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arXiv:1006.0804 · Communications in Theoretical Physics

An Anyon Model in a Toric Honeycomb Lattice

Long Liang, Yue Yu

Published Jan 1, 2011• 5 citations
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Abstract

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 ].

DOI
10.1088/0253-6102/55/1/16

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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 ].
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 present a solution of Kitaev's spin model on the honeycomb lattice and of related topologically ordered spin models. We employ a Jordan-Wigner-type fermionization and find that the Hamiltonian takes a BCS-type form, allowing the system to be solved by Bogoliubov transformation. Our fermionization does not employ nonphysical auxiliary degrees of freedom and the eigenstates we obtain are completely explicit in terms of the spin variables. The ground state is obtained as a BCS condensate of fermion pairs over a vacuum state which corresponds to the toric-code state with the same vorticity. We show in detail how to calculate all eigenstates and eigenvalues of the model on the torus. In particular, we find that the topological degeneracy on the torus descends directly from that of the toric-code, which now supplies four vacua for the fermions, one for each choice of periodic vs antiperiodic boundary conditions. The reduction of the degeneracy in the non-Abelian phase of the model is seen to be due to the vanishing of one of the corresponding candidate BCS ground states in that phase. This occurs in particular in the fully periodic vortex-free sector. The true ground state in this sector is exhibited and shown to be gapped away from the three partially antiperiodic ground states whenever the non-Abelian phase is gapped.
We study the gapped phase of the Kitaev model on the honeycomb lattice using perturbative continuous unitary transformations. The effective low-energy Hamiltonian is found to be an extended toric code with interacting anyons. High-energy excitations are emerging free fermions which are composed of hard-core bosons with an attached string of spin operators. The excitation spectrum is mapped onto that of a single particle hopping on a square lattice in a magnetic field. We also illustrate how to compute correlation functions in this framework. The present approach yields analytical perturbative results in the thermodynamical limit without using the Majorana or the Jordan-Wigner fermionization initially proposed to solve this problem.
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).
The exact solubility of the Kitaev-type spin honeycomb lattice model was proved by means of a Majorana fermion representation or a Jordan-Wigner transformation while the explicit form of the anyon in terms of Pauli matrices became not transparent. The nonabelian statistics of anyons and the fusion rules can only be expressed in indirect ways to Pauli matrices. We convert the ground state and anyonic excitations back to the forms of Pauli matrices and explicitly demonstrate the nonabelian anyonic statistics as well as the fusion rules. These results may instruct the experimental realization of the nonabelian anyons. We suggest a proof-in-principle experiment to verify the existence of the nonabelian anyons in nature.
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.