Localization and entanglement entropy in the Discrete Non-Linear Schrödinger Equation

Abstract: In this work we perform an accurate numerical study of the very peculiar thermodynamic properties of the localized high-energy phase of the Discrete Non-Linear Schr\"odinger Equation (DNLSE). A numerical sampling of the microcanonical ensemble done by means of Hamiltonian dynamics reveals a new and subtle relation between the presence of the localized phase and a non-trivial behaviour of entanglement entropy. Our finding that the entanglement entropy grows as $S_{\text{ent}}(N) \sim \log(N)$ beautifully encodes the lack of additivity in the DNLSE non-thermal localized phase and reveals how a property so far believed peculiar of purely quantum systems may characterize even certain classical frameworks.