Physical solutions of the Kitaev honeycomb model

We investigate the exact solution of the honeycomb model proposed by Kitaev and derive an explicit formula for the projector onto the physical subspace. The physical states are simply characterized by the parity of the total occupation of the fermionic eigenmodes. We consider a general lattice on a torus and show that the physical fermion parity depends in a nontrivial way on the vortex configuration and the choice of boundary conditions. In the vortex-free case with a constant gauge field we are able to obtain an analytical expression of the parity. For a general configuration of the gauge field the parity can be easily evaluated numerically, which allows the exact diagonalization of large spin models. We consider physically relevant quantities, as in particular the vortex energies, and show that their true value and associated states can be substantially different from the one calculated in the unprojected space, even in the thermodynamic limit.

Figure 1

Honeycomb lattice, with basis vectors ${\mathbf{e}}_{1,2}$. The directions of $x$, $y$, $z$ links are indicated, as well as the vector $\tau$ joining B and A sublattices (white and black dots, respectively). The most general torus on the lattice can be specified by $L_1{\mathbf{e}}_1$ and $L_2{\mathbf{e}}_2+M{\mathbf{e}}_1$ (here $L_1=4$, $L_2=M=2$). The numbers $1,2,...,16$ label the sites as described in the main text, see Eq. (8).

Figure 2

An illustration of the partitioning into $\Omega ,{\Omega }_{\pm }$ described in the main text. The vectors ${\mathbf{b}}_{1,2}$ define the reciprocal lattice and the hexagon is the first Brillouin zone. The four square points are wave vectors in $\Omega$ while the dots and crosses are in ${\Omega }_{+}$ and ${\Omega }_{-}$, respectively. In this example $L_1=L_2=6$ and $M=0$.

Figure 3

Physical (solid line) and unphysical (dashed line) finite size corrections to the thermodynamic energy of a vortex-free configuration $u_{ij}=1$ for $L_1=L_2=L$. The main plot (a) refers to the gapless phase with $M=0$, $J_x=J_y=J_z=1$, and ${\varepsilon }_0\simeq -1.5746$.[1] The inset (b) corresponds to the gapped phase with $M=1$, $J_x=J_y=0.2$, $J_z=1$, and ${\varepsilon }_0\simeq -1.0202$.

Figure 4

Physical (thick solid line) and unphysical (thick dashed line) excitation energy of two adjacent vortices. This plot refers to the gapless phase with $J_x=J_y=J_z=1$, $L_1=2L$, $L_2=M=L$, and a single $u_{ij}=-1$ with $ij$ being a $z$ link. Thin dashed lines extrapolate to the thermodynamic limit. The dots at $L=3$ (unphysical) and $L=13$ (physical) represent the largest system sizes with negative excitation energy.

Figure 5

Physical (solid line) and unphysical (dashed line) excitation energy of two adjacent vortices in the gapped phase with $J_x=J_y=0.1$, $J_z=1$, $L_1=L_2=L$, and a single $u_{ij}=-1$ with $ij$ being a $y$ link. (a) refers to $M=0$ while (b) to $M=1$.