Equivalence of Local Potential Approximations

TL;DR

This paper demonstrates that the local potential approximations of two different flow equations are exactly equivalent due to an underlying exact map, clarifying their relationship in the context of the exact renormalization group.

Contribution

It reveals the fundamental reason for the equivalence of local potential approximations and discusses limitations of the optimized cutoff in derivative expansions.

Findings

Exact map between Legendre and Wilson-Polchinski flow equations

Equivalence of local potential approximations explained

Optimized cutoff restricts derivative expansion to second order

Abstract

In recent papers it has been noted that the local potential approximation of the Legendre and Wilson-Polchinski flow equations give, within numerical error, identical results for a range of exponents and Wilson-Fisher fixed points in three dimensions, providing a certain ``optimised'' cutoff is used for the Legendre flow equation. Here we point out that this is a consequence of an exact map between the two equations, which is nothing other than the exact reduction of the functional map that exists between the two exact renormalization groups. We note also that the optimised cutoff does not allow a derivative expansion beyond second order.