Localization and ‘classical entanglement’ in the discrete non-linear Schrödinger equation

We perform a detailed numerical study of the very peculiar thermo- dynamic properties of the localized high-energy phase of the discrete non-linear Schrödinger equation (DNLSE). A numerical sampling of the microcanonical ensemble done by means of Hamiltonian dynamics reveals a new and subtle rela- tion between the presence of the localized phase and a property of the system that we have called classical entanglement. Our main finding is that a quantity defined for our classical system in perfect analogy with the entanglement entropy of quantum ones, and that we have therefore called Sent , grows with the system size N in the localized phase as Sent (N ) ∼ log(N ), therefore revealing the pres- ence of subtle non-local correlations between any finite portion of the system and the rest of it. This manifestation of classical entanglement beautifully captures the lack of system separability in the DNLSE localized phase, revealing how stat- istical correlations specific to the microcanonical ensemble and non-reproducible in the canonical one, may concur to determine a property totally analogous to the one produced by non-local quantum correlations.