The non-perturbative renormalization group in the ordered phase

TL;DR

This paper analyzes the non-perturbative renormalization group equations for a scalar field with Z2 symmetry in the ordered phase, highlighting the importance of the cut-off function's properties for physical discontinuities.

Contribution

It identifies specific mathematical conditions on the cut-off function necessary for correctly reproducing physical discontinuities in the ordered phase.

Findings

Sharp cut-off fails to satisfy the conditions in dimensions d<4.

Physical discontinuity of magnetic susceptibility is reproduced only with certain cut-off functions.

Conditions are explicitly derived for the cut-off function in the local potential approximation.

Abstract

We study some analytical properties of the solutions of the non perturbative renormalization group flow equations for a scalar field theory with $Z_2$ symmetry in the ordered phase, i.e. at temperatures below the critical temperature. The study is made in the framework of the local potential approximation. We show that the required physical discontinuity of the magnetic susceptibility $\chi(M)$ at $M=\pm M_0$ ($M_0$ spontaneous magnetization) is reproduced only if the cut-off function which separates high and low energy modes satisfies to some restrictive explicit mathematical conditions; we stress that these conditions are not satisfied by a sharp cut-off in dimensions of space $d<4$.