投影带排列统计量：高维非阿贝尔辫子的一个残余
Projective Ribbon Permutation Statistics: a Remnant of non-Abelian Braiding in Higher Dimensions
摘要：
在最近的一篇论文中，Teo和Kane提出了一个3D模型，其中缺陷支持马约拉纳费米子零模式。他们认为，交换和扭曲这些缺陷将在零模希尔伯特空间上实现一组R么正变换，这是辫子群对2D伊辛任意子作用的“幽灵”回忆。本文通过分析缓变带隙自由费米子哈密顿量中2n个缺陷组态的空间K_{2n}的拓扑，找到了支配这些缺陷统计的群T_{2n}：T_{2n} \equiv {π_1}（K_{2n}）$.我们发现群T_{2n} = Z \times T^ r_{2n}，其中“带状置换群”T^ r_{2n}是置换群S_{2n}的轻度增强：T^ r_{2n} \equiv \Z_2 \times E（（Z_2）^{2n} \rtimes S_{2n}）。这里，E（（Z_2）^{2n} \rtimes S_{2n}）是（Z_2）^{2n} \rtimes S_{2n}的“偶数部分”，即（Z_2）^{2n}中的元素的总奇偶性加上置换的奇偶性是偶数的那些元素。令人惊讶的是，R只是T_{2n}的一个投影表示，这是Wilczek提出的一种可能性。因此，Teo和Kane的缺陷实现了“投影带排列统计”，我们证明了它与局部性是一致的。我们将这一现象扩展到其他维度、余维度和对称类。由于它是一个重要的输入，我们的计算，我们回顾了带隙自由费米子系统的拓扑分类及其与波特周期的关系。
英文摘要：
In a recent paper, Teo and Kane proposed a 3D model in which the defects support Majorana fermion zero modes. They argued that exchanging and twisting these defects would implement a set R of unitary transformations on the zero mode Hilbert space which is a 'ghostly' recollection of the action of the braid group on Ising anyons in 2D. In this paper, we find the group T_{2n} which governs the statistics of these defects by analyzing the topology of the space K_{2n} of configurations of 2n defects in a slowly spatially-varying gapped free fermion Hamiltonian: T_{2n}\equiv {π_1}(K_{2n})$. We find that the group T_{2n}= Z \times T^r_{2n}, where the 'ribbon permutation group' T^r_{2n} is a mild enhancement of the permutation group S_{2n}: T^r_{2n} \equiv \Z_2 \times E((Z_2)^{2n}\rtimes S_{2n}). Here, E((Z_2)^{2n}\rtimes S_{2n}) is the 'even part' of (Z_2)^{2n} \rtimes S_{2n}, namely those elements for which the total parity of the element in (Z_2)^{2n} added to the parity of the permutation is even. Surprisingly, R is only a projective representation of T_{2n}, a possibility proposed by Wilczek. Thus, Teo and Kane's defects realize `Projective Ribbon Permutation Statistics', which we show to be consistent with locality. We extend this phenomenon to other dimensions, co-dimensions, and symmetry classes. Since it is an essential input for our calculation, we review the topological classification of gapped free fermion systems and its relation to Bott periodicity.
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