Feynman propagator, density matrices and Green functions for the inhomogeneous electron liquid generated by a bare Coulomb potential in two dimensions

Abstract

For a two-dimensional homogeneous electron gas, the canonical density matrix is well-known. This object is related to the Feynman propagator K(r,r′;t), where t is the time, by the transform β→it. From the free electron form of C(r,r′;β), the Green function follows in terms of the Bessel function K 0. When a bare Coulomb potential -Ze2/r is now “switched on”, one known property is the local density of states at the nucleus. This enables the imaginary part Im, G of the Green function at the nucleus to be determined as an explicit function of energy E and nuclear charge Ze. Off-diagonal information on Im, G will yield the real part of the Green function by using the Kramers–Krönig relation. The analysis of the two-dimensional Green function G into partial waves characterized by angular momentum quantum number ℓ is then considered. The imaginary part of G for ℓ = 0 is determined in terms of a hypergeometric function. The real part is again in principle accessible by invoking the Kramers–Krönig relation. From the relation between G and the Laplace transform of C with respect to β, information is also obtained on the ℓ = 0 partial wave component of the Slater sum and hence the Feynman propagator on the diagonal, in the limiting case Z→0.