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We present a metric-space approach to quantify the performance ofdensity-functional approximations for interacting many-body systems and toexplore the validity of the Hohenberg-Kohn-type theorem on fermionic lattices.This theorem demonstrates the existence of one-to-one mappings between particledensities, wave functions and external potentials. We then focus on thesequantities, and quantify how far apart in metric space the approximated andexact ones are. We apply our method to the one-dimensional Hubbard model fordifferent types of external potentials, and assess its validity on one of themost used approximations in density-functional theory, the local densityapproximation (LDA). We find that the potential distance may have a verydifferent behaviour from the density and wave function distances, in some caseseven providing the wrong assessments of the LDA performance trends. Weattribute this to the systems reaching behaviours which are borderline for theapplicability of the one-to-one correspondence between density and externalpotential. On the contrary the wave function and density distances behavesimilarly and are always sensitive to system variations. Our metric-basedmethod correctly predicts the regimes where the LDA performs fairly well andthe regimes where it fails. This suggests that our method could be a practicaltool for testing the efficiency of density-functional approximations.

Testing density-functional approximations on a lattice and the limits of the related Hohenberg-Kohn-type theorem