We study phonon relaxation in chains of particles coupled through polynomial-type pair-interaction potentials and obeying quantum dynamics. We present detailed calculations for the sixth-order potential and find that the wave-vector-dependent relaxation rate follows a power-law behavior, $\Gamma \left(q\right)\sim q^{\delta }$, with $\delta =5/3$, which is identical to that of the fourth-order potential. We argue through diagrammatic analysis that this is a generic feature of even-power potentials. Our earlier analysis has shown that $\delta =3/2$ when the leading-order term in the nonlinear potential is odd, suggesting that there are two universality classes for the phonon relaxation rates dependent on a simple property of the potential. This implies that the thermal conductivity $\kappa$ which diverges as a function of chain size $N$ as $\kappa \propto N^{\alpha }$ also has two universal behaviors, in that $\alpha =1-1/\delta$ as follows from a finite-size argument. We support these arguments by numerical calculations of conductivity for chains obeying classical dynamics for polynomial potentials of some even and odd powers.

• Received 23 May 2011

DOI:https://doi.org/10.1103/PhysRevE.84.041119