Interdimensional Scaling Laws

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, ($d-1\mathrm{}$)-dimensional mean field, ($d-1\mathrm{}$)-dimensional critical. By matching its behavior in the various regions we obtain equalities connecting the critical indices in $d$ and $d-1\mathrm{}$ 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: $\nu =\frac{2}{3}$, $\alpha =0\mathrm{}$, $\beta =\frac{3}{8}$, and $\gamma =\frac{5}{4}$.