Yasui Shigehiro, Hirono Yuji, Itakura Kazunori, Nitta Muneto
KEK Theory Center, Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 May;87(5):052142. doi: 10.1103/PhysRevE.87.052142. Epub 2013 May 30.
We extend our previous analysis on the exchange statistics of vortices having a single Dirac fermion trapped in each core to the case where vortices trap two Dirac fermions with U(2) symmetry. Such a system of vortices with non-Abelian Dirac fermions appears in color superconductors at extremely high densities and in supersymmetric QCD. We show that the exchange of two vortices having doublet Dirac fermions in each core is expressed by non-Abelian representations of a braid group, which is explicitly verified in the matrix representation of the exchange operators when the number of vortices is up to four. We find that the result contains the matrices previously obtained for the vortices with a single Dirac fermion in each core as a special case. The whole braid group does not immediately imply non-Abelian statistics of identical particles because it also contains exchanges between vortices with different numbers of Dirac fermions. However, we find that it does contain, as its subgroup, genuine non-Abelian statistics for the exchange of the identical particles, that is, vortices with the same number of Dirac fermions. This result is surprising compared with conventional understanding because all Dirac fermions are defined locally at each vortex, unlike the case of Majorana fermions for which Dirac fermions are defined nonlocally by Majorana fermions located at two spatially separated vortices.
我们将之前关于每个涡旋核心捕获单个狄拉克费米子的涡旋交换统计的分析扩展到涡旋捕获两个具有U(2)对称性的狄拉克费米子的情况。这样一个具有非阿贝尔狄拉克费米子的涡旋系统出现在极高密度的色超导和超对称量子色动力学中。我们表明，每个核心具有双重态狄拉克费米子的两个涡旋的交换由辫群的非阿贝尔表示来描述，当涡旋数量至多为四个时，在交换算符的矩阵表示中得到了明确验证。我们发现，该结果包含了之前针对每个核心具有单个狄拉克费米子的涡旋所得到的矩阵作为特殊情况。整个辫群并不立即意味着全同粒子的非阿贝尔统计，因为它还包含具有不同数量狄拉克费米子的涡旋之间的交换。然而，我们发现它确实包含作为其子群的全同粒子交换的真正非阿贝尔统计，即具有相同数量狄拉克费米子的涡旋。与传统理解相比，这个结果令人惊讶，因为所有狄拉克费米子都是在每个涡旋处局部定义的，这与马约拉纳费米子的情况不同，在马约拉纳费米子的情况下，狄拉克费米子是由位于两个空间分离的涡旋处的马约拉纳费米子非局部定义的。
