Wilson-Polchinski exact renormalization group equation for O(N) systems: Leading and next-to-leading orders in the derivative expansion

TL;DR

This paper investigates the convergence of the derivative expansion in the exact renormalization group equation for O(N) models by calculating critical exponents at leading and next-to-leading orders and comparing with perturbative results.

Contribution

It provides a detailed analysis of the derivative expansion's convergence properties for the Wilson-Polchinski RG equation applied to O(N) models, including a study of reparametrization invariance.

Findings

Critical exponents are estimated for N=1 to 20 in three dimensions.

Comparison shows the derivative expansion converges towards perturbative results.

Reparametrization invariance is examined and its preservation is discussed.

Abstract

With a view to study the convergence properties of the derivative expansion of the exact renormalization group (RG) equation, I explicitly study the leading and next-to-leading orders of this expansion applied to the Wilson-Polchinski equation in the case of the $N$-vector model with the symmetry $\mathrm{O}(N) $. As a test, the critical exponents $% \eta $ and $\nu $ as well as the subcritical exponent $\omega $ (and higher ones) are estimated in three dimensions for values of $N$ ranging from 1 to 20. I compare the results with the corresponding estimates obtained in preceding studies or treatments of other $\mathrm{O}(N) $ exact RG equations at second order. The possibility of varying $N$ allows to size up the derivative expansion method. The values obtained from the resummation of high orders of perturbative field theory are used as standards to illustrate the eventual convergence in each case. A peculiar attention is drawn on the preservation (or not) of the reparametrisation invariance.