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Physical review. B, Condensed matter

Drude weight, optical conductivity, and flux properties of one-dimensional Hubbard rings

Richard M. Fye, Marcio Jose Martins, Douglas J. Scalapino, J. Wagner, Werner R. Hanke

Published Oct 1, 1991• 164 citations
Publisher

Abstract

We study the Drude weight, optical conductivity, and flux-periodicity properties of one-dimensional Hubbard chains using Bethe-ansatz, Lanczos, and exact-diagonalization techniques. We find that the Drude weight D is unexpectedly negative for half-filled Hubbard rings with N=4n sites reflecting a strong paramagnetic response, while it is positive (diamagnetic) for half-filled rings of N=4n+2 sites. In both cases at half filling, D vanishes exponentially with N on a length scale set by the inverse of the gap for small U/t. Near half filling, we find that D approaches a constant, positive, diamagnetic value as N increases, indicating metallic behavior. Combining Bethe-ansatz results near half filling with the known U=0 and U=\ensuremath{\infty} limits and Lanczos finite-size extrapolations, we obtain a general picture of D in the thermodynamic limit as a function of band filling and the ratio U/t. We note similarities with the low-frequency integrated spectral weight of certain hole-doped and electron-doped high-${\mathit{T}}_{\mathit{c}}$ compounds. We investigate the finite-frequency optical conductivity, finding structure at \ensuremath{\omega}\ensuremath{\approxeq}U and, for one hole off half filling, at \ensuremath{\omega}\ensuremath{\propto}${\mathit{t}}^{2}$/U. We find that minima in the energy of a Hubbard ring enclosing a flux \ensuremath{\Phi} occur with changes of half a flux quantum rather than a full flux quantum when the Coulomb repulsion is turned on. Lastly, we discuss the relationship of the Drude weight computed from open chains to that obtained from rings, and also the effect of arbitrary phases at the ring boundary.

DOI
10.1103/physrevb.44.6909

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We study the Drude weight, optical conductivity, and flux-periodicity properties of one-dimensional Hubbard chains using Bethe-ansatz, Lanczos, and exact-diagonalization techniques. We find that the Drude weight D is unexpectedly negative for half-filled Hubbard rings with N=4n sites reflecting a strong paramagnetic response, while it is positive (diamagnetic) for half-filled rings of N=4n+2 sites. In both cases at half filling, D vanishes exponentially with N on a length scale set by the inverse of the gap for small U/t. Near half filling, we find that D approaches a constant, positive, diamagnetic value as N increases, indicating metallic behavior. Combining Bethe-ansatz results near half filling with the known U=0 and U=\ensuremath{\infty} limits and Lanczos finite-size extrapolations, we obtain a general picture of D in the thermodynamic limit as a function of band filling and the ratio U/t. We note similarities with the low-frequency integrated spectral weight of certain hole-doped and electron-doped high-Tc{\mathit{T}}_{\mathit{c}}Tc​ compounds. We investigate the finite-frequency optical conductivity, finding structure at \ensuremath{\omega}\ensuremath{\approxeq}U and, for one hole off half filling, at \ensuremath{\omega}\ensuremath{\propto}t2{\mathit{t}}^{2}t2/U. We find that minima in the energy of a Hubbard ring enclosing a flux \ensuremath{\Phi} occur with changes of half a flux quantum rather than a full flux quantum when the Coulomb repulsion is turned on. Lastly, we discuss the relationship of the Drude weight computed from open chains to that obtained from rings, and also the effect of arbitrary phases at the ring boundary.
We use the Bethe-ansatz equations to calculate the total and zero-frequency spectral weight in the optical conductivity of the half-filled one-dimensional Hubbard model as a function of the lattice size L and the on-site repulsion U. The zero-frequency spectral weight \ensuremath{\pi}D scales as L1/2{\mathit{L}}^{1/2}L1/2exp(-L/\ensuremath{\xi}) as L\ensuremath{\rightarrow}\ensuremath{\infty}. Near U=0, \ensuremath{\xi} varies as the inverse of the Lieb-Wu charge gap. In the strongly correlated regime (U\ensuremath{\gg}t), ${\ensuremath{\xi}}^{\mathrm{\ensuremath{-}}1}$=ln(U/t)-1.48. $D--- is negative when L is a multiple of 4, corresponding to a negative inductance. We give a physical explanation of our results in terms of a simple model of ring exchange. The finite-size corrections to the total spectral weight scale as ${\mathit{L}}^{\mathrm{\ensuremath{-}}2}$. We discuss the implications of our results for exact diagonalization calculations of the optical conductivity.
We compare the optical-conductivity properties of one-dimensional repulsive-U and attractive-U Hubbard rings. First we determine the Drude weight, or ``charge stiffness,'' D as a function of band filling and the ratio U/t, where t is the overlap ``hopping'' integral. We note similarities in the repulsive-U case to experimental data both from hole-doped and from electron-doped high-Tc{\mathit{T}}_{\mathit{c}}Tc​ materials. Then we consider the real part of the finite-frequency conductivity, ${\mathrm{\ensuremath{\sigma}}}_{1}$(\ensuremath{\omega}). In the attractive-U case, we find an unexpected feature in ${\mathrm{\ensuremath{\sigma}}}_{1}$(\ensuremath{\omega}) associated with an unpaired electron moving through a background of bound pairs.
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.
We study the Drude weight D and optical conductivity of the two-dimensional (2D) Hubbard model at half filling with staggered magnetic flux (SMF). When SMF being introduced, the hopping integrals are modulated by the magnetic flux. The optical sum rule, which is related to the mean kinetic energy of band electrons, is evaluated for this 2D Hubbard Hamiltonian. Our present result gives the dependence of the kinetic energy, D and the optical conductivity on SMF and U. At half filling D vanishes exponentially with system size. We also find in the frequency dependence of the optical conductivity, there is δ-function peak at ω ≈ 2|m|U and the incoherent excitations begin to present themselves extended to a higher energy region.
We investigate the optical conductivity of a Hubbard ring in presence of an impurity by means of exact diagonalization using the Lanczos algorithm. We concentrate thereby on the first excited, open shell state, i.e. on twisted boundary conditions. In the metallic phase a substantial part of the spectral weight lies in the Drude peak, σ(ω)=Dδ(ω)+σregσ(ω) = Dδ(ω)+ σ_{\rm reg}σ(ω)=Dδ(ω)+σreg​. In the non-interacting system, the Drude peak can be visualized in our calculations at ω=0ω= 0ω=0 even for finite chain lengths. Adding the impurity, the main peak is shifted to finite frequencies proportional to the impurity strength. The shift indicates the energy gap of the disturbed finite size system, also in the interacting system. Thus, we can pursue in the optical conductivity for finite metallic systems the energy gap. However, due to level crossing, the impurity-induced peak arises in the interacting system first when a certain impurity strength is exceeded. In the Mott insulating phase, the impurity leads to an impurity state within the gap.
We study photoinduced optical responses of one-dimensional strongly correlated electron systems. The optical conductivity spectra are calculated for the ground state and a photoexcited state in the one-dimensional Hubbard model at half filling by using the exact diagonalization method. It is found that, in the Mott insulator phase, the photoexcited state has large spectral weights including the Drude weight below the optical gap. As a consequence, the spectral weight above the optical gap is largely reduced. These results imply that a metallic state is induced by photoexcitation. Comparison between the photoexcited and hole-doped states shows that the photoexcitation is similar to chemical doping.