The self-consistent effective medium approximation (SEMA): New tricks from an old dog

The fact that the self-consistent effective medium approximation (SEMA) leads to incorrect values for the percolation threshold, as well as for the critical exponents which characterize that threshold, has led to a decline in using that approximation. In this article I argue that SEMA has the unique capability, which is lacking in other approximation schemes for macroscopic response of composite media, of leading to the discovery or prediction of new critical points. This is due to the fact that SEMA can often lead to explicit equations for the macroscopic response of a composite medium, even when that medium has a rather complicated character. In such cases, the SEMA equations are usually coupled and nonlinear, often even transcendental in character. Thus there is no question of finding exact solutions. Nevertheless, a useful ansatz, leading to a closed form asymptotic solution, can often be made. In this way, singularities in the macroscopic response can be identified from a theoretical or mathematical treatment of the physical problem. This is demonstrated for two problems of magneto-transport in a composite medium, where the SEMA equations are solved using asymptotic analysis, leading to new types of critical points and critical behavior.