Deep learning the Hohenberg-Kohn maps of Density Functional Theory

TL;DR

This paper demonstrates that deep learning can accurately approximate the Hohenberg-Kohn map in one-dimensional fermionic systems, revealing both the potential and limitations of data-driven approaches in density functional theory.

Contribution

The study introduces a supervised deep learning method to construct the Hohenberg-Kohn map from synthetic data, highlighting its effectiveness and challenges across different quantum phases.

Findings

Deep learning accurately models the Hohenberg-Kohn map in various phases.

Learning effectiveness decreases near quantum phase transitions.

Proposes a scheme for reconstructing observables from local density measurements.

Abstract

A striking consequence of the Hohenberg-Kohn theorem of density functional theory is the existence of a bijection between the local density and the ground-state many-body wave function. Here we study the problem of constructing approximations to the Hohenberg-Kohn map using a statistical learning approach. Using supervised deep learning with synthetic data, we show that this map can be accurately constructed for a chain of one-dimensional interacting spinless fermions, in different phases of this model including the charge ordered Mott insulator and metallic phases and the critical point separating them. However, we also find that the learning is less effective across quantum phase transitions, suggesting an intrinsic difficulty in efficiently learning non-smooth functional relations. We further study the problem of directly reconstructing complex observables from simple local density measurements, proposing a scheme amenable to statistical learning from experimental data.