… a stability consideration requires that ϕ be a rational multiple of 2π…

… when a charge q goes around a vortex v, it picks up the phase 2πqv, known as the Aharonov-Bohm phase…..

…. The most amazing thing about anyons is that they actually exist as excitations in some condensed matter systems….

…. The best studied example (both theoretically and experimentally) is the Laughlin state [8] in the fractional quantum Hall system at the filling factor ν = 1/3. It carries Abelian anyons with exchange phase ϕ = π/3 and electric charge ±1/3. It is the fractional value of the charge that was predicted in original Laughlin’s paper and confirmed by several methods, in particular by a shot noise measurement [9, 10]. The statistical phase is a subtler property which is deduced theoretically [11, 12]; a nontrivial experimental test has been performed recently using quasiparticle tunneling [13].

The notion of anyons assumes that the underlying state has an energy gap (at least for topologically nontrivial quasiparticles). Otherwise excitations are not localizable and braiding may not be defined. Note that if all excitations are gapped, then all equal-time correlators decay exponentially with distance

…Thus anyons carry some topological quantum numbers which make them stable:….

…. while in anyonic systems correlations decay exponentially due to the energy gap….

… But anyons, by virtue of their topological stability, must have some observable signatures….

…Thus, the hunt for anyons and topological order is a difficult endeavor. Why do we care?
First, because these are conceptually important phenomena, breaking some paradigms. In particular, consider these principles (which work well and provide important guidance in many cases):
1. Conservation laws come from symmetries (by Noether’s theorem or its quantum analogue);
2. Symmetries are initially present in the Hamiltonian (or Lagrangian), but may be spontaneously broken….

… A more practical reason to look for anyons is their potential use in quantum computing….

…Actually, the degeneracy is not absolute by very precise. It is lifted due to virtual particle tunneling across the torus, but this process is exponentially suppressed. Therefore the distance between ground energy levels is proportional to exp(−L/ξ), where L the linear size of the torus and ξ is some characteristic length, which is related to the gap in the excitation spectrum….
(note: torus = Planck size void created by Planck spheres.)

… In fact, a natural process that “distinguishes” them by multiplying by different factors is tunneling of a virtual ε-particle between the fixed σ-particles
(which is possible since σ × ε = σ). However, ε-particles are gapped, therefore this process is exponentially suppressed…
(note: the gap is the void in the Planck spheres.)

…. In order to implement a qubit, one needs four σ-particles. A logical |0i is represented by the quantum state |ξ1i that is obtained by creating the pairs (1, 2) and (3, 4) from the vacuum (see Fig. 2)….

… Indeed, the operators of counterclockwise exchange between two particles
(R-matrices) are related to the fusion rules by so-called hexagon equations and pentagon equation. We will see on concrete examples that these equations have only a finite number of solutions and therefore do not admit small deformations. In general, it is a nontrivial theorem known as Ocneanu rigidity….

… But for the meantime, finding and studying topological phases seems to be a very reasonable goal, also attractive from the fundamental science perspective….

…. The model has two phases (denoted by A and B ) which occur at different values of parameters. The exact solution is obtained by a reduction to free real fermions. Thus quasiparticles in the system may be characterized as fermions and Z2-vortices. Vortices and fermions interact by the Aharonov-Bohm factor equal to −1. In phase A the fermions have an energy gap, and the vortices are bosons that fall into two distinct superselection sectors. (Interestingly enough,
the two types of vortices have identical physical properties and are related to each other by a lattice translation.)…
(note: would this be cubic packing and hex. packing in my model)

p. 31 ….8.2 Unpaired Majorana modes
We now begin a rather lengthy derivation of the properties listed in Table 1, dealing only with anyons in the bulk….

An inspiring TYPO by yquantum
A theory must be tested in lavatories and physicist around the world to show valitiy.