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

Michael Freedman, Matthew B. Hastings, Chetan Nayak, Xiao Liang Qi, Kevin Walker, Zhenghan Wang

Research output: Contribution to journalArticlepeer-review

Abstract

In a recent paper, Teo and Kane [Phys. Rev. Lett.PRLTAO0031-900710.1103/ PhysRevLett.104.046401 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 T2n, which governs the statistics of these defects by analyzing the topology of the space K2n of configurations of 2n defects in a slowly spatially varying gapped free-fermion Hamiltonian: T2nπ1(K2n). We find that the group T2n=Z×T2nr, where the "ribbon permutation group" T2nr is a mild enhancement of the permutation group S2n: T2nrZ2×E((Z2)2nS2n). Here, E((Z 2)2nS2n) is the "even part" of (Z 2)2nS2n, 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 T2n, 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, 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.

Original languageEnglish
Article number115132
JournalPhysical Review B - Condensed Matter and Materials Physics
Volume83
Issue number11
DOIs
StatePublished - 23 Mar 2011
Externally publishedYes

ASJC Scopus subject areas

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics

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