbard spectra for U ≥W , where
W is the U = 0 band width. For these U values the per-
turbation series in t/U agrees well with the full Hubbard
Hamiltonian.
The derivation of the strong-coupling Hamiltonian by
means of a canonical transformation, written as an ex-
pansion in t/U , is well known [8,9]. Starting point is the
Hubbard model,
H = V + T = U
∑
i
ni,↑ni,↓ − t
∑
i,δ,σ
a†i,σai+δ,σ. (1)
Here ni,↑ is the occupation number of the original Hub-
bard fermions ai,σ, and δ is a vector connecting nearest
neighbors. After the transformation the states in the
lowest Hubbard sector (no doubly occupied sites) are de-
scribed to order t2/U by the strong-coupling Hamilto-
nian,
Hsc = −t
∑
i,δ,σ
c†i,σci+δ,σ −
J
2
∑
i,δ
(S˜i · S˜i+δ −
1
4
n˜in˜i+δ)
−
J
4
∑
i,δ 6=δ′,σ
(
c†i+δ,σn˜i,σ¯ci+δ′,σ − c
†
i+δ,σ¯c
†
i,σci,σ¯ci+δ′,σ
)
,
(2)
where J = 4t2/U . The motion of the above transformed
ci,σ fermions will leave the number of doubly occupied
sites unchanged, and S˜i and n˜i,σ are the spin and occu-
pation number of these new fermions. Simply neglecting
the ‘three-site term’, i.e. the second line of (2), one ob-
tains the t-J model.
Why the full strong-coupling model has received far
less attention than the t-J model is unclear. It might be
expected that the three-site term is very important in the
one-hole limit (and low doping region) since, unlike the
1
nearest-neighbor hopping, a direct next-nearest-neighbor
hopping process does not frustrate the spin system. How-
ever, it was shown by a numerical study [10], spin-polaron
approaches [11,12] and a linear-spin-wave self-consistent
Born (LSW-SCB) calculation [13] that the main effect of
this term on the quasiparticle dispersion is an increase of
the bandwidth of the dominant ‘quasiparticle peak’.
By virtue of its construction via a canonical transfor-
mation, the Hamiltonian (2) reproduces the eigenvalue
spectrum of the Hubbard model for large U . To obtain
further information, such as matrix elements or expec-
tation values, operators acting in the Hilbert space of
the Hubbard model have to be subjected to the same
canonical transformation. For the electron annihilation
operator this procedure leads to the expression [8,9],
Pai,σP = Pci,σP
+
t
U
∑
δ
P
(
n˜i,σ¯ci+δ,σ − c
†
i,σ¯ci,σci+δ,σ¯
)
P. (3)
P is defined as the projector on states with no dou-
ble occupancy of the transformed fermions (states with
〈V˜ 〉 = 0). The first-order correction terms (second line)
describe the dressing of the annihilation operator with
‘virtual’ charge fluctuations involving doubly-occupied
sites. While their prefactor t/U seems to suggest that
they represent only a minor correction, their impact on
the calculated spectra is dramatic – one reason being
that the summation over neighbors introduces the co-
ordination number z as an additional ‘hidden’ prefac-
tor. When comparing strong-coupling calculations di-
rectly with experiments it is therefore crucial to take
these corrections into account. For the optical conduc-
tivity such corrections were shown to lead to qualita-
tively different results [14]. Another consequence is a
rapid reshuﬄing of weight between the upper (UHB) and
lower-Hubbard band (LHB) as a function of hole doping,
observed in both one-particle and optical experiments.
The first-order corrections to transformed operators such
as Eq. (3) lead to modified expressions for sum rules that
consistently describe the rapid intensity changes observed
in experiments [5,6,8,9].
For the numerical calculations of the one-particle spec-
tra we implemented the strong-coupling model Eq. (2).
Electron removal is described by the operator Eq. (3).
This operator does not change double-occupancy and
only the LHB part of the Hubbard spectrum is pro-
duced. (There is, in fact, a similar independent proce-
dure to calculate the UHB response as well.) The t/U
term in Eq. (3) amounts to the removal of an electron
from a neighbor of i. This leads to an extra phase factor
exp(ik·δ). Because of this the total weight is enhanced at
Γ and reduced at (pi, pi) and Eq. (3) leads to the correct
behavior of n(k) (see Fig. 2 of Ref. [8]). Note that the
implementation of the first-order correction is straight-
forward and does not increase memory or CPU time in
FIG. 1. The one-particle spectrum of the two-dimensional
4x4 strong-coupling model (solid line) compared with the
Hubbard model (dotted line), at half ﬁlling (left) and for two
holes (right). U/t = 10. Only the LHB part of the Hubbard
spectrum is shown. Dashed line is the chemical potential in
the two-hole case. Hubbard data reproduced from Ref. [15]
a Lanczos-like diagonalization calculation. In a similar
way such operator corrections can be taken into account
in analytic approaches, for instance in the LSW-SCB cal-
culation of the one-hole spectrum.
In Fig. 1 we compare the two-dimensional 16-site 4x4
strong-coupling model with the LHB electron removal
and addition spectra of the Hubbard model, reproduced
from Ref. [15]. Note that U = 10 is comparable to the
bandwidth W = 8. No shift or scaling has been applied
to the spectra, showing that the value of the gap between
the UHB (not shown, starting at energy ≈ 8t) and LHB
is accurately described. (While there is no UHB in the
strong coupling model, the magnitude of the energy gap
still can be inferred from the relation ∆ = U − 2 · Emin,
with Emin ≈ 2t the minimal ionization energy of the
model.) The resulting gap of ≈ 6t is consistent with the
measured insulating gap of ≈ 2eV .
As can be seen, excitation energies and weights of the
dominant features in the spectra are reproduced correctly
by the strong-coupling approach. The agreement is best
for the physically most relevant low excitation energies,
but the strong coupling approach also gives a complete
description of the higher excitations, with energies not
much smaller than U . The total width is slightly overes-
timated by the second-order t2/U terms in the Hamilto-
nian, due especially to those peaks which obtain a large
energy correction by the three-site terms, for instance at
(pi, pi). There is no significant deterioration of the agree-
ment upon doping, demonstrating that the discrepancies
2
between the doping dependences of low energy sum-rules
for t− J and Hubbard originate from matrix element ef-
fects, rather than an incompleteness of the Hilbert space
of the strong coupling model. We explicitly checked that
second-order corrections to the Fermi operators lead to
only minor changes of the peak intensities. For U values
below 10 the agreement between Hubbard and its strong-
coupling limit gradually deteriorates, which we interpret
as a breaking down of the perturbation approach.
Stimulated by the good agreement in the 4x4 system
we now present large-U Hubbard spectra for clusters up
to 20 sites. Because the momenta of the 16 and 18 site
clusters nicely complement each other and are all on the
high-symmetry axis, we show a mix of these two clus-
ters in Figs. 2 and 3. The curves on the left show a scan
from (0, 0) to (pi, pi). The plots on the right go from (0, 0)
to (pi, 0) and from (pi, 0) to (pi, pi). We shifted all 4 × 4
spectra by the same small energy such that the lowest
peak at k = (0, 0) is at the same energy for both clus-
ters. For the two-hole case we shifted the t-J data by an
additional 0.3t to have the chemical potentials at roughly
the same position. The ground-state momentum is (0, 0)
in all cases (there is an extra degeneracy for 16-sites, 2
holes). In both figures the results are compared with the
“bare” t-J model.
As mentioned before there are substantial differ-
ences between t-J and strong coupling. At half-filling
the low energy ‘quasiparticle peaks’ are at higher
energies (implying a larger Hubbard gap) for the
FIG. 2. The momentum dependence of the elec-
tron-removal spectrum of the strong-coupling Hubbard model
(solid line) compared with the t-J model (dotted line). The
plot is a mix of data from the 16 and 18 site clusters.
U/t = 10. n = 1. The spectra for k = (pi/2, pi/2), (pi/2, 0),
(pi, 0) and (pi/2, pi) are calculated using the 2D 16-site cluster;
the others, using the 18-site cluster.
FIG. 3. Combined electron addition and removal spectra
for the two-hole ground state. See caption of Fig. 2.
t-J model and their dispersion has a smaller width.
Moreover, the momentum dependence of the weights is
qualitatively different. The t-J spectrum at (0, 0), and
indeed through much of the Brillouin zone, is more com-
pact than for large-U Hubbard. The strongest difference
is at (pi, pi). Here strong coupling shows a little peak at
high energy, while the t-J model has pronounced struc-
tures at a more intermediate energy. This reflects the
small value of nk for k = (pi, pi) in the Hubbard model,
which is reproduced by the first-order corrections to the
Fermi operator Eq. (3).
The spectra at half filling can be compared with re-
cent angle-resolved photoemission data on insulating
Sr2CuO2Cl2 [16]. A significant feature of these measure-
ments is the strong momentum dependence of the weights
of the spectra. The region in momentum space with the
most intense peaks is not centered around (pi/2, pi/2),
but displaced towards (0, 0). Beyond (pi/2, pi/2) the in-
tensity drops very sharply and seems to vanish around
(pi, pi) [17]. As shown in Fig. 4 these features are well
reproduced by the strong-coupling Hubbard model. In
contrast, the intensities in t-J are much more symmetric
around (pi/2, pi/2) and change only little in that part
of momentum space. On average the peaks in strong-
coupling Hubbard are more intense than those in the
t-J model, at the expense of the incoherent part of the
spectrum. Note that also in the t-J model there are
differences in weight between k and k + (pi, pi). This is
due to the coupling of the created hole to quantum-spin
fluctuations in the ground state of the Heisenberg anti-
ferromagnet [12]. Experimentally the strong changes at
the antiferromagnetic Brillouin-zone boundary look like
3
FIG. 4. Weights (proportional to the radius of the circles)
of the lowest energy peak for the strong-coupling Hubbard
model (left) and t-J model (right). Data collected from the
16, 18 and 20 site clusters. U/t = 10, n = 1.
a kind of ‘pseudo Fermi surface’: The spectral weight
observed drops significantly, but an actual Fermi-level
crossing does not take place. We note that the large
difference observed between (pi, 0) and (pi/2, pi/2) in the
photoemission measurements actually is in disagreement
with the near degeneracy of these momenta in both the
standard Hubbard and t-J model. As discussed by vari-
ous authors this may be explained by adding extra next-
neighbor hopping terms to the Hamiltonian [13,18].
The increased quasiparticle weight and the strong drop
of intensity when passing the half-filled Fermi surface
are consistent with a continuous cross-over to a small-U
spin-density-wave (SDW) like shadow-band scenario [19].
Surely Fig. 4 shows that such a strong drop in weight is
not necessarily a weak-coupling feature.
The interpretation of the doping dependence of the
spectral function is a subtle issue [20–23], and will not be
discussed here. For roughly 10% holes (Fig. 3) there are
intense features appearing in the electron addition spec-
trum for all momenta outside the magnetic zone. These
features are more intense than for t-J and because of
the sum rule the extra weight must come from the up-
per Hubbard band. This weight-transfer effect has been
clearly observed in O-1s X-ray absorption [5].
In conclusion, we have shown that the strong-coupling
approach is a good alternative to Hubbard calculations
even in the intermediate U regime relevant for the high-
Tc superconductors. Only the complete strong-coupling
model gives a faithful representation of the properties
of the Hubbard model. The corrections to t-J lead to
a smooth crossover to small U . For comparisons with
experiments such as angle-resolved photoemission, O-1s
X-ray absorption (and optical conductivity) it is impor-
tant to use the renormalized wavefunctions (rotated op-
erators), as these can lead to even qualitatively different
results.
We acknowledge stimulating discussions with George
A. Sawatzky, Jan Zaanen and Andrzej M. Oles´. HE is
supported by the Stichting voor Fundamenteel Onder-
zoek der Materie (FOM), which is financially supported
by the Nederlandse Organisatie voor Wetenschappelijk
Onderzoek (NWO). RE is supported by a postdoctoral
fellowship granted by the Commission of the European
Communities.
[1] P.W. Anderson, Science 235, 1196 (1987).
[2] F.C. Zhang and T.M. Rice, Phys. Rev. B 37, 3757 (1988).
[3] M.S. Hybertsen et al., Phys. Rev. B 41, 11068 (1990).
[4] L.F. Feiner, J.H. Jeﬀerson, and R. Raimondi, Phys. Rev.
B 53, 8751 (1996), and references therein.
[5] C.T. Chen et al., Phys. Rev. Lett. 66, 104 (1991).
[6] S. Uchida et al., Phys. Rev. B 43, 7942 (1991);
[7] E. Dagotto, Rev. Mod. Phys. 66, 763 (1994), and refer-
ences therein.
[8] H. Eskes and A. M. Oles´, Phys. Rev. Lett. 73, 1279
(1994); H. Eskes et al., Phys. Rev. B 50, 17980 (1994).
[9] A.B. Harris and R.V. Lange, Phys. Rev. 157, 295 (1967).
[10] K.J. von Szczepanski, et al., Phys. Rev. B 41, 2017
(1990).
[11] S.A. Trugman, Phys. Rev. B 37, 1597 (1988); ibid. Phys.
Rev. B 41, 892 (1990).
[12] R. Eder and K.W. Becker, Z. Phys. B 78, 219 (1990);
ibid. Phys. Rev. B 44, 6982 (1991).
[13] J. Ba la, A.M. Oles´ and J. Zaanen, Phys. Rev. B 52, 4597
(1995).
[14] W. Stephan and P. Horsch, Int. J. Mod. Phys. B6, 141
(1992). P. Horsch and W. Stephan, Phys. Rev. B 48,
10595 (1993).
[15] P.W. Leung et al., Phys. Rev.B 46, 11779 (1992).
[16] B.O. Wells et al., Phys. Rev. Lett. 74, 964 (1995).
[17] It is important to realize that other eﬀects can inﬂuence
the momentum dependence of intensities as well, such
as the changing angle of the escaping electron and the
actual underlying Cu-3d, O-2p orbital structure.
[18] See, for instance, A. Nazarenko et al., Phys. Rev. B 51,
8676 (1995).
[19] A.R. Kampf and J.R. Schrieﬀer, Phys. Rev. B 42, 7967
(1990).
[20] W. Stephan and P. Horsch, Phys. Rev. Lett. 66, 2258
(1991).
[21] R. Preuss, W. Hanke, and W. von der Linden, Phys. Rev.
Lett. 75, 1344 (1995).
[22] R. Eder, Y. Ohta, and T. Shimozato, Phys. Rev. B 50,
3350 (1994); R. Eder and Y. Ohta, to appear in Phys.
Rev. B.
[23] X.G. Wen and P.A. Lee, Phys. Rev. Lett. 76 503 (1996).
4


The origin of the apparent discrepancies between the one-particle spectra of the Hubbard and t-J models is revealed: Wave-function corrections, in addition to the three-site terms, should supplement the bare t-J model. In this way a quantitative agreement between the two models is obtained, even for the intermediate-ii values appropriate for the high-T-c cuprate superconductors. Numerical results for clusters of up to 20 sites are presented. The momentum dependence of the observed intensities in the photoemission spectra of Sr2CuO2Cl2 are well described by this complete strong-coupling approach

NARCIS 

Hubbard model versus t-J model: The one-particle spectrum

   University of GroningenHubbard model versus t-J modelEskes, H; Eder, RPublished in:Physical Review BDOI:10.1103/PhysRevB.54.R14226IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1996Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Eskes, H., & Eder, R. (1996). Hubbard model versus t-J model: The one-particle spectrum. PhysicalReview B, 54(20), 14226-14229. https://doi.org/10.1103/PhysRevB.54.R14226CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019arXiv:cond-mat/9609233 v1   24 Sep 1996Hubbard model versus t-J model: The one-particle spectrumHenk EskesInstitute Lorentz, Leiden University, P.O. Box 9506, 2300 RA Leiden, The NetherlandsRobert EderLaboratory of Applied and Solid-State Physics, Materials Science Centre,University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands(August 13, 2006)The origin of the apparent discrepancies between the one-particle spectra of the Hubbard and t-Jmodels is revealed: Wavefunction corrections, in addition to the three-site terms, should supplementthe bare t-J model. In this way a quantitative agreement between the two models is obtained,even for the intermediate-U values appropriate for the high-Tc cuprate superconductors. Numericalresults for clusters of up to 20 sites are presented. The momentum dependence of the observedintensities in the photoemission spectra of Sr2CuO2Cl2 are well described by this complete strong-coupling approach.71.27.+a,79.60.-i,71.10.+x,74.72.-hThe past ten years the interest in Hamiltonians mod-eling correlated electrons has greatly increased. This islargely due to the mysterious normal-state properties ofthe high-Tc cuprate superconductors and the realizationthat local correlations on the Cu sites are large in thesecompounds. The two most intensively studied Hamilto-nians are the Hubbard and t-J models. Although at firstsight such models may seem oversimplified, it has beenargued that the two dimensional square-lattice Hubbardmodel captures the essential dynamics of the cupratesin a window of a few eV around the chemical poten-tial [1–4]. The materials belong to the intermediate-Uregime, U/t ≈ 10, t ≈ 0.3−0.4 eV [3,4]. Further supportfor a single-band Hubbard approach comes from mea-sured one-particle and optical spectra of the cuprates.These show a dramatic reshuﬄing of intensity as a func-tion of doping on the scale of a few eV [5,6]. This char-acteristic behavior is a fingerprint of strong correlationsand is described well by the Hubbard model [5,7,8].Although the t-J model is frequently referred to as be-ing the large-U limit of the single-band Hubbard model,the (one-particle) spectra of the two models differ sig-nificantly in the intermediate-U regime of interest. (Fora recent review on numerical results, see Ref. [7]) Prob-ably because of this, the two models are often treatedas being “distinct”. One obvious difference concernsthe momentum occupation number nk at half filling,equal to the one-electron removal sum rule. For the t-Jmodel this quantity is trivially equal to 1/2, while for theintermediate-U Hubbard model it is a strongly varyingfunction of k. Even though in the infinite-U limit nktends to 1/2, as in the t-J model, the convergence to thisvalue by increasing U is very slow.In this Letter we show that these discrepancies can beattributed to the neglect of certain terms in the large-U perturbation theory. Specifically, the weight distribu-tions in the spectral functions show a high sensitivity tothe first-order renormalization of the wave functions. Thecomplete strong-coupling approach is found to quantita-tively reproduce the Hubbard spectra for U ≥W , whereW is the U = 0 band width. For these U values the per-turbation series in t/U agrees well with the full HubbardHamiltonian.The derivation of the strong-coupling Hamiltonian bymeans of a canonical transformation, written as an ex-pansion in t/U , is well known [8,9]. Starting point is theHubbard model,H = V + T = U∑ini,↑ni,↓ − t∑i,δ,σa†i,σai+δ,σ. (1)Here ni,↑ is the occupation number of the original Hub-bard fermions ai,σ, and δ is a vector connecting nearestneighbors. After the transformation the states in thelowest Hubbard sector (no doubly occupied sites) are de-scribed to order t2/U by the strong-coupling Hamilto-nian,Hsc = −t∑i,δ,σc†i,σci+δ,σ −J2∑i,δ(S˜i · S˜i+δ −14n˜in˜i+δ)−J4∑i,δ 6=δ′,σ(c†i+δ,σn˜i,σ¯ci+δ′,σ − c†i+δ,σ¯c†i,σci,σ¯ci+δ′,σ),(2)where J = 4t2/U . The motion of the above transformedci,σ fermions will leave the number of doubly occupiedsites unchanged, and S˜i and n˜i,σ are the spin and occu-pation number of these new fermions. Simply neglectingthe ‘three-site term’, i.e. the second line of (2), one ob-tains the t-J model.Why the full strong-coupling model has received farless attention than the t-J model is unclear. It might beexpected that the three-site term is very important in theone-hole limit (and low doping region) since, unlike the1nearest-neighbor hopping, a direct next-nearest-neighborhopping process does not frustrate the spin system. How-ever, it was shown by a numerical study [10], spin-polaronapproaches [11,12] and a linear-spin-wave self-consistentBorn (LSW-SCB) calculation [13] that the main effect ofthis term on the quasiparticle dispersion is an increase ofthe bandwidth of the dominant ‘quasiparticle peak’.By virtue of its construction via a canonical transfor-mation, the Hamiltonian (2) reproduces the eigenvaluespectrum of the Hubbard model for large U . To obtainfurther information, such as matrix elements or expec-tation values, operators acting in the Hilbert space ofthe Hubbard model have to be subjected to the samecanonical transformation. For the electron annihilationoperator this procedure leads to the expression [8,9],Pai,σP = Pci,σP+tU∑δP(n˜i,σ¯ci+δ,σ − c†i,σ¯ci,σci+δ,σ¯)P. (3)P is defined as the projector on states with no dou-ble occupancy of the transformed fermions (states with〈V˜ 〉 = 0). The first-order correction terms (second line)describe the dressing of the annihilation operator with‘virtual’ charge fluctuations involving doubly-occupiedsites. While their prefactor t/U seems to suggest thatthey represent only a minor correction, their impact onthe calculated spectra is dramatic – one reason beingthat the summation over neighbors introduces the co-ordination number z as an additional ‘hidden’ prefac-tor. When comparing strong-coupling calculations di-rectly with experiments it is therefore crucial to takethese corrections into account. For the optical conduc-tivity such corrections were shown to lead to qualita-tively different results [14]. Another consequence is arapid reshuﬄing of weight between the upper (UHB) andlower-Hubbard band (LHB) as a function of hole doping,observed in both one-particle and optical experiments.The first-order corrections to transformed operators suchas Eq. (3) lead to modified expressions for sum rules thatconsistently describe the rapid intensity changes observedin experiments [5,6,8,9].For the numerical calculations of the one-particle spec-tra we implemented the strong-coupling model Eq. (2).Electron removal is described by the operator Eq. (3).This operator does not change double-occupancy andonly the LHB part of the Hubbard spectrum is pro-duced. (There is, in fact, a similar independent proce-dure to calculate the UHB response as well.) The t/Uterm in Eq. (3) amounts to the removal of an electronfrom a neighbor of i. This leads to an extra phase factorexp(ik·δ). Because of this the total weight is enhanced atΓ and reduced at (pi, pi) and Eq. (3) leads to the correctbehavior of n(k) (see Fig. 2 of Ref. [8]). Note that theimplementation of the first-order correction is straight-forward and does not increase memory or CPU time inFIG. 1. The one-particle spectrum of the two-dimensional4x4 strong-coupling model (solid line) compared with theHubbard model (dotted line), at half ﬁlling (left) and for twoholes (right). U/t = 10. Only the LHB part of the Hubbardspectrum is shown. Dashed line is the chemical potential inthe two-hole case. Hubbard data reproduced from Ref. [15]a Lanczos-like diagonalization calculation. In a similarway such operator corrections can be taken into accountin analytic approaches, for instance in the LSW-SCB cal-culation of the one-hole spectrum.In Fig. 1 we compare the two-dimensional 16-site 4x4strong-coupling model with the LHB electron removaland addition spectra of the Hubbard model, reproducedfrom Ref. [15]. Note that U = 10 is comparable to thebandwidth W = 8. No shift or scaling has been appliedto the spectra, showing that the value of the gap betweenthe UHB (not shown, starting at energy ≈ 8t) and LHBis accurately described. (While there is no UHB in thestrong coupling model, the magnitude of the energy gapstill can be inferred from the relation ∆ = U − 2 · Emin,with Emin ≈ 2t the minimal ionization energy of themodel.) The resulting gap of ≈ 6t is consistent with themeasured insulating gap of ≈ 2eV .As can be seen, excitation energies and weights of thedominant features in the spectra are reproduced correctlyby the strong-coupling approach. The agreement is bestfor the physically most relevant low excitation energies,but the strong coupling approach also gives a completedescription of the higher excitations, with energies notmuch smaller than U . The total width is slightly overes-timated by the second-order t2/U terms in the Hamilto-nian, due especially to those peaks which obtain a largeenergy correction by the three-site terms, for instance at(pi, pi). There is no significant deterioration of the agree-ment upon doping, demonstrating that the discrepancies2between the doping dependences of low energy sum-rulesfor t− J and Hubbard originate from matrix element ef-fects, rather than an incompleteness of the Hilbert spaceof the strong coupling model. We explicitly checked thatsecond-order corrections to the Fermi operators lead toonly minor changes of the peak intensities. For U valuesbelow 10 the agreement between Hubbard and its strong-coupling limit gradually deteriorates, which we interpretas a breaking down of the perturbation approach.Stimulated by the good agreement in the 4x4 systemwe now present large-U Hubbard spectra for clusters upto 20 sites. Because the momenta of the 16 and 18 siteclusters nicely complement each other and are all on thehigh-symmetry axis, we show a mix of these two clus-ters in Figs. 2 and 3. The curves on the left show a scanfrom (0, 0) to (pi, pi). The plots on the right go from (0, 0)to (pi, 0) and from (pi, 0) to (pi, pi). We shifted all 4 × 4spectra by the same small energy such that the lowestpeak at k = (0, 0) is at the same energy for both clus-ters. For the two-hole case we shifted the t-J data by anadditional 0.3t to have the chemical potentials at roughlythe same position. The ground-state momentum is (0, 0)in all cases (there is an extra degeneracy for 16-sites, 2holes). In both figures the results are compared with the“bare” t-J model.As mentioned before there are substantial differ-ences between t-J and strong coupling. At half-fillingthe low energy ‘quasiparticle peaks’ are at higherenergies (implying a larger Hubbard gap) for theFIG. 2. The momentum dependence of the elec-tron-removal spectrum of the strong-coupling Hubbard model(solid line) compared with the t-J model (dotted line). Theplot is a mix of data from the 16 and 18 site clusters.U/t = 10. n = 1. The spectra for k = (pi/2, pi/2), (pi/2, 0),(pi, 0) and (pi/2, pi) are calculated using the 2D 16-site cluster;the others, using the 18-site cluster.FIG. 3. Combined electron addition and removal spectrafor the two-hole ground state. See caption of Fig. 2.t-J model and their dispersion has a smaller width.Moreover, the momentum dependence of the weights isqualitatively different. The t-J spectrum at (0, 0), andindeed through much of the Brillouin zone, is more com-pact than for large-U Hubbard. The strongest differenceis at (pi, pi). Here strong coupling shows a little peak athigh energy, while the t-J model has pronounced struc-tures at a more intermediate energy. This reflects thesmall value of nk for k = (pi, pi) in the Hubbard model,which is reproduced by the first-order corrections to theFermi operator Eq. (3).The spectra at half filling can be compared with re-cent angle-resolved photoemission data on insulatingSr2CuO2Cl2 [16]. A significant feature of these measure-ments is the strong momentum dependence of the weightsof the spectra. The region in momentum space with themost intense peaks is not centered around (pi/2, pi/2),but displaced towards (0, 0). Beyond (pi/2, pi/2) the in-tensity drops very sharply and seems to vanish around(pi, pi) [17]. As shown in Fig. 4 these features are wellreproduced by the strong-coupling Hubbard model. Incontrast, the intensities in t-J are much more symmetricaround (pi/2, pi/2) and change only little in that partof momentum space. On average the peaks in strong-coupling Hubbard are more intense than those in thet-J model, at the expense of the incoherent part of thespectrum. Note that also in the t-J model there aredifferences in weight between k and k + (pi, pi). This isdue to the coupling of the created hole to quantum-spinfluctuations in the ground state of the Heisenberg anti-ferromagnet [12]. Experimentally the strong changes atthe antiferromagnetic Brillouin-zone boundary look like3FIG. 4. Weights (proportional to the radius of the circles)of the lowest energy peak for the strong-coupling Hubbardmodel (left) and t-J model (right). Data collected from the16, 18 and 20 site clusters. U/t = 10, n = 1.a kind of ‘pseudo Fermi surface’: The spectral weightobserved drops significantly, but an actual Fermi-levelcrossing does not take place. We note that the largedifference observed between (pi, 0) and (pi/2, pi/2) in thephotoemission measurements actually is in disagreementwith the near degeneracy of these momenta in both thestandard Hubbard and t-J model. As discussed by vari-ous authors this may be explained by adding extra next-neighbor hopping terms to the Hamiltonian [13,18].The increased quasiparticle weight and the strong dropof intensity when passing the half-filled Fermi surfaceare consistent with a continuous cross-over to a small-Uspin-density-wave (SDW) like shadow-band scenario [19].Surely Fig. 4 shows that such a strong drop in weight isnot necessarily a weak-coupling feature.The interpretation of the doping dependence of thespectral function is a subtle issue [20–23], and will not bediscussed here. For roughly 10% holes (Fig. 3) there areintense features appearing in the electron addition spec-trum for all momenta outside the magnetic zone. Thesefeatures are more intense than for t-J and because ofthe sum rule the extra weight must come from the up-per Hubbard band. This weight-transfer effect has beenclearly observed in O-1s X-ray absorption [5].In conclusion, we have shown that the strong-couplingapproach is a good alternative to Hubbard calculationseven in the intermediate U regime relevant for the high-Tc superconductors. Only the complete strong-couplingmodel gives a faithful representation of the propertiesof the Hubbard model. The corrections to t-J lead toa smooth crossover to small U . For comparisons withexperiments such as angle-resolved photoemission, O-1sX-ray absorption (and optical conductivity) it is impor-tant to use the renormalized wavefunctions (rotated op-erators), as these can lead to even qualitatively differentresults.We acknowledge stimulating discussions with GeorgeA. Sawatzky, Jan Zaanen and Andrzej M. Oles´. HE issupported by the Stichting voor Fundamenteel Onder-zoek der Materie (FOM), which is financially supportedby the Nederlandse Organisatie voor WetenschappelijkOnderzoek (NWO). RE is supported by a postdoctoralfellowship granted by the Commission of the EuropeanCommunities.[1] P.W. Anderson, Science 235, 1196 (1987).[2] F.C. Zhang and T.M. Rice, Phys. Rev. B 37, 3757 (1988).[3] M.S. Hybertsen et al., Phys. Rev. B 41, 11068 (1990).[4] L.F. Feiner, J.H. Jeﬀerson, and R. Raimondi, Phys. Rev.B 53, 8751 (1996), and references therein.[5] C.T. Chen et al., Phys. Rev. Lett. 66, 104 (1991).[6] S. Uchida et al., Phys. Rev. B 43, 7942 (1991);[7] E. Dagotto, Rev. Mod. Phys. 66, 763 (1994), and refer-ences therein.[8] H. Eskes and A. M. Oles´, Phys. Rev. Lett. 73, 1279(1994); H. Eskes et al., Phys. Rev. B 50, 17980 (1994).[9] A.B. Harris and R.V. Lange, Phys. Rev. 157, 295 (1967).[10] K.J. von Szczepanski, et al., Phys. Rev. B 41, 2017(1990).[11] S.A. Trugman, Phys. Rev. B 37, 1597 (1988); ibid. Phys.Rev. B 41, 892 (1990).[12] R. Eder and K.W. Becker, Z. Phys. B 78, 219 (1990);ibid. Phys. Rev. B 44, 6982 (1991).[13] J. Ba la, A.M. Oles´ and J. Zaanen, Phys. Rev. B 52, 4597(1995).[14] W. Stephan and P. Horsch, Int. J. Mod. Phys. B6, 141(1992). P. Horsch and W. Stephan, Phys. Rev. B 48,10595 (1993).[15] P.W. Leung et al., Phys. Rev.B 46, 11779 (1992).[16] B.O. Wells et al., Phys. Rev. Lett. 74, 964 (1995).[17] It is important to realize that other eﬀects can inﬂuencethe momentum dependence of intensities as well, suchas the changing angle of the escaping electron and theactual underlying Cu-3d, O-2p orbital structure.[18] See, for instance, A. Nazarenko et al., Phys. Rev. B 51,8676 (1995).[19] A.R. Kampf and J.R. Schrieﬀer, Phys. Rev. B 42, 7967(1990).[20] W. Stephan and P. Horsch, Phys. Rev. Lett. 66, 2258(1991).[21] R. Preuss, W. Hanke, and W. von der Linden, Phys. Rev.Lett. 75, 1344 (1995).[22] R. Eder, Y. Ohta, and T. Shimozato, Phys. Rev. B 50,3350 (1994); R. Eder and Y. Ohta, to appear in Phys.Rev. B.[23] X.G. Wen and P.A. Lee, Phys. Rev. Lett. 76 503 (1996).4

Submitted to: Phys. Rev., B

Abstract: The origin of the apparent discrepancies between the one-particle spectra of the Hubbard and t-J models is revealed: Wavefunction corrections, in addition to the three-site terms, should supplement the bare t-J. In this way a quantitative agreement between the two models is obtained, even for the intermediate-$U$ values appropriate for the high-Tc cuprate superconductors. Numerical results for clusters of up to 20 sites are presented. The momentum dependence of the observed intensities in the photoemission spectra of Sr2CuO2Cl2 are well described by this complete strong-coupling approach.

Eskes, H.,

Eder, R.,

University of Groningen Digital Archive

Hubbard model versus t-J model : The one-particle spectrum

The origin of the apparent discrepancies between the one-particle spectra of the Hubbard and t-J models is revealed: Wave-function corrections, in addition to the three-site terms, should supplement the bare t-J model. In this way a quantitative agreement between the two models is obtained, even for the intermediate-ii values appropriate for the high-T-c cuprate superconductors. Numerical results for clusters of up to 20 sites are presented. The momentum dependence of the observed intensities in the photoemission spectra of Sr2CuO2Cl2 are well described by this complete strong-coupling approach.</p

ARTS repository - University of Groningen

   University of GroningenHubbard model versus t-J modelEskes, H; Eder, RPublished in:Physical Review BDOI:10.1103/PhysRevB.54.R14226IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1996Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Eskes, H., & Eder, R. (1996). Hubbard model versus t-J model: The one-particle spectrum. PhysicalReview B, 54(20), 14226-14229. https://doi.org/10.1103/PhysRevB.54.R14226CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 06-06-2022arXiv:cond-mat/9609233 v1   24 Sep 1996Hubbard model versus t-J model: The one-particle spectrumHenk EskesInstitute Lorentz, Leiden University, P.O. Box 9506, 2300 RA Leiden, The NetherlandsRobert EderLaboratory of Applied and Solid-State Physics, Materials Science Centre,University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands(August 13, 2006)The origin of the apparent discrepancies between the one-particle spectra of the Hubbard and t-Jmodels is revealed: Wavefunction corrections, in addition to the three-site terms, should supplementthe bare t-J model. In this way a quantitative agreement between the two models is obtained,even for the intermediate-U values appropriate for the high-Tc cuprate superconductors. Numericalresults for clusters of up to 20 sites are presented. The momentum dependence of the observedintensities in the photoemission spectra of Sr2CuO2Cl2 are well described by this complete strong-coupling approach.71.27.+a,79.60.-i,71.10.+x,74.72.-hThe past ten years the interest in Hamiltonians mod-eling correlated electrons has greatly increased. This islargely due to the mysterious normal-state properties ofthe high-Tc cuprate superconductors and the realizationthat local correlations on the Cu sites are large in thesecompounds. The two most intensively studied Hamilto-nians are the Hubbard and t-J models. Although at firstsight such models may seem oversimplified, it has beenargued that the two dimensional square-lattice Hubbardmodel captures the essential dynamics of the cupratesin a window of a few eV around the chemical poten-tial [1–4]. The materials belong to the intermediate-Uregime, U/t ≈ 10, t ≈ 0.3−0.4 eV [3,4]. Further supportfor a single-band Hubbard approach comes from mea-sured one-particle and optical spectra of the cuprates.These show a dramatic reshuffling of intensity as a func-tion of doping on the scale of a few eV [5,6]. This char-acteristic behavior is a fingerprint of strong correlationsand is described well by the Hubbard model [5,7,8].Although the t-J model is frequently referred to as be-ing the large-U limit of the single-band Hubbard model,the (one-particle) spectra of the two models differ sig-nificantly in the intermediate-U regime of interest. (Fora recent review on numerical results, see Ref. [7]) Prob-ably because of this, the two models are often treatedas being “distinct”. One obvious difference concernsthe momentum occupation number nk at half filling,equal to the one-electron removal sum rule. For the t-Jmodel this quantity is trivially equal to 1/2, while for theintermediate-U Hubbard model it is a strongly varyingfunction of k. Even though in the infinite-U limit nktends to 1/2, as in the t-J model, the convergence to thisvalue by increasing U is very slow.In this Letter we show that these discrepancies can beattributed to the neglect of certain terms in the large-U perturbation theory. Specifically, the weight distribu-tions in the spectral functions show a high sensitivity tothe first-order renormalization of the wave functions. Thecomplete strong-coupling approach is found to quantita-tively reproduce the Hubbard spectra for U ≥ W , whereW is the U = 0 band width. For these U values the per-turbation series in t/U agrees well with the full HubbardHamiltonian.The derivation of the strong-coupling Hamiltonian bymeans of a canonical transformation, written as an ex-pansion in t/U , is well known [8,9]. Starting point is theHubbard model,H = V + T = U∑ini,↑ni,↓ − t∑i,δ,σa†i,σai+δ,σ. (1)Here ni,↑ is the occupation number of the original Hub-bard fermions ai,σ, and δ is a vector connecting nearestneighbors. After the transformation the states in thelowest Hubbard sector (no doubly occupied sites) are de-scribed to order t2/U by the strong-coupling Hamilto-nian,Hsc = −t∑i,δ,σc†i,σci+δ,σ −J2∑i,δ(S̃i · S̃i+δ −14ñiñi+δ)−J4∑i,δ 6=δ′,σ(c†i+δ,σñi,σ̄ci+δ′,σ − c†i+δ,σ̄c†i,σci,σ̄ci+δ′,σ),(2)where J = 4t2/U . The motion of the above transformedci,σ fermions will leave the number of doubly occupiedsites unchanged, and S̃i and ñi,σ are the spin and occu-pation number of these new fermions. Simply neglectingthe ‘three-site term’, i.e. the second line of (2), one ob-tains the t-J model.Why the full strong-coupling model has received farless attention than the t-J model is unclear. It might beexpected that the three-site term is very important in theone-hole limit (and low doping region) since, unlike the1nearest-neighbor hopping, a direct next-nearest-neighborhopping process does not frustrate the spin system. How-ever, it was shown by a numerical study [10], spin-polaronapproaches [11,12] and a linear-spin-wave self-consistentBorn (LSW-SCB) calculation [13] that the main effect ofthis term on the quasiparticle dispersion is an increase ofthe bandwidth of the dominant ‘quasiparticle peak’.By virtue of its construction via a canonical transfor-mation, the Hamiltonian (2) reproduces the eigenvaluespectrum of the Hubbard model for large U . To obtainfurther information, such as matrix elements or expec-tation values, operators acting in the Hilbert space ofthe Hubbard model have to be subjected to the samecanonical transformation. For the electron annihilationoperator this procedure leads to the expression [8,9],Pai,σP = Pci,σP+tU∑δP(ñi,σ̄ci+δ,σ − c†i,σ̄ci,σci+δ,σ̄)P. (3)P is defined as the projector on states with no dou-ble occupancy of the transformed fermions (states with〈Ṽ 〉 = 0). The first-order correction terms (second line)describe the dressing of the annihilation operator with‘virtual’ charge fluctuations involving doubly-occupiedsites. While their prefactor t/U seems to suggest thatthey represent only a minor correction, their impact onthe calculated spectra is dramatic – one reason beingthat the summation over neighbors introduces the co-ordination number z as an additional ‘hidden’ prefac-tor. When comparing strong-coupling calculations di-rectly with experiments it is therefore crucial to takethese corrections into account. For the optical conduc-tivity such corrections were shown to lead to qualita-tively different results [14]. Another consequence is arapid reshuffling of weight between the upper (UHB) andlower-Hubbard band (LHB) as a function of hole doping,observed in both one-particle and optical experiments.The first-order corrections to transformed operators suchas Eq. (3) lead to modified expressions for sum rules thatconsistently describe the rapid intensity changes observedin experiments [5,6,8,9].For the numerical calculations of the one-particle spec-tra we implemented the strong-coupling model Eq. (2).Electron removal is described by the operator Eq. (3).This operator does not change double-occupancy andonly the LHB part of the Hubbard spectrum is pro-duced. (There is, in fact, a similar independent proce-dure to calculate the UHB response as well.) The t/Uterm in Eq. (3) amounts to the removal of an electronfrom a neighbor of i. This leads to an extra phase factorexp(ik·δ). Because of this the total weight is enhanced atΓ and reduced at (π, π) and Eq. (3) leads to the correctbehavior of n(k) (see Fig. 2 of Ref. [8]). Note that theimplementation of the first-order correction is straight-forward and does not increase memory or CPU time inFIG. 1. The one-particle spectrum of the two-dimensional4x4 strong-coupling model (solid line) compared with theHubbard model (dotted line), at half filling (left) and for twoholes (right). U/t = 10. Only the LHB part of the Hubbardspectrum is shown. Dashed line is the chemical potential inthe two-hole case. Hubbard data reproduced from Ref. [15]a Lanczos-like diagonalization calculation. In a similarway such operator corrections can be taken into accountin analytic approaches, for instance in the LSW-SCB cal-culation of the one-hole spectrum.In Fig. 1 we compare the two-dimensional 16-site 4x4strong-coupling model with the LHB electron removaland addition spectra of the Hubbard model, reproducedfrom Ref. [15]. Note that U = 10 is comparable to thebandwidth W = 8. No shift or scaling has been appliedto the spectra, showing that the value of the gap betweenthe UHB (not shown, starting at energy ≈ 8t) and LHBis accurately described. (While there is no UHB in thestrong coupling model, the magnitude of the energy gapstill can be inferred from the relation ∆ = U − 2 · Emin,with Emin ≈ 2t the minimal ionization energy of themodel.) The resulting gap of ≈ 6t is consistent with themeasured insulating gap of ≈ 2eV .As can be seen, excitation energies and weights of thedominant features in the spectra are reproduced correctlyby the strong-coupling approach. The agreement is bestfor the physically most relevant low excitation energies,but the strong coupling approach also gives a completedescription of the higher excitations, with energies notmuch smaller than U . The total width is slightly overes-timated by the second-order t2/U terms in the Hamilto-nian, due especially to those peaks which obtain a largeenergy correction by the three-site terms, for instance at(π, π). There is no significant deterioration of the agree-ment upon doping, demonstrating that the discrepancies2between the doping dependences of low energy sum-rulesfor t− J and Hubbard originate from matrix element ef-fects, rather than an incompleteness of the Hilbert spaceof the strong coupling model. We explicitly checked thatsecond-order corrections to the Fermi operators lead toonly minor changes of the peak intensities. For U valuesbelow 10 the agreement between Hubbard and its strong-coupling limit gradually deteriorates, which we interpretas a breaking down of the perturbation approach.Stimulated by the good agreement in the 4x4 systemwe now present large-U Hubbard spectra for clusters upto 20 sites. Because the momenta of the 16 and 18 siteclusters nicely complement each other and are all on thehigh-symmetry axis, we show a mix of these two clus-ters in Figs. 2 and 3. The curves on the left show a scanfrom (0, 0) to (π, π). The plots on the right go from (0, 0)to (π, 0) and from (π, 0) to (π, π). We shifted all 4 × 4spectra by the same small energy such that the lowestpeak at k = (0, 0) is at the same energy for both clus-ters. For the two-hole case we shifted the t-J data by anadditional 0.3t to have the chemical potentials at roughlythe same position. The ground-state momentum is (0, 0)in all cases (there is an extra degeneracy for 16-sites, 2holes). In both figures the results are compared with the“bare” t-J model.As mentioned before there are substantial differ-ences between t-J and strong coupling. At half-fillingthe low energy ‘quasiparticle peaks’ are at higherenergies (implying a larger Hubbard gap) for theFIG. 2. The momentum dependence of the elec-tron-removal spectrum of the strong-coupling Hubbard model(solid line) compared with the t-J model (dotted line). Theplot is a mix of data from the 16 and 18 site clusters.U/t = 10. n = 1. The spectra for k = (π/2, π/2), (π/2, 0),(π, 0) and (π/2, π) are calculated using the 2D 16-site cluster;the others, using the 18-site cluster.FIG. 3. Combined electron addition and removal spectrafor the two-hole ground state. See caption of Fig. 2.t-J model and their dispersion has a smaller width.Moreover, the momentum dependence of the weights isqualitatively different. The t-J spectrum at (0, 0), andindeed through much of the Brillouin zone, is more com-pact than for large-U Hubbard. The strongest differenceis at (π, π). Here strong coupling shows a little peak athigh energy, while the t-J model has pronounced struc-tures at a more intermediate energy. This reflects thesmall value of nk for k = (π, π) in the Hubbard model,which is reproduced by the first-order corrections to theFermi operator Eq. (3).The spectra at half filling can be compared with re-cent angle-resolved photoemission data on insulatingSr2CuO2Cl2 [16]. A significant feature of these measure-ments is the strong momentum dependence of the weightsof the spectra. The region in momentum space with themost intense peaks is not centered around (π/2, π/2),but displaced towards (0, 0). Beyond (π/2, π/2) the in-tensity drops very sharply and seems to vanish around(π, π) [17]. As shown in Fig. 4 these features are wellreproduced by the strong-coupling Hubbard model. Incontrast, the intensities in t-J are much more symmetricaround (π/2, π/2) and change only little in that partof momentum space. On average the peaks in strong-coupling Hubbard are more intense than those in thet-J model, at the expense of the incoherent part of thespectrum. Note that also in the t-J model there aredifferences in weight between k and k + (π, π). This isdue to the coupling of the created hole to quantum-spinfluctuations in the ground state of the Heisenberg anti-ferromagnet [12]. Experimentally the strong changes atthe antiferromagnetic Brillouin-zone boundary look like3FIG. 4. Weights (proportional to the radius of the circles)of the lowest energy peak for the strong-coupling Hubbardmodel (left) and t-J model (right). Data collected from the16, 18 and 20 site clusters. U/t = 10, n = 1.a kind of ‘pseudo Fermi surface’: The spectral weightobserved drops significantly, but an actual Fermi-levelcrossing does not take place. We note that the largedifference observed between (π, 0) and (π/2, π/2) in thephotoemission measurements actually is in disagreementwith the near degeneracy of these momenta in both thestandard Hubbard and t-J model. As discussed by vari-ous authors this may be explained by adding extra next-neighbor hopping terms to the Hamiltonian [13,18].The increased quasiparticle weight and the strong dropof intensity when passing the half-filled Fermi surfaceare consistent with a continuous cross-over to a small-Uspin-density-wave (SDW) like shadow-band scenario [19].Surely Fig. 4 shows that such a strong drop in weight isnot necessarily a weak-coupling feature.The interpretation of the doping dependence of thespectral function is a subtle issue [20–23], and will not bediscussed here. For roughly 10% holes (Fig. 3) there areintense features appearing in the electron addition spec-trum for all momenta outside the magnetic zone. Thesefeatures are more intense than for t-J and because ofthe sum rule the extra weight must come from the up-per Hubbard band. This weight-transfer effect has beenclearly observed in O-1s X-ray absorption [5].In conclusion, we have shown that the strong-couplingapproach is a good alternative to Hubbard calculationseven in the intermediate U regime relevant for the high-Tc superconductors. Only the complete strong-couplingmodel gives a faithful representation of the propertiesof the Hubbard model. The corrections to t-J lead toa smooth crossover to small U . For comparisons withexperiments such as angle-resolved photoemission, O-1sX-ray absorption (and optical conductivity) it is impor-tant to use the renormalized wavefunctions (rotated op-erators), as these can lead to even qualitatively differentresults.We acknowledge stimulating discussions with GeorgeA. Sawatzky, Jan Zaanen and Andrzej M. Oleś. HE issupported by the Stichting voor Fundamenteel Onder-zoek der Materie (FOM), which is financially supportedby the Nederlandse Organisatie voor WetenschappelijkOnderzoek (NWO). RE is supported by a postdoctoralfellowship granted by the Commission of the EuropeanCommunities.[1] P.W. Anderson, Science 235, 1196 (1987).[2] F.C. Zhang and T.M. Rice, Phys. Rev. B 37, 3757 (1988).[3] M.S. Hybertsen et al., Phys. Rev. B 41, 11068 (1990).[4] L.F. Feiner, J.H. Jefferson, and R. Raimondi, Phys. Rev.B 53, 8751 (1996), and references therein.[5] C.T. Chen et al., Phys. Rev. Lett. 66, 104 (1991).[6] S. Uchida et al., Phys. Rev. B 43, 7942 (1991);[7] E. Dagotto, Rev. Mod. Phys. 66, 763 (1994), and refer-ences therein.[8] H. Eskes and A. M. Oleś, Phys. Rev. Lett. 73, 1279(1994); H. Eskes et al., Phys. Rev. B 50, 17980 (1994).[9] A.B. Harris and R.V. Lange, Phys. Rev. 157, 295 (1967).[10] K.J. von Szczepanski, et al., Phys. Rev. B 41, 2017(1990).[11] S.A. Trugman, Phys. Rev. B 37, 1597 (1988); ibid. Phys.Rev. B 41, 892 (1990).[12] R. Eder and K.W. Becker, Z. Phys. B 78, 219 (1990);ibid. Phys. Rev. B 44, 6982 (1991).[13] J. Ba la, A.M. Oleś and J. Zaanen, Phys. Rev. B 52, 4597(1995).[14] W. Stephan and P. Horsch, Int. J. Mod. Phys. B6, 141(1992). P. Horsch and W. Stephan, Phys. Rev. B 48,10595 (1993).[15] P.W. Leung et al., Phys. Rev.B 46, 11779 (1992).[16] B.O. Wells et al., Phys. Rev. Lett. 74, 964 (1995).[17] It is important to realize that other effects can influencethe momentum dependence of intensities as well, suchas the changing angle of the escaping electron and theactual underlying Cu-3d, O-2p orbital structure.[18] See, for instance, A. Nazarenko et al., Phys. Rev. B 51,8676 (1995).[19] A.R. Kampf and J.R. Schrieffer, Phys. Rev. B 42, 7967(1990).[20] W. Stephan and P. Horsch, Phys. Rev. Lett. 66, 2258(1991).[21] R. Preuss, W. Hanke, and W. von der Linden, Phys. Rev.Lett. 75, 1344 (1995).[22] R. Eder, Y. Ohta, and T. Shimozato, Phys. Rev. B 50,3350 (1994); R. Eder and Y. Ohta, to appear in Phys.Rev. B.[23] X.G. Wen and P.A. Lee, Phys. Rev. Lett. 76 503 (1996).4

   University of GroningenHubbard model versus t-J modelEskes, H; Eder, RPublished in:Physical Review BDOI:10.1103/PhysRevB.54.R14226IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1996Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Eskes, H., & Eder, R. (1996). Hubbard model versus t-J model: The one-particle spectrum. PhysicalReview B, 54(20), 14226-14229. https://doi.org/10.1103/PhysRevB.54.R14226CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 05-12-2023arXiv:cond-mat/9609233 v1   24 Sep 1996Hubbard model versus t-J model: The one-particle spectrumHenk EskesInstitute Lorentz, Leiden University, P.O. Box 9506, 2300 RA Leiden, The NetherlandsRobert EderLaboratory of Applied and Solid-State Physics, Materials Science Centre,University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands(August 13, 2006)The origin of the apparent discrepancies between the one-particle spectra of the Hubbard and t-Jmodels is revealed: Wavefunction corrections, in addition to the three-site terms, should supplementthe bare t-J model. In this way a quantitative agreement between the two models is obtained,even for the intermediate-U values appropriate for the high-Tc cuprate superconductors. Numericalresults for clusters of up to 20 sites are presented. The momentum dependence of the observedintensities in the photoemission spectra of Sr2CuO2Cl2 are well described by this complete strong-coupling approach.71.27.+a,79.60.-i,71.10.+x,74.72.-hThe past ten years the interest in Hamiltonians mod-eling correlated electrons has greatly increased. This islargely due to the mysterious normal-state properties ofthe high-Tc cuprate superconductors and the realizationthat local correlations on the Cu sites are large in thesecompounds. The two most intensively studied Hamilto-nians are the Hubbard and t-J models. Although at firstsight such models may seem oversimplified, it has beenargued that the two dimensional square-lattice Hubbardmodel captures the essential dynamics of the cupratesin a window of a few eV around the chemical poten-tial [1–4]. The materials belong to the intermediate-Uregime, U/t ≈ 10, t ≈ 0.3−0.4 eV [3,4]. Further supportfor a single-band Hubbard approach comes from mea-sured one-particle and optical spectra of the cuprates.These show a dramatic reshuffling of intensity as a func-tion of doping on the scale of a few eV [5,6]. This char-acteristic behavior is a fingerprint of strong correlationsand is described well by the Hubbard model [5,7,8].Although the t-J model is frequently referred to as be-ing the large-U limit of the single-band Hubbard model,the (one-particle) spectra of the two models differ sig-nificantly in the intermediate-U regime of interest. (Fora recent review on numerical results, see Ref. [7]) Prob-ably because of this, the two models are often treatedas being “distinct”. One obvious difference concernsthe momentum occupation number nk at half filling,equal to the one-electron removal sum rule. For the t-Jmodel this quantity is trivially equal to 1/2, while for theintermediate-U Hubbard model it is a strongly varyingfunction of k. Even though in the infinite-U limit nktends to 1/2, as in the t-J model, the convergence to thisvalue by increasing U is very slow.In this Letter we show that these discrepancies can beattributed to the neglect of certain terms in the large-U perturbation theory. Specifically, the weight distribu-tions in the spectral functions show a high sensitivity tothe first-order renormalization of the wave functions. Thecomplete strong-coupling approach is found to quantita-tively reproduce the Hubbard spectra for U ≥ W , whereW is the U = 0 band width. For these U values the per-turbation series in t/U agrees well with the full HubbardHamiltonian.The derivation of the strong-coupling Hamiltonian bymeans of a canonical transformation, written as an ex-pansion in t/U , is well known [8,9]. Starting point is theHubbard model,H = V + T = U∑ini,↑ni,↓ − t∑i,δ,σa†i,σai+δ,σ. (1)Here ni,↑ is the occupation number of the original Hub-bard fermions ai,σ, and δ is a vector connecting nearestneighbors. After the transformation the states in thelowest Hubbard sector (no doubly occupied sites) are de-scribed to order t2/U by the strong-coupling Hamilto-nian,Hsc = −t∑i,δ,σc†i,σci+δ,σ −J2∑i,δ(S̃i · S̃i+δ −14ñiñi+δ)−J4∑i,δ 6=δ′,σ(c†i+δ,σñi,σ̄ci+δ′,σ − c†i+δ,σ̄c†i,σci,σ̄ci+δ′,σ),(2)where J = 4t2/U . The motion of the above transformedci,σ fermions will leave the number of doubly occupiedsites unchanged, and S̃i and ñi,σ are the spin and occu-pation number of these new fermions. Simply neglectingthe ‘three-site term’, i.e. the second line of (2), one ob-tains the t-J model.Why the full strong-coupling model has received farless attention than the t-J model is unclear. It might beexpected that the three-site term is very important in theone-hole limit (and low doping region) since, unlike the1nearest-neighbor hopping, a direct next-nearest-neighborhopping process does not frustrate the spin system. How-ever, it was shown by a numerical study [10], spin-polaronapproaches [11,12] and a linear-spin-wave self-consistentBorn (LSW-SCB) calculation [13] that the main effect ofthis term on the quasiparticle dispersion is an increase ofthe bandwidth of the dominant ‘quasiparticle peak’.By virtue of its construction via a canonical transfor-mation, the Hamiltonian (2) reproduces the eigenvaluespectrum of the Hubbard model for large U . To obtainfurther information, such as matrix elements or expec-tation values, operators acting in the Hilbert space ofthe Hubbard model have to be subjected to the samecanonical transformation. For the electron annihilationoperator this procedure leads to the expression [8,9],Pai,σP = Pci,σP+tU∑δP(ñi,σ̄ci+δ,σ − c†i,σ̄ci,σci+δ,σ̄)P. (3)P is defined as the projector on states with no dou-ble occupancy of the transformed fermions (states with〈Ṽ 〉 = 0). The first-order correction terms (second line)describe the dressing of the annihilation operator with‘virtual’ charge fluctuations involving doubly-occupiedsites. While their prefactor t/U seems to suggest thatthey represent only a minor correction, their impact onthe calculated spectra is dramatic – one reason beingthat the summation over neighbors introduces the co-ordination number z as an additional ‘hidden’ prefac-tor. When comparing strong-coupling calculations di-rectly with experiments it is therefore crucial to takethese corrections into account. For the optical conduc-tivity such corrections were shown to lead to qualita-tively different results [14]. Another consequence is arapid reshuffling of weight between the upper (UHB) andlower-Hubbard band (LHB) as a function of hole doping,observed in both one-particle and optical experiments.The first-order corrections to transformed operators suchas Eq. (3) lead to modified expressions for sum rules thatconsistently describe the rapid intensity changes observedin experiments [5,6,8,9].For the numerical calculations of the one-particle spec-tra we implemented the strong-coupling model Eq. (2).Electron removal is described by the operator Eq. (3).This operator does not change double-occupancy andonly the LHB part of the Hubbard spectrum is pro-duced. (There is, in fact, a similar independent proce-dure to calculate the UHB response as well.) The t/Uterm in Eq. (3) amounts to the removal of an electronfrom a neighbor of i. This leads to an extra phase factorexp(ik·δ). Because of this the total weight is enhanced atΓ and reduced at (π, π) and Eq. (3) leads to the correctbehavior of n(k) (see Fig. 2 of Ref. [8]). Note that theimplementation of the first-order correction is straight-forward and does not increase memory or CPU time inFIG. 1. The one-particle spectrum of the two-dimensional4x4 strong-coupling model (solid line) compared with theHubbard model (dotted line), at half filling (left) and for twoholes (right). U/t = 10. Only the LHB part of the Hubbardspectrum is shown. Dashed line is the chemical potential inthe two-hole case. Hubbard data reproduced from Ref. [15]a Lanczos-like diagonalization calculation. In a similarway such operator corrections can be taken into accountin analytic approaches, for instance in the LSW-SCB cal-culation of the one-hole spectrum.In Fig. 1 we compare the two-dimensional 16-site 4x4strong-coupling model with the LHB electron removaland addition spectra of the Hubbard model, reproducedfrom Ref. [15]. Note that U = 10 is comparable to thebandwidth W = 8. No shift or scaling has been appliedto the spectra, showing that the value of the gap betweenthe UHB (not shown, starting at energy ≈ 8t) and LHBis accurately described. (While there is no UHB in thestrong coupling model, the magnitude of the energy gapstill can be inferred from the relation ∆ = U − 2 · Emin,with Emin ≈ 2t the minimal ionization energy of themodel.) The resulting gap of ≈ 6t is consistent with themeasured insulating gap of ≈ 2eV .As can be seen, excitation energies and weights of thedominant features in the spectra are reproduced correctlyby the strong-coupling approach. The agreement is bestfor the physically most relevant low excitation energies,but the strong coupling approach also gives a completedescription of the higher excitations, with energies notmuch smaller than U . The total width is slightly overes-timated by the second-order t2/U terms in the Hamilto-nian, due especially to those peaks which obtain a largeenergy correction by the three-site terms, for instance at(π, π). There is no significant deterioration of the agree-ment upon doping, demonstrating that the discrepancies2between the doping dependences of low energy sum-rulesfor t− J and Hubbard originate from matrix element ef-fects, rather than an incompleteness of the Hilbert spaceof the strong coupling model. We explicitly checked thatsecond-order corrections to the Fermi operators lead toonly minor changes of the peak intensities. For U valuesbelow 10 the agreement between Hubbard and its strong-coupling limit gradually deteriorates, which we interpretas a breaking down of the perturbation approach.Stimulated by the good agreement in the 4x4 systemwe now present large-U Hubbard spectra for clusters upto 20 sites. Because the momenta of the 16 and 18 siteclusters nicely complement each other and are all on thehigh-symmetry axis, we show a mix of these two clus-ters in Figs. 2 and 3. The curves on the left show a scanfrom (0, 0) to (π, π). The plots on the right go from (0, 0)to (π, 0) and from (π, 0) to (π, π). We shifted all 4 × 4spectra by the same small energy such that the lowestpeak at k = (0, 0) is at the same energy for both clus-ters. For the two-hole case we shifted the t-J data by anadditional 0.3t to have the chemical potentials at roughlythe same position. The ground-state momentum is (0, 0)in all cases (there is an extra degeneracy for 16-sites, 2holes). In both figures the results are compared with the“bare” t-J model.As mentioned before there are substantial differ-ences between t-J and strong coupling. At half-fillingthe low energy ‘quasiparticle peaks’ are at higherenergies (implying a larger Hubbard gap) for theFIG. 2. The momentum dependence of the elec-tron-removal spectrum of the strong-coupling Hubbard model(solid line) compared with the t-J model (dotted line). Theplot is a mix of data from the 16 and 18 site clusters.U/t = 10. n = 1. The spectra for k = (π/2, π/2), (π/2, 0),(π, 0) and (π/2, π) are calculated using the 2D 16-site cluster;the others, using the 18-site cluster.FIG. 3. Combined electron addition and removal spectrafor the two-hole ground state. See caption of Fig. 2.t-J model and their dispersion has a smaller width.Moreover, the momentum dependence of the weights isqualitatively different. The t-J spectrum at (0, 0), andindeed through much of the Brillouin zone, is more com-pact than for large-U Hubbard. The strongest differenceis at (π, π). Here strong coupling shows a little peak athigh energy, while the t-J model has pronounced struc-tures at a more intermediate energy. This reflects thesmall value of nk for k = (π, π) in the Hubbard model,which is reproduced by the first-order corrections to theFermi operator Eq. (3).The spectra at half filling can be compared with re-cent angle-resolved photoemission data on insulatingSr2CuO2Cl2 [16]. A significant feature of these measure-ments is the strong momentum dependence of the weightsof the spectra. The region in momentum space with themost intense peaks is not centered around (π/2, π/2),but displaced towards (0, 0). Beyond (π/2, π/2) the in-tensity drops very sharply and seems to vanish around(π, π) [17]. As shown in Fig. 4 these features are wellreproduced by the strong-coupling Hubbard model. Incontrast, the intensities in t-J are much more symmetricaround (π/2, π/2) and change only little in that partof momentum space. On average the peaks in strong-coupling Hubbard are more intense than those in thet-J model, at the expense of the incoherent part of thespectrum. Note that also in the t-J model there aredifferences in weight between k and k + (π, π). This isdue to the coupling of the created hole to quantum-spinfluctuations in the ground state of the Heisenberg anti-ferromagnet [12]. Experimentally the strong changes atthe antiferromagnetic Brillouin-zone boundary look like3FIG. 4. Weights (proportional to the radius of the circles)of the lowest energy peak for the strong-coupling Hubbardmodel (left) and t-J model (right). Data collected from the16, 18 and 20 site clusters. U/t = 10, n = 1.a kind of ‘pseudo Fermi surface’: The spectral weightobserved drops significantly, but an actual Fermi-levelcrossing does not take place. We note that the largedifference observed between (π, 0) and (π/2, π/2) in thephotoemission measurements actually is in disagreementwith the near degeneracy of these momenta in both thestandard Hubbard and t-J model. As discussed by vari-ous authors this may be explained by adding extra next-neighbor hopping terms to the Hamiltonian [13,18].The increased quasiparticle weight and the strong dropof intensity when passing the half-filled Fermi surfaceare consistent with a continuous cross-over to a small-Uspin-density-wave (SDW) like shadow-band scenario [19].Surely Fig. 4 shows that such a strong drop in weight isnot necessarily a weak-coupling feature.The interpretation of the doping dependence of thespectral function is a subtle issue [20–23], and will not bediscussed here. For roughly 10% holes (Fig. 3) there areintense features appearing in the electron addition spec-trum for all momenta outside the magnetic zone. Thesefeatures are more intense than for t-J and because ofthe sum rule the extra weight must come from the up-per Hubbard band. This weight-transfer effect has beenclearly observed in O-1s X-ray absorption [5].In conclusion, we have shown that the strong-couplingapproach is a good alternative to Hubbard calculationseven in the intermediate U regime relevant for the high-Tc superconductors. Only the complete strong-couplingmodel gives a faithful representation of the propertiesof the Hubbard model. The corrections to t-J lead toa smooth crossover to small U . For comparisons withexperiments such as angle-resolved photoemission, O-1sX-ray absorption (and optical conductivity) it is impor-tant to use the renormalized wavefunctions (rotated op-erators), as these can lead to even qualitatively differentresults.We acknowledge stimulating discussions with GeorgeA. Sawatzky, Jan Zaanen and Andrzej M. Oleś. HE issupported by the Stichting voor Fundamenteel Onder-zoek der Materie (FOM), which is financially supportedby the Nederlandse Organisatie voor WetenschappelijkOnderzoek (NWO). 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B 52, 4597(1995).[14] W. Stephan and P. Horsch, Int. J. Mod. Phys. B6, 141(1992). P. Horsch and W. Stephan, Phys. Rev. B 48,10595 (1993).[15] P.W. Leung et al., Phys. Rev.B 46, 11779 (1992).[16] B.O. Wells et al., Phys. Rev. Lett. 74, 964 (1995).[17] It is important to realize that other effects can influencethe momentum dependence of intensities as well, suchas the changing angle of the escaping electron and theactual underlying Cu-3d, O-2p orbital structure.[18] See, for instance, A. Nazarenko et al., Phys. Rev. B 51,8676 (1995).[19] A.R. Kampf and J.R. Schrieffer, Phys. Rev. B 42, 7967(1990).[20] W. Stephan and P. Horsch, Phys. Rev. Lett. 66, 2258(1991).[21] R. Preuss, W. Hanke, and W. von der Linden, Phys. Rev.Lett. 75, 1344 (1995).[22] R. Eder, Y. Ohta, and T. Shimozato, Phys. Rev. B 50,3350 (1994); R. Eder and Y. Ohta, to appear in Phys.Rev. B.[23] X.G. Wen and P.A. Lee, Phys. Rev. Lett. 76 503 (1996).4

Hubbard model versus t-J model: The one-particle spectrum

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