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

2017/10/31 by Klaas J. H. Giesbertz, Klaas J.H. Giesbertz, Michael Ruggenthaler
Mathematics · Physics and Astronomy · #Basis (linear algebra) #Boson #Cold Atom Physics and Bose-Einstein Condensates #Fock space #Hamiltonian (control theory) #Hermitian matrix #Pauli exclusion principle #Physics of Superconductivity and Magnetism #Quantum many-body systems #Space (punctuation) #cond-mat.other #math-ph #math.MP #physics.atom-ph #physics.chem-ph #quant-ph

paper · pdf · doi:10.1016/j.physrep.2019.01.010

published as Physics Reports 806, 1-47 (2019) · 55 pages, 7 figures

openalex created_date 2017/11/10 · arxiv created 2019/02/01 · openalex publication_date 2019/02/12 · arxiv updated 2019/06/07 · openalex updated_date 2026/08/05

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.

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