← Back to Condensed matter and solid-state physics Book II

Anyon is a ScholarlyWiki page in the Quantum Collection about two-dimensional quasiparticle excitations whose exchange statistics can differ from both bosons and fermions.

Overview

An anyon is a quasiparticle excitation that can occur in effectively two-dimensional quantum systems and has exchange statistics more general than those of bosons or fermions. When two identical particle-like excitations are exchanged in three spatial dimensions, the quantum state can only remain the same or change sign. In two dimensions, the paths of localized excitations can braid around one another, allowing more general statistical phases or even transformations within a space of degenerate states.

Anyons are not elementary particles in empty space. They are emergent excitations of many-body quantum systems, especially topologically ordered phases such as some fractional Hall states. Their unusual statistics make them important in condensed-matter physics, quantum topology, and proposals for fault-tolerant quantum computation.

Key ideas

The distinction between bosons and fermions is based on what happens to a many-body wavefunction when two identical excitations are exchanged. Bosonic exchange gives a phase of $+ 1$ ; fermionic exchange gives a phase of $- 1$ . In two dimensions, the topology of exchange paths is richer, because one localized excitation can wind clockwise or counterclockwise around another in distinguishable ways.

For Abelian anyons, exchange can multiply the state by a general phase:

$ψ ⟼ e^{i θ} ψ,$

where $θ$ need not be only $0$ or $π$ . For non-Abelian anyons, exchange can act as a matrix on a degenerate state space, so the final state can depend on the order in which braids are performed.

Braiding

The word "anyon" reflects that, in two dimensions, suitable quasiparticle excitations can have exchange phases beyond the boson and fermion cases. The mathematical structure is described by the braid group rather than the ordinary permutation group. A braid records how excitation worldlines wind around each other in spacetime.

Braiding is robust because it depends on the topology of paths rather than on fine geometric details. This robustness is one reason non-Abelian anyons are studied for topological quantum information. Logical operations could be encoded by moving quasiparticles around one another, making some errors less sensitive to local disturbances.

Fractional Hall systems

The fractional Hall effect provides the most important physical setting for anyonic quasiparticles. In Laughlin states, quasiparticles can carry fractional electric charge and Abelian anyonic statistics. More complex fractional Hall states are candidates for non-Abelian anyons.

The connection arises because a strongly correlated electron fluid can have topological order. Its excitations are not simply individual electrons or holes. Instead, collective excitations can carry fractional quantum numbers and acquire characteristic phases when moved around one another.

Abelian and non-Abelian anyons

Abelian anyons acquire a phase under exchange. Their braiding operations commute, so changing the order of braids does not change the final operation except through the total phase. Many simple fractional Hall quasiparticles are described this way.

Non-Abelian anyons are more exotic. Their braiding operations generally do not commute, and exchanging particles can rotate the system within a degenerate Hilbert space. This property is the basis of topological quantum computation proposals, where information is stored nonlocally and manipulated by braiding.

Relation to topology

Anyons are closely connected with topological phases of matter. Their existence signals that a phase has long-range entanglement and topological order rather than only conventional symmetry breaking. Edge states, ground-state degeneracy, and fractionalized quasiparticles often appear together in these systems.

The study of anyons links condensed matter, quantum field theory, knot theory, and quantum information. Even when direct braiding experiments are difficult, signatures of anyonic physics can appear through interferometry, tunneling, thermal transport, and the structure of low-energy effective theories.

Physics:Quantum anyon

Overview

Key ideas

Braiding

Fractional Hall systems

Abelian and non-Abelian anyons

Relation to topology

See also

Table of contents (217 articles)

Index

Full contents

References

Navigation menu

Search