Fusion rules and vortices in $p_x+i{p}_y$ superconductors

The “half-quantum” vortices $\left(\sigma \right)$ and quasiparticles $\left(\psi \right)$ in a two-dimensional $p_x+i{p}_y$ superconductor obey the Ising-like fusion rules $\psi \times \psi =1$, $\sigma \times \psi =\sigma$, and $\sigma \times \sigma =1+\psi$. We explain how the physical fusion of vortex-antivortex pairs allows us to use these rules to read out the information encoded in the topologically protected space of degenerate ground states. We comment on the potential applicability of this fact to quantum computation.

Figure 1

(Color online) Phase changes due to the braid operation $T_1$: Before the braiding the phase ${\chi }_2\left(1\right)$ seen by vortex 1 due to vortex 2 is just more than 180°, and the phase ${\chi }_1\left(2\right)$ seen by vortex 2 due to vortex 1 is just more than 0°. After the braiding the phase ${\chi }_1\left(2\right)$ at vortex 2 due to vortex 1 is just more than 180°, but because vortex 1 remains below below the displaced vortex-2 branch cut, the phase ${\chi }_2\left(1\right)$ is now just more than 360°. Consequently ${\gamma }_1$ is replaced by ${\gamma }_2$, but ${\gamma }_2$ is replaced by $-{\gamma }_1$.