Diverging Exchange Force and Form of the Exact Density Matrix Functional.

Area of Science:

• Quantum Chemistry
• Condensed Matter Physics
• Computational Materials Science

Background:

• Reduced Density Matrix Functional Theory (RDMFT) is a powerful tool for studying many-body quantum systems.
• Simplifying RDMFT is crucial for accurate and efficient calculations of electronic structure.
• The exact form of the interaction functional in RDMFT remains a significant challenge.

Purpose of the Study:

• To simplify the reduced density matrix functional theory (RDMFT) for translationally invariant one-band lattice models.
• To uncover the underlying structure and properties of the interaction functional in RDMFT.
• To investigate the role of natural orbitals and geometric constraints in determining the exact functional.

Main Methods:

• Exploiting ab initio knowledge of natural orbitals to simplify RDMFT.
• Analyzing the dependence of the interaction functional on natural occupation numbers.
• Investigating the geometric properties of the polytope of N-representable one-matrices.
• Deriving the exact functional for the Hubbard square model as an illustration.

Main Results:

• The interaction functional depends solely on natural occupation numbers within each symmetry sector.
• The sets of pure and ensemble N-representable one-matrices coincide.
• The exact functional is strongly shaped by the geometry of the N-representable polytope.
• The gradient of the functional diverges on the boundary, indicating an 'exchange force' due to fermionic symmetry.

Conclusions:

• Natural orbitals provide a significant simplification for RDMFT.
• The geometric structure of N-representability is key to the exact interaction functional.
• The 'exchange force' is an emergent property of fermionic symmetry within RDMFT.