Localization in the Discrete Non-Linear Schr\"odinger Equation and geometric properties of the microcanonical surface

TL;DR

This paper links localization phenomena in the Discrete Non-Linear Schr"odinger Equation to geometric features of the microcanonical surface, revealing a phase transition related to the Laplacian's eigenvalues and a synchronization transition at low temperatures.

Contribution

It demonstrates that localization in DNLSE is connected to the geometry of the energy surface and identifies a phase transition in the Laplacian's eigenvalues, with an analysis of the fully connected model.

Findings

Localization correlates with geometric properties of the energy surface.

A phase transition occurs in the lowest eigenvalue of the Laplacian.

Synchronization transition observed at low temperatures in the fully connected model.

Abstract

It is well known that, if the initial conditions have sufficiently high energy density, the dynamics of the classical Discrete Non-Linear Schr\"odinger Equation (DNLSE) on a lattice shows a form of breaking of ergodicity, with a finite fraction of the total charge accumulating on a few sites and residing there for times that diverge quickly in the thermodynamic limit. In this paper we show that this kind of localization can be attributed to some geometric properties of the microcanonical potential energy surface, and that it can be associated to a phase transition in the lowest eigenvalue of the Laplacian on said surface. We also show that the approximation of considering the phase space motion on the potential energy surface only, with effective decoupling of the potential and kinetic partition functions, is justified in the large connectivity limit, or fully connected model. In this model we further observe a synchronization transition, with a synchronized phase at low temperatures.