Kitaev Honeycomb Spin Liquid

Bond Couplings

J_x (red bonds) 0.33

J_y (green bonds) 0.33

J_z (blue bonds) 0.34

Auto-sweep J_z through the transition

Live Readouts

Majorana gap Δ

0.000 J

Phase

gapless B-phase

Z2 flux defects (click a bond)

Legend

J_x bond

J_y bond

J_z bond

flipped (flux-defect) bond

|f(k)| dispersion surface, height = energy

How it works

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 mesh above the lattice — its height and colour are literally this function) and reports its minimum as the gap Δ.

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 zero at two Dirac points — a gapless algebraic spin liquid of free Majorana fermions.

Clicking a bond in the 3D 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.

Drag to orbit · scroll to zoom · click a bond to flip its Z2 flux

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