Interdimensional Scaling Laws from Spatially Anisotropic Interactions

The matching of leading singular contributions to the different mean-field or critical behaviors of a system with long-but-finite-ranged spatially anisotropic interactions leads to "scaling laws" connecting criticial exponents for different dimensions. The results are exact for mean-field theory (MFT) and spherical model exponents and predict MFT exponents for $d=4\mathrm{}$. From the known $d=2\mathrm{}$ Ising-model exponents, the scaling laws give $\alpha =0\mathrm{}$, $\beta =\frac{4}{11}$, $\gamma =\frac{14}{11}$, $\nu =\frac{2}{3}$ for $d=3\mathrm{}$. Agreement with previous results for the $d=2,3\mathrm{}$ excluded volume problem is also quite good. For the $X-Y$ and Heisenberg models the results predict $\gamma \approx 2.0 \mathrm{and} 2.4$, respectively, for $d=2\mathrm{}$.