Observing Golden Mean Universality Class in the Scaling of Thermal Transport

We address the issue of whether the golden mean $\left[\psi=(\sqrt{5}+1)/2 \simeq 1.618\right]$ universality class, as predicted by several theoretical models, can be observed in the dynamical scaling of thermal transport. Remarkably, we show estimate with unprecedented precision, that $\psi$ appears to be the scaling exponent of heat mode correlation in a purely quartic anharmonic chain. This observation seems somewhat deviation from the previous expectation and we explain it by the unusual slow decay of the cross-correlation between heat and sound modes. Whenever the cubic anharmonicity is included, this cross-correlation is gradually died out and another universality class with scaling exponent $\gamma =5\mathrm{/}3$, as commonly predicted by theories, seems recovered. However, this recovery is accompanied by two interesting phase transition processes characterized by a change of symmetry of the potential and a clear variation of the dynamic structure factor, respectively. Due to these transitions, an additional exponent close to $\gamma \simeq 1.580$ emerges. All these evidences suggest that, to gain a full prediction of the scaling of thermal transport, more ingredients should be taken into account.