Geometry of Generalized Density Functional Theories

TL;DR

This work develops a generalized theory of ground-state functionals that unifies DFT and RDMFT across fermionic, bosonic, and spin systems, using both abelian and momentum-map frameworks. It introduces a robust mathematical structure with three universal functionals, leverages purification and convex-analysis tools, and derives exact formulas for boundary/boundary-like forces that govern representability and the approach to domain facets. The translation-invariant bosonic lattice example serves as a concrete realization, revealing how domain geometry controls the behavior of the universal functional and exposing a general boundary-force mechanism that may inform improved functionals such as Piris NOFs. The momentum-map perspective further opens the door to nonabelian functional theories by linking representability to convex polytopes and symplectic geometry, providing a principled route to handle more complex systems beyond abelian cases.

Abstract

Density functional theory (DFT) is an indispensable ab initio method in both quantum chemistry and condensed matter physics. Based on recent advancements in reduced density matrix functional theory (RDMFT), a variant of DFT that is believed to be better suited for strongly correlated systems, we construct a mathematical framework generalizing all ground state functional theories, which in particular applies to fermionic, bosonic, and spin systems. Within the special class of such functional theories where the space of external potentials forms the Lie algebra of a compact Lie group, the $N$-representability problem is readily solved by applying techniques from the study of momentum maps in symplectic geometry, an approach complementary to Klyachko's famous solution to the quantum marginal problem. The ``boundary force'', a diverging repulsive force from the boundary of the functional's domain observed in previous works but only qualitatively understood in isolated systems, is studied quantitatively and extensively in our work. Specifically, we present a formula capturing the exact behavior of the functional close to the boundary. In the case where the Lie algebra is abelian, a completely rigorous proof of the boundary force formula based on Levy-Lieb constrained search is given. Our formula is a first step towards developing more accurate functional approximations, with the potential of improving current RDMFT approximate functionals such as the Piris natural orbital functionals (NOFs). All key concepts and ideas of our work are demonstrated in translation invariant bosonic lattice systems.

Figures (18)

• Figure 1: Illustration of the geometry of computing the pure and ensemble functionals. The sphere represents the set $\mathcal{P}$ of all pure states, and the ball is the set $\mathcal{E}$ of ensemble states. The value of the pure functional $\mathcal{F}_{\!p}$ at a density $\rho\in[-1,1]$ is defined to be the minimum of $\braket{\Gamma, W}$, where $\Gamma$ is a pure state mapping to $\rho$. The set of all admissible pure states $\Gamma$ is $(\iota^*)^{-1}(\rho)\cap \mathcal{P}$, which is nothing but the horizontal circle at height $\rho$ in the figure. Similarly, for the ensemble functional, the set of admissible ensemble states is the orange disk. We may think of the fixed interaction $W=\lambda X$ as a vector in $\mathbb{R}^3$, and in this case it points towards the positive $x$ axis if $\lambda > 0$ (red arrow). The minimizer of $\braket{\Gamma, W}$ is then the blue dot on the left for both pure and ensemble states, which explains why $\mathcal{F}_{\!p}$ and $\mathcal{F}_{\!e}$ agree.
• Figure 2: Structure of $N$-particle momentum space permanents and domains of the functional.
• Figure 3: Plots of the magnitude of the derivative of the functional for various combinations of $(N, P)$ with $d=3$. Whenever $N$ is not a multiple of $3$, all momenta are equivalent, so only the case $P=0$ is presented. If $N$ is a multiple of $3$, only $P=1$ and $P=2$ are equivalent, thus we present plots for $P=0$ and $P=1$. For $N=3,6,12$, the functional is affine on the black disc in the center of its domain, which is a degeneracy region (see Refs. penzGeometryDegeneracyPotential2023penzGeometricalPerspectiveSpin2024 for more detailed discussions).
• Figure 4: Illustration of the geometric objects involved in the analysis of the generalized BEC force. The upper facet is the facet labeled by $s$, i.e., the points on it satisfy $D^{(s)} = 0$. In the diagram, $m^{(\alpha)}$ and $m^{(\beta)}$ are the occupation number vectors of two permanents (arbitrarily chosen for the purpose of illustration) not lying on the facet. Their distances $D^{(s)}(m^{(\alpha)})$ and $D^{(s)}(m^{(\beta)})$ are indicated on the left. The repulsion strength is defined in terms of the values of the functional $\mathcal{F}_{\!p}$ along the straight path starting at a point $m_*$ on the facet (blue diamond marker on top) and extending inwards perpendicularly to the facet (red dotted line), i.e., in the direction of the normal vector $S^{(s)}$ (red arrow).
• Figure 5: Comparison of the approximate functional given by Eq. \ref{['eq:hubbard_approximate_functional']} (solid lines) for the Hubbard model with exact numerical values (square dots).

Theorems & Definitions (211)

• Definition 2.1
• Proposition 2.1
• proof
• Example 2.1
• Example 2.2
• Example 2.3
• Remark 2.1
• Remark 2.2
• Definition 2.2
• Definition 2.3