Investigations on the Kitaev Model and some of its generalisations

2011

Ph.D

HBNI

This thesis studies a frustrated quantum spin 1/2 model on a hexagonal lattice [A] which was originally proposed and analysed by A. Kitaev. This model was introduced for possible implementation in the field of topological quantum computation. It has anisotropic type nearest neighbour spin spin interaction which depends on the direction of the bonds. In Chapter 1 and 2 of this thesis, a brief introduction of the Kitaev model and review the relevant research done on it, is given. Though it was proposed with the view of application in quantum computation, the author is interested in many-body aspect of the Kitaev Model. To this end, an alternative method of the exact solution of this model using Jordan-Wigner fermionization has been studied. The ground state degeneracy of the system on a torus has been shown to be four all over the parameter space. These have been presented in Chapter 3. In Chapter 4, spin-spin correlation function has been calculated exactly. A spin operator is shown to be fractionalised into two static π(pi) fluxes and a dynamical Majorana fermion. Multi-spin correlations are also computed. The entanglement aspect of this model has been investigated in Chapter 5. In Chapter 6, the toric code limit ( Jz ≫ Jx, Jy ) of the Kitaev model has been studied in terms of gauge invariant Jordan-Wigner fermions. The stability of this spin model has been studied against Ising perturbation in Chapter 7. In Chapter 8 and 9, an extension of the 2D Kitaev model to 3 spatial dimensions has been presented and solved exactly. Various many body aspects and the low energy excitations of this 3D spin model have also been studied.