Self Consistent Screening Approximation For Critical Dynamics

TL;DR

This paper extends Bray's self-consistent screening approximation to analyze the critical dynamics of the $^4$ theory, providing numerical estimates of the dynamical exponent $z$ that align with recent simulations.

Contribution

It introduces a generalized approximation method for critical dynamics and constrains the scaling functions to compute the dynamical exponent $z$.

Findings

Numerical values of $z$ for $d=3$, $n=1,...,10$ are obtained.

The value $z \u2248 2.115$ for 3D Ising matches recent Monte-Carlo results.

The method provides consistent estimates across different ansatz.

Abstract

We generalise Bray's self-consistent screening approximation to describe the critical dynamics of the $\phi^4$ theory. In order to obtain the dynamical exponent $z$, we have to make an ansatz for the form of the scaling functions, which fortunately can be much constrained by general arguments. Numerical values of $z$ for $d=3$, and $n=1,...,10$ are obtained using two different ans\"atze, and differ by a very small amount. In particular, the value of $z \simeq 2.115$ obtained for the 3-d Ising model agrees well with recent Monte-Carlo simulations.