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We develop a rigorous and highly accurate technique for calculation of theBerry phase in systems with a quadratic Hamiltonian within the context of theKitaev honeycomb lattice model. The method is based on the recently foundsolution of the model which uses the Jordan-Wigner-type fermionization in anexact effective spin-hardcore boson representation. We specifically simulatethe braiding of two non-Abelian vortices (anyons) in a four vortex systemcharacterized by a two-fold degenerate ground state. The result of the braidingis the non-Abelian Berry matrix which is in excellent agreement with thepredictions of the effective field theory. The most precise results of oursimulation are characterized by an error on the order of $10^{-5}$ or lower. Weobserve exponential decay of the error with the distance between vortices,studied in the range from one to nine plaquettes. We also study its correlationwith the involved energy gaps and provide preliminary analysis of the relevantadiabaticity conditions. The work allows to investigate the Berry phase inother lattice models including the Yao-Kivelson model and particularly thesquare-octagon model. It also opens the possibility of studying the Berry phaseunder non-adiabatic and other effects which may constitute important sources oferrors in topological quantum computation.

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