Transverse Field $\gamma$-Matrix Spin Chains

TL;DR

This paper introduces a novel lattice spin model based on $ ext{Clifford algebra}$ with a transverse field, exhibiting a quantum phase transition from a valence bond solid to a critical phase described by an $SU(2)_1$ WZW theory.

Contribution

The paper presents a new spin model using $ ext{gamma matrices}$ with a symmetry-driven phase transition, connecting lattice models to conformal field theory.

Findings

Model exhibits a quantum phase transition controlled by a transverse field.

Transition is from a valence bond solid to an $SU(2)_1$ WZW critical phase.

The model's spins transform under a spinorial representation of $SO(4)$.

Abstract

We introduce a simple lattice spin model that is written in terms of the well-known four-dimensional $\gamma$-matrix representation of the Clifford algebra. The local spins with a four-dimensional Hilbert space transform in a spinorial $(1/2,0) \oplus (0,1/2)$ 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.