Localization Transition in the Discrete Non-Linear Schrödinger Equation

Explore a comprehensive lecture on the localization transition in the Discrete Non-Linear Schrödinger Equation, delivered by Giacomo Gradenigo at the Erwin Schrödinger International Institute for Mathematics and Physics. Delve into the detailed analysis of a first-order localization transition, where the localized phase is associated with the high energy region in parameter space. Discover how ensemble inequivalence affects the thermodynamic stability of this phase in the microcanonical ensemble. Learn about the explicit expression of the microcanonical entropy near the transition line at infinite temperature, obtained through large-deviation techniques. Understand the significance of subleading corrections to the microcanonical entropy in accounting for the first-order mechanism of the transition, computing its order parameter, and predicting negative temperatures in the localized phase. Examine these features as signatures of a thermodynamic phase with spontaneously broken translational symmetry, resulting from a condensation mechanism that leads to energy fluctuations deviating from equipartition. Gain insights into the formation of nonlinear localized excitations (breathers) containing a macroscopic fraction of the total energy.