Transverse Field γ-Matrix Spin Chains

We introduce a simple lattice spin model that is written in terms of the well-known four-dimensional γ-matrix representation of the Clifford algebra. The local spins with a four-dimensional Hilbert space transform in a spinorial (1/2,0) ⊕ (0,1/2) representation of $SO(4)$, a symmetry of our model. When studied on a chain, and as a function of a transverse field tuning parameter, our model undergoes a quantum phase transition from a valence bond solid phase to a critical phase that is described by an SU(2)₁ WZW field theory.