  Using finite size scaling arguments, the critical properties of a chain of
interacting anyons can be extracted from the low energy spectrum of a finite
system. In Phys. Rev. Lett. 98, 160409 (2007), Feiguin et al. showed that an
antiferromagnetic (AFM) chain of Fibonacci anyons on a torus is in the same
universality class as the tricritical Ising model, and that criticality is
protected by a topological symmetry. In the present paper we first review the
graphical formalism for the study of anyons on the disc and demonstrate how
this formalism may be consistently extended to the study of systems on surfaces
of higher genus. We then employ this graphical formalism to study finite rings
of interacting anyons on both the disc and the torus, and show that analysis on
the disc necessarily yields an energy spectrum which is a subset of that which
is obtained on the torus. For a critical Hamiltonian, one may extract from this
subset the scaling dimensions of the local scaling operators which respect the
topological symmetry of the system. Related considerations are also shown to
apply for open chains.
