Jordan–Wigner transformation for spin S = 1/2 models in D = 1, 2, 3

Oleg Derzhko, Johannes Richter, Taras Verkholyak

Oleg Derzhko, Taras Krokhmalskii

Michael Fowler

Zsolt Gulácsi

An exact Jordan–Wigner type of transformation is presented in 1D connecting spin-1/2 operators to spinful canonical Fermi operators. The transformation contains two free parameters allowing a broad interconnection possibility between spin models and fermionic models containing spinful Fermi operators.

Jianlong Fu

We explore the properties of the SO(3) Majorana representation of spin. Based on its nonlocal 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 anticommuting 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 a honeycomb lattice and the ${90}^{\ensuremath{\circ}}$ compass model on a 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.

Al.B. Zamolodchikov, M. I. Monastyrskiǐ