Exactly solvable model with Majorana fermions and non-Abelian anyons
Alexei Kitaev's 2006 honeycomb model places spin-1/2 particles on a honeycomb lattice with bond-dependent Ising couplings. It is exactly solvable by mapping spins to Majorana fermions via a Jordan-Wigner transformation: the Hamiltonian becomes free Majoranas hopping on the lattice threaded by a Z₂ gauge field.
The phase diagram has three gapped A-phases where one coupling dominates (topological order with Abelian Z₂ anyons), and a central gapped B-phase when all couplings are comparable and a small magnetic field h is applied. The B-phase hosts non-Abelian Ising anyons — the same as in the ν=5/2 fractional quantum Hall state — whose braiding implements topologically protected quantum gates.
The visualization shows the honeycomb lattice with colored bonds (x=red, y=green, z=blue) and animated Majorana fermion hopping amplitudes modulated by the local Z₂ flux configuration.