Phase transitions do not necessarily correspond to a symmetry-breaking phenomenon. This is the case of the Kosterlitz–Thouless (KT) phase transition in a two-dimensional classical XY model, a typical example of a transition stemming from a deeper phenomenon than a symmetry-breaking. Actually, the KT transition is a paradigmatic example of the successful application of topological concepts to the study of phase transition phenomena in the absence of an order parameter. Topology conceptually enters through the meaning of defects in real space. In the present work, the same kind of KT phase transition in a two-dimensional classical XY model is tackled by resorting again to a topological viewpoint, however focussed on the energy level sets in phase space rather than on topological defects in real space. Also from this point of view, the origin of the KT transition can be attributed to a topological phenomenon. In fact, the transition is detected through peculiar geometrical changes of the energy level sets which, after a theorem in differential topology, are direct probes of topological changes of these level sets.

相变不一定对应于对称破坏现象。二维经典XY模型中的Kosterlitz-Thouless（KT）相变就是这种情况，这是由比对称破坏更深的现象产生的典型转变示例。实际上，KT过渡是拓扑概念在没有阶数参数的情况下成功应用于相变现象研究的范例。拓扑从概念上是通过实际空间中缺陷的含义进入的。在目前的工作中，二维经典XY中的相同类型的KT相变该模型是通过再次求助于拓扑观点来解决的，但是重点在于相空间中的能级集，而不是实际空间中的拓扑缺陷。同样从这个观点来看，KT转变的起源可以归因于拓扑现象。实际上，通过能级集的特殊几何变化来检测跃迁，在微分拓扑中的一个定理之后，能级集的几何变化是这些能级组的拓扑变化的直接探查。