Interface Fluctuations in Disordered Systems:5−εExpansion and Failure of Dimensional Reduction

A functional renormalization-group $5-\epsilon$ expansion is used to calculate the exponent $\zeta$ which measures the roughness of an interface in a disorded Ising system as a function of length scale. Formal dimensional reduction is explicitly shown to break down at leading order in $\epsilon$ and field-theoretic methods fail. For random-field disorder $\zeta =\frac{\epsilon }{3}$ is obtained which resolves the discrepancy between various previous calculations; for random-bond disorder, $\zeta \approx 0.2083\epsilon$ to leading order.