$\mathrm{SU}\left(N\right)$ toric code and non-Abelian anyons

We construct an $\mathrm{SU}\left(N\right)$ toric code model describing the dynamics of $\mathrm{SU}\left(N\right)$ electric and magnetic fluxes on a two-dimensional torus. We show that the model has $N^2$ topologically distinct ground states $|{\psi }_0{\rangle }_{(\mathsf{p},\mathsf{q})}$, which are the loop states characterized by $Z_N\otimes Z_N$ center charges $(\mathsf{p},\mathsf{q}=0,1,2,...,N-1)$. We explicitly construct them in terms of coherent superpositions of all possible spin network states on a torus with Wigner coefficients as their amplitudes. All excited quasiparticle states with $\mathrm{SU}\left(N\right)$ electric charges and magnetic fluxes are constructed. We show that the braiding statistics of these $\mathrm{SU}\left(N\right)$ electric or magnetic quasiparticles or non-Abelian anyons are encoded in the Wigner rotation matrices.

Figure 1

(a) The $\mathrm{SU}\left(N\right)$ electric fields $E_{\pm }^a\left(l\right)$ and the conjugate link operators $U_{\alpha \beta }\left(l\right)$. The operators ${\mathcal{G}}^a\left(\vec{n}\right)$ and $\mathcal{W}\left(p\right)$ appearing in the Hamiltonian (9) are also shown. (b) Two noncontractible Wilson loops ${\gamma }_x$ and ${\gamma }_y$ on the torus ${\mathcal{T}}_2$.

Figure 2

Ground states of the SU(2) toric code, from (a) link states $\left|Z\right(l)\rangle$ to (b) site states ${|\xi \left(s\right)\rangle }_m^j$ to (c) loop states $|\vec{J}{\rangle }_s\equiv {|j_1,j_2,j_{12}=j_{34},j_3,j_4\rangle }_s$. The geometrical triangular constraints manifest on the dual lattice (solid lines) around the site $s$. Every lattice site has two triangular constraints: $\{j_1,j_2,j_{12}\}$ and $\{j_3,j_4,j_{34}=j_{12}\}$ must form triangles.

Figure 3

SU(2) toric code. (a) Electric fluxes on a four-plaquette toric code and (b) 12-$j$ Wigner coefficients involving eight triangular constraints as the amplitude for the SU(2) toric code ground state.

Figure 4

Construction of topologically nontrivial ground states, inserting $Z_N$ twist factors, shown as $•$, in the ground states: (a) $Z_N$ phase ${\eta }_y=e^{2\pi i\mathsf{q}/N}$ along ${\mathcal{S}}_x$ and (b) $Z_N$ phase ${\eta }_x=e^{2\pi i\mathsf{p}/N}$ along ${\mathcal{S}}_y$.

Figure 5

Creating electric charges at sites and magnetic vortices on plaquettes. The operator ${\mathrm{\Gamma }}_{\alpha \beta }^{\left(N\right)}(n^{\prime },n)$ creates electric charges in the fundamental representations at the two end points $n$ and $n^{\prime }$. The operator ${\mathrm{\Sigma }}_{\vec{\omega }}$ acts on all the vertical link fluxes $U(s;\widehat{2})\in \mathcal{S}$ (dark vertical links) and rotates all of them by the same angle $\omega (x_0,y_0)={\omega }_{s=0}\equiv \omega$. However, the rotation axes ${\widehat{\omega }}_{s=1},{\widehat{\omega }}_{s=2},...,{\widehat{\omega }}_{s=L}$ are different and related to ${\widehat{\omega }}_0$ by parallel transports (42). Note that the shapes and lengths of $\mathcal{L}$ and $\mathcal{S}$ are unphysical.

Figure 6

Taking a charge around closed loop $\mathcal{C}$ produces a non-Abelian rotation by the Wigner rotation matrix if a vortex is enclosed as shown in (49).

Figure 7

Deformation of the Dirac string $\mathcal{S}$ by gauge transformations.