Mean-field theory of the DNLS equation at positive and negative absolute temperatures

The discrete nonlinear Schrödinger (DNLS) model, owing to the existence of two conserved quantities, exhibits an equilibrium transition from a homogeneous phase at positive absolute temperature to a localized phase at negative absolute temperature. Here, we provide a mean-field (MF) theory of DNLS through a suitable approximation of the grand canonical partition func- tion, which makes it factorizable and can be used to describe the equilibrium state at positive temperatures and the metastable state at negative temperatures. Comparison of our MF results with numerically exact calculations shows that this approximation is good to excellent in the entire grand canonical phase dia- gram. Explicit approximate expressions for equilibrium observables are provided in the high-temperature limit. Our theory represents a clear advancement over the model that neglects the interaction between sites.