Step Heat Profile in Localized Chains

TL;DR

This paper investigates the steady-state profiles of energy or particle transport in strongly disordered one-dimensional systems, revealing a step-function profile with a width growing no faster than the square root of the chain length, confirmed by numerical simulations.

Contribution

It demonstrates that localized disordered chains exhibit a step-function profile in steady state, with a detailed analysis of the profile shape and breakdown of local equilibrium at the step.

Findings

Profile is a step-function in disordered chains.

Step width grows at most as √L with chain length.

Chaotic temperature profile at the step in harmonic oscillators.

Abstract

We consider two types of strongly disordered one-dimensional Hamiltonian systems coupled to baths (energy or particle reservoirs) at the boundaries: strongly disordered quantum spin chains and disordered classical harmonic oscillators. These systems are believed to exhibit localization, implying in particular that the conductivity decays exponentially in the chain length $L$. We ask however for the profile of the (very slowly) transported quantity in the steady state. We find that this profile is a step-function, jumping in the middle of the chain from the value set by the left bath to the value set by the right bath. The width of the step grows not faster than $\sqrt{L}$. This is confirmed by numerics on a disordered quantum spin chain of 9 spins and on much longer chains of harmonic oscillators. In the case of harmonic oscillators, we also observe a drastic breakdown of local equilibrium at the step, resulting in a chaotic temperature profile.