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

TL;DR

This work develops a symmetry-adapted reduced density matrix functional theory (RDMFT) for strongly correlated bosonic lattice systems by exploiting translational symmetry to reduce the functional from the full 1RDM to momentum-occupation numbers. The authors show that the domain of the universal ground-state functional is a convex polytope determined by $N$-representability, and that the functional exhibits a universal boundary divergence—the generalized BEC force—induced purely by geometry and constrained-search structure. They derive exact forms in the simplex setting and a general representation beyond the simplex, including a closed expression for the boundary force and its dependence on facet distances and interaction matrix elements. A complete Bose-Hubbard illustrative example validates the framework and demonstrates how geometry-informed approximations can yield accurate energies, suggesting a principled route to systematic, geometry-driven functionals for strongly correlated bosons.

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.

Figures (11)

• Figure 1: Conceptual overview of the symmetry-adapted functional-theoretical framework developed in this work. See text for details.
• Figure 2: Schematic illustration of the concept of scope: a functional theory is associated with a family of Hamiltonians $\hat{H}(\hat{h})=\hat{h}+\hat{W}$, where $\hat{h}$ varies within a physically relevant real vector space of Hermitian operators determined by the problem class and its symmetries. The set of admissible Hamiltonians $\{\hat{H}(\hat{h})\}$ thus forms an affine space (blue dashed line), referred to as the scope of the functional theory.
• Figure 3: Structure of $N$-particle momentum space configurations and domains of the functional.
• Figure 4: The universal functional $\mathcal{F}[\vec{n}]$ for $(d,N,P)=(3,3,1)$ with negative (left) and positive (right) coupling parameter $w$. The solid triangle is the domain of $\mathcal{F}$, while the dashed lines outline the convex hull of $(3,0,0), (0,3,0),$ and $(0,0,3)$. See text for details.
• Figure 5: Magnitude of the gradient $|\nabla_{\vec{n}}\mathcal{F}|$ for $d=3$ and various combinations of $(N,P)$. See text for more details.

Theorems & Definitions (2)

• Theorem 1
• proof