Properties and application of the SO(3) Majorana representation of spin: equivalence with the Jordan-Wigner transformation and exact Z2 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 Z2 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 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 compass model on square lattice. It is shown that under the SO(3) Majorana representation both spin models can be exactly mapped into Z2 gauge theory of spinons, with the standard form of Z2 Gauss law constraint.
Turn this paper into a lesson
ArcXiv compiles a structured reading guide from this paper's metadata: plain-English importance, contributions, prerequisite concepts, which sections to read first, flashcards, and a quiz. Grounded in the abstract, never invented.