Majorana representations of spin and an alternative solution of the Kitaev honeycomb model

Abstract

Based on the Dirac spinor representation of the SO(4) group, we discuss the relationship between three types of representation of spin in terms of Majorana fermions, namely the Kitaev representation, the SO(3) representation and the SO(4) chiral representation. Comparing the three types, we show that the Hilbert space of the SO(3) representation is different from the other two by requiring pairing of sites, but it has the advantage over the other two in that no unphysical states are involved. As an example of its application, we present a new alternative solution of the Kitaev honeycomb model. Our new solution involves no unphysical states which enables a systematic calculation of physical observables. Finally, we discuss an extension of the model to a more general exactly soluble $Z_2$ gauge theory interacting with complex fermions.

arXiv comment14 pages, 3 figure

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• Pseudo-fermion functional renormalization group for spin models
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• Soluble limit and criticality of fermions in Z2 gauge theories
• Exact chiral spin liquid state in a Kitaev type spin model
• Bilinear Majorana representations for spin operators with spin magnitudes $S>1/2$
• Correlation effects on the magnetization process of the Kitaev model
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• Dynamical response theory of interacting Majorana fermions and its application to generic Kitaev quantum spin liquids in a field
• Majorana Fermion Mean-Field Theories of Kitaev Quantum Spin Liquids
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• Nature of Topological Phase Transition of Kitaev Quantum Spin Liquids
• Exact weak bosonic zero modes in a spin/fermion chain
• Quantum Spin Liquid phases in Kitaev Materials
• Electric polarization near vortices in the extended Kitaev model
• A solvable embedding mechanism for one-dimensional spinless and Majorana fermions in higher-dimensional spin-1/2 magnets
• Microscopic theory of field-tuned topological transitions in the Kitaev honeycomb model
• Quantitative functional renormalization for three-dimensional quantum Heisenberg models
• «Anticommuting» ${\mathbb{Z}}_2$ quantum spin liquids
• Majorana Diagrammatics for Quantum Spin-1/2 Models