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Physical review. B, Condensed matter

Optical conductivity of the two-dimensional Hubbard model

Elbio R. Dagotto, Adriana Moreo, F. Ortolani, José A. Riera, Douglas J. Scalapino

Published May 1, 1992• 48 citations
Publisher

Abstract

The optical conductivity ${\mathrm{\ensuremath{\sigma}}}_{1}$(\ensuremath{\omega}) of the two-dimensional Hubbard model on finite clusters of 4\ifmmode\times\else\texttimes\fi{}4 and \ensuremath{\surd}10 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}10 sites is reported. Experimental features found in the high-${\mathit{T}}_{\mathit{c}}$ cuprate superconductors can be qualitatively reproduced by this model including the presence of mid-infrared spectral weight, a Drude peak whose intensity grows with doping and a total spectral weight that varies slowly as the system is doped away from half filling. We find that to reproduce qualitatively the main experimental features of ${\mathrm{\ensuremath{\sigma}}}_{1}$(\ensuremath{\omega}), we need to work with a coupling constant of the order of the bandwidth, i.e., U/t\ensuremath{\sim}8--10. Results for the negative-U Hubbard model are included for comparison.

DOI
10.1103/physrevb.45.10107

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The optical conductivity \ensuremathσ1{\mathrm{\ensuremath{\sigma}}}_{1}\ensuremathσ1​(\ensuremath{\omega}) of the two-dimensional Hubbard model on finite clusters of 4\ifmmode\times\else\texttimes\fi{}4 and \ensuremath{\surd}10 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}10 sites is reported. Experimental features found in the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ cuprate superconductors can be qualitatively reproduced by this model including the presence of mid-infrared spectral weight, a Drude peak whose intensity grows with doping and a total spectral weight that varies slowly as the system is doped away from half filling. We find that to reproduce qualitatively the main experimental features of \ensuremathσ1{\mathrm{\ensuremath{\sigma}}}_{1}\ensuremathσ1​(\ensuremath{\omega}), we need to work with a coupling constant of the order of the bandwidth, i.e., U/t\ensuremath{\sim}8--10. Results for the negative-U Hubbard model are included for comparison.
The optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ in the two-dimensional Hubbard model is examined by applying the exact diagonalization technique to small square clusters with periodic boundary conditions up to $\sqrt{20}\ifmmode\times\else\texttimes\fi{}\sqrt{20}$ sites. Spectral-weight distributions at half-filling and their doping dependence in the 20-site cluster are found to be similar to those in a $\sqrt{18}\ifmmode\times\else\texttimes\fi{}\sqrt{18}$ cluster, but different from $4\ifmmode\times\else\texttimes\fi{}4$ results. The results for the 20-site cluster enable us to perform a systematic study of the doping dependence of the spectral-weight transfer from the region of the Mott-gap excitation to lower-energy regions. We discuss the dependence of the Drude weight and the effective carrier number on the electron density at a large on-site Coulomb interaction.
A Monte Carlo maximum-entropy calculation of the optical conductivity of the infinite-dimensional Hubbard model is presented. We show that the optical conductivity displays the anomalies found in the cuprate superconductors, including a Drude width which grows linearly with temperature, a Drude weight which grows linearly with doping, and a temperature and doping-dependent mid-IR peak. These anomalies arise as a consequence of the dynamical generation of a quasiparticle band at the Fermi energy as T\ensuremath{\rightarrow}0, and are a generic property of the strongly correlated Hubbard model in all dimensions greater than one.
We study the Cu and O spectral density of states and the optical conductivity of CuO2{}_{2}2​ planes using an effective generalized one-band Hubbard model derived from the extended three-band Hubbard model. We solve exactly a square cluster of 10 unit cells and average the results over all possible boundary conditions, which leads to smooth functions of frequency. Upon doping, the Fermi energy jumps to Zhang-Rice states which are connected to the rest of the valence band (in contrast to an isolated band in the middle of the gap). The transfer of spectral weight depends on the parameters of the original three-band model not only through the one-band effective parameters but also through the relevant matrix elements. We discuss the evolution of the gap upon doping. The optical conductivity of the doped system shows a mid-infrared peak due to intraband transitions, a pseudogap, and a high-frequency part related to interband transitions. Its shape and integrated weight up to a given frequency (including the Drude weight) agree qualitatively with experiments in the cuprates for low to moderate doping levels, but significant deviations exist for doping x>0.3x>0.3x>0.3.
The optical conductivity, \ensuremath{\sigma}(\ensuremath{\omega}), of the two-dimensional one-band Hubbard model is calculated at finite temperature using exact diagonalization techniques on finite clusters. The in-plane dc resistivity, ${\mathrm{\ensuremath{\rho}}}_{\mathit{a}\mathit{b}}$, is also evalauted. We find that at large U/t and temperature T, ${\mathrm{\ensuremath{\rho}}}_{\mathit{a}\mathit{b}}$ is approximately linear with temperature. We also observed that \ensuremath{\sigma}(\ensuremath{\omega}) displays charge excitations, a midinfrared (MIR) band, and a Drude peak. The combination of the Drude peak and the MIR oscillator strengths leads to a conductivity that decays slower than 1/${\mathrm{\ensuremath{\omega}}}^{2}$ at energies smaller than the insulator gap near half filling.
The optical conductivity of the one-band Hubbard model is calculated using the ``dynamical cluster approximation'' implementation of dynamical mean-field theory for parameters appropriate to high-temperature copper-oxide superconductors. The calculation includes vertex corrections and the result demonstrates their importance. At densities of one electron per site, an insulating state is found with gap value and above-gap absorption consistent with measurements. As carriers are added the above-gap conductivity rapidly weakens and a three component structure emerges, with a low-frequency ``Drude'' peak, a mid-infrared absorption, and a remnant of the insulating gap. The mid-infrared feature obtained at intermediate dopings is shown to arise from a pseudogap structure in the density of states. On further doping the conductivity evolves to the Drude peak plus weakly frequency dependent tail structure expected for less strongly correlated metals.
The optical conductivity for the Hubbard and t-J models is calculated numerically by use of a Lanczos algorithm. One- and two-dimensional systems of up to 16 sites with periodic and open boundary conditions are examined for the t-J model, as well as Hubbard-model systems of up to 9 sites. In the one-dimensional case, the low-frequency part of the conductivity away from half filling is shown to consist almost entirely of a single \ensuremath{\delta} function (Drude peak) even for J of the order of t. This is a direct consequence of the decoupling of charge and spin in one dimension. In two dimensions there is an additional broad absorption band above the Drude peak, consistent with the experimentally observed midinfrared band in the copper oxide superconductors. The importance of vertex corrections is demonstrated by comparing the exact conductivity with the convolution integral of the exact single-particle Green's function.