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تسجيل مستخدم جديدProjective Ribbon Permutation Statistics: a Remnant of non-Abelian Braiding in Higher Dimensions
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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 {pi_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 Kanes 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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