Exact and approximate differential renormalization-group generators

In dealing with critical phenomena of complex systems that simulate realistic materials, the full structure of the renormalization group is often unnecessarily cumbersome. For approximate calculations and for systems with special properties, specialized generators are simpler to apply. We derive several such exact and approximate differential generators and solve a number of interesting practical problems to illustrate this approach: (i) We derive a new approximate differential generator based on the Wilson incomplete-integration generator. Using this generator we calculate for an $n$-component spin system the eigenvalues (critical-point exponents) associated with a critical point of arbitrary order $\mathcal{O}$ and "propagator exponent" $\tilde{\sigma }$ to first order in the expansion parameter ${\epsilon }_{\mathcal{O}}\left(\tilde{\sigma }\right)=d+\mathcal{O}(\tilde{\sigma }-d)$; this extends previous work for $\tilde{\sigma }=2\mathrm{}$, $\mathcal{O}$ arbitrary; $\tilde{\sigma }\le 2$, $\mathcal{O}=2\mathrm{}$ (long-rang forces); and $\tilde{\sigma }=4\mathrm{}$, $\mathcal{O}=2\mathrm{}$ (the "Lifshitz point"). Our results agree with those obtainable using an approximate generator based on the Wegner-Houghton equation. The cases $\tilde{\sigma }=2L$ ($L$ a positive integer ≥2) describe the onset of helical ordering for which $\left|\overrightarrow{\mathrm{k}}\right|\sim {\mathrm{}(-p)\mathrm{}}^{{\beta }_k}$, where ${\beta }_k=\frac{1}{2(L-1\mathrm{})\mathrm{}}+o\left({\epsilon }_{\mathcal{O}}^2\mathrm{}\right(2L\left)\right)$ and $p$ parametrizes the hypersurface of critical points. For $p>\mathrm{}0\mathrm{}$, the ordered phase is uniform; for $p<\mathrm{}0\mathrm{}$ there is spiral order. The point $p=0\mathrm{}$, at which such nonuniform ordering commences, we term a generalized Lifshitz point of Lifschitz character $L$. (ii) We consider the full Wilson and Wegner-Houghton generators in the paired spin-momenta limit and the $n\to \infty$ limit for even-order critical points. These limiting generators are identical for both full generators. This demonstrates that at least in these cases the Wilson and Wegner-Houghton generators agree exactly, without recourse to perturbation theory. These simple exact generators should provide "anchors" for calculations of exponents for higher-order critical points. (iii) We derive approximate generators which are suitable for compressible magnetic systems and more general systems with constraints for which the spin momenta are grouped in any arbitrary manner. We apply this to the case of a simple compressible magnet model and obtain the exact renormalization-group trajectories to order $\epsilon$ with