Projective Ribbon Permutation Statistics: a Remnant of non-Abelian Braiding in Higher Dimensions

TL;DR

This paper investigates the topological and algebraic structure of defect statistics in 3D fermionic systems, revealing a novel projective permutation group that generalizes braiding phenomena to higher dimensions.

Contribution

It identifies the group governing defect statistics in 3D fermionic systems as a projective ribbon permutation group, extending the concept of braiding to higher dimensions.

Findings

The group T_{2n} is Z x T^r_{2n}, with T^r_{2n} a mild extension of the permutation group.

Defects realize 'Projective Ribbon Permutation Statistics', a new form of non-Abelian statistics.

The phenomenon extends to other dimensions, co-dimensions, and symmetry classes.

Abstract

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 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.