Superconductivity exponents in two- and three-dimensional percolation

Abstract

The first transfer-matrix calculation of the superconductivity exponent $s$ of a random mixture of normal and superconducting elements is presented: The exponent $s$ is defined through the divergence of the conductivity $\Sigma$ as the critical fraction $p_c$ of superconducting elements is approached: $\Sigma \sim {(p-p_c)}^{-s}$. We obtain very accurate values for the exponents which disagree with the Alexander-Orbach conjecture as well as other conjectures. Our results are $\frac{s}{\nu }=0.977\mathrm{}\pm 0.010$ in two dimensions and $\frac{s}{\nu }=0.85\mathrm{}\pm 0.04$ in three dimensions.