Density-Wavefunction Mapping in Degenerate Current-Density-Functional Theory

We show that the particle density, $\rho (\mathbf{r})$, and the paramagnetic current density, ${\mathbf{j}}^p(\mathbf{r})$, are not sufficient to determine the set of degenerate ground-state wave functions. This is a general feature of degenerate systems where the degenerate states have different angular momenta. We provide a general strategy for constructing Hamiltonians that share the same ground state density, yet differ in degree of degeneracy. We then provide a fully analytical example for a noninteracting system subject to electrostatic potentials and uniform magnetic fields. Moreover, we prove that when $(\rho ,{\mathbf{j}}^p)$ is ensemble $(v,\mathbf{A})$-representable by a mixed state formed from $r$ degenerate ground states, then any Hamiltonian $H(v^{\mathrm{\prime }},{\mathbf{A}}^{\mathrm{\prime }})$ that shares this ground state density pair must have at least $r$ degenerate ground states in common with $H(v,\mathbf{A})$. Thus, any set of Hamiltonians that shares a ground-state density pair $(\rho ,{\mathbf{j}}^p)$ by necessity has at least have one joint ground state.