A standard two-dimensional (2D) XY model is well known to show a topological Berezinskii-Kosterlitz-Thouless (BKT) phase transition with the quasi-long-range-order (QLRO) BKT phase characterized by an algebraically decaying correlation function. It will be shown by spin-wave theory and Monte Carlo simulations that including higher-order terms can lead to dramatic changes in its critical behavior, including phase transition order or universality class and even the phase…