Statistical Mechanics of Anyons

1.首先研究了任意子的三体问题。半经典的二体配分函数精确地再现了第二维里系数，这一点早已为人所知。我们试图将其推广到三个任意子问题。通过引入一个以超球面坐标表示的射影算子，我们可以在对称群的不可约表示空间上将迹分离成一个子迹。这将被简短地报道。我们还研究了拓扑非平凡曲面上的多任意子问题。根据辫群分析，统计参数在柱面上被量子化。我们通过要求任意子波函数的唯一性来研究这个问题，发现对于统计参数的任何值，波函数都是很好地定义的。我们还研究了一个精确的多任意子波函数。在量子场论中，任意子在Chern-Simons规范理论中被实现为涡旋，研究一般系统中的涡旋结构是非常重要的。为此，我们发展了一种计算涡旋轴承系统在任意温度下的自由能的技术。作为初步应用，我们已将该技术应用于聚乙炔溶液和第二类超导体中的涡旋线。切恩-西蒙斯涡旋的应用正在进行中。

1. First, three-body problem of anyons is studied. It has been known for sometime that semiclassical two-body partition function reproduces exact second virial coefficient. We tried to generalize this for three-anyon problem. By introducing a projective operator written in terms of hyperspherical coordinate, we could separate the trace into a subtrace over the space of irreducible representation of the symmetric group. This will be reported shortly.2. We also looked at many-anyon problem on topologically nontrivial surface. According to the braid group analysis, the statistical parameter is quantized on a cylinder. We studied this problem by requiring the uniqueness of the anyon wave function and found that the wave function is well defined for any value of the statistical parameter. We also studied an exact multi-anyon wave function. We are seeking for a possibility that with a certain interaction the exact wave function may be obtained.3, In the quantum field theory, an anyon is realized as a vortex in Chern-Simons gauge theory and it is quite important to investigate the structure of the vortices in general system. For this purpose, we have developed a technique to compute the free energy of a vortex-bearing system at an arbitrary temperature. We have applied this technique to a solution in polyacetylene and a vortex line in a type II superconductor as a preliminary. Application to Chern-Simons vortex is under progress.