This dissertation consists of two independent parts: density functional theory (Part I), and nuclear quantum effects (Part II). Kohn-Sham density functional theory (KSDFT) is by far the most widely used electronic structure theory in condensed matter systems. The computational time of KSDFT increases rapidly with respect to the number of electrons in the system, which hinders its practical application to systems of large size. The central quantities in KSDFT are the electron density and the electron energy, which can be fully characterized by the diagonal elements and the nearest off-diagonal elements of the single particle density matrix. However, methods that are currently available require the calculation of the full density matrix. This procedure becomes highly inefficient for systems of large size. Part I of this dissertation develops a new method for solving KSDFT, which directly targets at the calculation of the diagonal and the nearest offdiagonal elements of the single particle density matrix. The new method is developed under the framework of Fermi operator expansion. The new method achieves the optimal expansion cost in the operator level. The electron density and the electron energy is then evaluated from a serires of Green’s functions by a new fast algorithm developed in this dissertation. This dissertation also develops a novel method for discretizing the Hamiltonian of the system that achieves high accuracy with a very small number of basis functions. Combining all these components together, we obtain a unified, accurate, efficient method to solve KSDFT for insulating and metallic systems. Nuclear quantum effects play an important role in a large variety of hydrogen bonded systems such as water and ice due to the small mass of protons (the nuclei of hydrogen atoms). The equilibrium proton dynamics is reflected in the quantum momentum distribution and is the focus of intense research. The standard open path integral formalism for computing the quantum momentum distribution requires the
This dissertation consists of two independent parts: density functional theory (Part I), and nuclear quantum effects (Part II). Kohn-Sham density functional theory (KSDFT) is by far the most widely used electronic structure theory in condensed matter systems. The computational time of KSDFT increases rapidly with respect to the number of electrons in the system, which hinders its practical application to systems of large size. The central quantities in KSDFT are the electron density and the electron energy, which can be fully characterized by the diagonal elements and the nearest off-diagonal elements of the single particle density matrix. However, methods that are currently available require the calculation of the full density matrix. This procedure becomes highly inefficient for systems of large size. Part I of this dissertation develops a new method for solving KSDFT, which directly targets at the calculation of the diagonal and the nearest offdiagonal elements of the single particle density matrix. The new method is developed under the framework of Fermi operator expansion. The new method achieves the optimal expansion cost in the operator level. The electron density and the electron energy is then evaluated from a serires of Green’s functions by a new fast algorithm developed in this dissertation. This dissertation also develops a novel method for discretizing the Hamiltonian of the system that achieves high accuracy with a very small number of basis functions. Combining all these components together, we obtain a unified, accurate, efficient method to solve KSDFT for insulating and metallic systems. Nuclear quantum effects play an important role in a large variety of hydrogen bonded systems such as water and ice due to the small mass of protons (the nuclei of hydrogen atoms). The equilibrium proton dynamics is reflected in the quantum momentum distribution and is the focus of intense research. The standard open path integral formalism for computing the quantum momentum distribution requires the
Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg, Kohn, and Sham [4,5] in the mid-1960s. According to this theory, the fundamental physical information about a many-body system is provided by single-particle densities in a three-dimensional space, which are obtained variationally within a time-independent framework. When compared with other previous formalisms, DFT presents two clear advantages: (1) it is able to treat many-body problems in a sufficiently accurate way and (2) it is computationally simple. This explains why it is one of the most widely used theories to deal with electronic structure-the electronic ground-state energy as a function of the position of the atomic nuclei determines molecular structures and solids, providing at the same time the forces acting on the atomic nuclei when they are not at their equilibrium positions. At present, DFT is used routinely to solve many problems in gas phase and condensed matter. Furthermore, it has made possible the development of accurate molecular dynamics schemes in which the forces are evaluated quantum mechanically ‘‘on the fly.’’ Nonetheless, DFT is a fundamental tool provided the systems studied are relatively large; for small systems, standard methods based on the use of the wave function render quite accurate results [6]. Moreover, it is also worth stressing that all practical applications of DFT rely on essentially uncontrolled approximations [7] (e.g., the local density approximation [4,5], the local spin-density approximation, or generalized gradient approximations [8]), and therefore the validity of DFT is conditioned to its capability to provide fairly good values of the experimental data.
Deriving accurate energy density functional is one of the central problems in condensed matter physics, nuclear physics, and quantum chemistry. We propose a novel method to deduce the energy density functional by combining the idea of the functional renormalization group and the Kohn–Sham scheme in density functional theory. The key idea is to solve the renormalization group flow for the effective action decomposed into the mean-field part and the correlation part. Also, we propose a simple practical method to quantify the uncertainty associated with the truncation of the correlation part. By taking the φ4 theory in zero dimension as a benchmark, we demonstrate that our method shows extremely fast convergence to the exact result even for the highly strong coupling regime.
This dissertation concerns the quantum many-body problem, which is the problem of predicting the properties of systems of several quantum particles from the first principles of quantum mechanics. Included under this umbrella are various problems of fundamental importance in quantum chemistry, condensed matter physics, and materials science. Of particular interest is the electronic structure problem, the problem of determining the state of the electrons in a system with fixed atomic nuclei. Since direct numerical solution of the many-body Schrödinger equation is intractable even for systems of moderate size, a diverse array of approximate methods has been developed. The broad goals of this dissertation are to improve the mathematical understanding of certain widely-used approximations, as well as to propose new methods. Roughly speaking, we consider three (overlapping) categories of methods: Green's function methods, embedding methods, and variational methods.One can understand Green's function methods in terms of many-body perturbation theory, which computes series expansions of physical quantities about a non-interacting reference system. These expansions can be expressed graphically in terms of Feynman diagrams, which can in turn be reorganized, in some cases, into an expansion in terms of so-called bold diagrams. Green's function methods can be specified by choosing a subset of bold diagrams to approximate the sum. At the same time, such methods can be understood in terms of an object known as the Luttinger-Ward (LW) functional, which admits a representation in terms of the bold diagrams. Many aspects of these constructions are purely formal, and indeed the existence of the fermionic LW functional as a single-valued functional has recently been called into question. To contribute to the understanding of these issues, we provide rigorous proofs of the combinatorial construction and analytic interpretation of the bold diagrams in the simplified setting of a classical field theory. In this setting we also provide a rigorous non-perturbative construction of the LW functional via convex duality and prove several key properties, including continuity up to the boundary of its domain and asymptotics in the limit of large interaction.Quantum embedding methods, meanwhile, view a large system as being composed of smaller fragments that are treated with high accuracy and embedded in the larger system in a mutually consistent way. Inspired by a connection between the boundary analysis of the LW functional and embedding, we perform similar analysis for the 1-RDM theory for fermionic systems, which is also developed via convex duality, illustrating a relation to fermionic embedding methods such as the density matrix embedding theory (DMET).Another embedding method of note is the dynamical mean-field theory (DMFT), which is at the same time a Green's function method that can be understood in terms of the LW functional. DMFT relies on the solution of impurity problems, which specify the embedding of an interacting system into a non-interacting bath. Underlying DMFT is a result about the sparsity pattern of the self-energy matrix for impurity problems, which to our knowledge has not been proved in the literature. We provide a rigorous proof of this result in various classical and quantum settings. We go on to investigate the fermionic DMFT in depth, identifying the key mathematical structures that appear in the algorithmic loop for solving it and using these to prove the well-posedness of this loop, in a certain sense.Finally, we introduce a suite of new approaches to the quantum many-body problem that provide variational lower bounds to the ground-state energy. These methods, which combine the themes of convexity and embedding, are based on novel convex relaxations of the variational principles for the ground-state energies of many-body systems. To begin, we recover a second-quantized version of the formalism of strictly correlated electrons (SCE), which yields an exact expression for the exchange-correlation functional in Kohn-Sham density functional theory in the limit of infinite Coulomb repulsion in terms of the solution of a multi-marginal optimal transport problem. We introduce a semidefinite relaxation method for approximately solving this problem and obtaining a lower bound for the ground-state energy. The ideas underlying this relaxation are generalized considerably, outside the context of SCE, to yield much tighter lower bounds, which we validate numerically for both quantum spin systems and fermionic systems. We also describe how these relaxation methods can be interpreted as embedding methods via convex duality.
If there were no Coulomb interaction among electrons, it would be relatively straightforward to solve the many-electron Schrödinger equation. It is, however, precisely this interaction that is at the heart of numerous fascinating phenomena in condensed matter physics such as superconductivity, Kondo physics, magnetism, etc. Due to the large number of electrons in a material being of the order of Avogadro's number, it is at present — and perhaps in the foreseeable future — not feasible or even desirable to solve the Schrödinger equation to obtain the many-electron wavefunction. Fortunately, a large number of important physical properties can be calculated without explicit knowledge of the wavefunction. Two of the most important formalisms for dealing with the many-electron problem which avoid a direct use of the many-electron wavefunction are the Green function and the density functional theory. Within the Kohn-Sham scheme the latter is used to calculate ground-state properties whereas the former for excitation spectra. The book presents the fundamentals of these two theories in detail with essential many-body tools, such as the occupation number representation and Grassmann algebra developed from scratch. Prior knowledge of many-body theory is not a prerequisite so that it is readable for final-year undergraduates and graduate students in physics and chemistry as well as researchers in the field of electronic structure and many-body theory. The book includes in the last chapter an exposition of a density-functional path for determining the Green function, a new formalism recently proposed by the author. The book should be a valuable companion for those embarking in the field of many-electron physics.
Abstract This article is a short introduction to the modern computational techniques used to tackle the many-body problem in materials. The aim is to present the basic ideas, using simple examples to illustrate strengths and weaknesses of each method. We will start from density-functional theory (DFT) and the Kohn–Sham construction—the standard computational tools for performing electronic structure calculations. Leaving the realm of rigorous density-functional theory, we will discuss the established practice of adopting the Kohn–Sham Hamiltonian as approximate model. After recalling the triumphs of the Kohn–Sham description, we will stress the fundamental reasons of its failure for strongly-correlated compounds, and discuss the strategies adopted to overcome the problem. The article will then focus on the most effective method so far, the DFT+DMFT technique and its extensions. Achievements, open issues and possible future developments will be reviewed. The key differences between dynamical (DFT+DMFT) and static (DFT+ U ) mean-field methods will be elucidated. In the conclusion, we will assess the apparent dichotomy between first-principles and model-based techniques, emphasizing the common ground that in fact they share.
A new approach for the calculation of ground states of many-electron systems is proposed via an integral formulation of the Hohenberg-Kohn-Sham density-functional theory. Only equations for the total electron density are involved; orbitals are not employed. Exchange and correlation effects are incorporated. In place of the set of single-electron equations, the total electron density is explicitly expressed in terms of the Kohn-Sham effective local potential through multidimensional integrations. The development is based on the first-order density matrix as obtained from the one-body Green's function in polygonal and Fourier path-integral representations. This might open up the possibility of ab initio calculations for molecules with very many electrons. It also provides explicit solutions to two long-standing problems: electron kinetic energy and momentum density as functionals of the total electron density. The formulation can also be used in calculations for general fermions.