## Fractional Spin

Image credit: University of California – Santa Barbara

The term “fractional” in the title of this post should be read as “any”.

Frank Wilczek called the quasi-particles that can have any spin anyons in a 1982 paper. The “any” in “anyon” refers to the spin angular momentum value of the quasi-particle.

Abstract of his paper (https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.49.957):

“Composites formed from charged particles and vortices in (2+1)-dimensional models, or flux tubes in three-dimensional models, can have any (fractional) angular momentum. The statistics of these objects, like their spin, interpolates continuously between the usual boson and fermion cases. How this works for two-particle quantum mechanics is discussed here. “

A copy of that paper can be found  here

Thank God, this is NOT about elementary particles which can have the following spin values only:

0 : Higgs boson is the only elementary particle that does not spin

1/2: Fermions (electron, neutrino, quark) impart $\frac{1}{2} \hbar$ amount of angular momentum when they interact with other particles (irrespective of their energy)

1: Bosons (photon, W, Z, gluon) impart $1 \hbar$ amount of angular momentum when they interact with other particles (irrespective of their energy)

3/2: Hypothetical. Not observed yet.

2: Hypothetical force-carrier of gravitation (graviton) is supposed to impart $2 \hbar$ amount of angular momentum when it interacts with objects.

Time for reflection

Quantum spin of an elementary particle is an intrinsic property. As I emphasized many times in my blogposts, we do not understand the intrinsic properties such as charge, spin and invariant mass. Quantum spin is probably the most mysterious intrinsic property.

The spin angular momentum of an elementary particle is independent of its energy. I find this very curious. Speaking of curious, I sometimes read my old blogpost titled “What is Spin?” and react internally saying “Wow! I was bold.” There are some speculative statements there but I still stand by them.

Spin effects in collective behavior of particles

The Pauli exclusion principle states that no two fermions (spin=1/2) can occupy the same quantum state. The Bose-Einstein condensation refers to the fact that multiple bosons (spin=1) can occupy the same quantum state. These principles are codifications of empirical facts. Quantum wavefunctions of a collection of fermions or bosons are devised such that they reflect these empirical facts. The statistics implied by those collective wavefunctions have been experimentally verified.

We can do the reverse as well. We can study the statistics of unusual collective entities (for example, 2-D arragements) and infer the spin of the constituents. This is how the spin of the anyons were inferred.

Physical realizations of anyons and Quantum Computing

Physical realizations of anyons can be used in quantum computing. This approach – topological quantum computing – is pursued by Microsoft. Anyon (more specifically, non-abelian anyon) qubits would be more stable compared to other physical realizations of qubits.

In the field of experimental realizations of non-abelian anyon qubits, it is difficult to follow who did what. This Nature magazine article from 2016 helps. There is also a Natalie Wolchover article titled “Forging a Qubit To Rule Them All” about Bob Willett’s contributions. But, I think the best article to read is the one Frank Wilczek wrote last year (2017) titled “Inside the Knotty World of Anyon Particles“.

At the time of this writing, it is not clear to me that there was a clear demonstration of non-abelian anyon qubit yet.

PS: The tutorial titled “Fractional quantum statistics” (from 2013) by T.H. Hansson is excellent.