Probability current of a two-dimensional Dirac particle ($\mathbf{j}=c{\Psi }^{†}({\sigma }_x,{\sigma }_y)\Psi$)

For a Hermitian local mass coefficient and $H=c\mathbf{\sigma }\cdot \mathbf{p}+V{\sigma }_z$, the density and current obey ${\partial }_t({\Psi }^{†}\Psi )+\nabla \cdot (c{\Psi }^{†}\mathbf{\sigma }\Psi )=0$. A normalized ${\sigma }_x=-1$ interface spinor carries $j_x=-c\rho$ and $j_y=0$.