A momentum conserving model with anomalous thermal conductivity in low dimension

TL;DR

This paper introduces a microscopic model demonstrating how momentum conservation leads to divergent thermal conductivity in one and two dimensions, while conductivity remains finite in three or more dimensions, clarifying the role of momentum in anomalous heat transport.

Contribution

The paper presents an exactly solvable model showing the impact of momentum conservation on thermal conductivity across different dimensions.

Findings

Thermal conductivity diverges in 1D and 2D when momentum is conserved.

Thermal conductivity is finite in 3D or with an on-site potential.

Explicit computation of energy current correlations supports the dimensional dependence.

Abstract

Anomalous large thermal conductivity has been observed numerically and experimentally in one and two dimensional systems. All explicitly solvable microscopic models proposed to date did not explain this phenomenon and there is an open debate about the role of conservation of momentum. We introduce a model whose thermal conductivity diverges in dimension 1 and 2 if momentum is conserved, while it remains finite in dimension $d\ge 3$. We consider a system of harmonic oscillators perturbed by a non-linear stochastic dynamics conserving momentum and energy. We compute explicitly the time correlation function of the energy current $C\_J(t)$, and we find that it behaves, for large time, like $t^{-d/2}$ in the unpinned cases, and like $t^{-d/2-1}$ when an on site harmonic potential is present. Consequently thermal conductivity is finite if $d\ge 3$ or if an on-site potential is present, while it is infinite in the other cases. This result clarifies the role of conservation of momentum in the anomalous thermal conductivity in low dimensions.