Deep Learning the Hohenberg-Kohn Maps of Density Functional Theory

Area of Science:

• Quantum Many-Body Physics
• Computational Physics
• Statistical Learning

Background:

• The Hohenberg-Kohn theorem establishes a unique mapping between a system's electron density and its ground-state wave function.
• Approximating this mapping is crucial for accurate predictions in density functional theory (DFT).
• Statistical learning offers a novel approach to constructing these complex functional relationships.

Purpose of the Study:

• To develop and evaluate a deep learning-based method for approximating the Hohenberg-Kohn map.
• To test the accuracy of this approach across different phases of a 1D interacting spinless fermion model.
• To explore the reconstruction of complex observables from local density measurements using machine learning.

Main Methods:

• Utilized supervised deep learning with synthetically generated data.
• Trained neural networks to learn the mapping from local density to the ground-state wave function.
• Investigated the model's performance across metallic, Mott insulator, and critical phases of 1D spinless fermions.

Main Results:

• Deep learning accurately approximated the Hohenberg-Kohn map for 1D spinless fermions in various phases.
• Learning effectiveness decreased across quantum phase transitions, indicating challenges with nonsmooth functional relations.
• A scheme was proposed for reconstructing complex observables from local density data, suitable for experimental application.

Conclusions:

• Deep learning shows promise for approximating fundamental mappings in quantum many-body physics.
• Challenges remain in learning across quantum phase transitions, highlighting the need for advanced learning strategies.
• The proposed method offers a potential pathway for extracting rich information from local density measurements in experiments.