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Density Matrix Embedding Theory: Foundations, Applications and Connection to Functional Theories

Teresa E. Reinhard

Published Apr 11, 2019• 0 citations
OpenAlex

Abstract

The Schrödinger equation describes the motion of the microscopic particles that constitute our world such as the electrons or atomic nuclei. Albeit being applicable to the smallest particle that we know of, it has observable consequences in the macroscopic world. It determines the conductivity of metals, it tells us which materials are magnetic and whether they show exotic behaviour such as super-conductivity. Unfortunately, solving the Schrödinger equation directly for any piece of material that is visible for the human eye is practically impossible. Already a grain of sand contains 1023 (that is written out 10.000.000.000.000.000.000.000) electrons and atomic nuclei. This means that only specifying the initial positions of the particles requires to safe an incredible amount of data; a procedure which is unfeasible for any human or computer. Due to the fundamental problem of applying quantum mechanics to practically relevant scenarios, a number of eective and approximate methods have been developed. In essence, they all try to reduce the dimension of the problem, i.e., the curse of the enormous amount of data required to simulate the Schrödinger equation. In this thesis, we try to analyze and expand one of those method called Density Matrix Embedding Theory (DMET). In a lot of physical system, especially when considering solid states, we can already learn a lot about its physics when describing its properties on a small fragment of the whole system. In a system with interacting particles though, we cannot simply consider just a subsystem and describe its properties without taking into account its interactions with the rest of the system. The basic idea of DMET is to divide the considered system into two parts called impurity and envi- ronment. The impurity is chosen to be so small that its wave function can be computed exactly. In the environment, only those degrees of freedom directly interacting with the impurity are considered and are included in our description. The physics on the environment itself is neglected. In the following, we will explain in detail how this can be done specically. In part I of this thesis, we will set the stage for the considered systems and present well-known and established methods to solve them. In the next part II, we will present DMET in great mathematical detail, which allows us to illustrate the advantages of DMET, but also some problems and drawbacks. We proceed by expanding DMET to the treatment of coupled electron-phonon system in part III and applying this new method to the Hubbard-Holstein model. Part of this work has been published in [53]. Finally, in part IV, we discuss some problems of DMET and, by combining DMET with functional theories, solve these problems. These insights, together with the extensive discussion of the DMET algorithm, will be published soon [59]. We illustrate this new method with an example system. This work will be published in paper [44] soon. We conclude this thesis by a summary and outlook (part V).

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The Schrödinger equation describes the motion of the microscopic particles that constitute our world such as the electrons or atomic nuclei. Albeit being applicable to the smallest particle that we know of, it has observable consequences in the macroscopic world. It determines the conductivity of metals, it tells us which materials are magnetic and whether they show exotic behaviour such as super-conductivity. Unfortunately, solving the Schrödinger equation directly for any piece of material that is visible for the human eye is practically impossible. Already a grain of sand contains 1023 (that is written out 10.000.000.000.000.000.000.000) electrons and atomic nuclei. This means that only specifying the initial positions of the particles requires to safe an incredible amount of data; a procedure which is unfeasible for any human or computer. Due to the fundamental problem of applying quantum mechanics to practically relevant scenarios, a number of eective and approximate methods have been developed. In essence, they all try to reduce the dimension of the problem, i.e., the curse of the enormous amount of data required to simulate the Schrödinger equation. In this thesis, we try to analyze and expand one of those method called Density Matrix Embedding Theory (DMET). In a lot of physical system, especially when considering solid states, we can already learn a lot about its physics when describing its properties on a small fragment of the whole system. In a system with interacting particles though, we cannot simply consider just a subsystem and describe its properties without taking into account its interactions with the rest of the system. The basic idea of DMET is to divide the considered system into two parts called impurity and envi- ronment. The impurity is chosen to be so small that its wave function can be computed exactly. In the environment, only those degrees of freedom directly interacting with the impurity are considered and are included in our description. The physics on the environment itself is neglected. In the following, we will explain in detail how this can be done specically. In part I of this thesis, we will set the stage for the considered systems and present well-known and established methods to solve them. In the next part II, we will present DMET in great mathematical detail, which allows us to illustrate the advantages of DMET, but also some problems and drawbacks. We proceed by expanding DMET to the treatment of coupled electron-phonon system in part III and applying this new method to the Hubbard-Holstein model. Part of this work has been published in [53]. Finally, in part IV, we discuss some problems of DMET and, by combining DMET with functional theories, solve these problems. These insights, together with the extensive discussion of the DMET algorithm, will be published soon [59]. We illustrate this new method with an example system. This work will be published in paper [44] soon. We conclude this thesis by a summary and outlook (part V).
In complex systems, it is often the case that the region of interest forms only one part of a much larger system. The idea of joining two different quantum simulations-a high level calculation on the active region of interest, and a low level calculation on its environment-formally defines a quantum embedding. While any combination of techniques constitutes an embedding, several rigorous formalisms have emerged that provide for exact feedback between the embedded system and its environment. These three formulations: density functional embedding, Green's function embedding, and density matrix embedding, respectively, use the single-particle density, single-particle Green's function, and single-particle density matrix as the quantum variables of interest. Many excellent reviews exist covering these methods individually. However, a unified presentation of the different formalisms is so far lacking. Indeed, the various languages commonly used, functional equations for density functional embedding, diagrammatics for Green's function embedding, and entanglement arguments for density matrix embedding, make the three formulations appear vastly different. In this Account, we introduce the basic equations of all three formulations in such a way as to highlight their many common intellectual strands. While we focus primarily on a straightforward theoretical perspective, we also give a brief overview of recent applications and possible future developments. The first section starts with density functional embedding, where we introduce the key embedding potential via the Euler equation. We then discuss recent work concerning the treatment of the nonadditive kinetic potential, before describing mean-field density functional embedding and wave function in density functional embedding. We finish the section with extensions to time-dependence and excited states. The second section is devoted to Green's function embedding. Here, we use the Dyson equation to obtain equations that parallel as closely as possible the density functional embedding equations, with the hybridization playing the role of the embedding potential. Embedding a high-level self-energy within a low-level self-energy is treated analogously to wave function in density functional embedding. The numerical computation of the high-level self-energy allows us to briefly introduce the bath representation in the quantum impurity problem. We then consider translationally invariant systems to bring in the important dynamical mean-field theory. Recent developments to incorporate screening and long-range interactions are discussed. The third section concerns density matrix embedding. Here, we first highlight some mathematical complications associated with a simple Euler equation derivation, arising from the open nature of fragments. This motivates the density matrix embedding theory, where we use the Schmidt decomposition to represent the entanglement through bath orbitals. The resulting impurity plus bath formulation resembles that of dynamical mean-field theory. We discuss the numerical self-consistency associated with using a high-level correlated wave function with a mean-field low-level treatment, and connect the resulting numerical inversion to that used in density functional embedding. We finish with perspectives on the future of all three methods.
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.
This thesis is divided into three chapters. In the first chapter we outline a simple and numerically inexpensive approach to describe the spectral features of the single-impurity Anderson model. The method combines aspects of the density matrix embedding theory (DMET) approach with a spectral broadening approach inspired by those used in numerical renormalization group (NRG) methods. At zero temperature for a wide range of U , the spectral function produced by this approach is found to be in good agreement with general expectations as well as more advanced and complex numerical methods such as DMRG-based schemes. The theory developed here is simply transferable to more complex impurity problems The second chapter outlines the density matrix embedding methodology in the context of electronic structure applications. We formulate analytical gradients for energies obtained from DMET focusing on two scenarios: RHF-in-RHF embedding and FCI-in-RHF embedding. The former involves solving the small embedded system at Restricted Hartree-Fock (RHF) level of theory. This serves to check the validity of the formulas by reproducing the RHF results on the full system for energies and gradients. The latter scenario employs full configuration interaction (FCI) as a high level solver for the small embedded system. Our results show that only Hellmann-Feynman terms, which involve derivatives of one- and two-electron terms in the atomic orbital basis, are required to calculate energy gradients in both cases. We applied our methodology to the problem of H 10 ring dissociation where the analytical gradients matched those obtained numerically. The gradient formulation is applicable to geometry optimization of strongly correlated molecules and solids. It can also be used in ab initio molecular dynamics where forces on nuclei are obtained from DMET energy gradients. In the final chapter we focus on the study of finite-temperature equilibrium properties of quantum systems in continuous space. We formulate an expansion of the partition function in continuous-time and use Monte Carlo to sample terms in the resulting infinite series. Such a strategy has been highly successful in quantum lattice models but has found scant application in off-lattice systems. The Monte Carlo estimate of the average energy of quantum particles in continuous space subject to simple model potentials is found to converge with low statistical error to the exact solutions even when very high perturbation order is required. We outline two ways in which the algorithm can be applied to more complex problems. First, by drawing an analogy between the formulation in continuous-time with the discretized, Trotter factorized version of standard path integral Monte Carlo (PIMC). This allows one to use the suite of standard PIMC moves to carry out the position sampling required to obtain the weight of each time configuration. Finally, we propose an alternate route by fitting the many-particle potential with multidimensional Gaussians which provides an analytical form for the position integrals.
This article provides the first mathematical analysis of the Density Matrix Embedding Theory (DMET) method. We prove that, under certain assumptions, (i) the exact ground-state density matrix is a fixed-point of the DMET map for non-interacting systems, (ii) there exists a unique physical solution in the weakly-interacting regime, and (iii) DMET is exact at first order in the coupling parameter. We provide numerical simulations to support our results and comment on the physical meaning of the assumptions under which they hold true. We show that the violation of these assumptions may yield multiple solutions of the DMET equations. We moreover introduce and discuss a specific N-representability problem inherent to DMET.
We introduce density matrix embedding theory (DMET), a quantum embedding theory for computing frequency-independent quantities, such as ground-state properties, of infinite systems. Like dynamical mean-field theory, DMET maps the bulk interacting system to a simpler impurity model and is exact in the noninteracting and atomic limits. Unlike dynamical mean-field theory, DMET is formulated in terms of the frequency-independent local density matrix, rather than the local Green's function. In addition, it features a finite, algebraically constructible bath of only one bath site per impurity site, with no bath discretization error. Frequency independence and the minimal bath make DMET a computationally simple and efficient method. We test the theory in the one-dimensional and two-dimensional Hubbard models at and away from half filling, and we find that compared to benchmark data, total energies, correlation functions, and metal-insulator transitions are well reproduced, at a tiny computational cost.
Quantum embedding is a divide and conquer strategy that aims at solving the electronic Schrödinger equation of sizeable molecules or extended systems. We establish in the present work a clearer and in-principle-exact connection between density matrix embedding theory (DMET) and density-functional theory (DFT) within the simple but nontrivial one-dimensional Hubbard model. For that purpose, we use our recent reformulation of single-impurity DMET as a Householder transformed density-matrix functional embedding theory (Ht-DMFET). On the basis of well-identified density-functional approximations, a self-consistent local potential functional embedding theory (LPFET) is formulated and implemented. Combining both LPFET and DMET numerical results with our formally exact density-functional embedding theory reveals that a single statically embedded impurity can in principle describe the density-driven Mott–Hubbard transition, provided that a complementary density-functional correlation potential (which is neglected in both DMET and LPFET) exhibits a derivative discontinuity (DD) at half filling. The extension of LPFET to multiple impurities (which would enable to circumvent the modeling of DDs) and its generalization to quantum chemical Hamiltonians are left for future work.