Bose-Einstein condensation and Berezinskii-Kosterlitz-Thouless transition in the two-dimensional nonlinear Schrödinger model

We analyze the Bose-Einstein condensation process and the Berezinskii-Kosterlitz-Thouless phase transition within the nonlinear Schr\"odinger model and their interplay with wave turbulence theory. By using numerical experiments we study how the condensate fraction and the first-order correlation function behave with respect to the mass, the energy, and the size of the system. By relating the free-particle energy to the temperature, we are able to estimate the Berezinskii-Kosterlitz-Thouless transition temperature. Below this transition we observe that for a fixed temperature the superfluid fraction appears to be size independent, leading to a power-law dependence of the condensate fraction with respect to the system size.