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arXiv:2002.06943 · Physical Review Letters

Reduced Density Matrix Functional Theory for Bosons

Carlos L. Benavides-Riveros, Jakob Wolff, Miguel A. L. Marques, Christian Schilling

Published May 7, 2020• 32 citations
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Abstract

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.

DOI
10.1103/physrevlett.124.180603

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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.
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 NNN-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\boldsymbol{w}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 NNN-representability problem leading to a generalization of Pauli's famous exclusion principle to bosonic mixed states.
Abstract Motivated by the Penrose–Onsager criterion for Bose–Einstein condensation we propose a functional theory for targeting low-lying excitation energies of bosonic quantum systems through the one-particle picture. For this, we employ an extension of the Rayleigh–Ritz variational principle to ensemble states with spectrum w and prove a corresponding generalization of the Hohenberg–Kohn theorem: the underlying one-particle reduced density matrix determines all properties of systems of N identical particles in their w -ensemble states. Then, to circumvent the v-representability problem common to functional theories, and to deal with energetic degeneracies, we resort to the Levy–Lieb constrained search formalism in combination with an exact convex relaxation. The corresponding bosonic one-body w -ensemble N-representability problem is solved comprehensively. Remarkably, this reveals a complete hierarchy of bosonic exclusion principle constraints in conceptual analogy to Pauli’s exclusion principle for fermions and recently discovered generalizations thereof.
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.
A Geometric Approach to Strongly Correlated Bosons: From NNN-Representability to the Generalized BEC Force
Building on recent advances in reduced density matrix theory, we develop a geometric framework for describing strongly correlated lattice bosons. We first establish that translational symmetry, together with a fixed pair interaction, enables an exact functional formulation expressed solely in terms of momentum occupation numbers. Employing the constrained-search formalism and exploiting a geometric correspondence between NNN-boson configuration states and their one-particle reduced density matrices, we derive the general form of the ground-state functional. Its structure highlights the omnipresent significance of one-body NNN-representability: (i) the domain is exactly determined by the NNN-representability conditions; (ii) at its boundary, the gradient of the functional diverges repulsively, thereby generalizing the recently discovered Bose-Einstein condensate (BEC) force; and (iii) an explicit expression for this boundary force follows directly from geometric arguments. These key results are demonstrated analytically for few-site lattice systems, and we illustrate the broader significance of our functional form in defining a systematic hierarchy of functional approximations.
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.
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 NNN-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.