Finite-Size Scaling and Correlation Lengths for Disordered Systems

Abstract

For a large class of $d$-dimensional disordered systems, we prove that if an appropriately defined finite-size scaling correlation length diverges at a nontrivial value of the disorder with an exponent $\ensuremath{\nu}$, then $\ensuremath{\nu}$ must satisfy the bound $\ensuremath{\nu}\ensuremath{\ge}\frac{2}{d}$. Given the assumption that such a correlation length can be defined, the result applies to, e.g., percolation, disordered magnets, and Anderson localization, both with and without interactions. For localization, this puts stringent constraints on scaling theories and interpretation of experiments.