Properties and application of the SO(3) Majorana representation of spin: equivalence with the Jordan-Wigner transformation and exact $Z_2$ gauge theories for spin models

Abstract

We explore the properties of the SO(3) Majorana representation of spin. Based on its non-local nature, it is shown that there is an equivalence between the SO(3) Majorana representation and the Jordan-Wigner transformation in one and two dimensions. From the relation between the SO(3) Majorana representation and one-dimensional Jordan-Wigner transformation, we show that application of the SO(3) Majorana representation usually results in $Z_2$ gauge structure. Based on lattice Chern-Simons gauge theory, it is shown that the anti-commuting link variables in the SO(3) Majorana representation make it equivalent to an operator form of compact ${\text{U(1)}}_1$ Chern-Simons Jordan-Wigner transformation in 2d. As examples of its application, we discuss two spin models, namely the quantum XY model on honeycomb lattice and the $90^{\circ }$ compass model on square lattice. It is shown that under the SO(3) Majorana representation both spin models can be exactly mapped into $Z_2$ gauge theory of spinons, with the standard form of $Z_2$ Gauss law constraint.

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• Towards analog quantum simulations of lattice gauge theories with trapped ions
• Exact chiral spin liquid state in a Kitaev type spin model
• Exact weak bosonic zero modes in a spin/fermion chain
• Quantifying the non-Abelian property of Andreev bound states in inhomogeneous Majorana nanowires
• Chiral spin liquid in a generalized Kitaev honeycomb model with ${\mathbb{Z}}_4$ 1-form symmetry
• Microscopic theory of field-tuned topological transitions in the Kitaev honeycomb model