Deriving density-matrix functionals for excited states

TL;DR

This paper develops the first approximations for the $oldsymbol{w}$-ensemble RDMFT, enabling excitation energy calculations and revealing divergence phenomena at the functional boundary in both fermionic and bosonic systems.

Contribution

It introduces the $oldsymbol{w}$-RDMFT framework, derives functional approximations, and explores divergence behavior at the functional boundary in excited states for fermions and bosons.

Findings

Gradient divergence at the functional boundary extends Bose-Einstein condensation force to excited states.

Derived functional approximations for symmetric Hubbard dimer and homogeneous Bose gas.

Revealed the importance of the generalized exclusion principle and universality limitations.

Abstract

We initiate the recently proposed $\boldsymbol{w}$-ensemble one-particle reduced density matrix functional theory ($\boldsymbol{w}$-RDMFT) by deriving the first functional approximations and illustrate how excitation energies can be calculated in practice. For this endeavour, we first study the symmetric Hubbard dimer, constituting the building block of the Hubbard model, for which we execute the Levy-Lieb constrained search. Second, due to the particular suitability of $\boldsymbol{w}$-RDMFT for describing Bose-Einstein condensates, we demonstrate three conceptually different approaches for deriving the universal functional in a homogeneous Bose gas for arbitrary pair interaction in the Bogoliubov regime. Remarkably, in both systems the gradient of the functional is found to diverge repulsively at the boundary of the functional's domain, extending the recently discovered Bose-Einstein condensation force to excited states. Our findings highlight the physical relevance of the generalized exclusion principle for fermionic and bosonic mixed states and the curse of universality in functional theories.