Density functionals and model Hamiltonians: Pillars of many-particle physics

Abstract

Density-functional theory (DFT) and model Hamiltonians are conceptually distinct approaches to the many-particle problem, which can be developed and applied independently. In practice, however, there are multiple connections between the two. This review focuses on these connections. After some background and introductory material on DFT and on model Hamiltonians, we describe four distinct, but complementary, connections between the two approaches: (i) the use of DFT as input for model Hamiltonians, in order to calculate model parameters such as the Hubbard U and the Heisenberg J. (ii) The use of model Hamiltonians as input for DFT, as in the LDA + U functional. (iii) The use of model Hamiltonians as theoretical laboratories to study aspects of DFT. (iv) The use of special formulations of DFT as computational tools for studying spatially inhomogeneous model Hamiltonians. We mostly focus on this fourth combination, model DFT, and illustrate it for the Hubbard model and the Heisenberg model. Other models that have been treated with DFT, such as the PPP model, the Gaudin–Yang $\delta$-gas model, the XXZ chain, variations of the Anderson and Kondo models and Hooke’s atom are also briefly considered. Representative applications of model DFT to electrons in crystal lattices, atoms in optical lattices, entanglement measures, dynamics and transport are described.

Introduction

When confronted with a very difficult problem, one can think of two distinct strategies to make progress. One is a frontal attack, trying to solve the original problem as it is, making sequences of approximations and simplifications along the way as they become necessary. Alternatively, one can replace the difficult problem from the outset by a simpler one that retains some similarity to the original problem but can be analyzed more easily, ideally without further approximations.

The quantum-mechanical many-particle problem is extraordinarily difficult, in particular when these particles are exposed to an inhomogeneous environment. Not surprisingly, variations of both basic strategies have therefore been employed in order to deal with the inhomogeneous many-body problem posed by the quantum description of interacting electrons in atoms, molecules and solids. The conventional formulation of density-functional theory is perhaps the most widely used representative of the first, ab initio, strategy, while the construction of model Hamiltonians is the principal representative of the second.

In order to motivate a review on density-functional theory and model Hamiltonians, we first need to recall a few pertinent facts about each of the two subjects. Their common aim is to provide understanding of the structure of matter and its properties. For the purpose of this review, we will take matter to mean ‘ordinary matter’, composed of interacting electrons and nuclei following the laws of (mostly nonrelativistic) quantum mechanics. This includes atoms, molecules, clusters, nanoscale aggregates, solids, surfaces, quantum liquids and condensates, among others, but excludes extraordinary states of matter such as nuclear matter and quark–gluon plasmas, as well as systems not requiring a quantum description, such as classical gases and liquids.

The quantum-mechanical many-electron many-nuclei problem is extraordinarily difficult to solve completely, and physics (joined by quantum chemistry and materials science) has employed both of the two basic strategies for dealing with it. In the model approach one replaces the full Hamiltonian of the interacting many-electron many-nuclei system by a simplified model Hamiltonian, retaining only the degrees of freedom expected to be relevant for a particular problem or class of problems. Perhaps the best-known such models are those of Ising, Heisenberg and Hubbard, which pervade all of solid-state and statistical physics.

The Hubbard model, in particular, is used very frequently to describe strongly interacting electrons in, e.g., cuprate and manganite systems. (Recently it is also increasingly being used to model systems of cold atoms in optical lattices.) This approach, typical of many-body theory, emphasizes the electron–electron interaction, but does not take into account in any detail the spatially inhomogeneous environment constituted by the crystal lattice and ignores most electronic and lattice degrees of freedom. In practice, a well-constructed model is most useful for qualitative understanding, but it does not permit the kind of material-specific quantitative prediction of system properties demanded by materials science and quantum chemistry.

In the ab initio approach, on the other hand, one starts from the full many-electron Hamiltonian, including the electron–electron interaction. The electron–nucleus interaction is normally accounted for by the lattice potential, describing the spatial arrangement of nuclei in the solid or molecule under study. Instead of attempting a complete solution for the resulting many-electron wavefunction one then focuses on simpler objects, such as single-particle Green’s functions, correlation functions, Slater determinants or reduced density matrices. The most widely used such simpler quantity is the particle density, which is the central variable in density-functional theory (DFT). While DFT is formally exact, in practice it requires approximations for the electron–electron interaction terms, and the most widely used approximations are adequate mainly for weak interactions. This approach, typical of electronic-structure simulations, thus emphasizes the spatially inhomogeneous environment, but is less reliable in the presence of strong particle–particle interactions.

Today, DFT is the standard methodology for quantitative electronic-structure calculations of solids and molecules, but its predictive and interpretative capabilities for strongly correlated systems are severely handicapped by the absence of suitable approximations and computational methodologies, as well as the difficulties of extracting relevant information on particle–particle correlations exclusively from the single-particle density.

Since the early days of the quantum mechanical many-body problem, the model Hamiltonian and the ab initio approach have coexisted, each within its own regime of applicability. Detailed reviews in the journal literature and many books have been devoted to analyzing and describing each approach in great detail. Typically, a researcher works with one of the two approaches, and tends to address problems for which the chosen approach is suitable. Misunderstandings between workers representing each of the two communities abound, and wrong statements made by users of one on the capabilities and limitations of the other can be found even in recent research literature. Nevertheless, the density-functional approach and the model Hamiltonian approach have more than once met fruitfully over the past decades. The intention of the present review is to provide a description of these meeting points.

Schematically, we classify these meeting points in four broad somewhat overlapping classes, summarized in Table 1.

(i) DFT as input for model Hamiltonians. Here DFT calculations, or other first-principles methods, are used to calculate model parameters, such as the Hubbard U or the Heisenberg J, for a given material or class of materials. These parameters are then adopted as constants in subsequent model calculations. DFT here serves as an underlying microscopic theory fixing model parameters whose values are determined by phenomena occurring at higher energy scales than are of interest for the problem under study, while the model Hamiltonian itself addresses the remaining low-energy degrees of freedom.

(ii) Model Hamiltonians as input for DFT. Here insight gained from the study of model Hamiltonians is used to improve approximations within DFT. A typical example is the local-density approximation (LDA) to DFT, whose performance for, e.g., transition-metal oxides can be improved by adding a term involving the Hubbard U parameter. The resulting LDA + U approach has recently been generalized do include dynamic many-body effects, and given rise to the very popular and successful dynamical mean-field theory (DMFT).

(iii) Model Hamiltonians used to study DFT. Here the model Hamiltonian is used not to describe a certain class of materials, but serves as a theoretical laboratory for investigating fundamental questions and properties of DFT for a simpler Hamiltonian, constituting a better controlled environment than that provided by the first-principles Hamiltonian. Questions such as the meaning of the Kohn–Sham eigenvalues and Fermi surface, the accuracy of local and semilocal functionals, the band-gap problem, and $v$-representability, among others, have been investigated in this way.

(iv) DFT used to study model Hamiltonians. Here concepts and computational tools developed within DFT for ab initio calculations are adapted for model Hamiltonians in regimes or situations where traditional many-body methods encounter difficulties. A typical example is the presence of spatial inhomogeneity (e.g., nonequivalent lattice sites or confining potentials), breaking translational invariance. Traditional analytical and numerical methods for model Hamiltonians are hard (and sometimes impossible) to use in the absence of simplifying symmetries, while DFT, due to its genesis in the inhomogeneous many-electron problem, provides a set of tools that allow to make progress also for inhomogeneous models.

All four meeting points are different in scope and in details, each has met with its own successes and failures, and some of them have given rise to rather voluminous literature. However, to best of our knowledge they have never been systematically classified in this way, or otherwise discussed or described in parallel and in a language that would be accessible for readers who are specialists in one of the two basic approaches but not the other. In this review we provide such a comparison and description, following the above classification of the four basic combinations. In Section  2 we provide necessary background on model Hamiltonians and DFT. Section  3 is devoted to combination (i), Section  4 to combination (ii), Section  5 to combination (iii), and Sections  6 DFT used to study model Hamiltonians: concepts and terminology, 7 DFT for the Hubbard model, 8 DFT for the Heisenberg model, 9 DFT for other model Hamiltonians, 10 Dynamics and transport to combination (iv), with Section  6 covering general aspects, and Sections  7 DFT for the Hubbard model, 8 DFT for the Heisenberg model, 9 DFT for other model Hamiltonians, 10 Dynamics and transport describing applications to specific models and problems. The larger space devoted to combination (iv) is intentional, as it is the least reviewed of the four basic combinations.

A review of these four combinations is timely from the methodological point of view, as recently significant progress has been made in all of them, but largely independently of what is being done simultaneously in the others. At this stage a unified description of the two basic approaches and their four meeting points can be expected to be useful for readers working on rather different problems in distinct communities. It is also timely from the point of view of recent experimental advances that have brought into focus the need for combinations of model approaches with ab initio approaches. To give just two examples: The presence of transition-metal oxides in nanoscale systems requires an understanding of strong-correlation physics in the first-principles description of e.g. electronic transport through a nanoscale junction; and the realization of degenerate gases of cold atoms in optical lattices requires an understanding of the behavior of the Hubbard model in the presence of confining potentials. In both cases spatial inhomogeneity coexists with strong particle–particle interactions, and methods combining model approaches with ab initio simulations emerge as promising ways to make progress.

In the following sections, we first provide necessary background on DFT and on model Hamiltonians, and then describe, one by one, the four basic combinations listed above, with special emphasis on combination (iv). Necessarily, a review of this nature must be incomplete and partial, and we apologize in advance to all the authors whose beautiful and important work we could not include. Even within the limited scope of this work, the existing literature is rather extensive, and our bibliography is intended to be a starting point, not an exhaustive list. Similarly, the treatment given to the different subjects covered here is biased by our personal background and interests, although we have made an effort to address the largest possible number of readers, from different scientific communities.