Particle diagrams and embedded many-body random matrix theory

Area of Science:

• Quantum mechanics
• Statistical physics
• Condensed matter physics

Background:

• Many-body random matrix theories (RMT) are crucial for understanding complex quantum systems.
• Existing methods for calculating statistical quantities in RMT can be computationally intensive.
• Feynman-like diagrams offer a visual and systematic approach to complex calculations.

Purpose of the Study:

• To introduce a novel method utilizing Feynman-like diagrams for calculating statistical quantities in embedded many-body RMT.
• To simplify existing techniques for analyzing the behavior of quantum systems.
• To investigate the statistical properties of interacting many-body systems.

Main Methods:

• Application of Feynman-like diagrams to calculate statistical moments of the level density.
• Analysis of an m-body system with k fermions or bosons interacting via a random Hermitian potential.
• Consideration of the limit where the number of single-particle states approaches infinity.

Main Results:

• The method successfully calculates the fourth, sixth, and eighth moments of the level density.
• A transition is observed from semicircular to Gaussian level density moments when 2k = m.
• The domain of the 2nth moment is found to be naturally divided into n subdomains based on k and m.

Conclusions:

• The Feynman-like diagram method is a powerful and simplifying alternative for RMT calculations.
• The study elucidates the statistical behavior of interacting quantum systems, highlighting critical transitions.
• The findings provide a deeper understanding of the structure and properties of many-body random matrix ensembles.