Observational entropic study of Anderson localization

TL;DR

This study investigates how observational entropy behaves during localization-delocalization transitions in the Aubry-André model, revealing distinct growth patterns and scaling laws in different phases, thus providing insights into quantum thermodynamics.

Contribution

It demonstrates the behavior of observational entropy across phases in the Aubry-André model, highlighting its growth, saturation, and scaling properties during localization-delocalization transitions.

Findings

In the delocalized phase, observational entropy saturates rapidly with coarse-grain size.

In the localized phase, entropy growth is logarithmic and follows an area law.

Entropy increases logarithmically with time after a quantum quench in the delocalized phase.

Abstract

The notion of the thermodynamic entropy in the context of quantum mechanics is a controversial topic. While there were proposals to refer von Neumann entropy as the thermodynamic entropy, it has it's own limitations. The observational entropy has been developed as a generalization of Boltzmann entropy, and it is presently one of the most promising candidates to provide a clear and well-defined understanding of the thermodynamic entropy in quantum mechanics. In this work, we study the behaviour of the observational entropy in the context of localization-delocalization transition for one-dimensional Aubrey-Andr\'e (AA) model. We find that for the typical mid-spectrum states, in the delocalized phase the observation entropy grows rapidly with coarse-grain size and saturates to the maximal value, while in the localized phase the growth is logarithmic. Moreover, for a given coarse-graining, it increases logarithmically with system size in the delocalized phase, and obeys area law in the localized phase. We also find the increase of the observational entropy followed by the quantum quench, is logarithmic in time in the delocalized phase as well as at the transition point, while in the localized phase it oscillates. Finally, we also venture the self-dual property of the AA model using momentum space coarse-graining.