Deep Learning the Functional Renormalization Group.

Area of Science:

• Condensed Matter Physics
• Quantum Field Theory
• Computational Physics

Background:

• The Hubbard model is a key theoretical tool for understanding strongly correlated electron systems.
• The functional renormalization group (FRG) is a powerful method for studying quantum many-body systems, but it involves complex, high-dimensional calculations.
• Characterizing the scale-dependent four-point vertex function is computationally intensive.

Purpose of the Study:

• To apply data-driven dimensionality reduction techniques to the FRG flow of the two-dimensional Hubbard model.
• To develop and validate an artificial intelligence (AI) approach for efficiently learning the FRG dynamics.
• To extract compact representations of four-point vertex functions for correlated electrons.

Main Methods:

• Dimensionality reduction of the scale-dependent four-point vertex function using a deep learning architecture.
• Employing a neural ordinary differential equation solver within a low-dimensional latent space.
• Utilizing dynamic mode decomposition (DMD) to analyze the learned FRG dynamics.

Main Results:

• The AI architecture successfully learned the FRG dynamics, accurately delineating magnetic and d-wave superconducting regimes.
• DMD analysis confirmed that a small number of modes capture the essential FRG dynamics.
• The study demonstrated the efficient extraction of compact representations of the four-point vertex functions.

Conclusions:

• AI methods, particularly neural ODEs, offer an efficient way to handle complex FRG calculations in condensed matter physics.
• This approach provides a pathway to overcome the computational challenges associated with many-electron problems.
• The extracted compact representations are vital for advancing quantum field theoretical methods.