Subdiffusion in one-dimensional Hamiltonian chains with sparse interactions

TL;DR

This paper rigorously proves that energy and particle transport in certain disordered one-dimensional Hamiltonian chains is subdiffusive due to rare insulating regions, highlighting the impact of sparse interactions on transport properties.

Contribution

It introduces a rigorous analysis of subdiffusive transport in disordered chains with sparse interactions, a novel setting compared to typical models with uniform interactions.

Findings

Transport is slower than diffusive due to Griffiths regions.

The diffusion constant is proven to be finite in a non-integrable model.

Rare non-interacting regions cause localization effects.

Abstract

We establish rigorously that transport is slower than diffusive for a class of disordered one-dimensional Hamiltonian chains. This is done by deriving quantitative bounds on the variance in equilibrium of the energy or particle current, as a function of time. The slow transport stems from the presence of rare insulating regions (Griffiths regions). In many-body disordered quantum chains, they correspond to regions of anomalously high disorder, where the system is in a localized phase. In contrast, we deal with quantum and classical disordered chains where the interactions, usually referred to as anharmonic couplings in classical systems, are sparse. The system hosts thus rare regions with no interactions and, since the chain is Anderson localized in the absence of interactions, the non-interacting rare regions are insulating. Part of the mathematical interest of our model is that it is one of the few non-integrable models where the diffusion constant can be rigorously proven not to be infinite.