Rigorous Calculations of Non-Abelian Statistics in the Kitaev Honeycomb Model

Abstract

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 $10^{-5}$ 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.

27 pages, 9 figures, 3 appendices

References in corpus (11)

• Non-Abelian Anyons and Topological Quantum Computation
• Non-Abelian statistics and topological quantum information processing in 1D wire networks
• An exact chiral spin liquid with non-Abelian anyons
• Splitting of Majorana modes due to intervortex tunneling in a p + ip superconductor
• The sign of the overlap of HFB wave functions
• Numerical Analysis of Quasiholes of the Moore-Read Wavefunction
• Non-adiabatic Effects in the Braiding of Non-Abelian Anyons in Topological Superconductors
• A Description of Kitaev's Honeycomb Model with Toric-Code Stabilizers
• Topological degeneracy and vortex manipulation in Kitaev's honeycomb model
• Non-Abelian statistics as a Berry phase in exactly solvable models
• Zero energy and chiral edge modes in a p-wave magnetic spin model

Cited by in corpus (11)

• A Short Introduction to Topological Quantum Computation
• Momentum polarization: an entanglement measure of topological spin and chiral central charge
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• Exact resummation of the Holstein-Primakoff expansion and differential equation approach to operator square-roots
• Identifying Non-Abelian Topological Order through Minimal Entangled States
• Kitaev spin models from topological nanowire networks
• Focus on topological quantum computation
• Anyon dynamics in field-driven phases of the anisotropic Kitaev model
• Digital Quantum Simulation of the Kitaev Quantum Spin Liquid
• Dynamically Induced Topology and Quantum Monodromies in a Proximity Quenched Gapless Wire
• Compatibility of Braiding and Fusion on Wire Networks