Interdimensional Scaling Laws

Abstract

We consider in detail the critical behavior of a $d$-dimensional system that is finite in one of its dimensions. Such a system can exhibit four different types of critical behavior: $d$-dimensional mean field, $d$-dimensional critical, ($d\ensuremath{-}1$)-dimensional mean field, ($d\ensuremath{-}1$)-dimensional critical. By matching its behavior in the various regions we obtain equalities connecting the critical indices in $d$ and $d\ensuremath{-}1$ dimensions. Assuming the usual two-dimensional critical indices of the Ising model, these relations lead to the following indices for the three-dimensional Ising model: $\ensuremath{\nu}=\frac{2}{3}$, $\ensuremath{\alpha}=0$, $\ensuremath{\beta}=\frac{3}{8}$, and $\ensuremath{\gamma}=\frac{5}{4}$.