The Analyticity Breakdown for Frenkel-Kontorova Models in Quasi-periodic Media: Numerical Explorations

TL;DR

This paper numerically investigates the analyticity breakdown transition in 1D quasi-periodic media, revealing smooth transition surfaces, anisotropic scaling, and phenomena similar to other KAM problems.

Contribution

It introduces efficient numerical algorithms to study the analyticity breakdown in quasi-periodic media, providing new insights into the transition's nature and scaling behavior.

Findings

Transition occurs on a smooth surface

Scaling relations are observed near breakdown

Scaling is highly anisotropic with directional derivative blow-up

Abstract

We study numerically the "analyticity breakdown" transition in 1-dimensional quasi-periodic media. This transition corresponds physically to the transition between pinned down and sliding ground states. Mathematically, it corresponds to the solutions of a functional equation losing their analyticity properties. We implemented some recent numerical algorithms that are efficient and backed up by rigorous results so that we can compute with confidence even close to the breakdown. We have uncovered several phenomena that we believe deserve a theoretical explanation: A) The transition happens in a smooth surface. B) There are scaling relations near breakdown. C) The scaling near breakdown is very anisotropic. Derivatives in different directions blow up at different rates. Similar phenomena seem to happen in other KAM problems.