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 Wess-Zumino-Witten field theory.