  We prove a classification result for ground state solutions of the critical
Dirac equation on $\mathbb{R}^n$, $n\geq2$. By exploiting its conformal
covariance, the equation can be posed on the round sphere $\mathbb{S}^n$ and
the non-zero solutions at the ground level are given by Killing spinors, up to
conformal diffeomorphisms. Moreover, such ground state solutions of the
critical Dirac equation are also related to the Yamabe equation for the sphere,
for which we crucially exploit some known classification results.
