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 $\ensuremath{\epsilon}\ensuremath{\equiv}{\ensuremath{\epsilon}}_{2}(2)=4\ensuremath{-}d$.
We give the first nonlinear solution of renormalization-group equations. This solution, based on the differential generator of Wegner and Houghton, exhibits an explicit mixing of (or crossover between) critical and mean-field behavior. The solution is given for all values of the spin dimension
n and to first order in $\ensuremath{\epsilon}\ensuremath{\equiv}4\ensuremath{-}d$, where
d is the lattice dimensionality.
We present a detailed derivation of the one-particle-irreducible (1PI) differential renormalization-group generators originally developed by Nicoll and Chang and by Chang, Nicoll, and Young. We illustrate the machinery of the irreducible formulation by calculating to order ${\ensuremath{\epsilon}}^{2}$ the characteristic time exponent
z for the time-dependent Ginsburg-Landau model in the cases of conserved and nonconserved order parameter. We then calculate both
z and $\ensuremath{\eta}$ to order ${\ensuremath{\epsilon}}^{2}$ by applying to the 1PI generator an extension of the operator expansion technique developed by Wegner for the Wilson smooth-cutoff renormalization-group generator.
The non-perturbative Wegner-Houghton renormalization group is analyzed by the local potential approximation in O(N) scalar theories in d-dimensions (3 ≤ d ≤ 4). The leading critical exponents ν are calculated in order to investigate the effectiveness of the local potential approximation by comparing them with other non-perturbative methods. We show analytically that the local potential approximation gives the exact exponents up to O(ε) in ε-expansion and the leading in 1/N-expansion. We claim that this approximation offers fairly accurate results in the whole range of the parameter space of N and d. It is a great advantage of our method that no diverging expansions appear in the procedure.
Abstract Actual computations of fixed points and eigenvalues usually involve approximations, often crude ones, except for a very limited number of simple cases such as the one-dimensional Ising model of the previous chapter. In real- and momentum-space renormalization group theory, there are no general prescriptions to systematically improve the degree of the approximation with a modest amount of effort. There are established methods to systematically improve precision, but they usually need a large amount of numerical calculations. The scope of the present chapter is modest as we limit ourselves to basic examples, including the epsilon expansion about the Gaussian fixed-point of the Landau-Ginzburg-Wilson model. Finally, the last section illustrates the extension of the renormalization group framework to study quantum phase transitions.