The clear distinction between metals and insulators is a well established and appartently simple concept which can be clarified by studying the behavior of the electrical conductivity. However, a theoretical description of the processes leading to one of the two extremes is no trivial task. The widely accepted band structure theory relies on an one-electron picture to characterize the metallic/insulating behavior of a material. However, in spite of the fact that it can be used to make successful predictions, this description does not capture one of the essential elements characterizing the nature of an electronic quantum system, i.e. the electron correlation. The key to the different behaviors is clearly contained in the correlated many- electron wave function, but the analysis of such a high-dimensional object in configuration space can be quite cumbersome. Therefore the development of powerful tools to simplify this task is necessary. The total position-spread (TPS) tensor, which was derived by Walter Kohn’s theory of the insulating state, plays a key role in the field of metal-insulator transitions (MITs). This tool provides information about the electron localization, which is strongly connected to the electrical conductivity. Therefore, in this work particular attention was paid to the calculation of the TPS tensor of small and extended systems. Moreover, part of this thesis focuses on a new formalism, which allows the study of spin mobility and spin entanglement and provides therefore deeper insight into the effects of electron correlation. In the framework of quantum information theory (QIT), the quantum entanglement can be quantified by means of measures such as the von Neumann entropy. Also, the analysis of the elements of the two-orbital reduced density matrix provides an overview of the complexity of the many-electron wave function. These insights were used in this work for the analysis of the metallic/insulating character of a system. However, in order to perform such investigastions, the problem of achieving an accurate description of the many- electron wave function has to be dealt with. Despite the tremedous progress in the efficiency of quantum-chemical methods, the treatement of extended systems using standard approaches is still a difficult (if not unfeasible) task. Therefore the development and testing of sophisticated, low-scaling methods is currently the subject of many investigations. In this work, two approaches were employed: 1) the ab-initio density matrix renormalization group (DMRG) approach which, based on a tensor product ansatz, provides very precise results and exploits QIT to improve its performance; 2) the method of increments (MoI) which, using a many-body expansion in terms of localized orbitals, yields accurate correlation energies. By using both methods on strongly correlated one-dimesional model systems, we worked on a new formalism of the method of increments which can be used to describe the whole dissociation curve without inconvenient discontinuities. Finally, aiming to calculate the total position-spread tensor of extended systems, we worked on the application of the MoI for quantities other than the correlation energy.
The clear distinction between metals and insulators is a well established and appartently simple concept which can be clarified by studying the behavior of the electrical conductivity. However, a theoretical description of the processes leading to one of the two extremes is no trivial task. The widely accepted band structure theory relies on an one-electron picture to characterize the metallic/insulating behavior of a material. However, in spite of the fact that it can be used to make successful predictions, this description does not capture one of the essential elements characterizing the nature of an electronic quantum system, i.e. the electron correlation. The key to the different behaviors is clearly contained in the correlated many- electron wave function, but the analysis of such a high-dimensional object in configuration space can be quite cumbersome. Therefore the development of powerful tools to simplify this task is necessary. The total position-spread (TPS) tensor, which was derived by Walter Kohn’s theory of the insulating state, plays a key role in the field of metal-insulator transitions (MITs). This tool provides information about the electron localization, which is strongly connected to the electrical conductivity. Therefore, in this work particular attention was paid to the calculation of the TPS tensor of small and extended systems. Moreover, part of this thesis focuses on a new formalism, which allows the study of spin mobility and spin entanglement and provides therefore deeper insight into the effects of electron correlation. In the framework of quantum information theory (QIT), the quantum entanglement can be quantified by means of measures such as the von Neumann entropy. Also, the analysis of the elements of the two-orbital reduced density matrix provides an overview of the complexity of the many-electron wave function. These insights were used in this work for the analysis of the metallic/insulating character of a system. However, in order to perform such investigastions, the problem of achieving an accurate description of the many- electron wave function has to be dealt with. Despite the tremedous progress in the efficiency of quantum-chemical methods, the treatement of extended systems using standard approaches is still a difficult (if not unfeasible) task. Therefore the development and testing of sophisticated, low-scaling methods is currently the subject of many investigations. In this work, two approaches were employed: 1) the ab-initio density matrix renormalization group (DMRG) approach which, based on a tensor product ansatz, provides very precise results and exploits QIT to improve its performance; 2) the method of increments (MoI) which, using a many-body expansion in terms of localized orbitals, yields accurate correlation energies. By using both methods on strongly correlated one-dimesional model systems, we worked on a new formalism of the method of increments which can be used to describe the whole dissociation curve without inconvenient discontinuities. Finally, aiming to calculate the total position-spread tensor of extended systems, we worked on the application of the MoI for quantities other than the correlation energy.
The many-body correlation between electrons is the origin of many fascinating phenomena in condensed matter systems, such as high temperature superconductivity, superfluidity, fractional quantum Hall effect, and Mott insulator. Strongly correlated systems have been an important subject of condensed matter physics for several decades, especially after the discovery of high temperature cuprate superconductors. In this thesis, we apply first principles quantum Monte Carlo (QMC) method to several representative systems to study the electron correlations in transition metal oxides (vanadium dioxide) and low dimensional electronic systems (graphene and graphene-like two dimensional systems). Vanadium dioxide (VO2) is a paradigmatic example of a strongly correlated system that undergoes a metal-insulator transition at a structural phase transition. To date, this transition has necessitated significant post-hoc adjustments to theory in order to be described properly. We apply first principles quantum Monte Carlo (QMC) to study the structural dependence of the properties of VO2. Using this technique, we simulate the interactions between electrons explicitly, which allows for the metal-insulator transition to naturally emerge, importantly without ad-hoc adjustments. The QMC calculations show that the structural transition directly causes the metal-insulator transition and a change in the coupling of vanadium spins. This change in the spin coupling results in a prediction of a momentum-independent magnetic excitation in the insulating state. While two-body correlations are important to set the stage for this transition, they do not change significantly when VO2 becomes an insulator. These results show that it is now possible to account for electron correlations in a quantitatively accurate way that is also specific to materials. Electron correlation in graphene is unique because of the interplay of the Dirac cone dispersion of pi electrons with long range Coulomb interaction. The random phase approximation predicts no metallic screening at long distances and low energies because of the zero density of states at Fermi level. It is thus interesting to see how screening takes place in graphene at different length scales. We addressed this problem by computing the structure factor S(q) and S(q, ω) of freestanding graphene using ab initio fixed-node diffusion Monte Carlo and the random phase approximation. The X-ray measured structure factor is reproduced very accurately using both techniques, provided that sigma-bonding electrons are included in the simulations. Strong dielectric screening from sigma electrons are observed, which redshifts the pi plasmons resonance frequency at long distance and reduces the effective interactions between pi electrons at short distance. The short distance screening makes suspended graphene a weakly correlated semimetal which otherwise would be an insulator. The third piece of works is dedicated to studying the low energy excitation of many-body systems using extended Koopmans' theorem (EKT). The EKT provides a straight forward way to compute charge excitation spectra, such as ionization potentials, electron affinities from any level of theory. We implemented the EKT within the QMC framework, and performed systematic benchmark studies of ionization potentials of the second- and third-row atoms, and closed- and open-shell molecules. We also applied it to compute the quasiparticle band structure of solids (graphene). For complex correlated systems, identifying relevant low energy physics degrees of freedom is extremely important to understanding the system's collective behavior at different length scales. In this sense, bridging the realistic systems to lower energy effective lattice models that involve fewer but important degrees of freedom is significant to understanding correlated systems. We have formulated three ab initio density matrix based downfolding (AIDMD) methods to downfold the ab initio systems into effective lattice models. We have demonstrated the successfulness of these methods by applying them to molecules (H2) and periodic systems (hydrogen chain and graphene).
Electronic structure theory faces many computational challenges in transition metal chemistry. Usually, density functional theory is the method of choice for theoretical studies on transition metal complexes and clusters mostly because it is the only feasible one, although its results are not systematically improvable. By contrast, multireference ab initio methods could provide a correct description of the electronic structure, but are limited to small molecules because of the tremendous computational resources required. In recent years, conceptually new ab initio methods emerged that turned out to be promising for theoretical coordination chemistry. We review and discuss two efficient parametrization schemes for the electronic wave function, the matrix product states and the complete-graph tensor network states. Their advantages are demonstrated at example transition metal complexes. Especially, tensor network states might provide the key to accurately describe strongly correlated and magnetic molecular systems in transition metal chemistry.
The study of low-dimensional, strongly-interacting quantum systems has been of interest in the past few decades due to new and exotic physics that show great potential in the advancement of fundamental physics as well as technological applications. In order to investigate such systems, a vast number of techniques have been developed throughout the years, all of which have their respective strengths and weaknesses. In the world of one-dimensional (1D) quantum systems, the density-matrix renormalization group (DMRG) algorithm and matrix product states (MPS) have proven to be powerful and reliable tools that provide invaluable information while providing a wide range of data that support both theoretical and experimental studies. Coupled with the development and refinement of tensor networks - an ansatz for quantum many-body wavefunctions, in particular MPS, the DMRG algorithm has seen a spike of improvements that has pushed the boundaries of its application into more sophisticated and complex systems.This thesis broadly covers the study of several 1D many-body quantum systems represented by infinite MPS (iMPS) - a class of ansatz that represent translationally-invariant, 1D many-body quantum states. More specifically, it is divided into two main parts.The first part investigates the 1D topological Kondo insulator (TKI). This is an effective model designed to investigate how strong interactions between conduction electrons and localized moments result in a symmetry-protected topological (SPT) phase, specifically the Haldane phase. By studying the effect of the electron repulsion as well as the competition between the local and non-local couplings between the electrons and the local moments, it is found that a topological phase transition from the SPT phase to a topologically trivial phase occurs with the assistance of electron interaction. The properties of both topologically trivial and nontrivial groundstates were revealed via means of quantum information theoretic quantities such as the entanglement entropy, entanglement spectrum and non-local order parameters. These quantities unveil the nature, structure and the effect that the different interactions have on the groundstate properties. Besides the phases, the properties of the critical point was studied and classified. By obtaining the critical exponents and central charge, the universality class of this transition was found to be that of 1D free fermions/bosons. Furthermore, the presence of electron interactions was found not to alter this universality class.The second part of this thesis is a project that stems from the necessity to overcome the inherent limitation of an iMPS in representing a system at a cusp of a quantum phase transition (QPT), specifically, in determining the critical point and exponents. In the thermodynamic limit, the laws of statistical mechanics dictates that the system’s correlation length diverges at the critical point. However, when representing a critical state with an iMPS, the finite bond dimension implies that the correlation length of the iMPS remains finite, albeit large. This presents a fundamental challenge in extracting critical data since the critical exponents are defined for systems in the thermodynamic limit. To overcome this shortcoming, finite-entanglement scaling (FES) - the analog of finite-size scaling, has previously been used to locate the critical point and extract the critical exponents for systems represented by finite-bond dimensional iMPS. While FES serves its purpose sufficiently well, the computational cost is high for just about any moderately complex, real-world system. To overcome such difficulties, the second part of this thesis combines new ideas of using higher-order cumulants of the order parameter with previously-known iMPS techniques to determine the critical point and extract the exponents, thus relating statistical mechanics of QPTs and tensor networks of critical states. Some of the main methods used and developed here are FES, the application of the Binder cumulant in iMPS and the new cumulant exponent relation. Putting these tools together provides a scheme that can be used to efficiently locate the critical point and extract the critical exponents at a lower computational cost than previously-known methods. This is shown through comparisons of several exemplary models such as the 1D and quasi-1D transverse field Ising model, the 1D TKI, the S = 1 Heisenberg chain with single-ion anisotropy and the 1D Bose-Hubbard model.
Quantum-mechanical calculations based on Kohn–Sham density functional theory (DFT) played a significant role in accurately predicting various aspects of materials behavior over the past decade. The Kohn–Sham approach to DFT reduces the many-body Schrodinger (eigen value) problem of interacting electrons into an equivalent problem of noninteracting electrons in an effective mean field that is governed by electron-density. Despite the reduced computational complexity of Kohn–Sham DFT, large-scale DFT calculations are still computationally very demanding with the resulting computational complexity scaling cubically with number of atoms in a given materials system. Numerical algorithms with reduced scaling behavior which are robust, computationally efficient and scalable on parallel computing architectures are always desirable to enable simulations at larger scales and on more complex systems. Following this line of thought, this study explores the use of tensor structured methods for ab-initio numerical solution of Kohn–Sham equations arising in DFT calculations. Earlier studies on tensor-structured methods have been quite successful in the accurate calculation of Hartree and the nonlocal exchange operators arising in the Hartree–Fock equations. A recent investigation of low-rank Tucker-type decomposition of the electron-density of large aluminum clusters (obtained from the finite-element discretization of orbital free DFT) shows the exponential decay of approximation error with respect to Tucker rank (number of tensor-basis functions in Tucker type representation). The results also indicate a smaller Tucker rank for the accurate representation of the electron density and is only weakly dependent on the system sizes studied. The promising success of tensor-structured techniques in resolving the electronic structure of material systems has enabled us to take a step further. In this study, we propose a systematic way of computing a globally adapted Tucker-type basis for solving the Kohn–Sham DFT problem by using a separable approximation of the Kohn–Sham Hamiltonian. Further, the resulting Kohn–Sham eigenvalue problem is projected into the aforementioned Tucker basis and is solved for ground-state energy using a self-consistent field iteration. The rank of the resulting Tucker representation and the computational complexity of the calculation are examined on representative benchmark examples involving metallic and insulating systems.
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.
This PhD thesis conducts a focused study of strongly correlated materials with localized electron orbitals. We have studied two real materials (LuNiO3 and VO2) and one model system, i.e., the Anderson impurity model. The thesis is divided into two parts. In the first part, we use the DFT+U method, to calculate the equilibrium phase transitions of LuNiO3 as an example of rare-earth nickelates under a substrate-induced strain, and the nonequilibrium phase transitions of VO2 as an example of narrow-gap Mott insulators under laser-pulse-induced photoexcitations. The effect of adding U is manifested in both materials as the change of band structure in response to the change of orbital occupancies of electrons, i.e., the soft band effect. This effect brings about competitions of electrons between different orbitals and gives rise to multiple metastable states. In the second part, we go beyond band theory and study how we can use the density matrix renormalization group (DMRG) method based on matrix product states (MPS) to perform real-time evolutions of the Anderson impurity model, towards the goal of building a real-time impurity solver for the nonequilibrium dynamical mean-field theory (DMFT). We study both the quenched and periodically driven single-impurity Anderson models (SIAM) and have obtained some complexity results based on a non-standard implementation of DMRG (the 4-MPS method) we developed in the thesis. We find in the star geometry of the SIAM that the ordering of the bath orbitals in the MPS affects dramatically the increase of entanglement entropy and thus the computational complexity of the simulation.