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 $\ensuremath{\Sigma}$ as the critical fraction ${p}_{c}$ of superconducting elements is approached: $\ensuremath{\Sigma}\ensuremath{\sim}{(p\ensuremath{-}{p}_{c})}^{\ensuremath{-}s}$. We obtain very accurate values for the exponents which disagree with the Alexander-Orbach conjecture as well as other conjectures. Our results are $\frac{s}{\ensuremath{\nu}}=0.977\ifmmode\pm\else\textpm\fi{}0.010$ in two dimensions and $\frac{s}{\ensuremath{\nu}}=0.85\ifmmode\pm\else\textpm\fi{}0.04$ in three dimensions.