Functional truncations for the solution of the nonperturbative RG equations

Abstract

Abstract We consider the Wetterich exact renormalization group (RG) equation. Approximate closed equations are obtained from it, applying certain truncation schemes for the effective average action. These equations are solved either purely numerically or by certain extra truncations for the potential and related quantities, called the functional truncations. Traditionally, the functional truncations consist of truncated expansions in powers of ρ ˜ − ρ ˜ 0 , where ρ ˜ ∝ ϕ 2 , φ is the averaged order parameter, and ρ ˜ 0 corresponds to the minimum of the dimensionless potential u k ( ρ ˜ ) , depending on the infrared cut-off scale k. We propose a new approach of functional truncations, using the expansion u k ( ρ ˜ ) − u k ( 0 ) = ( 1 − s ) − μ u 1 , k s + u 2 , k s 2 + ⋯ , where s = ρ ˜ / ( ρ ˜ 0 + ρ ˜ ) , ρ ˜ 0 is an optimization parameter and µ is the exponent, describing the ρ ˜ → ∞ asymptotic. The newly developed method provides accurate estimates of the critical exponents η, ν and also ω. In the case of the local potential approximation (LPA), the discrepancy with the purely numerical solution is 0.000 15 for ν and 0.0012 for ω at the s 13 order of truncation. We show that this method is advantageous for estimations beyond the LPA, especially, for the critical exponent ω. In particular, we have obtained η = 0.0454 ( 1 ) , ν = 0.6292 ( 2 ) , and ω = 0.8606 ( 30 ) for the equation originally derived in 2020 J. Phys. A: Math. Theor. 53 415002 within a new truncation scheme for the effective action. The approach can also be used for the solution of other nonperturbative RG models that have a broad range of applications.