Finite-size effects on the optical conductivity of a half-filled Hubbard ring.

Abstract

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 ${\mathit{L}}^{1/2}$exp(-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.