Projective ribbon permutation statistics: A remnant of non-Abelian braiding in higher dimensions

In a recent paper, Teo and Kane Phys. Rev. Lett. 104 046401 (2010) proposed a three-dimensional (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 two dimensions. 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≡π₁(K_2n). We find that the group T_2n=Z×T_2n^r, where the “ribbon permutation group” T_2n^r is a mild enhancement of the permutation group S_2n: T_2n^r≡Z₂×E((Z₂)²ⁿ⋊S_2n). Here, E((Z₂)²ⁿ⋊S_2n) is the “even part” of (Z₂)²ⁿ⋊S_2n, namely, those elements for which the total parity of the element in (Z₂)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 [e-print arXiv:hep-th/9806228]. 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, codimensions, and symmetry classes. We note that our analysis applies to 3D networks of quantum wires supporting Majorana fermions; thus, these networks are not required to be planar. Because it is an essential input for our calculation, we review the topological classification of gapped free-fermion systems and its relation to Bott periodicity.