One-body reduced density-matrix functional theory in finite basis sets at elevated temperatures

TL;DR

This paper reviews the theoretical foundations of one-body reduced density-matrix functional theory in finite basis sets, addressing both fermionic and bosonic systems at elevated temperatures and variable particle numbers, highlighting mathematical subtleties and conditions for rigorous formulation.

Contribution

It provides a rigorous, self-contained presentation of 1RDM functional theory in finite basis sets, including fermionic and bosonic cases, and discusses mathematical challenges and restrictions for bosonic systems.

Findings

Fermionic 1RDM theory is straightforward due to finite-dimensional Fock space.

Bosonic 1RDM theory requires restrictions due to infinite-dimensional Fock space.

Mathematical subtleties arise in infinite-dimensional cases, affecting the formulation of 1RDM and DFT.

Abstract

In this review we provide a rigorous and self-contained presentation of one-body reduced density-matrix (1RDM) functional theory. We do so for the case of a finite basis set, where density-functional theory (DFT) implicitly becomes a 1RDM functional theory. To avoid non-uniqueness issues we consider the case of fermionic and bosonic systems at elevated temperature and variable particle number, i.e, a grand-canonical ensemble. For the fermionic case the Fock space is finite-dimensional due to the Pauli principle and we can provide a rigorous 1RDM functional theory relatively straightforwardly. For the bosonic case, where arbitrarily many particles can occupy a single state, the Fock space is infinite-dimensional and mathematical subtleties (not every hermitian Hamiltonian is self-adjoint, expectation values can become infinite, and not every self-adjoint Hamiltonian has a Gibbs state) make it necessary to impose restrictions on the allowed Hamiltonians and external non-local potentials. For simple conditions on the interaction of the bosons a rigorous 1RDM functional theory can be established, where we exploit the fact that due to the finite one-particle space all 1RDMs are finite-dimensional. We also discuss the problems arising from 1RDM functional theory as well as DFT formulated for an infinite-dimensional one-particle space.