Abstract
Kitaev's sixteenfold way is a classification of exotic topological orders in which $\mathbb{Z}_2$ gauge theory is coupled to Majorana fermions of Chern number $C$. The $16$ distinct topological orders within this class, depending on $C \, \mathrm{mod} \, 16$, possess a rich variety of Abelian and non-Abelian anyons. We realize more than half of Kitaev's sixteenfold way, corresponding to Chern numbers $0$, $\pm 1$, $\pm 2$, $\pm 3$, $\pm 4$, and $\pm 8$, in an exactly solvable generalization of the Kitaev honeycomb model. For each topological order, we explicitly identify the anyonic excitations and confirm their topological properties. In doing so, we observe that the interplay between lattice symmetry and anyon permutation symmetry may lead to a "weak supersymmetry" in the anyon spectrum. The topological orders in our honeycomb lattice model could be directly relevant for honeycomb Kitaev materials, such as $\alpha$-RuCl$_3$, and would be distinguishable by their specific quantized values of the thermal Hall conductivity.
Highlights
Topological order is an important cornerstone of modern condensed matter physics which facilitates a classification of gapped phases of matter beyond the classical paradigm of spontaneous symmetry breaking [1]
For Abelian topological orders, these braiding operations act on a single quantum state, while for non-Abelian topological orders, they act on a set of degenerate quantum states within an internal space spanned by the anyons themselves
We emphasize that this fermionic symmetry is not a supersymmetry in the conventional sense because it relates the vortex excitations e and m that are both bosons in terms of their self statistics [8]
Summary
Topological order is an important cornerstone of modern condensed matter physics which facilitates a classification of gapped phases of matter beyond the classical paradigm of spontaneous symmetry breaking [1]. We study an exactly solvable generalization [39] of the Kitaev model that respects all symmetries of the honeycomb lattice and realizes more than half of the topological orders in Kitaev’s sixteenfold way, corresponding to Majorana Chern numbers 0, ±1, ±2, ±3, ±4, and ±8. These topological orders contain both Abelian and nonAbelian anyons with a rich variety of fusion and braiding rules, and are experimentally distinguishable by their different quantized values of the thermal Hall conductivity. Since the additional four-spin interactions of our generalized Kitaev model arise naturally from time-reversal-symmetric perturbations [39], in the same way as the three-spin interactions in the original Kitaev model arise from an external magnetic field, we believe that the |C| > 1 topological orders described in this work are likely to be realized in spin-orbit-coupled honeycomb magnets, such as α-RuCl3
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