Density–wave-function mapping in degenerate current-density-functional theory

We show that the particle density $\ensuremath{\rho}(\mathbf{r})$ 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 pair 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 $(\ensuremath{\rho},{\mathbf{j}}^{p})$ is ensemble $(v,\mathbf{A})$-representable by a mixed state formed from $r$ degenerate ground states, then any Hamiltonian $H({v}^{\ensuremath{'}},{\mathbf{A}}^{\ensuremath{'}})$ 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 $(\ensuremath{\rho},{\mathbf{j}}^{p})$ by necessity has to have at least one joint ground state.