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arXiv:2210.09180 · New Journal of Physics

The Kitaev honeycomb model on surfaces of genus g ≥ 2

John Brennan, Jiří Vala

Published Apr 4, 2018• 2 citations
arXivHTMLPDF

Abstract

We present a construction of the Kitaev honeycomb lattice model on an arbitrary higher genus surface. We first generalize the exact solution of the model based on the Jordan–Wigner fermionization to a surface with genus g = 2, and then use this as a basic module to extend the solution to lattices of arbitrary genus. We demonstrate our method by calculating the ground states of the model in both the Abelian doubled phase and the non-Abelian Ising topological phase on lattices with the genus up to g = 6. We verify the expected ground state degeneracy of the system in both topological phases and further illuminate the role of fermionic parity in the Abelian phase.

DOI
10.1088/1367-2630/aabb95

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We present a construction of the Kitaev honeycomb lattice model on an arbitrary higher genus surface. We first generalize the exact solution of the model based on the Jordan–Wigner fermionization to a surface with genus g = 2, and then use this as a basic module to extend the solution to lattices of arbitrary genus. We demonstrate our method by calculating the ground states of the model in both the Abelian doubled phase and the non-Abelian Ising topological phase on lattices with the genus up to g = 6. We verify the expected ground state degeneracy of the system in both topological phases and further illuminate the role of fermionic parity in the Abelian phase.
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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.
The Kitaev honeycomb lattice system is an important model of topological materials whose phase diagram exhibits both abelian and non-abelian topological phases. The latter, a so-called Ising phase, is related to topological superconductors. Its quasiparticle excitations, which are formed by Majorana fermions attached to vortices, show non-abelian fractional statistics and are known as Ising anyons. We investigate dislocation defects in the Ising phase of the Kitaev honeycomb model. After introducing them to the system, we accordingly generalize our solution of this model to the situation with the defects. The important part of this effort is developing an appropriate Jordan-Wigner fermionization procedure. It is expected that the presence of defects manifests itself by the formation of fermionic zero-energy modes around the defect end points. We numerically confirm this expectation and further investigate properties of these modes. The computational potential of our technique is demonstrated for both diagonalization and dynamical simulations. The latter focuses on the process of fusion of the vortex zero-energy modes with the Majorana fermions attached to the defect. This process simulates fusion of non-abelian Ising anyons.
We develop a rigorous and highly accurate technique for calculation of the Berry phase in systems with a quadratic Hamiltonian within the context of the Kitaev honeycomb lattice model. The method is based on the recently found solution of the model which uses the Jordan-Wigner-type fermionization in an exact effective spin-hardcore boson representation. We specifically simulate the braiding of two non-Abelian vortices (anyons) in a four vortex system characterized by a two-fold degenerate ground state. The result of the braiding is the non-Abelian Berry matrix which is in excellent agreement with the predictions of the effective field theory. The most precise results of our simulation are characterized by an error on the order of 10−510^{-5}10−5 or lower. We observe exponential decay of the error with the distance between vortices, studied in the range from one to nine plaquettes. We also study its correlation with the involved energy gaps and provide preliminary analysis of the relevant adiabaticity conditions. The work allows to investigate the Berry phase in other lattice models including the Yao-Kivelson model and particularly the square-octagon model. It also opens the possibility of studying the Berry phase under non-adiabatic and other effects which may constitute important sources of errors in topological quantum computation.
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 develop a rigorous and highly accurate technique for the calculation of the Berry phase in systems with a quadratic Hamiltonian within the context of the Kitaev honeycomb lattice model. The method is based on the recently found solution of the model that uses the Jordan–Wigner-type fermionization in an exact effective spin-hardcore boson representation. We specifically simulate the braiding of two non-Abelian vortices (anyons) in a four-vortex system characterized by a twofold degenerate ground state. The result of the braiding is the non-Abelian Berry matrix, which is in excellent agreement with the predictions of the effective field theory. The most precise results of our simulation are characterized by an error of the order of 10−5 or lower.We observe exponential decay of the error with the distance between vortices, studied in the range of one to nine plaquettes. We also study its correlation with the involved energy gaps and provide a preliminary analysis of the relevant adiabaticity conditions. The work allows one to investigate the Berry phase in other lattice models including the Yao–Kivelson model and particularly the square–octagon model. It also opens up the possibility of studying the Berry phase under nonadiabatic and other effects that may constitute important sources of errors in topological quantum computation.