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.
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.
Ab-initio calculations are dominated by density functional theory (DFT), which provides an efficient and accurate description of the electronic structure for most materials. For materials with strong correlations, however, many of the available density functionals yield poor results. Most well known is the case of transition metal oxides, for which most density functionals produce a qualitatively incorrect description. There is a quest to improve the description by borrowing from methods specifically designed for strongly correlated materials. One of the method which is widely used to study strongly correlated electron system in the thermodynamic limit is the dynamical mean-field theory (DMFT). For calculations pertaining to real materials, hybrid approaches like DFT+DMFT are commonly used. We consider reduced density-matrix functional theory (rDMFT) to be a useful framework for a rigorous formulation of such hybrid theories. The link from rDMFT to many-particle wave functions has been established by Levy's constrained-search algorithm on the one hand. The link to many-body perturbation theory and Green's function, on the other hand, has been provided via the Luttinger-Ward functional. The development in the field of rDMFT proceeded analogously to the development of DFT. In order to avoid the full complexity of an explicit many-body description, most density-matrix functionals are not extracted from the exact expressions i.e. via Levy's constrained-search or Green's function based methods . Rather, one proceeds analogously to the development of density functionals, namely by searching models for the density-matrix functional, that capture the most essential physical effects while having an algebraic dependence on the density matrix. Among such models, the most prominent ones are the Müller functional, Sharma functional and Marques-Lathiotakis functional. One of the major arguments in favor of density-matrix functionals is that one of the most simple functionals, the Müller functional, seems to provide a correct description of the bond-dissociation problem, for which common density functionals fail. Despite the successes, the common density-matrix functionals also reproduce a number of features in a qualitatively incorrect manner. Several such discrepancies of the standard state of the art density-matrix functionals are discussed in details in this thesis. The objective of this thesis is to study the performance of the class of commonly used model density-matrix functionals and to set the stage for the desire to develop the framework for an approximate scheme to evaluate new class of functionals from exact Green's function based method.
An implementation of the single-site density matrix renormalization group (DMRG) approach for the quantum-chemical application to calculate approximate as well as exact electronic ground-state energies and wave functions of atoms and molecules is presented.The algorithm is formulated in terms of the variational and sizeconsistent class of wave functions, matrix product states, and their operator analogues, matrix product operators.A procedure to construct quantum-chemical Hamiltonian operators with long-range Coulomb interactions as matrix product operators is developed.The many-electron Hamiltonian operators are represented as sums and products of one-electron operators.For the efficient matrix product operator construction of quantum-chemical Hamiltonians, different basis spaces including discrete position and momentum space as well as molecular and atomic orbitals are investigated.Discrete position space yields the best matrix product operator size scaling, but introduces artificial long-range couplings in the kinetic energy term of the Hamiltonian which are a consequence of the DMRG algorithm in terms of matrix product states.The huge amount of discrete space points needed for an accurate description poses a problem because of the electron-electron interaction.Therefore, a momentum space or Fourier series expansion of the electron-electron interaction is shown which makes it possible to compress its matrix product operator representation.DMRG applied to quantum-chemical Hamiltonians in matrix product operator form can serve as means to directly (without mean-field treatment in advance) investigate quantum-chemical problems and find the analytically exact ground-state solution of small atoms and molecules.The next basis space analyzed in detail is the atomic orbital basis.The matrix product operator construction and the direct application of the DMRG method in this basis is examined.A physical compression scheme based on a singular value decomposition of the electron-electron interaction in the atomic orbital basis is demonstrated.Furthermore, DMRG calculations in the atomic orbital basis for certain systems where mean-field theory yields qualitatively wrong results and where higher-order approximative correction methods like truncated configuration interaction and coupled cluster cannot fully capture strong electron correlation effects are compared to results obtained in the molecular orbital basis.The comparison shows that atomic orbitals describe the physical situation well and thus lead to a tremendous reduction of the matrix product state dimension and the computational effort.i
Strong correlations arising from interactions between particles are responsible for many phenomena in condensed matter physics, from superconductivity to heavy fermions. An accurate theoretical description of such complex systems requires to solve so-called many-body problems, which are inherently difficult to tackle due to the exponential growth of the underlying Fock space with the size of the system. Usually, single-particle descriptions or perturbation theory at first orders fail to provide satisfactory results in correlated regimes, and pertinent Ansätze or cutting edge numerical methods are required to solvethese problems.Among the wide variety of strongly correlated electronic systems, quantum impurity systems are of central interest. These systems, where a few localized degrees of freedom that undergo strong interactions hybridize with a larger system of free particles, have been widely studied for decades. Beyond their relevance to study dilute dynamic impurities in metals, quantum dots, or more complex nanostructures, they are exploited in a broader class of problems through dynamical mean field theory (DMFT), that is designed for fully interacting lattice problems. Despite the effort devoted to studying of these problems,some questions remain open and new approaches are required to address them.In this thesis, quantum impurity problems are tackled with the use of natural orbitals, a representation rooted in the framework of quantum chemistry. The numerical renormalization group, the standard non-perturbative method to treat these systems, is employed to demonstrate the hierarchical structure of correlations unveiled by natural orbitals. An ansatz for the wavefunction of the ground state is then proposed, separating explicitely a small set of correlated orbitals to a large uncorrelated one, leaving the problem to a few-body complexity. This simplification brought new insights for simulating quantumimpurity problems, and a new algorithm is developed to generate an optimized set of correlated orbitals independently of existing methods, going beyond their usual limitations.In the first place, the algorithm is used to compute the screening cloud in large one-dimensional chains and in a realistic 2D square lattice coupled to a localized adatom. This observable is of great interest to understand the screening mechanism of the magnetic moment of the impurity that plays a central role in transport properties, for unidimensional systems or in bulk metals experiencing RKKY interactions. Spatial correlations were found to be highly non-symmetric, breaking the s-wave picture. In one direction, corre-lations vanish way faster than the usual isotropic correlation length, while on the other one they spread further than this same expected length. This anisotropy suggests that impurities in bulk systems may interact differently depending on their relative positions with respect to the lattice.Taking further advantage of the efficiency of this new algorithm allowed us to perform a statistical study on large systems to investigate correlations around a quantum impurity in a disordered host, averaging over a large amount of realizations, which was not possible with previous approaches. The question motivating this study was the fate of the correlation cloud spreading into the disordered electronic bath. Although spatial correlations are expected to vanish when the disorder increases, long range correlationssurvive to some configurations of the potential, inducing a bimodal structure of probability distribution functions of relevant scales.Finally, these disordered problems are investigated in the non-interacting situation with random matrix theory, which reproduced qualitatively the results obtained before. The joint distribution of one-particle orbitals energies and amplitudes is calculated, which allows to compute any disordered averaged local correlation functions, and calculations in the large-N limit are presented.
Reduced density-matrix functional theory (RDMFT) provides a variational route to electronic correlations beyond conventional density-functional approximations, but explicit evaluations of density-matrix functionals still scale exponentially with the number of active one-particle states. We formulate and assess a real-space-decomposed density-functional plus reduced-density-matrix-functional (DF+RDMF) scheme in which the Coulomb interaction is partitioned locally in real space and the RDMF correction is evaluated only for the strongly correlated part of the interaction. The resulting local density-matrix functionals are further compressed using the adaptive cluster approximation (ACA), which performs a unitary rotation of the bath subspace before truncation and therefore preserves the local interaction while reducing the number of explicitly correlated bath states. As a molecular test case, we consider the bending potential of carbon suboxide, C3O2. While semilocal PBE favors a linear molecule, the DF+RDMF/ACA correction stabilizes a bent configuration in qualitative agreement with the quasilinear behavior inferred from spectroscopy. The approach provides a systematic embedding hierarchy for combining density functionals with explicitly correlated density-matrix corrections in extended or spatially inhomogeneous systems.
In materials with strong electron correlation the proper treatment of local atomic physics described by orbital occupations is crucial. Reduced density-matrix functional theory is a natural extension of density functional theory for systems that are dominated by orbital physics. We review the current state of reduced density-matrix functional theory (RDMFT). For atomic structure relaxations or ab-initio molecular dynamics the combination of density functional theory (DFT) and dynamical mean-field theory (DMFT) possesses a number of disadvantages, like the cumbersome evaluation of forces. We therefore describe a method, DFT+RDMFT, that combines many-particle effects based on reduced density-matrix functional theory with a density functional-like framework. A recent development is the construction of density-matrix functionals directly from many-particle theory such as methods from quantum chemistry or many-particle Green’s functions. We present the underlying exact theorems and describe current progress towards quantitative functionals.
Recent developments in methods and computational power render it possible to realistically simulate nanoscopic systems such as surfaces, two-dimensional materials, and nanodots including strong electronic correlations. Nanoscopic structuring enables the tailoring of the electronic structure which can be the basis of future electronic devices. This thesis addresses method developments and applications at the interface of ab-initio methods and model based many-body methods for the case of nanoscopically structured systems with strong correlations. In contrast to bulk materials, low-dimensional materials exhibit long-range interactions due to reduced screening. In this work, the general question how these long-range interactions affect electronic properties is investigated. To this end, a variational approach which approximates models with long-range interactions by models with only local interactions is introduced. For the case of an ab-initio derived model of graphene it is found that nonlocal interactions stabilize the semimetallic phase. The quality of this approach is discussed using a simple test case. Realistic models of interacting impurities embedded in an extended solid involve a large amount of bath sites and low symmetries regarding the impurity, which renders an exact treatment impossible. A variational algorithm is presented which optimizes corresponding exactly solvable effective models. Thus, the method is a proposition to an unambiguous solution of the so-called bath-discretization problem in exact diagonalization approaches to the Anderson impurity model. The method is benchmarked for a simple test case and applied to realistic models of Co atoms in Cu hosts and Fe atoms on alkali surfaces. Finally, the (001) surface of Cr is investigated by incorporating local correlation effects into a material realistic description derived from density functional theory. To this end, the LDA DMFT method is used to calculate spectral functions, which are compared to spectroscopic experimental data. So far open experimental features are thereby clarified. Cr(001) exemplifies a situation where correlation effects are determined by the geometric structure of a material: While correlations effects are weak in bulk Cr, they are key for the electronic structure of the Cr(001) surface.