A Simple XY Model for Cascade Transfer

Abstract: We propose a modified XY model in which cascade transfer emerges from spatially local interactions, where the spin corresponds to the "velocity" of a turbulent field. For this model, we theoretically calculate the scale-to-scale energy flux and the equal time correlation function in $d$ dimensions. The result indicates an inverse energy cascade with the non-Kolmogorov energy spectrum proportional to $k^{-3}$. We also numerically confirm the result for the cases $d=2$ and $d=3$. We thus conclude that the cascade transfer in our model represents a different universality class from standard fluid turbulence.