A Geometric Approach to Strongly Correlated Bosons: From $N$-Representability to the Generalized BEC Force

TL;DR

This paper introduces a geometric framework for strongly correlated lattice bosons, emphasizing the role of N-representability and a generalized BEC force, with analytical demonstrations and implications for functional approximations.

Contribution

It develops an exact functional formulation based on momentum occupation numbers using geometric and symmetry considerations, extending the concept of the BEC force.

Findings

The domain of the functional is exactly determined by N-representability conditions.

The functional's gradient diverges at the boundary, revealing a generalized BEC force.

Analytical results are demonstrated for few-site lattice systems.

Abstract

Building on recent advances in reduced density matrix theory, we develop a geometric framework for describing strongly correlated lattice bosons. We first establish that translational symmetry, together with a fixed pair interaction, enables an exact functional formulation expressed solely in terms of momentum occupation numbers. Employing the constrained-search formalism and exploiting a geometric correspondence between $N$-boson configuration states and their one-particle reduced density matrices, we derive the general form of the ground-state functional. Its structure highlights the omnipresent significance of one-body $N$-representability: (i) the domain is exactly determined by the $N$-representability conditions; (ii) at its boundary, the gradient of the functional diverges repulsively, thereby generalizing the recently discovered Bose-Einstein condensate (BEC) force; and (iii) an explicit expression for this boundary force follows directly from geometric arguments. These key results are demonstrated analytically for few-site lattice systems, and we illustrate the broader significance of our functional form in defining a systematic hierarchy of functional approximations.