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## "Majorana fermions and non-Abelian statistics in three dimensions", Jeffrey Teo, Charles Kane, 2010

*Reviewed November 7, 2023*

*Citation:* Teo, Jeffrey CY, and Charles L. Kane. "Majorana fermions and non-Abelian statistics in three dimensions." Physical review letters 104.4 (2010): 046401.

*Web:* https://arxiv.org/abs/0909.4741

*Tags:* Majorana-fermions, Hardware, Higher-dimensional

This paper shows that Majorana fermions with non-abelian statistics can
appear as bulk point excitations in 3+1 dimensional topological phases of matter.
This is in contrast to the standard theory of topological phases, which works in 2+1D.
It is somewhat surprising that non-abelian anyons can exist in 3+1D, but it is
important to remember that the standard argument that all particles in 3+1D
are Fermionic or Bosonic only applies to abelian anyons.

The starting point for this paper is the Bogoliubov de Gennes (BdG)
description of a 2+1D paired condensate. This is exactly the sort of system in
which normally Majorana fermions will appear. Given bulk volumes V, the authors
then classify bulk Hamiltonians H which restrict to BdG Hamiltonians on the boundary,
and satisfy nice properties such as particle-hole symmetry, as is typical for
superconductors.

After a detailed analysis using Chern-Simmons theory, a general gapped Hamiltonian is constructed.
The form of the Hamiltonian can be given a second motivation, from considering "a BdG
Hamiltonian describing ordinary and topological insulators coexisting with superconductivity" .

After this setup, the authors examine quasiparticles (zero modes) in this model.
They dub their quasiparticles to be "hedgehogs", which can be though of just like vorticies or skyrmions.
They are shown to behave as Majorana fermions. The proof of this fact as a straightforward
consequence of particle-hole symmetry.