Three types of representation of spin in terms of Majorana fermions and an alternative solution of the Kitaev honeycomb model

Based on the Dirac spinor representation of the SO(4) group, we discuss the relationship between three types of representation of spin in terms of Majorana fermions, namely the Kitaev representation, the SO(3) representation, and the SO(4) chiral representation. Comparing the three types, we show that the Hilbert space of the SO(3) representation is different from the other two by requiring a pairing of sites, but it has the advantage over the other two in that no unphysical states are involved. As an example of its application, we present an alternative solution of the Kitaev honeycomb model. Our solution involves no unphysical states which enables a systematic calculation of physical observables. Finally, we discuss an extension of the model to a more general exactly soluble $Z_2$ gauge theory interacting with complex fermions.

Figure 1

The Kitaev model on the honeycomb lattice. ${\mathit{e}}_1=a(-\frac{1}{2},-\frac{\sqrt{3}}{2})$ and ${\mathit{e}}_2=a(\frac{1}{2},-\frac{\sqrt{3}}{2})$ are the primitive translations, with $a$ being the lattice constant. The two sublattices denoted in the main text by A and B are shown by blue and red dots, respectively. The sites of each of the two sublattices form a diamond lattice, as shown by the green dotted line for the A sublattice.

Figure 2

The diamond lattice corresponding to the original honeycomb lattice, with unit vectors ${\mathit{e}}_1,{\mathit{e}}_2$. The matter (complex) fermion lives on the lattice sites, labeled by the black dots. The $Z_2$ gauge connection ${\tilde{\sigma }}^z$ lives on the bonds, labeled by the red dots. One of the plaquettes is shown by the blue dotted lines and one of the star operators used in the constraint (40) is shown by the green dotted lines.

Figure 3

The schematic ground state phase diagram of the generalized $Z_2$ gauge theory with matter fermions given by Eq. (54) is shown for isotropic couplings $J_x=J_y=J_z$ as a function of $K<0$ and $h$.