Parafermions in the tau-2 model II

Understand

Many years ago Baxter introduced an inhomogeneous two-dimensional classical spin model, now called the $\tau_2(t)$ model with free boundary conditions, and he specialized the resulting quantum spin-chain Hamiltonian in a special limit to a simple clock Hamiltonian.

• Recently, Fendley showed that this clock Hamiltonian can be expressed in terms of free "parafermions." Baxter followed this up by showing that this construction generalizes to the more general $\tau_2(t)$ model, provided some conjectures hold.
• In this paper, we will compare the different notations and approaches enabling us to express the Hamiltonians in terms of projection operators as introduced by Fendley.
• By examining the properties of the raising operators, we are then able to prove the last unproven conjecture in Baxter's paper left in our previous paper.