Topological anyonic molecule statistics

Question

In the last chapter of John Preskill's Lecture Notes for Physics 219: Quantum Computation (PDF), he mentions the following on pg.12.

This behavior is compatible with the spin-statistics connection: the angular momentum $J$ of the $n$-anyon molecule satisfies

$$e^{-2\pi i J_n} = e^{-2\pi i n^2 J} = e^{i n^2 \theta}.$$

For example, consider a molecule of two anyons, and imagine rotating the molecule counterclockwise by $2\pi$. Not only does each anyon in the molecule rotate by $2\pi$; in addition one of the anyons revolves around the other. One revolution is equivalent to two successive exchanges, so that the phase generated by the revolution is $e^{i2\theta}$. The total effect of the two rotations and the revolution is the phase

$$\exp[i(\theta + \theta + 2\theta)] = e^{i4\theta}.$$

Not only does each anyon in the molecule rotate by $2\pi$; in addition one of the anyons revolves around the other. One revolution is equivalent to two successive exchanges so that the phase generated by the revolution is $e^{i2\theta}$.

How is this possible? A molecule here is just a localized region of two close anyons in the given space. Rotating the "molecule" should just be equal to revolving the two anyons one complete orbit about each other, that must give me a phase of $e^{i2\theta}$. What is meant by

Not only does each anyon in the molecule rotate by $2\pi$...

As far as I know, "rotation" of an anyon is no permutation in Topological QC. What's the concept involved here with additional $e^{i2\theta}$ phase that we are getting?

Answer 1

The spin-statistics theorem requires a particle's wave function to acquire the same phase when it is rotated by an angle of $2 \pi$ about itself and when exchanged with an identical (indistinguishable) particle. For example, a fermion acquires a phase of $-1$ both in exchange and self-rotation.

Thus, when we think of an anyon as a flux-charge entity, we have to think of it also as an extended object having an inner structure of a spinning body, or equivalently, when it rotates about its axis, its own charge rotates around its own flux.

The self-rotation explains the doubling of the topological phase of the anyon molecule.

This property is explained, possibly more clearly, by Pachos, (sections 2.1.2 and 4.1.6). The following figure explains the phase acquired in self-rotation is taken from section 2.1.2:

Pachos description of the figure:

(a) Anyons can be described effectively as composite particles with an attached magnetic flux, and a ring with electric charge, $q.$ When anyon 1 moves around anyon 2 along loop $C$ its charge circulates the flux of the other anyon and the Aharonov-Bohm effect gives rise to a non-trivial statistical phase.

(b) When the composite particle rotates around itself by $2\pi$ it again acquires a phase factor as its charge circulates its flux, which can be attributed to a spin.

Abelian and non-Abelian anyons can be described by means of mathematical objects called ribbon categories. Pachos in section 4.1.6 (figure 4.9), shows how this ribbon model explains the spin statistics theorem, in an example when creation of an identical anyon-anti-anyon pair, their exchange, then their annihilation back to the vacuum is identical to replacing the exchange by a rotation of one anyon about itself.

Pachos description of the figure:

An anyon $q$ and an anti-anyon $\bar{q}$ denoted with their worldribbons are pair-created, exchanged and then fused to the vacuum. The exchange process can be continuously deformed to rotating one of the anyons around itself by $2\pi$. This equivalence can be nicely verified with a belt.