Statistical Mechanics of Anyons

1.首先，研究了任意子的三体问题。半经典两体配分函数可以精确地再现第二维里系数，这一点已经知道了一段时间。我们试图将其推广到三个任意子问题。通过引入一个用超球坐标表示的射影算子，我们可以将迹分解为对称群的不可约表示空间上的子迹。这将在短时间内报告。2.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，将统计参数量子化在圆柱上。我们通过要求任意子波函数的唯一性来研究这个问题，发现对于任何统计参数值，波函数都是定义良好的。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，研究涡旋的结构在一般系统中是非常重要的。为此，我们已经开发了一种技术来计算在任意温度下的涡轴承系统的自由能。我们已经将这种技术应用于聚乙炔溶液和II型超导体中的涡旋线作为初步研究。应用于Chern-Simons涡旋的工作正在进行中。

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.