Microscopic models for Kitaev's sixteenfold way of anyon theories

Abstract

In two dimensions, the topological order described by ${\mathbb{Z}}_2$ gauge theory coupled to free or weakly interacting fermions with a nonzero spectral Chern number $\nu$ is classified by $\nu \mathrm{mod}16$ as predicted by Kitaev [Ann. Phys. 321, 2 (2006)]. Here we provide a systematic and complete construction of microscopic models realizing this so-called sixteenfold way of anyon theories. These models are defined by $\Gamma$ matrices satisfying the Clifford algebra, enjoy a global $\mathrm{SO}(\nu )$ symmetry, and live on either square or honeycomb lattices depending on the parity of $\nu$. We show that all these models are exactly solvable by using a Majorana representation and characterize the topological order by calculating the topological spin of an anyonic quasiparticle and the ground-state degeneracy. The possible relevance of the $\nu =2$ and $\nu =3$ models to materials with Kugel-Khomskii-type spin-orbital interactions is discussed.

arXiv comment6+9 pages, 2+1 figures, published version

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• Fractionalized quantum criticality in spin-orbital liquids from field theory beyond the leading order
• Kitaev spin-orbital bilayers and their moiré superlattices
• Effect of ring-exchange interactions in the extended Kitaev honeycomb model
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• Vison crystals, chiral and crystalline phases in the Yao-Lee model
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• Microscopic Roadmap to a Yao-Lee Spin-Orbital Liquid
• Critical properties of metallic and deconfined quantum phase transitions in Dirac systems
• Bosonization of Majorana modes and edge states
• Anyon condensation and confinement transition in a Kitaev spin liquid bilayer
• Resonating valence bond realization of spin-1 non-Abelian chiral spin liquid on the torus
• Chiral conformal field theory for topological states and the anyon eigenbasis on the torus
• Variational Tensor Network Operator
• Chiral spin liquid in a ${\mathbb{Z}}_3$ Kitaev model
• Symmetry-protected topological phases, conformal criticalities, and duality in exactly solvable SO($n$) spin chains
• Exact deconfined gauge structures in the higher-spin Yao-Lee model: a quantum spin-orbital liquid with spin fractionalization and non-Abelian anyons
• Quantum Loop States in Spin-Orbital Models on the Honeycomb Lattice
• Fractionalized fermionic multicriticality in anisotropic Kitaev spin-orbital liquids
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• Transverse Field $\gamma$-Matrix Spin Chains
• Partons from stabilizer codes
• Exactly solvable spin liquids in Kitaev bilayers and moiré superlattices