We propose a systematic procedure for the approximation of density functionals in density functional theory that consists of two parts. First, for the efficient approximation of a general density functional, we introduce an efficient ansatz whose non-locality can be increased systematically. Second, we present a fitting strategy that is based on systematically increasing a reasonably chosen set of training densities. We investigate our procedure in the context of strongly correlated fermions on a one-dimensional lattice in which we compute accurate training densities with the help of matrix product states. Focusing on the exchange-correlation energy, we demonstrate how an efficient approximation can be found that includes and systematically improves beyond the local density approximation. Importantly, this systematic improvement is shown for target densities that are quite different from the training densities.
We propose a systematic procedure for the approximation of density functionals in density functional theory that consists of two parts. First, for the efficient approximation of a general density functional, we introduce an efficient ansatz whose non-locality can be increased systematically. Second, we present a fitting strategy that is based on systematically increasing a reasonably chosen set of training densities. We investigate our procedure in the context of strongly correlated fermions on a one-dimensional lattice in which we compute accurate training densities with the help of matrix product states. Focusing on the exchange-correlation energy, we demonstrate how an efficient approximation can be found that includes and systematically improves beyond the local density approximation. Importantly, this systematic improvement is shown for target densities that are quite different from the training densities.
We propose a systematic procedure for the approximation of density functionals in density functional theory that consists of two parts. First, for the efficient approximation of a general density functional, we introduce an efficient ansatz whose non-locality can be increased systematically. Second, we present a fitting strategy that is based on systematically increasing a reasonably chosen set of training densities. We investigate our procedure in the context of strongly correlated fermions on a one-dimensional lattice in which we compute accurate training densities with the help of matrix product states. Focusing on the exchange-correlation energy, we demonstrate how an efficient approximation can be found that includes and systematically improves beyond the local density approximation. Importantly, this systematic improvement is shown for target densities that are quite different from the training densities.
Strong electronic correlations are at the heart of many interesting phenomena. For the theoretical description of these materials, a proper treatment of the local atomic physics is required. We propose a novel approach combining functionals of the electron density and functionals of the one-particle reduced density matrix to improve this description. The proposed method has a solid foundation in reduced density-matrix functional theory and no double-counting problem arises. It employs a decomposition of the electron-electron interaction in real space. The interaction close to the correlated orbitals, for example, the partially filled 3d-orbitals of transition-metal ions, is described with a density-matrix functional and otherwise with a local or semi-local density functional. We propose to evaluate the density-matrix functional from Levy’s constrained search problem, i.e., via a constrained minimization over an ensemble of many-particle wave functions. In contrast to approximate parametrized density-matrix functionals, this evaluation allows us to systematically improve the functional towards the exact result. In situations where the one-particle basis is too large to evaluate the density-matrix functional from Levy’s constrained minimization problem, we apply a series of approximations that each can be converged to the exact result: the first approximation step is the local approximation of the density-matrix functional proposed by Blöchl, Walther and Pruschke. For the density-matrix functionals within the local interactions, we propose the adaptive cluster approximation (ACA) that systematically truncates non-interacting one-particle states and drastically reduces the computational effort. The resulting density-matrix functional for a local interaction and a small number of non-interacting one-particle states is then evaluated with the constrained minimization problem. We explore different parametrizations of many-particle wave functions in this constrained minimization problem. A parametrization based on a configuration-interaction-like ansatz is shown to converge rapidly if suitable selection criterion for the Slater determinants is chosen. An impurity-bath-separation ansatz is shown to be suitable for single-impurity Anderson models. It is shown that the constrained minimization can be solved for matrix product states with a DMRG-like iterative minimization. Furthermore, we show that Gutzwiller-Jastrow-correlated wave functions can be used with a quantum Monte Carlo procedure as many-particle wave functions. Finally, we formulate an algorithm for the evaluation of the density-matrix functional on near-term quantum computers. Results from the execution of the algorithm on an existing quantum computer with transmon qubits are presented. The proposed approach combining density functionals and density-matrix functionals is implemented in the CP-PAW code based on the projector augmented-wave formalism. We present results for the dissociation curve of the hydrogen molecule as the prototypical case of strong static correlation and show that static correlation is well described with the new approach. We show results for the nonmagnetic state of the transition-metaoxide NiO that is described qualitatively wrong with the DFT+U method. The proposed method properly describes the nonmagnetic state of NiO as an insulator and predicts a qualitatively correct spectral function.
Abstract Recently a novel approach to find approximate exchange–correlation functionals in density-functional theory was presented (Mordovina et al 2019 J. Chem. Theory Comput. 15 5209), which relies on approximations to the interacting wave function using density-matrix embedding theory (DMET). This approximate interacting wave function is constructed by using a projection determined by an iterative procedure that makes parts of the reduced density matrix of an auxiliary system the same as the approximate interacting density matrix. If only the diagonal of both systems are connected this leads to an approximation of the interacting-to-non-interacting mapping of the Kohn–Sham approach to density-functional theory. Yet other choices are possible and allow to connect DMET with other density-functional theories such as kinetic-energy density functional theory or reduced density-matrix functional theory. In this work we give a detailed review of the basics of the DMET procedure from a density-functional perspective and show how both approaches can be used to supplement each other. We do not present a specific realization of combining density-functional methods with DMET but rather provide common grounds to facilitate future developments that encompass both approaches. We do so explicitly for the case of a one-dimensional lattice system, as this is the simplest setting where we can apply DMET and the one that was originally presented. Among others we highlight how the mappings of density-functional theories can be used to identify uniquely defined auxiliary systems and projections in DMET and how to construct approximations for different density-functional theories using DMET inspired projections. Such alternative approximation strategies become especially important for density-functional theories that are based on non-linearly coupled observables such as kinetic-energy density-functional theory, where the Kohn–Sham fields are no longer obtainable by functional differentiation of an energy expression, or for reduced density-matrix functional theories, where a straightforward Kohn–Sham construction is not feasible.
A density-functional theory (DFT) of lattice fermion models is presented, which uses the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ as basic variable. A simple, explicit approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ of the Hubbard model is derived from exact dimer results, scaling properties of $W[\ensuremath{\gamma}]$ and known limits. Systematic tests on the one-dimensional chain show a remarkable agreement with the Bethe-ansatz exact solution for all interaction regimes and band fillings. New results are obtained for the ground-state energy and charge-excitation gap in two dimensions. A successful description of strong electron correlations within DFT is achieved.
In this Thesis, we develop and apply an accurate method to determine the ground-state properties of strongly-correlated electrons in the framework of lattice model Hamiltonians. In lattice density-functional theory (LDFT) the basic variable is the one-particle density matrix γ. From a generalized HK theorem, the ground state energy Egs[γgs] = minγ E[γ] is obtained by minimizing the energy-functional E[γ] over all the physical or representable γ. The energy functional can be divided into two contributions: the kinetic-energy functional T [γ], which linear dependence on γ is exactly known, and the correlation-energy functional W [γ], which dependence on γ is not explicitly known. Finding accurate approximations for W [γ] constitutes the actual challenge of this thesis. Part of this work is based on previous studies which derive an approximation of W [γ] for the Hubbard Hamiltonian based on scaling hypothesis and an exact analytical results for the Hubbard dimer. However, this approach is limited to spin independent and homogeneous systems. In order to extend the scope of LDFT, we have developed three different approaches to derive W [γ] in order to study broken symmetry problems. First we extend the previous scaling functional to system having charge transfer. By a systematic study of the dependence of W [γ] on the charge distribution we find similar scaling properties as for the homogeneous case. An extension to the Hubbard model on bipartite lattices is then derived and applied to finite and infinite systems with repulsive and attractive interactions. The high accuracy of this functional is demonstrated. Nevertheless, this approach is difficult to transfer to more complex systems, since the calculation of W [γ] applies to the system as a whole. To overcome this problem, we derive another approximation based on local scaling properties. This functional is formulated on-site so that it can be applied to any ordered or disordered Hamiltonian with on-site interaction. As applications we study the metal-insulator transition in the ionic Hubbard model in oneand two-dimensions, as well as the one-dimensional Hubbard chain with first and second nearest neighbor hoppings. Finally, we develop a numerical approach of W [γ], on the basis on exact diagonalizations of an effective many-body Hamiltonian corresponding to a cluster surrounded by an effective medium. This effective Hamiltonian depends on the density matrix γ and allows to derive approximation to W [γ] which improve systematically with increasing cluster size. The formulation is spin-dependent and allows a straightforward generalization to multi-orbital correlated systems such spd-Hamiltonians. In addition it take into account the effects of short-range charge and spin fluctuations in the functional. The accuracy of the method is demonstrated for the Hubbard model by comparison with the Bethe-Ansatz solution (1D) and quantum Monte Carlo simulations (2D). Finally, an outlook or relevant future developments of the theory is provided.
Over the past two decades, many studies in the density functional theory context revealed new aspects and properties of strongly correlated superfluid quantum systems in numerous configurations that can be simulated in experiments. This was made possible by the generalization of the local density approximation to superfluid systems by Bulgac [Phys. Rev. C 65, 051305(R) (2002); Phys. Rev. A 76, 040502(R) (2007)]. In the present work, we propose an extension of the superfluid local density approximation, systematically improvable and applicable to a large range of many-body quantum problems getting rid of the fitting procedures of the functional parameters. It turns out that only the knowledge of the density dependence of the quasiparticle properties, namely, the chemical potential, the effective mass, and the pairing gap function, are enough to obtain an explicit and accurate local functional of the densities without any adjustment a posteriori. This opens the way toward an effective field theory formulation of the density functional theory in the sense that we obtain a universal expansion of the functional parameters entering in the theory as a series in pairing gap function. Finally, we discuss possible applications of the developed approach allowing precise analysis of experimental observations. In that context, we focus our applications on the static structure properties of superfluid vortices.