Interacting anyons in topological quantum liquids: The golden chain

TL;DR

This work develops an interacting chain of Fibonacci anyons as an anyonic analogue of the Heisenberg model, revealing gapless critical behavior protected by topological symmetry. The authors map the model to the RSOS representation of the 2D tricritical Ising model and show an exact correspondence to a c=7/10 CFT, confirmed by finite-size spectra and entanglement entropy analysis. The results connect 1D anyonic chains to established 2D CFTs, provide a concrete mechanism for topological protection of gaplessness, and suggest avenues for exploring non-Fermi liquid behavior in anyonic networks and higher-k generalizations. These insights have implications for topological quantum computation and the broader understanding of non-Abelian anyon liquids.

Abstract

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.

Figures (4)

• Figure 1: a) Illustration of the Fibonacci chain with $L$$\tau$-anyons. b) The fusion path. c) Definition of the $F$-matrix.
• Figure 2: (Color online) Entropy scaling for interacting Fibonacci anyons arranged along an open (open squares) or periodic chain (closed circles) versus the system size $L$. Logarithmic fits (solid lines) give central charge estimates of $c_{\rm PBC}=0.701 \pm 0.001$ and $c_{\rm OBC}=0.70 \pm 0.01$ respectively, where for the open boundary conditions only the values for the 5 largest systems have been taken into account due to large finite-size effects.
• Figure 3: Transfer matrix of the RSOS model.
• Figure 4: Energy spectra for periodic Fibonacci chains of size $L=36$ and $L=37$. The spectra have been rescaled and shifted such that the two lowest eigenvalues match the conformal field theory assignments. The open boxes indicate the positions of the primary fields of the $c=7/10$ conformal field theory. The open circles give the positions of multiple descendant fields as indicated. While we find excellent agreement in general, finite-size effects lead to small discrepancies for the higher energy states. The solid line is a cosine-fit of the dispersion which serves as a guide to the eye.