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

The Discrete Non Linear Schrödinger (DNLS) model, due to the existence of two conserved quantities, displays an equilibrium transition between a homogeneous phase at positive absolute temperature and a localized phase at negative absolute temperature. Here, we provide a mean-field theory of DNLS through a suitable approximation of the grandcanonical partition function which makes it factorizable. By comparing our mean-field results with numerically exact ones, we show that this approximation is semi-quantitatively correct in the whole grandcanonical phase diagram, becoming increasingly accurate in proximity of the transition line and exact along the line itself. Our mean-field theory suggests that the passage from stable positive-temperature to metastable negative-temperature states is smooth and allows for a more accurate description of the metastable region.

💡 Research Summary

The paper presents a comprehensive mean‑field (MF) treatment of the discrete nonlinear Schrödinger (DNLS) equation, a lattice model that conserves both the total mass (or norm) and the energy. Because of these two conserved quantities, the DNLS exhibits an equilibrium transition between a homogeneous phase at positive absolute temperature (T > 0) and a localized phase at negative absolute temperature (T < 0), the latter being associated with the spontaneous formation of high‑amplitude breathers. The authors’ central idea is to replace the product of neighboring site amplitudes, (\sqrt{c_n c_{n+1}}), by a factorized form (q\sqrt{c_n}) where (q=\langle\sqrt{c}\rangle) is a self‑consistent average. This approximation turns the original Hamiltonian
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