Optical-conductivity properties of one-dimensional Hubbard rings: Repulsive-Uand attractive-Ucases

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-${\mathit{T}}_{\mathit{c}}$ materials. Then we consider the real part of the finite-frequency conductivity, ${\mathrm{\sigma }}_1$(ω). In the attractive-U case, we find an unexpected feature in ${\mathrm{\sigma }}_1$(ω) associated with an unpaired electron moving through a background of bound pairs.