$\Gamma$-matrix generalization of the Kitaev model

We extend the Kitaev model defined for the Pauli matrices to the Clifford algebra of $\Gamma$ matrices, taking the $4\times 4$ representation as an example. On a decorated square lattice, the ground state spontaneously breaks time-reversal symmetry and exhibits a topological phase transition. The topologically nontrivial phase carries gapless chiral edge modes along the sample boundary. On the three-dimensional (3D) diamond lattice, the ground states can exhibit gapless 3D Dirac-cone-like excitations and gapped topological insulating states. Generalizations to even higher rank $\Gamma$ matrices are also discussed.

Figure 1

(Color online) The decorated square lattice (right) with linked diagonal bonds for the Hamiltonian of Eq. (8). Each unit cell (left) consists four sites ($A$, $B$, $C$, and $D$), ten links, and six plaquettes including two squares and four triangles. The bonds are classified into five types as marked with labels $a$ which run from 1 to 5 and $1^{\prime }$ to $5^{\prime }$.

Figure 2

(Color online) When the magnetic and structural unit cells coincide, there are five ${\mathbb{Z}}_2$ gauge degrees of freedom, which we take to be ${\sigma }_{1–5}$ as shown, with $u_{AD}={\sigma }_1$, etc. The other $u_{ij}$ are positive for links where the black arrow points from $i$ to $j$.

Figure 3

(Color online) Total energies (per site) versus $\alpha ={\text{tan}}^{-1}\left(J_D/J\right)$ for our decorated square lattice model. Curves for all possible flux configurations are shown. The flux configurations for the two lowest-lying total energy states are twofold degenerate owing to time reversal, which reverses the flux in the odd-membered loops. The third-lowest-lying total energy state has an eightfold degenerate flux configuration. At $\alpha =\frac{1}{2}\pi$ the horizontal and vertical bond strengths vanish and the system becomes a set of disconnected dimers.

Figure 4

(Color online) The band structure of Eq. (38) at $J_D/J>0$. (a) $J_D/J=1$ with a massive Dirac spectrum at $\left(\pi ,\pi \right)$ with the gap value $2J_D$; (b) $J_D/J=2\sqrt{2}$ where a gapless Dirac cone appears; and (c) $J_D/J=3.5$. The gap value at (0,0) is $2\left|J_D-2\sqrt{2}J\right|$. A topological phase transition occurs from a topological nontrivial phase at $J_D/J<2\sqrt{2}$ to a trivial phase at $J_D/J>2\sqrt{2}$.

Figure 5

(Color online) (a) The gapless edge modes of $\eta$ fermions across the band gap appear in open boundary samples in the topologically nontrivial phase at $J_D/J<2\sqrt{2}$. (b) The edge modes become nontopological at $J_D/J>2\sqrt{2}$.

Figure 6

(Color online) The 3D diamond lattice with the nearest separation $a$ and two sublattices $A$ (filled circles) and $B$ (hollow circles). The unit vectors ${\hat{\mathit{e}}}_{1,2,3,4}$ are defined from each $A$ site to its four $B$ neighbors (see text).

Figure 7

(Color online) A noncoplanar hexagonal ring within the diamond lattice.

Figure 8

(Color online) First Brillouin zone for the fcc structure.