Localization in the Discrete Non-Linear Schroedinger Equation and geometric properties of the microcanonical surface

In this talk I will present some recent results regarding the relaxation to equilibrium in the classical Discrete Non-Linear Schroedinger Equation (DNLSE). This model is well known to show a high-energy-density phase in which ensemble equivalence breaks down, and in which dynamical localization takes place: the charge accumulates on few sites and the exploration of phase space is dramatically slowed down. So far, however, the attention to the DNLSE was focused mainly on 1d geometries, where it is common for non-harmonic chains to present weak ergodicity breaking due to quasi-integrability issues and solitonic modes. In contrast, I will show how also in the mean-field, fully-connected model equilibrium is reached dynamically only after a time exponentially large in the system size. To this end, I will introduce an infinite-temperature expansion that justifies the simplification of keeping only the potential energy term at high energy density. The problem then reduces to the dynamics of a particle on the equipotential surface, whose geometry is non-trivial because of the presence of two conservation laws. I will present exact Morse-theoretic results on the structure of such hypersurface, elucidating how localization emerges when the motion of the particle can be considered Brownian. This in turn implies a phase transition in the lowest eigenvalue (i.e. the gap) of the Laplacian on said surface.