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Description of Kitaev’s honeycomb model with toric-code stabilizers

Kells, G. and Slingerland, J.K. and Vala, J. (2009) Description of Kitaev’s honeycomb model with toric-code stabilizers. Physical Review B, 80 (125415). pp. 1-10. ISSN 1098-0121

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Abstract

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.

Item Type: Article
Additional Information: Original article published in Physical Review B Vol.80 No.125415(2009) ©2009 The American Physical Society.
Keywords: Kitaev; honeycomb model; toric-code; stabilizers;
Subjects: Science & Engineering > Experimental Physics
Science & Engineering > Mathematics & Statistics
Item ID: 2068
Identification Number: DOI: 10.1103/PhysRevB.80.125415
Depositing User: Dr. Joost Slingerland
Date Deposited: 10 Aug 2010 14:07
Journal or Publication Title: Physical Review B
Publisher: American Physical Society
Refereed: Yes
URI:

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