Drastic fall-off of the thermal conductivity for disordered lattices in the limit of weak anharmonic interactions

TL;DR

This paper rigorously demonstrates that in disordered harmonic lattices with weak anharmonic interactions, thermal conductivity diminishes faster than any polynomial in the interaction strength, indicating a drastic fall-off in heat transport.

Contribution

The study provides a rigorous proof that thermal conductivity vanishes faster than any polynomial in the weak interaction parameter, revealing a non-perturbative decay in disordered lattices.

Findings

Thermal conductivity decays faster than any polynomial in the interaction strength.

The result applies to disordered chains and classical spin chains.

Thermal transport becomes negligible as anharmonic interactions weaken.

Abstract

We study the thermal conductivity, at fixed positive temperature, of a disordered lattice of harmonic oscillators, weakly coupled to each other through anharmonic potentials. The interaction is controlled by a small parameter $\epsilon > 0$. We rigorously show, in two slightly different setups, that the conductivity has a non-perturbative origin. This means that it decays to zero faster than any polynomial in $\epsilon$ as $\epsilon\rightarrow 0$. It is then argued that this result extends to a disordered chain studied by Dhar and Lebowitz, and to a classical spins chain recently investigated by Oganesyan, Pal and Huse.