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

TL;DR

This paper demonstrates that in degenerate current-density-functional theory, the particle and paramagnetic current densities do not uniquely determine the ground-state wave functions, and explores the implications for Hamiltonian degeneracies and representability.

Contribution

It provides a general strategy for constructing Hamiltonians with the same ground-state density but different degeneracies, and proves conditions for shared ground states in degenerate systems.

Findings

Density and current densities do not uniquely determine degenerate ground states.

Constructed Hamiltonians can share the same density but differ in degeneracy.

Shared ground-state densities imply at least one common ground state among Hamiltonians.

Abstract

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 $H(v',\mathbf{A}')$ 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.