Approximate Renormalization Group Based on the Wegner-Houghton Differential Generator

We give an approximate renormalization-group formulation which parallels that of Wilson. The group generator represents the momentum-independent limit of the differential generator of Wegner and Houghton. The eigenfunctions near the Gaussian point are computed for all spin dimensions $n$ and lattice dimensions $d$, including $d=2\mathrm{}$. The nontrivial fixed-point Hamiltonian in dimensions near $d=\frac{2\mathcal{O}}{(\mathcal{O}-1)}$, together with the eigenvalues near that nontrivial fixed point, are found explicitly to first order in ${\epsilon }_{\mathcal{O}}\equiv \mathcal{O}\mathrm{}(2-d)+d$ for all values of $n$ and the order $\mathcal{O}$. Odd-dominated Ising systems and corresponding expansions in ${\epsilon }_{\mathcal{O}-\frac{1}{2}}$ are also treated.