Temperature-dependent free-electron susceptibility for one, two, and three dimensions

Abstract

The analytical forms of free-electron susceptibilities ${\ensuremath{\chi}}_{d}(q)$ and their range functions ${\ensuremath{\chi}}_{d}(r)$ are derived at nonzero temperature starting from the Green's function representation by properly evaluating the contributions from the poles of free-electron temperature Green's function for each dimension $d=1,$ 2, and 3. The present formalism produces not only ${\ensuremath{\chi}}_{d}(q)$ and ${\ensuremath{\chi}}_{d}(r)$ which show more accurate temperature-dependent behavior than our previous results for $d=1$ and 3, but also temperature-dependent two-dimensional ${\ensuremath{\chi}}_{2}(q)$ and ${\ensuremath{\chi}}_{2}(r)$ for a wide range of temperature. Our analytical results show that irrespective of dimension, the singular behavior of ${\ensuremath{\chi}}_{d}(q)$ at $q=\ifmmode\pm\else\textpm\fi{}{2k}_{F}$ becomes suppressed at nonzero temperature as the singular points transit to complex wave vectors ${2k}_{0}^{\ifmmode\pm\else\textpm\fi{}},$ and this transition causes ${\ensuremath{\chi}}_{d}(r)$ to be exponentially damped with common damping factor ${e}^{\ensuremath{-}2{\ensuremath{\eta}}_{0}\mathrm{sin}{\ensuremath{\varphi}}_{r}{k}_{F}r}\ensuremath{\sim}{e}^{\ensuremath{-}\ensuremath{\pi}{T}^{\ensuremath{'}}{k}_{F}r}$ for low enough temperature, where the exponent of the damping factor corresponds to an imaginary part of wave vectors ${2k}_{0}^{\ifmmode\pm\else\textpm\fi{}}.$ We also show that the causality relation of the response function is essential in understanding the correct behavior of ${\ensuremath{\chi}}_{d}(q)$ and ${\ensuremath{\chi}}_{d}(r)$ for all dimensions.