Topological Field Theory and Superfluid Thin Film

目前的研究表明，二维阿贝尔希格斯模型在某些参数区域成为拓扑场理论，并且在另一个参数区域包含XY模型，这是一个有效的超流体薄膜理论。本文采用重整化群的方法研究了阿贝尔希格斯模型，明确了该模型的全局结构，特别是拓扑激发和自旋波的行为。为此，找出阿贝尔希格斯模型与大质量正弦-戈登模型等效这一事实至关重要。在此基础上，采用惯有动量截断的重整化群方法对大质量正弦-戈登模型进行了研究，并通过数值方法得到了递推方程的解。结果表明重整化群有两条固定的线。一个对应于XY模型的低温相，另一个对应于拓扑场相。作为今年获得的另一个结果，我们证明了阿贝尔希格斯模型等效于具有四费米相互作用的二维QED。从阿贝尔希格斯模型的知识出发，澄清了该理论的真空结构，并明确计算了相关函数。

Present study showed that the 2-dimensional Abelian Higgs model becomes a topological field theory at some parameter region, and also it containes the XY model at another parameter region, which is an effective theory for superfluid thin film. In the present year, we studied the Abelian Higgs model by a method of the renormalization group, and clarified the global structure of this model, especially the behavior of the topological exciations and the spin wave. To this end, it was crucial to find out the fact that the Abelian Higgs model is equivalent to the massive sine-Gordon model. Then the massive sine-Gordon model was studied by the renormalization-group method with the usual momentum cut off, and solutions to the recursion equations were obtained by numerical method. The solutions show that there are two fixed line of the renormalization group. One corresponds to low-temperature phase of the XY model and the other is the topological-field phase.As another result obtained this year, we proved that the Abelian Higgs model is equivalent to the 2-dimensional QED with the four-fermi interaction. From the knowledge of the Abelian Higgs model, the vacuum structure of this theory was clarified and correlation functions were explicitly calculated.