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Magnon and Hole Excitations in the Two-Dimensional Half-filled Hubbard Model

Magnon and Hole Excitations in the Two-Dimensional Half-?lled Hubbard Model
Weihong Zheng1 , Rajiv R. P. Singh2 , Jaan Oitmaa1 , Oleg P. Sushkov1 Chris J. Hamer1

School of Physics, University of New South Wales, Sydney, NSW 2052, Australia 2 Department of Physics, University of California, Davis, CA 95616 (Dated: February 2, 2008)

arXiv:cond-mat/0501029v1 [cond-mat.str-el] 3 Jan 2005

Spin and hole excitation spectra and spectral weights are calculated for the half-?lled Hubbard model, as a function of t/U . We ?nd that the high energy spin spectra are sensitive to charge ?uctuations. The energy di?erence ?(π, 0) ? ?(π/2, π/2), which is negative for the Heisenberg model, changes sign at a fairly small t/U ≈ 0.053(5). The hole bandwidth is proportional to J, and considerably larger than in the t-J models. It has a minimum at (π/2, π/2) and a very weak dispersion along the antiferromagnetic zone boundary. A good ?t to the measured spin spectra in La2 CuO4 at T = 10K is obtained with the parameter values U = 3.1eV, t = 0.35eV.
PACS numbers: 75.10.Jm



Underdoped phases of high temperature superconducting materials and the nature of the metal insulator transition upon doping a Mott-insulating antiferromagnet remain central topics of research in condensed matter physics. Some puzzles extend all the way to the undoped stoichiometric insulating materials. Results such as the antiferromagnetic zone-boundary magnon excitations probed in inelastic neutron scattering1,2 , the twomagnon excitations probed in Raman scattering3 and the one-hole excitations probed in angle-resolved photoemission spectroscopy4 contiune to surprise us. The question of whether some of these anomalies are connected to the pseudogap phase of the weakly doped materials remains a topic of debate. An important question is the extent to which conventional approaches, based on ordered antiferromagnetic phases, can explain the observed spectra and spectral weights and to what extent the interpretation of data necessitates the introduction of novel ideas such as spinliquids and spin-charge separation. The low energy longwavelength spin excitations of the antiferromagnet are well described by the non-linear sigma model5 . However, the high energy zone-boundary spin excitations necessarily require a microscopic lattice model. The antiferromagnetic insulator, without charge ?uctuations, is represented by the Heisenberg model, and the excitation spectrum of this model has been the subject of several controlled numerical studies6,7,8 . It is clear that the antiferromagnetic zone-boundary spectrum of La2 CuO4 does not agree with that of the Heisenberg model. In particular, in the Heisenberg model, the magnon energy difference ?(π, 0) ? ?(π/2, π/2) is negative but it is found to be positive for La2 CuO4 . This result will be worse if second neighbor antiferromagnetic interactions are included. It has been suggested that one way to reconcile the di?erence is by invoking ring-exchange terms1,2,9,10 , which arise due to charge ?uctuations11 .

Here, we present systematic numerical calculations of the magnon and hole spectra and spectral weights of the Hubbard model as a function of t/U . First, we focus on the magnons. Earlier the magnon spectra were studied by mean-?eld theory12 and by a Quantum Monte Carlo Simulation combined with the Single Mode Approximation13 , neither of which are expected to be quantitatively accurate for small t/U . Our calculations show that the zone-boundary magnon energies are very sensitive to charge ?uctuations and the di?erence ?(π, 0) ? ?(π/2, π/2) changes sign at a relatively small t/U value of 0.053(5). The magnon spectra of La2 CuO4 and the spectral weights are well described by the Hubbard model as discussed below. The calculated hole spectra, on the other hand, are qualitatively similar to previous theoretical studies of Hubbard and t-J models14,15 . The hole-bandwidth is suppressed at large U by a factor of t/U , although we ?nd it to still be much larger than in the corresponding t-J models. The minimum is at (π/2, π/2) with a weak dispersion along the antiferromagnetic zone boundary. Hence, these results cannot be used to ?t the observed ARPES spectra in the undoped cuprate materials4 . Although same-sublattice hopping terms can allow better ?ts to the dispersion, the anomalous spectral weights remain harder to explain4 . We note that a complete understanding of the ARPES experiments may require a multi-band model, as well as inclusion of dielectric and charging e?ects. To carry out an Ising type expansion16,17 for this system at T = 0 we consider the Hubbard-Ising model with the following Hamiltonian: H = H0 + λH1 H0 = J/4
ij z +h(?1)i σi ] z z [J(σi σj ? ij z z (σi σj + 1) + i 1 [U (ni↑ ? 2 )(ni↓ ? 1 ) 2

H1 =

+ 1)/4 + t(c? cjσ + h.c.)] iσ

2 We have extended the linked cluster method16 to the spectra of the Hubbard-Ising models. At λ = 0, the model has a very simple excitation spectrum. Above the two ground states, all single spin-?ip states (or all states with a single hole for the hole spectra) are degenerate with each other. We construct an orthogonality transformation, which order by order in powers of λ, decouples these single spin-?ip states from the rest of the Hilbert space. This procedure amounts to ?nding the combination of one-?ip states with others that remains an eigenstate as λ changes from zero. For a translationally invariant system the resulting block diagonal Hamiltonian in the one particle subspace is diagonalized by Fourier transformation. To set the normalizations for our calculations, we begin with the de?nition for the dynamic structure factor
FIG. 1: (Color online) Magnon dispersion curve along selected directions in the Brillouin zone for various values of U/t, expressed in units of Je? = 4t2 /U . Results for the Heisenberg antiferromagnet are shown as a red curve.

S αβ (q, ω) =

1 2πN

exp[i(ωt + q · (ri ? rj ))]
i,j ?∞

β α Sj (t)Si (0) dt


We de?ne the single magnon contribution to the dynamical transverse structure factor as S XX (q, ω)+ S Y Y (q, ω) = As δ(ω ? ?s (q))+ Bs (q, ω) (2) Here, ?s (q) gives the dispersion for the magnons and As (q) de?nes the weights for the magnons. The quantity Bs (q, ω) de?nes the multiparticle contributions. Series expansions have been calculated for the spectra ?s (q) and the spectral weights As (q) up to order λ11 . Similarly, for the hole-excitations, we de?ne the spectral function A(q, ω) = 1 2πN

exp[i(ωt + q · (ri ? rj ))]
i,j ?∞

c? (t)ci,σ (0) dt j,σ
FIG. 2: Magnon energies at two wavevectors along the antiferromagnetic zone boundary as a function of t/U .


We de?ne the single hole contribution to the spectral function as A(q, ω) = Ah δ(ω ? ?h (q)) + Bh (q, ω) (4)


z (?1)i σi

z where σi = ni↑ ? ni↓ , and λ is the expansion parameter. The Ising interaction J is, in principle, an adjustable parameter but here is chosen to be 4t2 /U . The strength of the staggered ?eld h can be varied to improve convergence. Note that the full Hubbard model is recovered at λ = 1, at which point the extra terms cancel between H0 and H1 . On the other hand for λ < 1, there is an Isinglike anisotropy in the system, which favors a Neel state and induces a gap in the spectrum. The limit λ = 0 corresponds to the Ising model, with the usual N?el states e being the two unperturbed ground states.

Series expansions are calculated for the hole spectra ?h (q) and the spectral weights Ah (q) up to order λ11 . The gap in the spectrum closes at λ = 1, when spin rotational symmetry is restored in the model. This causes power-law singularities in certain properties of the model18 . Hence, the λ = 1 limit needs to be dealt with by series extrapolation methods. We use the method of integrated di?erential approximants, well known from the study of critical phenomena19 , to calculate various properties at λ = 1. In Figure 1, we show the calculated magnon dispersion, in units of Je? = 4t2 /U , along selected directions in the Brillouin zone, for several values of t/U . The results8 for the Heisenberg model are also shown. While the long wavelength spin-wave velocity is gradually reduced with

3 by U/t = 8.75 ± 0.7 and Je? = 162 ± 3 meV, while that at 295K is ?tted best by U/t = 10.5 ± 0.7 and Je? = 152 ± 3 meV. These results suggest that as the temperature is increased the e?ective exchange constant decreases whereas the e?ective U/t ratio increases. Assuming that the 10K data are essentially at T = 0, we obtain bare parameters of U = 3.1eV and t = 0.35eV. The spectral weights are only measured at 295K, hence we show a ?t to the calculated spectral weights at the larger U/t ratio. In any case, the spectral weights are not very sensitive to the U/t ratio. The relative ?t is excellent. To get a measure of the absolute ?t, we ?rst note that the integrated one-magnon spectral weight over the entire zone was20 found from experiment to be 0.36 ± 0.09, in the normalization where total transverse spectral weight is 0.5. With this normalization, our more accurate calculations for the Heisenberg model give an integrated transverse one-magnon spectral weight of 0.419(2). The one-magnon spectral weights decrease slightly with decreasing U/t, and are also much less accurate, giving 0.40(8) for U/t = 10. Hence, the results agree with experiments well within the uncertainties. We should note, however, that this agreement is very much dominated by the 1/q dependent behavior near the antiferromagnetic wavevector (where q is the deviation from the antiferromagnetic wavevector), which does not depend much on the microscopic model. The best place to look for multimagnon excitations and a more sensitive comparison with microscopic models is the spectral weight along the antiferromagnetic zone boundary. For the Heisenberg model, the multiparticle spctral weight is largest at (π, 0), where it is about 40 percent of the total transverse spectral weight. We hope our work may motivate more accurate measurements of multiparticle spectral weights along the antiferromagnetic zone boundary. The data on two other systems of square-lattice antiferromagnets (CuDCOO)2 · 4D2 O (CFTD)21 and Cu(II) spins of Sr2 Cu3 O4 Cl2 22 have much larger U/t ratios and are well ?tted by the Heisenberg model. Allowing U to vary, (CuDCOO)2 · 4D2 O (CFTD) can be well ?tted by the Hubbard model with U/t = 50 and U = 3.9eV. On the other hand, the experimental data on Sr2 Cu3 O4 Cl2 have substantial uncertainties and anisotropies, so one can not get a reliable estimate for U/t. If one assumes the U value for it is about 4eV, one estimates U/t ≈ 40. In Figure 5, we show the single hole excitation spectra along selected directions in momentum space at different values of t/U . Plots are also shown for the t-J model23 , with comparable t/J values. It is evident that the band-width scales with J but is much larger than in the t-J model. The reason for this is the e?ective samesublattice hopping generated in the Hubbard model in the order t2 /U which is not included in the t-J model. In all cases, the minimum of the hole energy remains at (π/2, π/2) and the dispersion along the line (π/2, π/2) to (π, 0) remains weak. As mentioned above, this is in contrast to the observed single hole dispersion4 in the material Sr2 CuO2 Cl2 . Furthermore, our calculated spec-

FIG. 3: (Color online) Fits to the spectra of the material La2 CuO4 : experimental spectra at 10K (open symbols) ?t by U/t = 8.75±.7, Je? = 162±3meV (green curve); experimental spectra at 295K (solid symbols) ?t with U/t = 10.5 ± .7, Je? = 152 ± 3meV (red curve).

FIG. 4: (Color online) Comparison of magnon spectral weight g 2 As for U/t = 10.5 with neutron scattering intensity1 for La2 CuO4 at 295K.

increasing t/U , other dramatic changes arise along the antiferromagnetic zone boundary. The magnon energy at wavevector (π, 0) at ?rst rises brie?y before it begins to decrease with increasing t/U . On the other hand, the magnon energy at (π/2, π/2) decreases sharply with increasing t/U . Both of these are plotted in Figure 2, where one can see that they cross at a relatively small value of t/U = 0.053(5). In Figures 3 and 4 we show ?ts to the magnon spectra and spectral weight of La2 CuO4 . Since we have not done any ?nite temperature calculations, we try to ?t the spectra at di?erent temperatures by e?ective U and t values. We ?nd that the spectrum at 10K is ?tted well

4 tral weights at (π/2, π/2) and (π, 0) are very similar. It has been argued that even small same-sublattice hopping terms can signi?cantly change the shape of the dispersion curves and bring them closer to those observed in ARPES measurements4 , they may also help reconcile the measured spectral weights in the undoped cuprates24 . In conclusion, we ?nd that allowing for charge ?uctuations by making t/U ?nite allows us to understand the antiferromagnetic zone-boundary excitations in the material La2 CuO4 very well. However, it appears that the simple one-band Hubbard model considered here is unable to explain the single hole spectra in the undoped cuprate materials. We would like to thank R. Coldea for useful discussion and sending us the experimental data. This work is supported by the Australian Research Council and the US National Science Foundation grant number DMR0240918 (RRPS). We are grateful for the computing resources provided by the Australian Partnership for Advanced Computing (APAC) National Facility and by the Australian Centre for Advanced Computing and Communications (AC3).

FIG. 5: (Color online) Single hole dispersion in the squarelattice half-?lled Hubbard model along selected directions in momentum space for several values of t/U (squares). Also shown as crosses is the dispersion for the t-J model with comparable t/Je? ratios.

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