Nonexistence of a Hohenberg-Kohn variational principle in total current-density-functional theory

For a many-electron system, whether the particle density $\rho \left(\mathbf{r}\right)$ and the total current density $\mathbf{j}\left(\mathbf{r}\right)$ are sufficent to determine the one-body potential $V\left(\mathbf{r}\right)$ and vector potential $A\left(\mathbf{r}\right)$ is still an open question. For the one-electron case, a Hohenberg-Kohn theorem exists formulated with the total current density. Here we show that the generalized Hohenberg-Kohn energy functional ${\mathcal{E}}_{V_0,{\mathbf{A}}_0}(\rho ,\mathbf{j})=\langle \psi (\rho ,\mathbf{j}),H(V_0,{\mathbf{A}}_0)\psi (\rho ,\mathbf{j})\rangle$ can be minimal for densities that are not the ground-state densities of the fixed potentials $V_0$ and ${\mathbf{A}}_0$. Furthermore, for an arbitrary number of electrons and under the assumption that a Hohenberg-Kohn theorem exists formulated with $\rho$ and $\mathbf{j}$, we discuss the possibility of a variational principle in total current-density-functional theory such as that of Hohenberg-Kohn.