The Kitaev honeycomb model puts a spin-½ on every site of a honeycomb lattice, with an exchange interaction whose axis depends on the bond direction:
H = -Jx Σ(x-bonds) σᵢˣσⱼˣ -Jy Σ(y-bonds) σᵢʸσⱼʸ -Jz Σ(z-bonds) σᵢᶻσⱼᶻ
Kitaev showed this exactly-solvable model fractionalizes each spin into an itinerant Majorana fermion coupled to a static Z2 gauge field living on the bonds. Fixing the flux-free gauge sector, the Majoranas disperse on the honeycomb Brillouin zone with energy
E(k) = ±2|f(k)|, f(k) = Jz + Jx e^(i k·a1) + Jy e^(i k·a2)
a1, a2 are the honeycomb's primitive lattice vectors. This simulator evaluates |f(k)| on a live grid over the Brillouin zone (the heatmap panel below the lattice — brightness is literally this function) and reports its exact minimum as the gap Δ.
Note on the gap formula: as k ranges over the Brillouin zone, the two phase angles k·a1 and k·a2 sweep independently over the full circle, so |f(k)| is the magnitude of a sum of three free-angle vectors of fixed lengths Jx, Jy, Jz. The minimum such magnitude has a closed form from the triangle inequality: Δ = 2·max(0, 2·max(Jx,Jy,Jz) − (Jx+Jy+Jz)). This exact expression (verified here against a brute-force grid search over 400×400 angle pairs, agreement to grid resolution) is what this simulator reports — the original 3D sibling instead thresholds a coarse 44×44 surface-mesh minimum against an arbitrary 3%-of-ΣJ cutoff, which mis-tags points very close to the true boundary; the formula above is exact at every slider position.
- When one coupling dominates the other two (|Jz| > |Jx| + |Jy|, and cyclic permutations), the minimum of |f(k)| stays strictly positive — a gapped, Abelian phase whose excitations are toric-code-like anyons.
- Inside the triangle where every coupling is smaller than the sum of the other two (the isotropic point Jx=Jy=Jz sits deep inside it), |f(k)| touches exactly zero at two Dirac points — a gapless algebraic spin liquid of free Majorana fermions.
Clicking a bond on the lattice flips its Z2 gauge variable uᵢⱼ → −uᵢⱼ, exactly the real Kitaev-model move that creates a bound pair of Z2 vortices on the two hexagonal plaquettes touching that bond (each vortex costs a finite energy set by the gap Δ — this is why a gapless spin liquid can never be perturbatively adiabatically connected to a conventional magnet). The defect counter tracks how many bonds you've flipped away from the flux-free ground-state sector.