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arXiv:cond-mat/9412084 · Physical review. B, Condensed matter

Optical conductivity of the infinite-dimensional Hubbard model

Mark Jarrell, James K. Freericks, Thomas Pruschke

Published May 1, 1995• 61 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.51.11704

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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.

arXiv:cond-mat/9401060•Physical review. B, Condensed matter•Jul 1, 1994•17 citations
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.
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-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 ${\mathrm{\ensuremath{\sigma}}}_{1}$(\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.
Exact formulas for the optical conductivity and the Hall conductivity of the two dimensional Hubbard model are derived in terms of an effective theory description of the local moment fluctuation in the system. In this framework, the quantum Monte Carlo simulation of the electromagnetic response of such a strongly correlated electron system becomes sign-problem-free in many physically relevant cases. In particular, it is sign-problem-free when we assume the widely used Millis-Monien-Pines form for the phenomenological susceptibility in the effective action of the fluctuating local moment, even though these local moments are now subjected to Landau damping as a result of their coupling to the itinerant quasiparticle on the fermi surface. This is true more generally when a φ4φ^{4}φ4 term is included in the effective action and is thus not restricted to the Gaussian limit. Here we demonstrate the power of this framework by studying the effect of thermal fluctuation of the local moment on the optical conductivity σxx(ω)σ^{xx}(ω)σxx(ω) and the Hall conductivity σxy(ω)σ^{xy}(ω)σxy(ω) of the cuprate superconductors. Both σxx(ω)σ^{xx}(ω)σxx(ω) and σxy(ω)σ^{xy}(ω)σxy(ω) calculated are found to exhibit a two-component structure, with a Drude component at low energy and a mid-infrared component at higher energy. Depending on the relative importance of the hole pocket and the electron pocket on the reconstructed fermi surface and the coupling strength to the local moment, the Drude component in Imσxy(ω)\mathrm{Im}σ^{xy}(ω)Imσxy(ω) can be either positive or negative.(full-length abstract can be found in the main text.)
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.
Results for the optical conductivity and resistivity of the Hubbard model in infinite spatial dimensions are presented. At half-filling we observe a gradual crossover from a metal with a Drude peak at ω = 0 in the optical conductivity to an insulator as a function of U for temperatures above the antiferromagnetic phase transition. When doped, the "insulator" becomes a Fermi liquid with a corresponding temperature dependence of the optical conductivity and resistivity. We find a T 2 -coefficient in the low-temperature resistivity which suggests that the carriers in the system acquire a considerable mass-enhancement due to the strong local correlations. At high temperatures, a crossover into a semi-metallic regime takes place.
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.