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 i
nteracting. 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. I also find higher Hamiltonians that commute with Baxters 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.
We show how to compute the exact partition function for lattice statistical-mechanical models whose Boltzmann weights obey a special crossing symmetry. The crossing symmetry equates partition functions on different trivalent graphs, allowing a transf
ormation to a graph where the partition function is easily computed. The simplest example is counting the number of nets without ends on the honeycomb lattice, including a weight per branching. Other examples include an Ising model on the Kagome lattice with three-spin interactions, dimers on any graph of corner-sharing triangles, and non-crossing loops on the honeycomb lattice, where multiple loops on each edge are allowed. We give several methods for obtaining models with this crossing symmetry, one utilizing discrete groups and another anyon fusion rules. We also present results indicating that for models which deviate slightly from having crossing symmetry, a real-space decimation (renormalization-group-like) procedure restores the crossing symmetry.
We explain how (perturbed) boundary conformal field theory allows us to understand the tunneling of edge quasiparticles in non-Abelian topological states. The coupling between a bulk non-Abelian quasiparticle and the edge is due to resonant tunneling
to a zero mode on the quasiparticle, which causes the zero mode to hybridize with the edge. This can be reformulated as the flow from one conformally-invariant boundary condition to another in an associated critical statistical mechanical model. Tunneling from one edge to another at a point contact can split the system in two, either partially or completely. This can be reformulated in the critical statistical mechanical model as the flow from one type of defect line to another. We illustrate these two phenomena in detail in the context of the nu=5/2 quantum Hall state and the critical Ising model. We briefly discuss the case of Fibonacci anyons and conclude by explaining the general formulation and its physical interpretation.
