## "Projective ribbon permutation statistics: A remnant of non-Abelian braiding in higher dimensions", Michael Freedman, Zhenghan Wang, Cheten Nayak, et al., 2011

#### Reviewed November 7, 2023

Citation: Freedman, Michael, et al. "Projective ribbon permutation statistics: A remnant of non-Abelian braiding in higher dimensions." Physical Review B 83.11 (2011): 115132.

Tags: Higher-dimensional, Monoidal-categories

This is a follow up paper to

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

In the original paper, two relatively unknown authors give a 4-page construction of Majorana zero modes in 3+1D. It takes a 40 page highly technical paper from an all-star team of mathematicians and physicists to explain how that's possible.

The standard argument about spin statistics goes as follows. If you flip particles once, you will get some phase. Doing that flip twice, you get back where you started. Hence, the phase factor must square to one, hence it must be +-1: +1 for Boson, -1 for Fermion. This breaks down in two ways. The first way is that when you flip twice, you don't get back where you started. You get a non-trivial braid in spacetime, which is only homotopic to the identity map in 3 or more dimensions.

The second way the standard argument is that it assumes that an interchange can only result in a phase factor. A priori, however, an interchange could result in any action on the underlying Hilbert space. In 2+1D, these higher dimensional representations are exactly non-abelian anyons. The only abelian anyons in 3+1D are Fermions or Bosons, but what about non-abelian anyons? These hypothetical multi-dimensional spin statistics in 3+1D are known as parastatistics. A series of two papers

>Doplicher, Sergio, Rudolf Haag, and John E. Roberts. "Local observables and particle statistics I." Communications in Mathematical Physics 23 (1971): 199-230.
>Doplicher, Sergio, Rudolf Haag, and John E. Roberts. "Local observables and particle statistics II." Communications in Mathematical Physics 35 (1974): 49-85.

showed that under a strong locality condition the only possible statistics are bosonic or fermionic, and this is often cited as saying that "the only possibilities for spin statistics are bosonic or fermionic". However, the locality condition imposed is very strong. Precisely, it states that all expectation values of all observables should approach the vacuum expectation values, uniformly, when the region of measurement is moved away from the origin. This breaks, for example, if the electromagnetic force is considered. This is because Gauss' law says that the expectation value of the electric charge can be measured as the flux of the field strength trough a sphere of arbitrarily large radius. Seeing as all of our systems are based on electromagnetism, this paper is not enough to ensure that all particles will be bosonic or fermionic. This is especially true if one considers quasiparticles, where the highly entangled nature of the system can lead to non-local interactions core to the non-abelianness of the anyon.

What makes the present model so interesting is that it does not violate locality, but still expresses non-abelian statistics. This feels like it should be impossible by the results of Doplicher-Haag-Roberts, but in fact it is not. The particles in the DHR theory don't give a representation of the braid group, but they give give a representation of a richer group than the symmetric group. They give a representation of the group of particle/antiparticle pairs, where the particles/antiparticles have ribbons attached between them tracking their trajectories through spacetime.

This paper re-states the Teo-Kane model in a more elementary lattice based way, describes the general relevant theory using the classification of gapped free fermion Hamiltonians, gives a toy model for understanding the group of particle/antiparticle pairs with ribbons attached between them, and shows how the key "braidless" operations of the Teo-Kane model can be implemented.

A more compelling argument for why all particles in 3+1D must be fermions are bosons is found in Muger's paper,

>MÃ¼ger, Michael. "Abstract duality theory for symmetric tensor *-categories." Philosophy of Physics (2007): 865-922.

In 3+1D, the braiding on the fusion category corresponding to your particles will be symmetric. Muger proves that every symmetric fusion category is equivalent to Rep(G) for some compact group G. This means that your particles will be following some gauge theory G. Now, the braidings on Rep(G) are well-known as classified: they corresponding to twisting by some central element which squares to the identity. In a sense, they are classified by adding some fermions to the theory. The choice of central element tells you which particles should be fermions and which ones should be bosons. By far the most pedagogical treatment of this subject is Simon's book,

> Simon, Steven H. Topological Quantum. Oxford University Press, 2023.

Unrelated to the main content, this paper is a great place to learn about the classification of gapped free-fermion Hamiltonians. Other good references (including the original one) are as follows:

> Kitaev, Alexei. "Periodic table for topological insulators and superconductors." AIP conference proceedings. Vol. 1134. No. 1. American Institute of Physics, 2009.
> Chiu, Ching-Kai, et al. "Classification of topological quantum matter with symmetries." Reviews of Modern Physics 88.3 (2016): 035005.