Free parafermions

Understand

The spectrum of the quantum Ising chain can be found by expressing the spins in terms of free fermions.

• An analogous transformation exists for clock chains with $Z_n$ symmetry, but is of less use because the resulting parafermionic operators remain interacting.
• Nonetheless, Baxter showed that a certain non-hermitian (but PT-symmetric) clock Hamiltonian is "free", in the sense that the entire spectrum is found in terms of independent energy levels, with the striking feature that there are $n$ possibilities for occupying each level.
• Here I show this directly explicitly finding shift operators obeying a $Z_n$ generalization of the Clifford algebra.