Critical exponent for the magnetization of the weakly coupled $\phi^4_4$ model

TL;DR

This paper rigorously proves that the magnetization in the weakly coupled $\,\phi^4_4$ model exhibits a logarithmic correction to mean field scaling as the magnetic field approaches zero, confirming a physicists' conjecture.

Contribution

The authors provide the first rigorous proof of the logarithmic correction to mean field behavior in the critical magnetization of the $\,\phi^4_4$ model.

Findings

Magnetization scales as $(h\log h^{-1})^{1/3}$ as $h \to 0$

Logarithmic correction to mean field scaling confirmed

Uses Gawedzki and Kupiainen's critical theory construction and cluster expansion

Abstract

We consider the weakly coupled $\phi^4 $ theory on $\mathbb Z^4 $, in a weak magnetic field $h$, and at the chemical potential $\nu_c $ for which the theory is critical if $h=0$. We prove that, as $h\to 0$, the magnetization of the model behaves as $(h\log h^{-1})^{\frac 13} $, and so exhibits a logarithmic correction to mean field scaling behavior. This result is well known to physicists, but had never been proven rigorously. Our proof uses the classic construction of the critical theory by Gawedzki and Kupiainen, and a cluster expansion with large blocks.