Pedestrian's way to Baxter's Bethe ansatz for the periodic XYZ chain

A chiral coordinate Bethe ansatz method is developed to study the periodic XYZ chain. We construct a set of chiral vectors with fixed number of kinks. All vectors are factorized and have simple structures. Under roots of unity conditions, the Hilbert space has an invariant subspace, and our vectors form a basis of this subspace. We propose a Bethe ansatz solely based on the action of the Hamiltonian on the chiral vectors, avoiding the use of transfer matrix techniques. This allows us to parametrize the expansion coefficients and derive the homogeneous Bethe ansatz equations, whose solutions give the exact energies and eigenstates. Our analytic results agree with earlier approaches, notably by Baxter, and are supported by numerical calculations.