Analytic properties of the specific macroscopic conductivity of a composite medium

Bergman has observed that the specific conductivity of a two-component composite medium, regarded as a function of the conductivity ratio of the two components, has a branch cut on the negative real axis (see AIP Conf. Proc., no.40, p.46 (1978)). He has further proposed that the singular behaviour of the conductivity near the conduction threshold can be understood in terms of the p-dependence of this branch cut. It is shown that the parametric representation of the singular part of Sigma contains a branch cut which behaves in agreement with Bergman's predictions. The amplitude of the discontinuity of Sigma is calculated for the two-dimensional square lattice by an adaptation of the position-space renormalisation method, and explicitly for 5*5 two-dimensional squares. The resulting spectral function is found to have structure unrelated to the critical behaviour.