On the continuity of phase transition of three-dimensional square-lattice XY models

We study the continuity of magnetization at the phase transition of the ferromagnetic XY model in the three-dimensional square lattice with the nearest neighborhood interaction. We assume that, at the critical temperature, with probability 1, for every edge in the infinite directed graph generated by the random path representation, finitely many edges exist so that they form a finite loop. Then, we prove that the phase transition is continuous at the critical temperature. The main technical contribution is to find a switching lemma to establish a bijection between equally weighted graphs.