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

The analytical forms of free-electron susceptibilities chi(d)(q) and their range functions 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 chi(d)(q) and 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 chi(2)(q) and chi(2)(r) for a wide range of temperature. Our analytical results show that irrespective of dimension, the singular behavior of chi(d)(q) at q = +/-2k(F) becomes suppressed at nonzero temperature as the singular points transit to complex wave vectors 2k(0)(+/-) and this transition causes chi(d)(r) to be exponentially damped with common damping factor e(-2 eta 0sin phi rkfr)similar to e(e-pi T'kFr) for low enough temperature, where the exponent of the damping factor corresponds to an imaginary part of wave vectors 2k(0)(+/-). We also show that the causality relation of the response function is essential in understanding the correct behavior of chi(d)(q) and chi(d)(r) for all dimensions. [S0163-1829(99)12505-0].