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 (BEC) or systems of strongly correlated electrons. In particular, for homogeneous BECs 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.
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 (BEC) or systems of strongly correlated electrons. In particular, for homogeneous BECs 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.
We initiate the recently proposed w w -ensemble one-particle reduced density matrix functional theory ( w 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 w 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.
Density functional theory constitutes the workhorse of modern electronic structure calculations due to its favourable computational cost despite the fact that it usually fails to describe strongly correlated systems. A particularly promising approach to overcome those difficulties is reduced density matrix functional theory (RDMFT): It abandons the complexity of the N-particle wave function and at the same time explicitly allows for fractional occupation numbers. It is the goal of this thesis to initiate and establish a bosonic RDMFT for both ground state and excited state energy calculations. Motivated by the Onsager and Penrose criterion which identifies RDMFT as a particularly suitable approach to describe Bose-Einstein condensates (BECs), we derive the universal functional for a homogeneous BEC in the Bogoliubov regime. Remarkably, the gradient of the universal functional is found to diverge repulsively in the regime of complete condensation. This introduces the new concept of a BEC force, which provides a universal explanation for quantum depletion since it is merely based on the geometry of quantum states. In the second part of the thesis, we propose and work out an ensemble RDMFT targeting excitations in bosonic quantum systems. This endeavour further highlights the potential of convex analysis for the development of functional theories in the future. Indeed by resorting to several concepts from convex analysis, we succeeded to provide a comprehensive foundation of w-ensemble RDMFT for bosons which is further based on a generalization of the Ritz variational principle and a constrained search formalism. In particular, we solve the emerging N-representability problem leading to a generalization of Pauli's famous exclusion principle to bosonic mixed states.
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.
Based on a generalization of Hohenberg-Kohn's theorem, we propose a ground state theory for bosonic quantum systems. Since it involves the one-particle reduced density matrix γ as a variable but still recovers quantum correlations in an exact way it is particularly well suited for the accurate description of Bose-Einstein condensates. As a proof of principle we study the building block of optical lattices. The solution of the underlying v-representability problem is found and its peculiar form identifies the constrained search formalism as the ideal starting point for constructing accurate functional approximations: The exact functionals F[γ] for this N-boson Hubbard dimer and general Bogoliubov-approximated systems are determined. For Bose-Einstein condensates with N_{BEC}≈N condensed bosons, the respective gradient forces are found to diverge, ∇_{γ}F∝1/sqrt[1-N_{BEC}/N], providing a comprehensive explanation for the absence of complete condensation in nature.
For translationally invariant one-band lattice models, we exploit the ab initio knowledge of the natural orbitals to simplify reduced density matrix functional theory (RDMFT). Striking underlying features are discovered. First, within each symmetry sector, the interaction functional F depends only on the natural occupation numbers $\mathbit{n}$. The respective sets PN1 and EN1 of pure and ensemble N-representable one-matrices coincide. Second, and most importantly, the exact functional is strongly shaped by the geometry of the polytope ${\mathcal{E}}_{N}^{1}\ensuremath{\equiv}{\mathcal{P}}_{N}^{1}$, described by linear constraints ${D}^{(j)}(\mathbit{n})\ensuremath{\ge}0$. For smaller systems, it follows as $\mathcal{F}[\mathbit{n}]=\ensuremath{\sum}_{i,{i}^{\ensuremath{'}}}{\overline{V}}_{i,{i}^{\ensuremath{'}}}\sqrt{{D}^{(i)}(\mathbit{n}){D}^{({i}^{\ensuremath{'}})}(\mathbit{n})}$. This generalizes to systems of arbitrary size by replacing each D(i) by a linear combination of ${{D}^{(j)}(\mathbit{n})}$ and adding a nonanalytical term involving the interaction $\stackrel{^}{V}$. Third, the gradient $d\mathcal{F}/d\mathbit{n}$ is shown to diverge on the boundary $\ensuremath{\partial}{\mathcal{E}}_{N}^{1}$, suggesting that the fermionic exchange symmetry manifests itself within RDMFT in the form of an ``exchange force.'' All findings hold for systems with a nonfixed particle number as well and $\stackrel{^}{V}$ can be any p-particle interaction. As an illustration, we derive the exact functional for the Hubbard square.
In this work, we explore the performance of approximations to electron correlation in reduced density-matrix functional theory (RDMFT) and of approximations to the observables calculated within this theory. Our analysis focuses on the calculation of total energies, occupation numbers, removal/addition energies, and spectral functions. We use the exactly solvable Hubbard dimer at 1/4 and 1/2 fillings as test systems. This allows us to analyze the underlying physics and to elucidate the origin of the observed trends. For comparison, we also report the results of the GW approximation, where the self-energy functional is approximated, but no further hypothesis is made concerning the approximations of the observables. In particular, we focus on the atomic limit, where the two sites of the dimer are pulled apart and electrons localize on either site with equal probability, unless a small perturbation is present: this is the regime of strong electron correlation. In this limit, using the Hubbard dimer at 1/2 filling with or without a spin-symmetry-broken ground state allows us to explore how degeneracies and spin-symmetry breaking are treated in RDMFT. We find that, within the used approximations, neither in RDMFT nor in GW, the signature of strong correlation is present, when looking at the removal/addition energies and spectral function from the spin-singlet ground state, whereas both give the exact result for the spin-symmetry broken case. Moreover, we show how the spectroscopic properties change from one spin structure to the other.