Statistical Mechanics of Anyons

任意子统计力学

• 批准号: 02640222
• 负责人: NAKAHARA Mikio
• 金额: $ 1.22万
• 依托单位: Shizuoka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1990
• 资助国家: 日本
• 起止时间: 1990 至 1991
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-02640222/
• 关键词: Anyon; Virial Expansion; Superconductivity; Vortex Line; Free Energy; Heat Kernel; Ginzburg Landau Expansion; Chern-Simons Gauge Theory; ヴイリアル展開; ヴィリアル係数; 組み紐群; ギンズブルグランダウ展開

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.

1. 首先，研究了任意子的三体问题。人们早就知道，半经典二体配分函数可以精确地再现秒维里系数。我们试着把它推广到3元问题。通过引入一个用超球坐标表示的射影算子，我们可以将对称群的不可约表示空间上的轨迹分离成子轨迹。这件事不久将作报告。我们还研究了拓扑非平凡曲面上的任意问题。根据编织群分析，在圆柱体上量化统计参数。我们通过要求任意波函数的唯一性来研究这个问题，发现波函数对于任意统计参数的值都是定义良好的。我们还研究了一个精确的多任意子波函数。我们正在寻找一种可能性，即在一定的相互作用下，可以得到精确的波函数。在量子场论中，任意子在chen - simons规范理论中被实现为涡旋，研究一般系统中涡旋的结构是非常重要的。为此，我们开发了一种计算任意温度下涡轴承系统自由能的技术。我们已经将该技术初步应用于聚乙炔溶液和II型超导体中的涡旋线。陈-西蒙斯涡的应用正在进行中。

项目成果

M.Nakahara: "Structure of a Vortex in Extreme Type II Superconductors at Low Temperatures" Phys.Rev.B.

M.Nakahara：“低温下极端 II 型超导体中的涡流结构”Phys.Rev.B。

M. Nakahara: "Ginzburg Landau Expansion at Low Temperatures" Prog. Theor. Phys.86. 315-320 (1991)

M. Nakahara：“低温下的金茨堡朗道膨胀”Prog。