Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2005.03655 · Physical review. B./Physical review. B

Parity of Chern numbers in the Kitaev honeycomb model and the sixteenfold way

Jean-Noël Fuchs, Sourabh Patil, Julien Vidal

Published Sep 15, 2020• 17 citations
arXivHTMLPDF

Abstract

In two dimensions, topological phases of free Majorana fermions coupled to a ${\mathbb{Z}}_{2}$ gauge field are known to be classified according to the Chern number $\ensuremath{\nu}\ensuremath{\in}\mathbb{Z}$. Its value mod 16 specifies the type of anyonic excitations. In this paper, we investigate triangular vortex configurations (and their dual) in the Kitaev honeycomb model and show that fourteen of these sixteen phases can be obtained by adding a time-reversal symmetry-breaking term. Missing phases are $\ensuremath{\nu}=\ifmmode\pm\else\textpm\fi{}7$. More generally, we prove that any periodic vortex configuration with an odd number of vortices per geometric unit cell can only host even Chern numbers whereas odd Chern numbers can be found in other cases.

DOI
10.1103/physrevb.102.115130

Related papers

Microscopic models for Kitaev's sixteenfold way of anyon theories

Sreejith Chulliparambil, Urban F. P. Seifert, Matthias Vojta C +2 more

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 $\ensuremath{\nu}$ is classified by $\ensuremath{\nu}\phantom{\rule{0.28em}{0ex}}\mathrm{mod}\phantom{\rule{0.28em}{0ex}}16$ 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 $\mathrm{\ensuremath{\Gamma}}$ matrices satisfying the Clifford algebra, enjoy a global $\mathrm{SO}(\ensuremath{\nu})$ symmetry, and live on either square or honeycomb lattices depending on the parity of $\ensuremath{\nu}$. 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 $\ensuremath{\nu}=2$ and $\ensuremath{\nu}=3$ models to materials with Kugel-Khomskii-type spin-orbital interactions is discussed.

arXiv:2005.13683•Physical review. B./Physical review. B•Nov 11, 2020•93 citations

Coupled Majorana modes in a dual vortex of the Kitaev honeycomb model

Anonymous, Jean-Noël Fuchs

The Kitaev model is exactly solvable in terms of Majorana fermions hopping on a honeycomb lattice and coupled to a static $\mathbb{Z}_2$ gauge field, giving the possibility of $π$-vortices in hexagonal plaquettes. In the vortex-full sector and in the presence of a time-reversal-breaking three-spin term of strength $κ$, the energy spectrum is gapped and the ground state possesses an even Chern number. An isolated vortex-free plaquette acts as a ``dual vortex'' and binds a fermionic mode at finite energy $ε$ in the bulk gap. This mode is equivalent to two coupled Majorana zero modes located on the same dual vortex. In a continuum approximation, we analytically compute the Majorana wavefunctions and their coupling $ε$ in the two limits of small or large $κ$. The analytical approach is confirmed by numerical perturbation theory directly on the lattice. The latter is in excellent agreement with the full numerics on a finite-size system. We contrast our results with states bound to an isolated vortex in a topological superconductor with even Chern number.

arXiv:2601.19458•Physical review. B./Physical review. B•Aug 11, 2026•0 citations

Effective models for dense vortex lattices in the Kitaev honeycomb model

David J. Alspaugh, Jean-Noël Fuchs, Anna Ritz-Zwilling +1 more

We introduce low-energy effective models for dense configurations of vortices in the Kitaev honeycomb model. Specifically, we consider configurations of vortices in which vortex-free plaquettes form triangular lattices against a vortex-full background. Depending on the vortex density, these ``dual'' configurations belong to either one of two families classified by translation and inversion symmetry. As a function of a time-reversal symmetry breaking term, one family exhibits gapped phases with even Chern numbers separated by extended gapless phases, while the other exhibits gapped phases with even or odd Chern numbers, separated by critical points. We construct an effective model for each family, determine the parameters of these models by fitting the integrated density of states, and reproduce energy spectra and Chern numbers of the Kitaev honeycomb model. We also derive phase diagrams and determine these models' validity.

arXiv:2312.14729•Physical review. B./Physical review. B•Mar 4, 2024•2 citations

Mapping vortices to anyons in toric code phases of generalized Kitaev models

Li Ern Chern, Roderich Moessner, Claudio Castelnovo

We present a comprehensive theory of mapping flux excitations, or vortices, to electric and magnetic particles in the toric code phases of generalizations of the Kitaev honeycomb model in two spatial dimensions. Our method, which is formulated with the Majorana fermion representation, utilizes the fusion rule of the Abelian anyons and the physical constraint on the fermion parity, and applies to generic model parameters including any perturbative limit. Not only are we able to reproduce the known mapping scheme in the dimer limit, we also derive the conditions for the invariance of anyon species of individual vortices. We prove that the mapping of anyons is left unchanged by any continuous evolution of model parameters that does not close the fermion gap in both the vortex-free and two-vortex sectors, which enables precise demarcations between multiple regimes associated with different maps within a single phase characterized by a trivial Chern number. We illustrate our theory via extensive computations for a number of selected models, in particular those defined on the square-octagon lattice and the honeycomb lattice with a Kekulé structure. We also demonstrate that distinct mappings of anyons can nevertheless exhibit the same weak symmetry breaking, and further argue that they belong to the same symmetry-enriched topological order.

arXiv:2607.12027•Jul 13, 2026•0 citations

Lacing topological orders in two dimensions: Exactly solvable models for Kitaev's sixteen-fold way

Jin-Tao Jin, Jian-Jian Miao, Yi Zhou

A family of two-dimensional (2D) spin-1/2 models have been constructed to realize Kitaev’s sixteen-fold way of anyon theories. Defining a one-dimensional (1D) path through all the lattice sites, and performing the Jordan-Wigner transformation with the help of the 1D path, we find that such a spin-1/2 model is equivalent to a model with \nu ν species of Majorana fermions coupled to a static \mathbb{Z}_2 ℤ 2 gauge field. Here each species of Majorana fermions gives rise to an energy band that carries a Chern number \mathcal{C}=1 𝒞 = 1 , yielding a total Chern number \mathcal{C}=\nu 𝒞 = ν . It has been shown that the ground states are three (four)-fold topologically degenerate on a torus, when \nu ν is an odd (even) number. These exactly solvable models can be achieved by quantum simulations.

arXiv:2209.09054•SciPost Physics•May 1, 2023•2 citations

Exact chiral-spin-liquid state in a Kitaev-type spin model

Jianlong Fu

We study a frustrated two-dimensional lattice spin model with Kitaev-type interaction. The lattice is obtained from the honeycomb lattice by replacing half of its sites with triangles. Using the SO(3) Majorana representation of spin, the model is exactly mapped into a ${Z}_{2}$ lattice gauge theory of complex fermions with standard Gauss law constraint. We show that the ground state of the model is a chiral spin liquid, and it has gapless excitations. The Chern number of the ground state is $\ifmmode\pm\else\textpm\fi{}1$, implying the existence of a chiral edge mode. The vortex excitation of the model is bound with a Majorana zero mode, such mode behaves as non-Abelian anyons.

arXiv:1908.03123•Physical review. B./Physical review. B•Nov 19, 2019•12 citations
In two dimensions, topological phases of free Majorana fermions coupled to a Z2{\mathbb{Z}}_{2}Z2​ gauge field are known to be classified according to the Chern number \ensuremathν\ensuremath∈Z\ensuremath{\nu}\ensuremath{\in}\mathbb{Z}\ensuremathν\ensuremath∈Z. Its value mod 16 specifies the type of anyonic excitations. In this paper, we investigate triangular vortex configurations (and their dual) in the Kitaev honeycomb model and show that fourteen of these sixteen phases can be obtained by adding a time-reversal symmetry-breaking term. Missing phases are \ensuremathν=\ifmmode±\else\textpm\fi7\ensuremath{\nu}=\ifmmode\pm\else\textpm\fi{}7\ensuremathν=\ifmmode±\else\textpm\fi7. More generally, we prove that any periodic vortex configuration with an odd number of vortices per geometric unit cell can only host even Chern numbers whereas odd Chern numbers can be found in other cases.
In two dimensions, the topological order described by Z2{\mathbb{Z}}_{2}Z2​ gauge theory coupled to free or weakly interacting fermions with a nonzero spectral Chern number $\ensuremath{\nu}$ is classified by $\ensuremath{\nu}\phantom{\rule{0.28em}{0ex}}\mathrm{mod}\phantom{\rule{0.28em}{0ex}}16$ 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 $\mathrm{\ensuremath{\Gamma}}$ matrices satisfying the Clifford algebra, enjoy a global $\mathrm{SO}(\ensuremath{\nu})$ symmetry, and live on either square or honeycomb lattices depending on the parity of $\ensuremath{\nu}$. 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 $\ensuremath{\nu}=2$ and $\ensuremath{\nu}=3$ models to materials with Kugel-Khomskii-type spin-orbital interactions is discussed.
The Kitaev model is exactly solvable in terms of Majorana fermions hopping on a honeycomb lattice and coupled to a static Z2\mathbb{Z}_2Z2​ gauge field, giving the possibility of πππ-vortices in hexagonal plaquettes. In the vortex-full sector and in the presence of a time-reversal-breaking three-spin term of strength κκκ, the energy spectrum is gapped and the ground state possesses an even Chern number. An isolated vortex-free plaquette acts as a ``dual vortex'' and binds a fermionic mode at finite energy εεε in the bulk gap. This mode is equivalent to two coupled Majorana zero modes located on the same dual vortex. In a continuum approximation, we analytically compute the Majorana wavefunctions and their coupling εεε in the two limits of small or large κκκ. The analytical approach is confirmed by numerical perturbation theory directly on the lattice. The latter is in excellent agreement with the full numerics on a finite-size system. We contrast our results with states bound to an isolated vortex in a topological superconductor with even Chern number.
We introduce low-energy effective models for dense configurations of vortices in the Kitaev honeycomb model. Specifically, we consider configurations of vortices in which vortex-free plaquettes form triangular lattices against a vortex-full background. Depending on the vortex density, these ``dual'' configurations belong to either one of two families classified by translation and inversion symmetry. As a function of a time-reversal symmetry breaking term, one family exhibits gapped phases with even Chern numbers separated by extended gapless phases, while the other exhibits gapped phases with even or odd Chern numbers, separated by critical points. We construct an effective model for each family, determine the parameters of these models by fitting the integrated density of states, and reproduce energy spectra and Chern numbers of the Kitaev honeycomb model. We also derive phase diagrams and determine these models' validity.
We present a comprehensive theory of mapping flux excitations, or vortices, to electric and magnetic particles in the toric code phases of generalizations of the Kitaev honeycomb model in two spatial dimensions. Our method, which is formulated with the Majorana fermion representation, utilizes the fusion rule of the Abelian anyons and the physical constraint on the fermion parity, and applies to generic model parameters including any perturbative limit. Not only are we able to reproduce the known mapping scheme in the dimer limit, we also derive the conditions for the invariance of anyon species of individual vortices. We prove that the mapping of anyons is left unchanged by any continuous evolution of model parameters that does not close the fermion gap in both the vortex-free and two-vortex sectors, which enables precise demarcations between multiple regimes associated with different maps within a single phase characterized by a trivial Chern number. We illustrate our theory via extensive computations for a number of selected models, in particular those defined on the square-octagon lattice and the honeycomb lattice with a Kekulé structure. We also demonstrate that distinct mappings of anyons can nevertheless exhibit the same weak symmetry breaking, and further argue that they belong to the same symmetry-enriched topological order.
A family of two-dimensional (2D) spin-1/2 models have been constructed to realize Kitaev’s sixteen-fold way of anyon theories. Defining a one-dimensional (1D) path through all the lattice sites, and performing the Jordan-Wigner transformation with the help of the 1D path, we find that such a spin-1/2 model is equivalent to a model with \nu ν species of Majorana fermions coupled to a static \mathbb{Z}_2 ℤ 2 gauge field. Here each species of Majorana fermions gives rise to an energy band that carries a Chern number \mathcal{C}=1 𝒞 = 1 , yielding a total Chern number \mathcal{C}=\nu 𝒞 = ν . It has been shown that the ground states are three (four)-fold topologically degenerate on a torus, when \nu ν is an odd (even) number. These exactly solvable models can be achieved by quantum simulations.
We study a frustrated two-dimensional lattice spin model with Kitaev-type interaction. The lattice is obtained from the honeycomb lattice by replacing half of its sites with triangles. Using the SO(3) Majorana representation of spin, the model is exactly mapped into a Z2{Z}_{2}Z2​ lattice gauge theory of complex fermions with standard Gauss law constraint. We show that the ground state of the model is a chiral spin liquid, and it has gapless excitations. The Chern number of the ground state is $\ifmmode\pm\else\textpm\fi{}1$, implying the existence of a chiral edge mode. The vortex excitation of the model is bound with a Majorana zero mode, such mode behaves as non-Abelian anyons.