Study Of2D-3D XY Model

In this paper we discuss XY model of different dimensions and it’s appli-cations. In3D-XY model, Under critical temperature, long range order exists through phase transition. This is the brief physics of spontaneous symmetry breaking. In2D system, we have KT transition. At low temperature, the order parameter is zero,which means, there is no spontaneous symmetry breaking, how-ever, there exist topology order, which means+/-vortex have to emerge in pairs, in another word, they arc entangled to cach other. So, here we find a example of topology order in low dimension. We apply this model to low dimension super-fluid and insulator-mental-superconductor transition. We use layer2D-XY model to discuss the crossover between KT transition to λ transition, that is, crossover between2D-3D systems. We assume that there exists the Josephson coupling among the layers, which directly related to phase coherence. In real world all system have finite size, no true critical behavior is expected. This phenomenon is called finite size effect. By studying the2D-3D XY model,we get a example of this very phenomenon. We discuss a typical example:finite size scaling in4He. In the framework of Renormalization group, crossover between different dimension can also discussed. In the end, we discuss emergent phenomenon and symmetry in the process of phase transition, trying to explain the origin of universality of fundamental excitation in many-body system in topology point of view.