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 Zn 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 Zn generalization of the Clifford algebra. I also find higher Hamiltonians that commute with Baxter’s and prove their spectrum comes from the same set of energy levels. This thus provides an explicit notion of a “free parafermion”. A byproduct is an elegant method for the solution of the Ising/Kitaev chain with spatially varying couplings.

http://arxiv.org/abs/1310.6049
Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)