Collective States of Interacting Fibonacci Anyons

We show that chains of interacting Fibonacci anyons can support a wide variety of collective ground states ranging from extended critical, gapless phases to gapped phases with ground-state degeneracy and quasiparticle excitations. In particular, we generalize the Majumdar-Ghosh Hamiltonian to anyonic degrees of freedom by extending recently studied pairwise anyonic interactions to three-anyon exchanges. The energetic competition between two- and three-anyon interactions leads to a rich phase diagram that harbors multiple critical and gapped phases. For the critical phases and their higher symmetry end points we numerically establish descriptions in terms of two-dimensional conformal field theories. A topological symmetry protects the critical phases and determines the nature of gapped phases.

Figure 1

Illustration of (a) the alternating chain and (b) the generalized Majumdar-Ghosh chain with a three-anyon interaction term. Shaded enclosures indicate the fusion products that are energetically biased by the Hamiltonian.

Figure 2

The phase diagram of our anyonic Majumdar-Ghosh chain on the circle parameterized by $\theta$, with pairwise fusion term $J_2=\cos \theta$ and three-particle fusion term $J_3=\sin \theta$. Besides extended critical phases around the exactly solvable points ($\theta =0$, $\pi$) that can be mapped to the tricritical Ising model and the 3-state Potts model, there are two gapped phases (gray filled). The phase transitions (red circles) out of the tricritical Ising phase exhibit higher symmetries and are both described by CFTs with central charge $4/5$. In the gapped phases exact ground states are known at the positions marked by the stars. In the lower left quadrant a small sliver of an incommensurate phase occurs and a phase which has $Z_4$ symmetry. These latter two phases also appear to be critical.

Figure 3

Energy spectrum at the $S_3$-symmetric point ($\theta \cong 0.176\pi$) of the Majumdar-Ghosh chain. The energies have been shifted and rescaled so that the two lowest eigenvalues match the CFT scaling dimensions. The open boxes indicate the positions of the primary fields of the parafermionic subset of the $c=4/5$ CFT; fields with topological flux $\tau$ are marked. The open circles give the positions of multiple descendant fields as indicated.

Figure 4

Schematic energy spectra in the gapped phase for $0.19\le \theta /\pi \le 0.30$. Below a continuum of scattering states (gray shaded) a quasiparticle band forms. The plot combines data with $24\le L\le 36$. Solid lines are a guide to the eye.