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.

6+9 pages, 2+1 figures, published version

References in corpus (9)

• Non-Abelian Anyons and Topological Quantum Computation
• Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models
• Exact results for spin dynamics and fractionization in the Kitaev Model
• Topological characterization of quantum phase transitions in a S=1/2 spin model
• A $\Gamma$-matrix generalization of the Kitaev model
• Three-dimensional topological phase on the diamond lattice
• Projected BCS states and spin Hamiltonians for the SO(n)_1 Wess-Zumino-Witten model
• Dynamics of a two-dimensional quantum spin-orbital liquid: spectroscopic signatures of fermionic magnons
• Parity of Chern numbers in the Kitaev honeycomb model and the sixteenfold way

Cited by in corpus (36)

• Fractionalized fermionic quantum criticality in spin-orbital Mott insulators
• Unveiling the S=3/2 Kitaev Honeycomb Spin Liquids
• Unifying Kitaev magnets, kagome dimer models and ruby Rydberg spin liquids
• Flux crystals, Majorana metals, and flat bands in exactly solvable spin-orbital liquids
• Topological Disorder Parameter
• Quantum phases of two-dimensional ${\mathbb{Z}}_2$ gauge theory coupled to single-component fermion matter
• 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
• Quantum liquids of the S=3/2 Kitaev honeycomb and related Kugel-Khomskii models
• Vison crystals, chiral and crystalline phases in the Yao-Lee model
• Phase diagrams of SO($N$) Majorana-Hubbard models: Dimerization, internal symmetry breaking, and fluctuation-induced first-order transitions
• Microscopic Roadmap to a Yao-Lee Spin-Orbital Liquid
• Critical properties of metallic and deconfined quantum phase transitions in Dirac systems
• Resonating valence bond realization of spin-1 non-Abelian chiral spin liquid on the torus
• Bosonization of Majorana modes and edge states
• Anyon condensation and confinement transition in a Kitaev spin liquid bilayer
• Chiral conformal field theory for topological states and the anyon eigenbasis on the torus
• Exact deconfined gauge structures in the higher-spin Yao-Lee model: a quantum spin-orbital liquid with spin fractionalization and non-Abelian anyons
• 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
• Variational Tensor Network Operator
• The spin-orbital Kitaev model: from kagome spin ice to classical fractons
• Multipolar spin liquid in an exactly solvable model for $j_{\mathrm{eff}}=\frac{3}{2}$ moments
• Magnetic order through Kondo coupling to quantum spin liquids
• Fractionalized fermionic multicriticality in anisotropic Kitaev spin-orbital liquids
• Lacing topological orders in two dimensions: exactly solvable models for Kitaev's sixteen-fold way
• Projectively implemented altermagnetism in an exactly solvable quantum spin liquid
• Wegner's Ising gauge spins versus Kitaev's Majorana partons: Mapping and application to anisotropic confinement in spin-orbital liquids
• Non-Abelian topological order with SO(5)${}_1$ chiral edge states
• Three-dimensional spin-orbital liquids
• Transverse Field $\gamma$-Matrix Spin Chains
• Thermal and quantum fluctuations in extended Kitaev-Yao-Lee spin-orbital model
• Emergent spacetime supersymmetry at 2D fractionalized quantum criticality
• Partons from stabilizer codes
• Exactly solvable spin liquids in Kitaev bilayers and moiré superlattices