Interacting Anyons in Topological Quantum Liquids: The Golden Chain

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 (2D) conformal field theory with central charge $c=\frac{7}{10}$. An exact mapping of the anyonic chain onto the 2D tricritical Ising model is given using the restricted-solid-on-solid representation of the Temperley-Lieb algebra. The gaplessness of the chain is shown to have topological origin.

Figure 1

(a) Illustration of the Fibonacci chain with $L$ $\tau$-anyons. (b) The fusion path. (c) Definition of the $F$ matrix.

Figure 2

Transfer matrix of the RSOS model.

Figure 3

Energy spectra for periodic chains of size $L$. Energies are rescaled and shifted such that the two lowest eigenvalues match the CFT assignments. Open boxes indicate positions of primary fields of the $c=\frac{7}{10}\mathrm{CFT}$. Open circles give positions of descendant fields as indicated. As a guide to the eye the solid line is a cosine-fit of the dispersion.