Field theory of bi- and tetracritical points: Relaxational dynamics

TL;DR

This paper investigates the relaxational dynamical critical behavior of systems with $O(n_ ext|) imes O(n_ot)$ symmetry using renormalization group methods, revealing different universality classes and dynamic scaling behaviors.

Contribution

It provides a two-loop order calculation of dynamical critical behavior for systems with complex symmetry, identifying three distinct dynamical universality classes and analyzing their scaling properties.

Findings

Strong dynamic scaling at Heisenberg and biconical fixed points.

Weak dynamic scaling near the decoupled fixed point.

Presence of a very small transient exponent (~0.0044) for $n_ ext|=1$, $n_ot=2$ in 3D.

Abstract

We calculate the relaxational dynamical critical behavior of systems of $O(n_\|)\oplus O(n_\perp)$ symmetry by renormalization group method within the minimal subtraction scheme in two loop order. The three different bicritical static universality classes previously found for such systems correspond to three different dynamical universality classes within the static borderlines. The Heisenberg and the biconical fixed point lead to strong dynamic scaling whereas in the region of stability of the decoupled fixed point weak dynamic scaling holds. Due to the neighborhood of the stability border between the strong and the weak scaling dynamic fixed point corresponding to the static biconical and the decoupled fixed point a very small dynamic transient exponent, of $\omega_v^{{\cal B}}=0.0044$, is present in the dynamics for the physically important case $n_\|=1$ and $n_\perp=2$ in $d=3$.