  Finding the precise correspondence between lattice operators and the
continuum fields that describe their long-distance properties is a largely open
problem for strongly interacting critical points. Here we solve this problem
essentially completely in the case of the three-state Potts model, which
exhibits a phase transition described by a strongly interacting 'parafermion'
conformal field theory. Using symmetry arguments, insights from integrability,
and extensive simulations, we construct lattice analogues of nearly all the
relevant and marginal physical fields governing this transition. This
construction includes chiral fields such as the parafermion. Along the way we
also clarify the structure of operator product expansions between order and
disorder fields, which we confirm numerically. Our results both suggest a
systematic methodology for attacking non-free field theories on the lattice and
find broader applications in the pursuit of exotic topologically ordered phases
of matter.
