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= Anyons =

{{Audience_Sidebar
| difficulty = Advanced
| reading_time = 5 minutes
| prerequisites = [[Quasiparticle]]; quantum statistics (bosons vs fermions); some 2D topology.
| if_too_advanced_see = [[Quasiparticle]]
| if_you_want_the_math_see = This page
}}

{{Notation
| signature = Non-relativistic; SI / natural units.
| units = ℏ = 1 in derivations; θ = exchange phase angle.
}}

'''Anyons''' are 2D [[Quasiparticle|quasiparticles]] with exchange statistics that '''interpolate continuously between bosons and fermions'''. Their existence is a special feature of two-dimensional systems and reflects the fact that the topology of particle exchange is fundamentally different in 2D from 3D.

Anyons were proposed theoretically by Frank Wilczek (1982, ''Physical Review Letters'' 49: 957). They were experimentally observed as fractional-quantum-Hall quasiparticles (Bartolomei et al. 2020) and may form the basis of topological quantum computation.

In the [[Psionics|psionic framework]], anyons are a candidate constituent of 2D-confined ψ field configurations. The framework's relevance is exploratory — anyons are included for completeness of the quasiparticle survey.

== Exchange statistics in 2D vs 3D ==

In 3D, the exchange of two identical particles can be parameterised by a phase factor:

:<math>|\psi(1, 2)\rangle = e^{i\theta}\,|\psi(2, 1)\rangle</math>

Topology demands θ = 0 (bosons) or θ = π (fermions) — only two possibilities, because two exchanges in 3D return the state to itself with no additional structure.

In 2D, '''exchange paths have topological inequivalence''': the braid group is non-trivial. A continuous range of exchange phases is permitted:

θ ∈ [0, 2π)

Particles with such fractional statistics are '''anyons'''.

== Abelian anyons ==

The simplest case: '''Abelian anyons''' have a single exchange phase θ. The framework was elaborated by:

* Leinaas, Myrheim (1977) — first theoretical proposal.
* Wilczek (1982) — gave them the name "anyons" and the formal description.

Abelian-anyon statistics are seen in:

* '''Fractional Quantum Hall states''' (Laughlin 1983) — quasiparticles at filling ν = 1/3, 1/5, etc.
* '''Bartolomei et al. (2020, ''Science'' 368: 173)''' — direct interferometric observation of anyonic statistics in a ν = 1/3 quantum-Hall state.

== Non-Abelian anyons ==

A more powerful generalisation: '''non-Abelian anyons''' have exchange statistics that mix multiple internal states. The exchange operation is matrix-valued rather than a single phase.

Non-Abelian anyons are predicted in:

* '''ν = 5/2 quantum-Hall states''' (Moore-Read 1991).
* '''p<sub>x</sub> + ip<sub>y</sub> superconductors''' (Read-Green 2000).
* '''Topological-superconductor wire junctions''' (Kitaev 2001; Majorana fermions).

Experimental confirmation of non-Abelian anyons is contested: Majorana-fermion signatures in semiconductor-superconductor wires have been reported and retracted; the field remains active.

== Topological quantum computation ==

A central application: non-Abelian anyons can encode quantum information '''topologically''' — the qubit state is encoded in the global braiding history of the anyons, not in any local property. This makes the qubit '''robust to local noise''' — a major advantage for quantum computing.

Microsoft Station Q has been pursuing non-Abelian-anyon topological qubits for ~ 20 years.

== Anyons in psionic context ==

The framework's interest in anyons is exploratory:

* '''2D-confined ψ configurations''' — if the ψ field is confined to a 2D substrate (e.g. a thin biological membrane, a 2D solid-state device), its quasiparticle excitations could in principle have anyonic statistics.
* '''Topological protection''' — anyon-based ψ encoding would be robust against local noise, a desirable property for ψ-information storage.
* '''Connection to [[Soliton_Solutions_of_Psi_Field|soliton solutions]]''' — 2D solitons can carry anyonic statistics under certain conditions.

These are speculative connections; the framework does not currently use anyons as a central mechanism. They are noted here for completeness.

== Sanity checks ==

* '''3D limit''' → only bosons (θ = 0) and fermions (θ = π); recover standard quantum mechanics. ✓
* '''Trivial 2D system''' (no Chern-Simons-like coupling) → only bosons and fermions. ✓
* '''Fractional Quantum Hall ν = 1/3''' → θ = π/3 anyon statistics; verified experimentally. ✓
* '''ψ → 0''' (in framework) → anyon physics intact; no ψ-coupling. ✓ ([[Sanity_Check_Limits]] §5.)

== See Also ==

* [[Quasiparticle]]
* [[Skyrmions]]
* [[Soliton_Solutions_of_Psi_Field]]

== References ==

* Wilczek, F. (1982). "Quantum mechanics of fractional-spin particles." ''Physical Review Letters'' 49: 957.
* Leinaas, J. M., Myrheim, J. (1977). "On the theory of identical particles." ''Il Nuovo Cimento B'' 37: 1–23.
* Moore, G., Read, N. (1991). "Nonabelions in the fractional quantum Hall effect." ''Nuclear Physics B'' 360: 362–396.
* Bartolomei, H., et al. (2020). "Fractional statistics in anyon collisions." ''Science'' 368: 173–177.
* Nayak, C., Simon, S. H., Stern, A., Freedman, M., Das Sarma, S. (2008). "Non-Abelian anyons and topological quantum computation." ''Reviews of Modern Physics'' 80: 1083–1159.

[[Category:Psionics]]
[[Category:Quasiparticles]]
[[Category:Topology]]

Anyons - Revision history

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to ; lint-clean per tools/wiki_latex_lint.py

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to $; lint-clean per tools/wiki_latex_lint.py$