Fig. 1

Classical ground states of the Kitaev model on the honeycomb lattice. The lattice has three types of NN bonds, labeled by “x”, “y”, and “z”. a General form of ground states. The numbers 1–13 label the spin sites, and the shaded 3×1 vectors below each number give the corresponding Cartesian components (x, y, and z) of the spins (colored red, green, and blue, respectively). Here, κ = −sgn(K), S _i  = (a _i , b _i , c _i ) or κ(a _i , b _i , c _i ) if i belongs to the A or B sublattice, and \(a_i^2 + b_i^2 + c_i^2 = S^2\). b The so-called Cartesian states of BSS correspond to the states where only one of the Cartesian component is finite. These states map to dimer coverings, with (yellow) dimers representing satisfied bonds. The spin orientation of each dimer is described by an Ising-like variable η = ±1 (colored according to the non-vanishing Cartesian component of the two spins shared by the bond). The shaded hexagon has the shortest loop with no dimers