Universality classes for phonon relaxation and thermal conduction in one-dimensional vibrational systems

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, Γ(q)∼q(δ), with δ=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 δ=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 κ which diverges as a function of chain size N as κ∝N(α) also has two universal behaviors, in that α=1-1/δ 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.