Boltzmann-type approach to transport in weakly interacting one-dimensional fermionic systems

TL;DR

This paper develops a Boltzmann-type approach to study transport in weakly interacting one-dimensional fermionic systems, revealing diffusive behavior with finite diffusion coefficients when next-nearest neighbor hopping is present, and anomalous diffusion otherwise.

Contribution

It introduces a linear quantum Boltzmann equation framework for analyzing transport in weakly interacting 1D fermionic models, including non-integrable cases.

Findings

Finite diffusion coefficients for non-zero next-nearest neighbor hopping

Diverging diffusion coefficient in the integrable case without next-nearest neighbor hopping

Evidence of anomalous diffusive behavior with slow current decay

Abstract

We investigate transport properties of one-dimensional fermionic tight binding models featuring nearest and next-nearest neighbor hopping, where the fermions are additionally subject to a weak short range mutual interaction. To this end we employ a pertinent approach which allows for a mapping of the underlying Schr\"odinger dynamics onto an adequate linear quantum Boltzmann equation. This approach is based on a suitable projection operator method. From this Boltzmann equation we are able to numerically obtain diffusion coefficients in the case of non-vanishing next-nearest neighbor hopping, i.e., the non-integrable case, whereas the diffusion coefficient diverges without next-nearest neighbor hopping. For the latter case we analytically investigate the decay behavior of the current with the result that arbitrarily small parts of the current relax arbitrarily slowly which suggests anomalous diffusive transport behavior within the scope of our approach.