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

TL;DR

This paper investigates the SO(3) Majorana representation of spin, demonstrating its equivalence to the Jordan-Wigner transformation and $Z_{2}$ gauge theories, and applies it to exactly map specific spin models into gauge theories.

Contribution

It establishes the equivalence between the SO(3) Majorana representation and Jordan-Wigner transformation, revealing its $Z_{2}$ gauge structure and applying it to exact mappings of spin models.

Findings

Equivalence between SO(3) Majorana and Jordan-Wigner transformations in 1D and 2D.

Representation leads to $Z_{2}$ gauge structures in spin models.

Exact mapping of quantum XY and compass models into $Z_{2}$ gauge theories.

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.