Anomalous transport and phonon renormalization in a chain with transverse and longitudinal vibrations

TL;DR

This paper investigates quantum thermal transport in a coupled atomic chain with both longitudinal and transverse vibrations, revealing mode-dependent phonon relaxation rates and a sublinear size dependence of thermal conductivity.

Contribution

It introduces a quantum mechanical model for coupled atomic chains with anharmonic interactions, deriving mode-specific phonon relaxation rates and their impact on thermal conductivity scaling.

Findings

Longitudinal phonon relaxation rate scales as q^{3/2}.

Transverse phonon relaxation rate scales as q^2.

Thermal conductivity scales as the square root of chain length.

Abstract

We study thermal transport in a chain of coupled atoms, which can vibrate in longitudinal as well as transverse directions. The particles interact through anharmonic potentials upto cubic order. The problem is treated quantum mechanically. We first calculate the phonon frequencies self-consistently taking into account the anharmonic interactions. We show that for all the modes, frequencies must have linear dispersion with wave-vector $q$ for small $q$ irrespective of their bare dispersions. We then calculate the phonon relaxation rates $\Gamma_i(q)$, where $i$ is the polarization index of the mode, in a self-consistent approximation based on second order perturbation diagrams. We find that the relaxation rate for the longitudinal phonon, $\Gamma_x(q) \propto q^{3/2}$, while that for the transverse phonon $\Gamma_y(q) \propto q^2$. The consequence of these results on the thermal conductivity $\kappa(N)$ of a chain of $N$ particles is that $\kappa(N) \propto N^{1/2}$.