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

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 ${90}^{\circ }$ 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.

Figure 1

The honeycomb lattice and the diamond lattice. The original spins in the quantum XY model are defined on the sites of the honeycomb lattice; the three types of bonds are labeled by vectors ${\widehat{e}}_1,{\widehat{e}}_2$, and ${\widehat{e}}_3$, respectively. The $A$ sublattice of the honeycomb lattice is formed by the red dots which in turn form the diamond lattice, whose bonds are denoted by the red dashed lines. The unit vectors of the diamond lattice is ${\widehat{a}}_1$ and ${\widehat{a}}_2$. After defining the staggered fermion, the link variables form a horizontal zigzag chain. The sites in one of the chains can be marked by integer numbers $2k-1$, $2k$,..., with $A$ sublattice sites labeled by even numbers. Link variables on each zigzag chain are mapped into spin variables defined on the ${\widehat{a}}_1$ bonds of the diamond lattice, labeled by black dots. Spins corresponding to the same zigzag chain form a horizontal line, which is the black dashed line.

Figure 2

The diamond lattice, with unit vector ${\widehat{a}}_1$ and ${\widehat{a}}_2$. Vector $\widehat{\delta }$ is defined to be ${\widehat{a}}_1-{\widehat{a}}_2$. The spins from the link variables of the original honeycomb lattice are denoted by black dots in the ${\widehat{a}}_1$ bonds; the original horizontal zigzag chains in the honeycomb lattice become the horizontal black dashed lines. The treatment of the constraint (66) for site $\widehat{i}$ in the middle of the (green dashed) block ABCD involves the spins in the half-infinite block CDEF. Duality transformation for each spin chain labeled by the black dashed line introduces spin variables whose positions are denoted by black crosses. The Gauss law constraint for the $Z_2$ gauge theory involves the staggered fermion and the four spin operators enclosed in ABCD.

Figure 3

The square lattice, with unit vectors ${\widehat{e}}_x$ and ${\widehat{e}}_y$. Spins in the ${90}^{\circ }$ compass are defined on the sites of the square lattice. Under SO(3) Majorana representation, we pair up the green bonds to form complex fermion. After the pairing, the lattice breaks into $A$ sublattice labeled by the red dots, and $B$ sublattice labeled by the blue dots. Complex fermion is defined on the $A$ sublattice, which then forms a rectangle lattice. The unit vectors of the rectangle lattice are labeled by $\widehat{x}$ and $\widehat{y}$.

Figure 4

The rectangle lattice, with unit vectors $\widehat{x}$ and $\widehat{y}$, note that we have shrunk the $\widehat{y}$ to half of its length to achieve a clearer look. The complex fermion lives on the sites of the rectangle lattice, denoted by the red dots. The spin variables $\tau$ are defined on the bonds of the lattice, which are the black dots. In the $Z_2$ gauge theory Hamiltonian (94), the hopping of complex fermions defined on points A and B couples to the spins on sites 1, 2, and 3. The dual lattice can be defined by connecting the centers of plaquettes of the original lattice. Part of the dual lattice is shown by the green dashed lines.