Observing golden-mean universality class in the scaling of thermal transport

We address the issue of whether the golden-mean $[\psi =(\sqrt{5}+1)/2\simeq 1.618]$ universality class, as predicted by several theoretical models, can be observed in the dynamical scaling of thermal transport. Remarkably, we show strong evidence that $\psi$ appears to be the scaling exponent of heat mode correlation in a purely quartic anharmonic chain. This observation seems to somewhat deviate 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 gradually dies out and another universality class with scaling exponent $\gamma =5/3$, 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 this evidence suggests that, to gain a full prediction of the scaling of thermal transport, more ingredients should be taken into account.

Figure 1

The purely quartic anharmonic chain: (a) [(c)] ${\rho }_Q(m,t)$ [rescaled ${\rho }_Q(m,t)]$ for three typical long times; (b) [(d)] The height $H_c^Q \left[H_s^Q\right]$ of the central (side) peak of ${\rho }_Q(m,t)$ vs $t$. The inset in panel (c) is a zoom for the right-side peaks; the inset in panel (d) shows the same plot as panel (d) but with a longer time $t=1900$.

Figure 2

The purely quartic anharmonic chain: the $3\times 3$ correlation functions of heat and sound modes, for three long times (the same below): $t=600$ (dotted), $t=800$ (dashed), and $t=1000$ (solid).

Figure 3

The purely quartic anharmonic chain: Rescaled $C_{00}(m,t)$ for three long times. The left (right) inset shows the height $H_c^{c_{00}}$ ($H_s^{c_{00}}$) of the central (side) peak of $C_{00}(m,t)$ vs $t$.

Figure 4

The purely quartic anharmonic chain: (a) Temperature profiles for four long $L$. (b) Measured $L$ dependence of $\kappa$.

Figure 5

Potentials of cubic-plus-quartic anharmonic chain for different $A$.

Figure 6

Cubic-plus-quartic anharmonic chains: Rescaled ${\rho }_Q(m,t)$ ($T=0.5$) for three typical long times (here and below, the same as Fig. 1) for (a) $A=0.25$, (b) $A=0.75$, (c) $A=1$, and (d) $A=1.75$, with measured average pressures $\langle F\rangle \simeq -0.176$, $\langle F\rangle \simeq -0.543$, $\langle F\rangle \simeq -0.744$, and $\langle F\rangle \simeq -1.503$, respectively. The insets are used as a zoom for the side parts (right).

Figure 7

Cubic-plus-quartic anharmonic chains: $\gamma$ vs $A$. The error bars are smaller than the symbol size.

Figure 8

The cubic-plus-quartic anharmonic chain ($A=0.25$): the same plot as Fig. 2.

Figure 9

Cubic-plus-quartic anharmonic chains: ${\rho }_g(m,t)$ ($T=0.5$) for three typical long times for (a) $A=0.25$, (b) $A=0.75$, (c) $A=1$, and (d) $A=1.75$. The inset in panel (d) shows the rescaled central parts of ${\rho }_g(m,t)$ under formula (3) with $\gamma =5/3$.