We review the Exact Renormalization Group equations of Wegner and Houghton in an approximation which permits both numerical and analytical studies of nonperturbative renormalization flows. We obtain critical exponents numerically and with the local polynomial approximation (LPA), and discuss the advantages and shortcomings of these methods, and compare our results with the literature. In particular, convergence of the LPA is discussed in some detail. We finally integrate the flows numerically and find a
c-function which determines these flows to be gradient in this approximation.
Equations related to the Polchinski version of the exact renormalization group (RG) equations for scalar fields which extend the local potential approximation to first order in a derivative expansion, and which maintain reparameterization invariance, are postulated. Reparameterization invariance ensures that the equations determine the anomalous dimension η unambiguously and the equations are such that the result is exact to O (ε 2 ) in an ε-expansion for any multi-critical fixed point. It is also straightforward to determine η numerically. When the dimension d = 3 numerical results for a wide range of critical exponents are obtained in theories with O ( N ) symmetry, for various N and for ranges of η, are obtained within the local potential approximation. The associated η, which follows from the derivative approximation described here, is found for various N . The large N limit of the equations is also analysed. A corresponding discussion is also given in a perturbative RG framework and scaling dimensions for derivative operators are calculated to first order in ε.
The Polchinski version of the exact renormalisation group equations is applied to multicritical fixed points, which are present for dimensions between two and four, for scalar theories using both the local potential approximation and its extension, the derivative expansion. The results are compared with the epsilon expansion by showing that the non linear differential equations may be linearised at each multicritical point and the epsilon expansion treated as a perturbative expansion. The results for critical exponents are compared with corresponding epsilon expansion results from standard perturbation theory. The results provide a test for the validity of the local potential approximation and also the derivative expansion. An alternative truncation of the exact RG equation leads to equations which are similar to those found in the derivative expansion but which gives correct results for critical exponents to order ε and also for the field anomalous dimension to order ε 2. An exact marginal operator for the full RG equations is also constructed.
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. The nontrivial fixed-point Hamiltonian in dimensions near $d=\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)}$, together with the eigenvalues near that nontrivial fixed point, are found explicitly to first order in ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}\mathcal{O}(2\ensuremath{-}d)+d$ for all values of
n and the order
O. Odd-dominated Ising systems and corresponding expansions in ${\ensuremath{\epsilon}}_{\mathcal{O}\ensuremath{-}\frac{1}{2}}$ are also treated.
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
O and "propagator exponent" $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}$ to first order in the expansion parameter ${\ensuremath{\epsilon}}_{\mathcal{O}}(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}})=d+\mathcal{O}(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}\ensuremath{-}d)$; this extends previous work for $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=2$,
O arbitrary; $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}\ensuremath{\le}2$,
O=2 (long-rang forces); and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=4$,
O=2 (the "Lifshitz point"). Our results agree with those obtainable using an approximate generator based on the Wegner-Houghton equation. The cases $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=2L$ (
L a positive integer \ensuremath{\ge}2) describe the onset of helical ordering for which $|\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}|\ensuremath{\sim}{(\ensuremath{-}p)}^{{\ensuremath{\beta}}_{k}}$, where ${\ensuremath{\beta}}_{k}=\frac{1}{2(L\ensuremath{-}1)}+o({\ensuremath{\epsilon}}_{\mathcal{O}}^{2}(2L))$ and
p parametrizes the hypersurface of critical points. For
p>0, the ordered phase is uniform; for
p<0 there is spiral order. The point
p=0, 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 $n\ensuremath{\rightarrow}\ensuremath{\infty}$ 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 $\ensuremath{\epsilon}$ with $\ensuremath{\epsilon}\ensuremath{\equiv}{\ensuremath{\epsilon}}_{2}(2)=4\ensuremath{-}d$.