Quantum Spin Liquid — Kitaev Honeycomb

Fractionalized excitations: Majorana fermions & Z₂ flux visons on the honeycomb lattice

Kitaev's Honeycomb Model

Alexei Kitaev (2006) showed that spin-1/2 particles on a honeycomb lattice with bond-directional Ising interactions can be solved exactly using a Jordan-Wigner transformation to Majorana fermions.

H = -Jx Σx σxσx - Jy Σy σyσy
-Jz Σz σzσz

Each plaquette carries a conserved Z₂ flux W_p = ±1. The ground state has W_p = +1 everywhere. Excited flux excitations are visons — created in pairs.

W_p = σx₁σy₂σz₃σx₄σy₅σz₆ = ±1

Three phases: A phase (gapped, abelian anyons) when one coupling dominates; B phase (gapless, non-abelian with TRS breaking) in the interior triangle.

Phase: A_z (gapped)
Toric code topological order
W_p = +1 (ground state)
W_p = −1 (vison)
Majorana hopping bond