Self-consistent effective-medium approximations with path integrals

We study effective-medium approximations for linear composite media by means of a path integral formalism with replicas. We show how to recover the Bruggeman and Hori-Yonezawa effective-medium formulas. Using a replica-coupling ansatz, these formulas are extended into ones which have the same percolation thresholds as those of the Bethe lattice and Potts model of percolation, and critical exponents s=0 and t=2 in any space dimension d>/=2. Like the Bruggeman and Hori-Yonezawa formulas, the obtained formulas are exact to second order in the weak-contrast and dilute limits. The dimensional range of validity of the four effective-medium formulas is discussed, and it is argued that out formulas are of better relevance than the classical ones in dimensions d=3,4 for systems obeying the nodes-links-blobs picture, such as random-resistor networks.