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.
The honeycomb lattice Kitaev model H K with two kinds of Wen–Toric-code four-body interactions H W T is investigated exactly using a new fermionization method, and the ground state phase diagram is obtained. Six kinds of three-body interactions are also considered. A Hamiltonian equivalent to the honeycomb lattice Kitaev model is also introduced. The fermionization method is generalized to two-dimensional systems, and the two-dimensional Jordan–Wigner transformation is obtained as a special case of this formula. The model H K + H W T is symmetric in four-dimensional space of coupling constants, and the anyon type excitations appear in each phase.
We study an anyon model in a toric honeycomb lattice. The ground states and the low-lying excitations coincide with those of Kitaev toric code model and then the excitations obey mutual semionic statistics. This model is helpful to understand the toric code of anyons in a more symmetric way. On the other hand, there is a direct relation between this toric honeycomb model and a boundary coupled Ising chain array in a square lattice via Jordan—Wigner transformation. We discuss the equivalence between these two models in the low-lying sector and realize these anyon excitations in a conventional fermion system. The analysis for the ground state degeneracy in the last section can also be thought of as a complementarity of our previous work [ Phys. A: Math. Theor. 43 (2010) 105306 ].
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.
The Kitaev model on a honeycomb lattice with bond-dependent Ising interactions offers an exactly solvable model of a quantum spin liquid (QSL) with fractionalized excitations: gapped
Z2 fluxes and gapless linearly dispersing Majorana fermions in the isotropic limit (
Kx=Ky=Kz). We explore the phase diagram along two axes, an external magnetic field
h, applied out of plane of the honeycomb, and anisotropic interactions
Kz larger than the other two (${K}_{x}={K}_{y}\ensuremath{\equiv}K$). For ${K}_{z}/K\ensuremath{\gg}2$ and
h=0, the matter Majorana fermions have the largest gap, and the system is described by a gapped
Z2 toric code. One of the central questions we address is whether the fractionalized excitations in the different phases have sharp signatures that can be detected in experiments. We show that while the response to single-spin excitations is broad, the spectral function corresponding to two-spin excitations across a bond has sharp signatures that can be attributed to specific anyons. In the toric code regime, the $\ensuremath{\epsilon}=e\ifmmode\times\else\texttimes\fi{}m$ fermion, formed from the bosonic Ising electric (
e) and magnetic (
m) charges, disperses along a specific one-dimensional direction that provides a fingerprint of fractionalization. At lower
Kz in the center of the Abelian phase, in a regime we dub the primordial fractionalized regime, the field generates a hybridization between the $\ensuremath{\epsilon}$ fermion and the Majorana matter fermion, resulting in a $\ensuremath{\psi}$ fermion which too has a distinct quasi-one-dimensional dispersion. All the other phases in the field-anisotropy plane are naturally obtained from this primordial soup. These highly constrained fractonlike dispersions can be observable by inelastic light and neutron scattering, thereby providing ``smoking gun'' signatures of fractionalization in the QSL phase. Our analysis is based on calculations of susceptibilities, topological entanglement entropy, and excitation dynamics, obtained using exact diagonalization and density matrix renormalization group, and supported by perturbation theory.
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).