New small RG parameter

Effective medium approach in the renormalization group theory of phase transitions

“…(In the limit of weak coupling all approximations reproduce correctly the lowest-order perturbative terms so, on the one hand, the perturbation theory justifies them in this limit, on the other hand, makes them superfluous.) The gradient expansion [22,23] frequently invoked to substantiate LPA is inapplicable in lattice models, so only heuristic arguments, such as those presented below, can be given to qualitatively understand, at least qualitatively, why LPA gives reasonable quantitative approximation to for such models, as will be illustrated by numerical calculations in section 5.…”

“…Thus, approximation ∆ ij ≈ δ ij in (24) may be quite consistent in this particular case. Besides, the gradient expansion [22,23] which in the momentum space is equivalent to expansion in powers of k m may also be invoked to substantiate LPA in the region of small momenta.…”

Exact renormalization group equations: an introductory review

“…How to deal with integro-differential equations is not known in general. In the case of the RG equations, one often had recourse to perturbative expansions such as the usual perturbation in powers of the coupling but also the famous ε-expansion (where ε = 4−d), the 1/N -expansion or expansions in the dimensionality 2 − d, let us mention also an expansion exploiting the smallness of the critical exponent 17 η [50].…”

Critical exponents predicted by grouping of Feynman diagrams in 4 model

“…Nowadays, an explicit ǫ-expansion is available up to the fifth order [10,11,12,13]. The 1/n expansion of critical exponents [3,4,14] is based on a similar idea with the only essential difference that 1/n appears as an expansion parameter (at large n) instead of ǫ. Alternatively, it has been proposed [15] to expand the critical exponents in terms of the renormalized coupling constant at a fixed dimension d = 3. Later this method has been developed by several authors [16,17,18,19,20,21].…”