Anyons as Quasiparticles in Quantum Matter

Anyons are superselection sectors of a two-dimensional gapped phase whose worldlines realize the braid group rather than merely the permutation group. An Abelian anyon contributes a phase under exchange; a non-Abelian anyon acts by a matrix on a degenerate fusion space. Electric charge is additional symmetry data and does not determine statistics.

Required background. Linking, braiding, and framing supplies worldline topology; topological field theory supplies the infrared setting; intrinsic topological order supplies superselection sectors and long-range entanglement.

Exchange and the braid group

For $n$ identical particles in the plane, neighboring counterclockwise exchanges $\sigma_i$ obey

$$\sigma_i\sigma_{i+1}\sigma_i =\sigma_{i+1}\sigma_i\sigma_{i+1}, \qquad \sigma_i\sigma_j=\sigma_j\sigma_i\quad (|i-j|>1).$$

Unlike permutations, $\sigma_i^2$ need not equal one. A one-dimensional unitary representation gives $\sigma_i=e^{i\theta}$: $\theta=0$ is bosonic, $\theta=\pi$ fermionic, and other values anyonic. Clockwise exchange gives the inverse phase. A full counterclockwise braid of one identical Abelian anyon around another contributes $e^{2i\theta}$ Leinaas and Myrheim 1977.

The measured phase also contains dynamical and Aharonov–Bohm pieces. Calling an interference phase “statistics” requires subtracting or independently calibrating them.

Fusion and topological spin

Anyon types fuse according to

$$a\times b=\sum_c N_{ab}^{c}\,c,$$

where $N_{ab}^{c}$ counts independent fusion channels. Antiparticle $\bar a$ satisfies $N_{a\bar a}^{1}>0$. Quantum dimensions are the positive solution $d_ad_b=\sum_cN_{ab}^{c}d_c$, and $\mathcal D=\sqrt{\sum_a d_a^2}$. Abelian anyons have $d_a=1$ and unique fusion outcomes; a sector with $d_a>1$ is non-Abelian.

The topological spin $\vartheta_a=e^{2\pi i h_a}$ is the phase for a $2\pi$ framed rotation. For an Abelian anyon it equals the exchange phase in the standard framing. Mutual monodromy $M_{ab}$ is the full braid of $a$ around $b$ and depends on their fusion channel in a non-Abelian theory. Fusion rules alone do not determine these phases; associativity and braiding data are also needed Nayak et al. 2008, §§ II.B–II.C.

Physical quasiparticles

A topological sector is defined modulo local excitations. In an electronic system, attaching an electron changes electric charge and fermion parity but not the intrinsic anyon label after the transparent local fermion is treated consistently. Quasiparticles must be separated farther than the correlation length and manipulated slower than the bulk gap but faster than poisoning or decoherence. At finite separation their fusion space splits exponentially, and disorder can pin or create unwanted quasiparticles.

Fractional charge, shot-noise scaling, edge exponents, topological spin, and braiding are different observables. A complete identification matches them to one consistent topological theory.

Exercise

An Abelian anyon has counterclockwise exchange phase $e^{i\pi/3}$. What is the phase for a full counterclockwise braid around an identical anyon?

Solution

A full braid is two exchanges, so the statistical contribution is $e^{2i\pi/3}$. Electromagnetic and dynamical phases in an interferometer must still be removed before this factor can be inferred.

References

• Jon Magne Leinaas and Jan Myrheim, “On the Theory of Identical Particles,” Il Nuovo Cimento B 37 (1977) 1–23, doi:10.1007/BF02727953.
• Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, “Non-Abelian Anyons and Topological Quantum Computation,” Reviews of Modern Physics 80 (2008) 1083–1159, doi:10.1103/RevModPhys.80.1083.