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arXiv:1209.1010 · Physical Review B

Hubbard-model description of the high-energy spin-weight distribution in La 2 CuO 4

J. M. P. Carmelo, M. A. N. Araújo, Steven R. White, M. J. Sampaio

Published Aug 20, 2012• 8 citations
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Abstract

The spectral-weight distribution in recent neutron scattering experiments on the parent compound La${}_{2}$CuO${}_{4}$ (LCO), which are limited in energy range to about 450 meV, is studied in the framework of the Hubbard model on the square lattice with effective nearest-neighbor transfer integral $t$ and on-site repulsion $U$. Our study combines a number of numerical and theoretical approaches, including, in addition to standard treatments, density matrix renormalization group calculations for Hubbard cylinders and a suitable spinon approach for the spin excitations. The latter spin-$\frac{1}{2}$ spinons are the spins of the rotated electrons that singly occupy sites. These rotated electrons are mapped from the electrons by a uniquely defined unitary transformation, in which rotated-electron single and double occupancy are good quantum numbers for finite interaction values. Our results confirm that the $U/8t$ magnitude suitable to LCO corresponds to intermediate $U$ values smaller than the bandwidth $8t$, which we estimate to be $8t\ensuremath{\approx}2.36$ eV for $U/8t\ensuremath{\approx}0.76$. This confirms the unsuitability of the conventional linear spin-wave theory. Our theoretical studies provide evidence for the occurrence of ground-state $d$-wave spinon pairing in the half-filled Hubbard model on the square lattice. This pairing applies only to the rotated-electron spin degrees of freedom, but it could play a role in a possible electron $d$-wave pairing formation upon hole doping. We find that the higher-energy spin spectral weight extends to about 566 meV and is located at and near the momentum $[\ensuremath{\pi},\ensuremath{\pi}]$. The continuum weight energy-integrated intensity vanishes or is extremely small at momentum $[\ensuremath{\pi},0]$. This behavior of this intensity is consistent with that of the spin waves observed in recent high-energy neutron scattering experiments, which are damped at the momentum $[\ensuremath{\pi},0]$. We suggest that future LCO neutron scattering experiments scan the energies between 450 and 566 meV and momenta around $[\ensuremath{\pi},\ensuremath{\pi}]$.

DOI
10.1103/physrevb.86.064520

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Spin-only descriptions of the half-filled one-band Hubbard model are relevant for a wide range of Mott insulators. In addition to the usual Heisenberg exchange, many other types of interactions, including ring exchange, appear in the effective Hamiltonian in the intermediate coupling regime. In order to improve on the quantitative description of magnetic excitations in the insulating antiferromagnetic phase of copper-oxide (cuprate) materials, and to be consistent with band-structure calculations and photoemission experiments on these systems, we include second- and third-neighbor hopping parameters, ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$, into the Hubbard Hamiltonian. A unitary transformation method is used to find systematically the effective Hamiltonian and any operator in the spin-only representation. The results include all closed four-hop electronic pathways in the canonical transformation. The method generates many ring exchange terms that play an important role in the comparison with experiments on La2CuO4{\text{La}}_{2}{\text{CuO}}_{4}La2​CuO4​. Performing a spin-wave analysis, we calculate the magnon dispersion as a function of UUU, ttt, ${t}^{\ensuremath{'}}$, and ${t}^{\ensuremath{''}}$. The four parameters are estimated by fitting the magnon dispersion to the experimental results of Coldea et al. [Phys. Rev. Lett. 86, 5377 (2001)] for La2CuO4{\text{La}}_{2}{\text{CuO}}_{4}La2​CuO4​. The ring exchange terms are found essential, in particular to determine the relative sign of ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$, with the values found in good agreement with independent theoretical and experimental estimates for other members of the cuprate family. The zero-temperature sublattice magnetization is calculated using these parameters and also found to be in good agreement with the experimental value estimated by Lee et al. [Phys. Rev. B 60, 3643 (1999)]. We find a value of the interaction strength $U\ensuremath{\simeq}8t$ consistent with Mott insulating behavior.
A description of the Hubbard model on a square lattice consistent with its global SO(3)×SO(3)×U(1)SO(3)\times SO(3)\times U(1)SO(3)×SO(3)×U(1) symmetry
In this paper a description of the Hubbard model on the square lattice with nearest-neighbor transfer integral ttt, on-site repulsion UUU, and Na2≫1N_a^2\gg 1Na2​≫1 sites consistent with its exact global SO(3)×SO(3)×U(1)SO(3)\times SO(3)\times U(1)SO(3)×SO(3)×U(1) symmetry is constructed. Our studies profit from the interplay of that recently found global symmetry of the model on any bipartite lattice with the transformation laws under a suitable electron - rotated-electron unitary transformation of a well-defined set of operators and quantum objects. For U/4t>0U/4t>0U/4t>0 the occupancy configurations of these objects generate the energy eigenstates that span the one- and two-electron subspace. Such a subspace as defined in this paper contains nearly the whole spectral weight of the excitations generated by application onto the zero-spin-density ground state of one- and two-electron operators. Our description involves three basic objects: charge ccc fermions, spin-1/2 spinons, and ηηη-spin-1/2 ηηη-spinons. Alike in chromodynamics the quarks have color but all quark-composite physical particles are color-neutral, the ηηη-spinon (and spinons) that are not invariant under that transformation have ηηη spin 1/2 (and spin 1/2) but are part of ηηη-spin-neutral (and spin-neutral) 2ν2ν2ν-ηηη-spinon (and 2ν2ν2ν-spinon) composite ηνηνην fermions (and sνsνsν fermions) where ν=1,2,...ν=1,2,...ν=1,2,... is the number of ηηη-spinon (and spinon) pairs.The description introduced here is consistent with a Mott-Hubbard insulating ground state with antiferromagnetic long-range order for half filling at x=0x=0x=0 hole concentration and a ground state with short-range spin order for a well-defined range of finite hole concentrations x>0x>0x>0.
In this paper we review recent results on the preliminary applications of the new-found extended global SO(3) × SO(3) × U(1) symmetry of the Hubbard model on a bipartite lattice. Our results refer to the particular case of the bipartite square lattice. Specifically, we review a general description for such a model with nearest-neighbor transfer integral t and on-site repulsion U on a square lattice with N2a 1 sites consistent with its extended global symmetry. It refers to three types of elementary objects whose occupancy configurations generate the state representations of the model extended global symmetry. Such objects emerge from a suitable electron-rotated-electron unitary transformation. An application to the spin spectrum of the parent compound La2CuO4 is shortly reviewed.
In the first paper of this series it was found that the ηηη-spin 1/2 holons, spin 1/2 spinons, and ccc pseudoparticles whose occupancy configurations describe the energy eigenstates of the one-dimensional Hubbard model emerge from the electron - rotated-electron unitary transformation. In this second paper we discuss and clarify how the relation of the electrons to the above objects can be used in a program for evaluation of finite-energy few-electron spectral functions. As a first step, here we characterize the dominant holon and spinon microscopic physical processes that originate more than 99% of the few-electron spectral weight. These dominant processes are related to exact selection rules for the values of the number of holons and spinons generated or annihilated by application onto a ground state of rotated-electron operators. We also generalize the concepts of a lower Hubbard band and upper Hubbard bands to all values of on-site Coulombian repulsion.
Using strong-coupling perturbation theory for the Hubbard model, explicit expressions are obtained for the integrated weights, energy positions, and widths of the upper and lower Hubbard bands separately, both for the optical and photoemission spectrum. In one dimension all expressions can be explicitly evaluated using the large-U Bethe-ansatz wave function. The k\ensuremath{\rightarrow}-dependent moments for the one-particle spectrum are compared with the closely related two-pole ansatz. The strong momentum dependence of the spectra demonstrates the importance of the k\ensuremath{\rightarrow}-dependent correlation functions in the spectral moments, which are often neglected. In order to estimate corrections due to the neglect of higher-order terms (in t/U), all results are compared with numerical-diagonalization data for intermediate-U values. The sum-rule results show a rapid spectral-weight transfer from the high- to the low-energy regime upon doping, similar to what is observed experimentally in one-particle and optical spectra of Cu (high-Tc{\mathit{T}}_{\mathit{c}}Tc​) and Ni compounds. Such fast weight redistributions as a function of the carrier density are a natural consequence of the strong correlations in these materials. The redistribution of intensity in the optical conductivity is connected with the dilution of the spin system by the added holes. For the one-particle spectrum the decrease of weight in the upper Hubbard band away from half filling can be understood by state counting at U=\ensuremath{\infty}, and the effect is strongly enhanced for small doping by both the first- and second-order terms in t/U. Altogether, the one-particle and optical spectra show the importance of the three-site hopping term. Therefore the t-J model does not represent well the spectral properties of the Hubbard model at large U. The local spin order is essential and determines both the k\ensuremath{\rightarrow} dependence of the one-particle spectrum and the intensities in the optical spectrum.