A $\Gamma $-matrix generalization of the Kitaev model

We extend the Kitaev model defined for the Pauli-matrices to the Clifford algebra $\Gamma $-matrices by taking the 4$\times $4 representation as an example. In a 2D decorated square lattice, the ground state spontaneously breaks time-reversal symmetry and exhibits a topological phase transition. The topologically non-trivial phase carries gapless chiral edge modes along the sample boundary. In the 3D diamond lattice, the ground states can exhibit gapless 3D Dirac cone- like excitations and gapped topological insulating states. The generalizations to even higher rank $\Gamma $-matrices are also discussed.