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

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

Figure 1

Orbital energies ${\varepsilon }_{n,m}$ [see Eq. (5)] for $\omega =0.8$ as a function of the magnetic field strength $B\ge 0$. Orbital energies with $m=0$ are shown as dotted black curves, $m=\pm 1$ as dashed blue curves, $m=\pm 2$ as solid red curves, $m=\pm 3$ as dashed dot magenta curves, and $m=\pm 4$ as solid green curves.

Figure 2

Two Hamiltonians have different sets of degenerate ground states (indicated by ellipses). Suppose the density matrices ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ are ground-state density matrices of $H(v_1,{\mathbf{A}}_1)$ and $H(v_2,{\mathbf{A}}_2)$, respectively. Assume further that they map to the same density, ${\mathrm{\Gamma }}_1\mapsto (\rho ,{\mathbf{j}}_p)$ and ${\mathrm{\Gamma }}_2\mapsto (\rho ,{\mathbf{j}}_p)$. Then it follows that ${\mathrm{\Gamma }}_1$ is also a ground-state density matrix of $H(v_2,{\mathbf{A}}_2)$ and that ${\mathrm{\Gamma }}_2$ is also a ground-state density matrix of $H(v_1,{\mathbf{A}}_1)$. Thus, both ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ are located in the intersection of the two ellipses.