Repulsively diverging gradient of the density functional in the Reduced Density Matrix Functional Theory

TL;DR

This paper demonstrates the universal presence of a diverging repulsive gradient in the density functional of RDMFT for both bosonic and fermionic systems, with implications for strongly correlated quantum systems and electron transfer calculations.

Contribution

It establishes the universality of the Bose-Einstein condensation force and introduces a variational approach to approximate the density functional near condensed states.

Findings

Universal divergence of the density functional gradient in bosonic RDMFT.

Existence of a similar repulsive gradient in fermionic RDMFT near Hartree-Fock states.

Approximate functional performs well in strongly correlated electron transfer simulations.

Abstract

The Reduced Density Matrix Functional Theory (RDMFT) is a remarkable tool for studying properties of ground states of strongly interacting quantum many body systems. As it gives access to the one-particle reduced density matrix of the ground state, it provides a perfectly tailored approach to studying the Bose-Einstein condensation or systems of strongly correlated electrons. In particular, for homogeneous Bose-Einstein condensates as well as for the Bose-Hubbard dimer it has been recently shown that the relevant density functional exhibits a repulsive gradient (called the Bose-Einstein condensation force) which diverges when the fraction of non-condensed bosons tends to zero. In this paper, we show that the existence of the Bose-Einstein condensation force is completely universal for any type of pair-interaction and also in the non-homogeneous gases. To this end, we construct a universal family of variational trial states which allows us to suitably approximate the relevant density functional in a finite region around the set of the completely condensed states. We also show the existence of an analogous repulsive gradient in the fermionic RDMFT for the $N$-fermion singlet sector in the vicinity of the set of the Hartree-Fock states. Finally, we show that our approximate functional may perform well in electron transfer calculations involving low numbers of electrons. This is demonstrated numerically in the Fermi-Hubbard model in the strongly correlated limit where some other approximate functionals are known to fail.