Geometry of Generalized Density Functional Theories Solves -Representability Problem Using Symplectic Geometry

Density functional theory forms a cornerstone of modern physics, enabling scientists to model the behaviour of materials with remarkable accuracy, but extending its power to strongly correlated systems remains a significant challenge. Chih-Chun Wang from Ludwig-Maximilians-Universität München, alongside colleagues, now presents a comprehensive mathematical framework that generalises all ground state functional theories, encompassing systems of fermions, bosons, and spins. This work solves a long-standing problem concerning the ‘representability’ of density functions, utilising techniques from symplectic geometry to provide a robust solution, and importantly, offers a precise formula for the ‘boundary force’, a previously understood but poorly quantified phenomenon. By rigorously defining the behaviour of this force, the team lays the groundwork for developing more accurate approximations within reduced density matrix functional theory, potentially leading to substantial improvements in modelling complex materials.

Ab initio methods are central to both quantum chemistry and condensed matter physics. Building on recent advancements in reduced density matrix functional theory, a variant of density functional theory believed to be better suited for strongly correlated systems, the researchers construct a generalized mathematical framework encompassing all ground state functional theories, with applications to systems of fermions, bosons, and spins. This framework offers a unified approach to understanding their behaviour and addresses limitations inherent in conventional approaches. A key achievement lies in solving the long-standing “representability problem” within this framework, utilising techniques from symplectic geometry to determine which functions can accurately represent the ground state of a physical system.

Quantum State Reconstruction via Marginal Problems

This work focuses on understanding the validity of quantum states through the quantum marginal problem. The research explores conditions under which a reduced description of a quantum system accurately represents its behaviour, drawing on concepts from mathematics and physics. The team investigates the geometric structure of quantum states, utilising tools from symplectic geometry, Lie groups, and convex geometry to analyse their properties, providing a rigorous mathematical foundation for understanding entanglement and other quantum phenomena.

Representability and Boundary Force in Functional Theory

This work presents a groundbreaking advancement in functional theories, extending beyond traditional density functional theory. Scientists also provide a quantitative understanding of the “boundary force”, a repulsive force appearing in functional theories, and derive a precise formula describing its behaviour. This formula, rigorously proven in certain cases using constrained search methods, represents a crucial step towards developing more accurate approximations for complex systems, potentially improving existing methods such as natural orbital functionals. Researchers demonstrated these key concepts using translationally invariant bosonic lattice systems, validating the approach for realistic physical scenarios. Further investigation revealed connections to the theory of momentum maps from symplectic geometry, providing a powerful tool for analyzing functional behaviour. Perturbative calculations support a conjecture mirroring the boundary force formula, suggesting broader applicability of the findings.

Ground State Theory, Representability and Boundary Force

This work presents a significant advancement in the theoretical framework underpinning density functional theory, a cornerstone of modern computational physics and chemistry. A key achievement lies in solving the long-standing “representability problem”, utilising techniques from symplectic geometry to determine which functions can accurately represent the ground state of a physical system. The team also provides a quantitative understanding of the “boundary force”, a repulsive force appearing in functional theories, and derives a precise formula describing its behaviour. The concepts and techniques developed were successfully demonstrated using translationally invariant bosonic lattice systems, validating the approach for realistic physical scenarios. The authors acknowledge that further research is needed to extend the applicability of the derived formula to more general cases.