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:1005.5103 · Journal of Statistical Mechanics Theory and Experiment

Kitaev model and dimer coverings on the honeycomb lattice

Michael Kamfor, Sébastien Dusuel, Julien Vidal, Kai Phillip Schmidt

Published Aug 12, 2010• 24 citations
arXivHTMLPDF

Abstract

We consider an extension of the Kitaev honeycomb model based on arbitrary dimer coverings satisfying matching rules. We focus on three different dimer coverings having the smallest unit cells for which we calculate the ground-state phase diagram. We also study one- and two-vortex properties for these coverings in the Abelian phases and show that vortex–vortex interactions can be attractive or repulsive. These qualitative differences are confirmed analytically by high-order perturbative expansions around the isolated-dimer limit. Similarities and differences with the original Kitaev honeycomb model are discussed.

DOI
10.1088/1742-5468/2010/08/p08010

Related papers

Perturbative approach to an exactly solved problem: Kitaev honeycomb model

Julien Vidal, Kai Phillip Schmidt, Sébastien Dusuel

We analyze the gapped phase of the Kitaev honeycomb model perturbatively in the isolated-dimer limit. Our analysis is based on the continuous unitary transformations method, which allows one to compute the spectrum as well as matrix elements of operators between eigenstates at high order. The starting point of our study consists of an exact mapping of the original honeycomb spin system onto a square-lattice model involving an effective spin and a hard-core boson. We then derive the low-energy effective Hamiltonian up to order 10 which is found to describe an interacting-anyon system, contrary to the order 4 result which predicts a free theory. These results give the ground-state energy in any vortex sector and thus also the vortex gap, which is relevant for experiments. Furthermore, we show that the elementary excitations are emerging free fermions composed of a hard-core boson with an attached spin- and phase-operator string. We also focus on observables and compute, in particular, the spin-spin correlation functions. We show that they admit a multiplaquette expansion that we derive up to order 6. Finally, we study the creation and manipulation of anyons with local operators, show that they also create fermions, and discuss the relevance of our findings for experiments in optical lattices.

arXiv:0809.1553•Physical Review B•Dec 29, 2008•75 citations

Zero Temperature Phase Diagram of the Kitaev Model

C. Nash, Denjoe O’Connor

We show that the zero-temperature phase diagram of the vortex free sector of the Kitaev model is in one-to-one correspondence with that of the classical dimer model on the same lattice. We find that the model generically has three distinct phases. On a honeycomb lattice with a 3 x 3 fundamental domain all three phases are accessible. As the couplings are varied, there are two distinct transitions. The new transition is one to a gapped phase that opens up in the interior of the B phase.

arXiv:0812.0099•Physical Review Letters•Apr 9, 2009•10 citations

Ground-state properties of the K−Γ model on a honeycomb lattice

Takuto Yamada, Takafumi J. Suzuki, Sei-ichiro Suga

We investigate the ground-state properties of the $K\ensuremath{-}\mathrm{\ensuremath{\Gamma}}$ model on a honeycomb lattice using series expansions and numerical exact diagonalizations, where the model includes Kitaev ($K$) and symmetric off-diagonal ($\mathrm{\ensuremath{\Gamma}}$) interactions. Starting from the weakly interacting dimers on the specific bond, we strengthen the interdimer interactions to the isotropically interacting system. We show that depending on $\mathrm{\ensuremath{\Gamma}}$ and $K$, the dimer state survives up to the isotropically interacting system, where the phase transition occurs or obeys a phase transition to a magnetically ordered state at an anisotropic interaction. The results are summarized in the phase diagram. We also show that the Kekul\'e dimerized state is unstable in the isotropic $K\ensuremath{-}\mathrm{\ensuremath{\Gamma}}$ model.

arXiv:2004.09622•Physical review. B./Physical review. B•Jul 13, 2020•13 citations

Kitaev's honeycomb lattice model

Jiannis K. Pachos

In this chapter, we consider Kitaev's honeycomb lattice model (Kitaev, 2006). This is an analytically tractable spin model that gives rise to quasiparticles with Abelian as well as non-Abelian statistics. Some of its properties are similar to the fractional quantum Hall effect, which has been studied experimentally in great detail even though it evades exact analytical treatment (Moore and Read, 1991). Due to its simplicity, the honeycomb lattice model is likely to be the first topological spin model to be realised in the laboratory, e.g., with optical lattice technology (Micheli et al. , 2006). Understanding its properties can facilitate its physical realisation and can provide a useful insight into the mechanisms underlining topological insulators and the fractional quantum Hall effect. The honeycomb lattice model comprises interacting spin-½ particles arranged on the sites of a honeycomb lattice. It is remarkable that such a simple model can support a rich variety of topological behaviours. For certain values of its couplings, Abelian anyons emerge that behave like the toric code anyons. For another coupling regime, non-Abelian anyons emerge that correspond to the Ising anyons. The latter are manifested as vortex-like configurations of the original spin model that can effectively be described by Majorana fermions. These are fermionic fields that are antiparticles of themselves. They were first introduced in the context of high-energy physics (Majorana, 1937) and become increasingly important in the analysis of solid state phenomena (Wilczek, 2009).

Cambridge University Press eBooks•Apr 12, 2012•0 citations

Phase Diagram of the Kitaev-type Model on a Decorated Honeycomb Lattice in the Isolated Dimer Limit

Joji Nasu, Yukitoshi Motome

An effective model in the isolated dimer limit of the Kitaev-type model on a decorated honeycomb lattice is investigated at finite temperature. The ground state of this model is exactly shown to be a chiral spin liquid with spontaneous breaking of time reversal symmetry. We elab- orate the finite-temperature phase diagram by using the mean-field approximation and Monte Carlo simulation. We find that the phase transition between the high-temperature paramagnetic phase and the low-temperature chiral spin liquid phase is always of second order in the Monte Carlo results, although a tricritical point appears in the mean-field phase diagram. The finite-size scaling analysis of the Monte Carlo data suggests that the phase transition belongs to the two-dimensional Ising universality class.

arXiv:1506.05678•Physics Procedia•Jan 1, 2015•1 citation

Perturbative study of the Kitaev model with spontaneous time-reversal symmetry breaking

Sébastien Dusuel, Kai Phillip Schmidt, Julien Vidal +1 more

We analyze the Kitaev model on the triangle-honeycomb lattice whose ground state has recently been shown to be a chiral spin liquid. We consider two perturbative expansions: the isolated-dimer limit containing Abelian anyons and the isolated-triangle limit. In the former case, we derive the low-energy effective theory and discuss the role played by multiplaquette interactions. In this phase, we also compute the spin-spin correlation functions for any vortex configuration. In the isolated-triangle limit, we show that the effective theory is, at lowest nontrivial order, the Kitaev honeycomb model at the isotropic point. We also compute the next-order correction which opens a gap and yields non-Abelian anyons.

arXiv:0804.4775•Physical Review B•Sep 5, 2008•59 citations
We consider an extension of the Kitaev honeycomb model based on arbitrary dimer coverings satisfying matching rules. We focus on three different dimer coverings having the smallest unit cells for which we calculate the ground-state phase diagram. We also study one- and two-vortex properties for these coverings in the Abelian phases and show that vortex–vortex interactions can be attractive or repulsive. These qualitative differences are confirmed analytically by high-order perturbative expansions around the isolated-dimer limit. Similarities and differences with the original Kitaev honeycomb model are discussed.
We analyze the gapped phase of the Kitaev honeycomb model perturbatively in the isolated-dimer limit. Our analysis is based on the continuous unitary transformations method, which allows one to compute the spectrum as well as matrix elements of operators between eigenstates at high order. The starting point of our study consists of an exact mapping of the original honeycomb spin system onto a square-lattice model involving an effective spin and a hard-core boson. We then derive the low-energy effective Hamiltonian up to order 10 which is found to describe an interacting-anyon system, contrary to the order 4 result which predicts a free theory. These results give the ground-state energy in any vortex sector and thus also the vortex gap, which is relevant for experiments. Furthermore, we show that the elementary excitations are emerging free fermions composed of a hard-core boson with an attached spin- and phase-operator string. We also focus on observables and compute, in particular, the spin-spin correlation functions. We show that they admit a multiplaquette expansion that we derive up to order 6. Finally, we study the creation and manipulation of anyons with local operators, show that they also create fermions, and discuss the relevance of our findings for experiments in optical lattices.
We investigate the ground-state properties of the $K\ensuremath{-}\mathrm{\ensuremath{\Gamma}}$ model on a honeycomb lattice using series expansions and numerical exact diagonalizations, where the model includes Kitaev (KKK) and symmetric off-diagonal ($\mathrm{\ensuremath{\Gamma}}$) interactions. Starting from the weakly interacting dimers on the specific bond, we strengthen the interdimer interactions to the isotropically interacting system. We show that depending on $\mathrm{\ensuremath{\Gamma}}$ and KKK, the dimer state survives up to the isotropically interacting system, where the phase transition occurs or obeys a phase transition to a magnetically ordered state at an anisotropic interaction. The results are summarized in the phase diagram. We also show that the Kekul\'e dimerized state is unstable in the isotropic $K\ensuremath{-}\mathrm{\ensuremath{\Gamma}}$ model.
In this chapter, we consider Kitaev's honeycomb lattice model (Kitaev, 2006). This is an analytically tractable spin model that gives rise to quasiparticles with Abelian as well as non-Abelian statistics. Some of its properties are similar to the fractional quantum Hall effect, which has been studied experimentally in great detail even though it evades exact analytical treatment (Moore and Read, 1991). Due to its simplicity, the honeycomb lattice model is likely to be the first topological spin model to be realised in the laboratory, e.g., with optical lattice technology (Micheli et al. , 2006). Understanding its properties can facilitate its physical realisation and can provide a useful insight into the mechanisms underlining topological insulators and the fractional quantum Hall effect. The honeycomb lattice model comprises interacting spin-½ particles arranged on the sites of a honeycomb lattice. It is remarkable that such a simple model can support a rich variety of topological behaviours. For certain values of its couplings, Abelian anyons emerge that behave like the toric code anyons. For another coupling regime, non-Abelian anyons emerge that correspond to the Ising anyons. The latter are manifested as vortex-like configurations of the original spin model that can effectively be described by Majorana fermions. These are fermionic fields that are antiparticles of themselves. They were first introduced in the context of high-energy physics (Majorana, 1937) and become increasingly important in the analysis of solid state phenomena (Wilczek, 2009).
An effective model in the isolated dimer limit of the Kitaev-type model on a decorated honeycomb lattice is investigated at finite temperature. The ground state of this model is exactly shown to be a chiral spin liquid with spontaneous breaking of time reversal symmetry. We elab- orate the finite-temperature phase diagram by using the mean-field approximation and Monte Carlo simulation. We find that the phase transition between the high-temperature paramagnetic phase and the low-temperature chiral spin liquid phase is always of second order in the Monte Carlo results, although a tricritical point appears in the mean-field phase diagram. The finite-size scaling analysis of the Monte Carlo data suggests that the phase transition belongs to the two-dimensional Ising universality class.
We analyze the Kitaev model on the triangle-honeycomb lattice whose ground state has recently been shown to be a chiral spin liquid. We consider two perturbative expansions: the isolated-dimer limit containing Abelian anyons and the isolated-triangle limit. In the former case, we derive the low-energy effective theory and discuss the role played by multiplaquette interactions. In this phase, we also compute the spin-spin correlation functions for any vortex configuration. In the isolated-triangle limit, we show that the effective theory is, at lowest nontrivial order, the Kitaev honeycomb model at the isotropic point. We also compute the next-order correction which opens a gap and yields non-Abelian anyons.