  We discuss generalizations of quantum spin Hamiltonians using anyonic degrees
of freedom. The simplest model for interacting anyons energetically favors
neighboring anyons to fuse into the trivial (`identity') channel, similar to
the quantum Heisenberg model favoring neighboring spins to form spin singlets.
Numerical simulations of a chain of Fibonacci anyons show that the model is
critical with a dynamical critical exponent z=1, and described by a
two-dimensional conformal field theory with central charge c=7/10. An exact
mapping of the anyonic chain onto the two-dimensional tricritical Ising model
is given using the restricted-solid-on-solid (RSOS) representation of the
Temperley-Lieb algebra. The gaplessness of the chain is shown to have
topological origin.
