In 3D, exchanging two identical particles can only multiply the wavefunction by ±1 — bosons or fermions, no other choice. In 2D the topology of paths is richer: exchanging twice is no longer the identity, just a knot. The wavefunction can pick up any phase (abelian anyons) or even a unitary matrix (non-abelian anyons). This is the math behind topological quantum computing.
why 2D is topologically special
Consider two identical particles in d-dimensional space. The configuration space — pairs of distinct points modulo exchange — has a fundamental group:
• In d ≥ 3: π₁ = ℤ/2. Two exchanges = identity. The wavefunction transforms by ±1 ⇒ bosons or fermions. No other option.
• In d = 2: π₁ = ℤ (or richer for n > 2 particles, the braid group B_n). Two exchanges form a loop around the other particle, which is not contractible. The phase can be anything.
Abelian anyons: under exchange, |ψ⟩ → e^(iθ) |ψ⟩ for some θ ∈ [0, 2π). Bosons (θ=0), fermions (θ=π), and the spectrum between. Semions at θ = π/2, Fibonacci anyons related to golden-ratio braid, and so on.
Non-abelian anyons: the wavefunction lives in a degenerate ground-state manifold, and braids act as matrices, not just phases. The order of braids matters — like rotations in 3D. Quantum information stored in this degeneracy is topologically protected: only large-scale braiding (not local noise) can change it.
Experimental discovery: fractional quantum Hall states (Tsui, Stormer, Laughlin Nobel 1998) host anyons at filling fractions ν = 1/3, 5/2, etc. The ν = 5/2 plateau is conjectured to host Ising anyons with non-abelian braiding. 2020-2023: experiments (Manfra/Halperin groups) saw fractional statistics directly.
Topological quantum computing (Kitaev 1997, Freedman 1998): encode a qubit in the degenerate ground state of a non-abelian anyon system. Gates = braids of anyons. Microsoft's Station Q has spent two decades chasing Majorana fermions, an Ising-anyon-like quasiparticle, as the basis of a fault-tolerant quantum computer. The wires never touch the qubit directly — the computation is performed by topology alone.