Non-existence of a Hohenberg-Kohn Variational Principle in Total Current Density Functional Theory

For a many-electron system, whether the particle density $\rho(\mathbf{r})$ and the total current density $\mathbf{j}(\mathbf{r})$ are sufficient to determine the one-body potential $V(\mathbf{r})$ and vector potential $\mathbf{A}(\mathbf{r})$, 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 $\mathord{\cal E}_{V_0,\mathbf{A}_0}(\rho,\mathbf{j}) = \langle \psi(\rho,\mathbf{j}),H(V_0,\mathbf{A}_0)\psi(\rho,\mathbf{j})\rangle$ 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 show that a variational principle for Total Current Density Functional Theory as that of Hohenberg-Kohn for Density Functional Theory does not exist. The reason is that the assumed map from densities to the vector potential, written $(\rho,\mathbf{j})\mapsto \mathbf{A}(\rho,\mathbf{j};\mathbf{r})$, enters explicitly in $\mathord{\cal E}_{V_0,\mathbf{A}_0}(\rho,\mathbf{j})$.