Static and dynamical properties of doped Hubbard clusters

Abstract

We study the t-J and the Hubbard models at zero temperature using exact-diagonalization techniques on \ensuremath{\surd}10 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}10 and 4\ifmmode\times\else\texttimes\fi{}4 sites clusters. Quantum Monte Carlo simulation results on larger lattices are also presented. All electronic fillings have been analyzed for the three models. We have measured equal-time correlation functions corresponding to various types of order (ranging from ``standard'' staggered spin order to more ``exotic'' possibilities like chiral order), as well as various dynamical properties of these models. Upper bounds for the critical hole doping (${\mathit{x}}_{\mathit{c}}$), where long-range antiferromagnetic order disappears, are presented. It was found that ${\mathit{x}}_{\mathit{c}}$ is very small in agreement with experiments for the high-${\mathit{T}}_{\mathit{c}}$ superconductors. For example, in the t-J model, ${\mathit{x}}_{\mathit{c}}$0.08 at J/t=0.4. However, short-distance spin correlations are important up to much higher dopings producing a sharp well-defined spin-wave-like peak in S(q=(\ensuremath{\pi},\ensuremath{\pi}),\ensuremath{\omega}). Regarding the possibility of phase separation in the Hubbard model, we have studied the behavior of the density of particles, 〈n〉, as a function of the chemical potential, using the Lanczos method on a 4\ifmmode\times\else\texttimes\fi{}4 Hubbard cluster, finding no indications of phase separation for any value of U/t. Then, we conclude that the t-J model at small J/t should not phase separate.In order to compare theoretical predictions with photoemission experiments, we evaluated the electronic density of states N(\ensuremath{\omega}), of the Hubbard and t-J models at several doping fractions. We found that upon doping the antiferromagnetic gap is filled for U\ensuremath{\sim}8t or smaller. The chemical potential moves across the insulating gap as one goes from electron to hole doping of the half-filled cluster, in agreement with x-ray absorption experiments but at variance with photoemission experiments. We have also calculated the optical conductivity, ${\mathrm{\ensuremath{\sigma}}}_{1}$(\ensuremath{\omega}), of the Hubbard and t-J models at all dopings on 4\ifmmode\times\else\texttimes\fi{}4 clusters. Results are compared with experiments and the weight of the Drude peak is presented as a function of couplings and dopings. Spectral weight found at small frequencies is associated with the mid-infrared band observed experimentally in ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_{4}$, and with the states filling the insulating gap in photoemission experiments. An overall good agreement with experiments in the normal state was found. Regarding the possibility of superconductivity n these models, we have studied s-, d-, and p-wave pairing correlations. Naively, the d-wave channel seems enhanced near half filling while the extended s-wave channel seems enhanced from half filling up to 40% doping.However, we found that the enhancement comes from short-distance effects and, thus, no numerical indications of superconductivity were found in these models. We emphasize the importance of analyzing the pairing correlations as a function of distance to distinguish between short- and long-distance effects in the susceptibilities. We also observed that spiral order is enhanced at small J/t and low doping. Uniform chiral rder is suppressed by dynamical holes while staggered chiral order may be enhanced, although with a small plaquette order parameter. We conclude with the observation that the simple one-band Hubbard model with intermediate values of U\ensuremath{\sim}8t may account for many of the ``anomalous'' properties of the normal state of the high-${\mathit{T}}_{\mathit{c}}$ superconductors.