Uniqueness of density-to-potential mapping for fermionic lattice systems

TL;DR

This paper proves that for fermionic lattice systems, the ground-state density uniquely determines the external potential, extending the Hohenberg-Kohn theorem, and introduces a practical method for potential inversion from density.

Contribution

It establishes the uniqueness of the density-to-potential mapping for fermionic lattice systems, including excited states, and develops an inversion scheme for practical applications.

Findings

Unique density-to-potential mapping for ground and excited states.

Inversion scheme for potential from density demonstrated.

Results applicable to Hubbard and similar lattice models.

Abstract

We demonstrate that, for a fermionic lattice system, the ground-state particle density uniquely determines the external potential except for the sites corresponding to nodes of the wave function, and the limiting case where the Pauli exclusion principle completely determines the occupation of all sites. Our fundamental finding completes, for this general class of systems, the one-to-one correspondence between ground states, their densities, and the external potential at the base of the Hohenberg-Kohn theorem. Moreover we demonstrate that the mapping from wave function to potential is unique not just for the ground state, but also for excited states. To illustrate our findings, we develop a practical inversion scheme to determine the external potential from a given density. Our results hold for a general class of lattice models, which includes the Hubbard model.