This chapter provides an introduction to the theory of strongly correlated electrons and discusses some of the ideas that presently are being pursued to try to resolve this dilemma. The Hubbard Model is by now surely the most infamous or notorious model problem in solid state physics because the ratio of number of papers to understanding of this model must be an all time record. Yet it is a model which describes the competition between the kinetic energy and the Coulomb interaction in the simplest way. Its origin lies in work of Van Vleck. One tried and true method when have to deal with strong interactions is to look for the right mean-field or Hartree-Fock method. This method has of course been widely applied to the Hubbard Model. The one-dimesional Hubbard model with t not equal to zero for nearest neighbours was solved by Lieb and Wu using Bethe-Ansatz methods.
This chapter provides an introduction to the theory of strongly correlated electrons and discusses some of the ideas that presently are being pursued to try to resolve this dilemma. The Hubbard Model is by now surely the most infamous or notorious model problem in solid state physics because the ratio of number of papers to understanding of this model must be an all time record. Yet it is a model which describes the competition between the kinetic energy and the Coulomb interaction in the simplest way. Its origin lies in work of Van Vleck. One tried and true method when have to deal with strong interactions is to look for the right mean-field or Hartree-Fock method. This method has of course been widely applied to the Hubbard Model. The one-dimesional Hubbard model with t not equal to zero for nearest neighbours was solved by Lieb and Wu using Bethe-Ansatz methods.
Introduction. Solvable Models, Rigorous Results, and Advances in Formalism: The Hubbard Model: Some Rigorous Results and Open Problems E.H. Lieb. On the Bethe Ansatz Soluble Degenerate Hubbard Model H. Frahm, A. Schadschneider. Infinite in all Dimensions: Large Coupling, High Dimensions, Many Components: The Mott Transition in Infinite Dimensions: Old Ideas and Some Surprises G. KIotliar, M.J. Rozenberg. The Hubbard Model with Infinite Interaction: Magnetic Properties V.Ya. Krivnov, et al. Fermi Liquid versus Luttinger Liquid: Two Particle Scattering and Orthogonality Catastrophe in the Hubbard Model W. Metzner. Conservation Laws in Normal Metals: Luttinger Liquid vs. Fermi Liquid C. Di Castro, et al. Perturbative, Mean Field, Variational, and Numerical Studies: Hartree-Fock and RPA Studies of the Hubbard Model G. Guinea, et al. The Wavefunction Renormalization Constant for the One- and Two-band Hubbard Hamiltonians in Two Dimensions E. Louis, et al. Experiments and Physical Applications: Electron Spectroscopy and Hubbard: Issues and Opportunities J.W. Allen, et al. On Electrical Properties of Chalcogenide Glassy Semiconductors in the Framework of Hubbard Model with Negative Correlation Energy S.D. Savransky. 34 additional articles. Index.
The purpose of this opening chapter is threefold: to introduce the Hubbard model, to discuss its origin and significance and to give a brief summary of its history. Rather than beginning with more general and historical considerations we will start with a concrete albeit somewhat technical discussion of how the Hubbard model arises as an effective description of electronic degrees of freedom in solids. On the origin of the Hubbard model The Hubbard model is named after John Hubbard, who in a series of influential articles [201–206] introduced the Hamiltonian in order to model electronic correlations in narrow energy bands and proposed a number of approximate treatments of the associated many-body problem. Our following discussion of how the Hubbard Hamiltonian arises in an approximate description of interacting electrons in a solid loosely parallels Hubbard's original work. We will assume that the reader is familiar with the basic concepts of solid state theory (see e.g. [25, 509]) and with the formalism of second quantization (e.g. [283]). For further reading we refer to the original literature [188, 201, 233] and to the monographs [27, 158, 498]. A solid consists of ions and electrons condensed in a three-dimensional crystalline structure. Since the ions are much heavier than the electrons, it is often a good phenomenological starting point for the exploration of the electronic properties of solids to think of the ions as forming a static lattice.
The Hubbard model was originally introduced as the simplest system which may exhibit an insulating (Mott) state. This state is the result of strong electron–electron interactions. In this chapter we consider the Hubbard model at half-filling. The main goal here is the study of the magnetic properties of its ground state. Apart from an exact solution in one dimension, no exact results are available for this problem. This leads to the use of several approximations. The most popular one, and the oldest, is the mean-field theory (MFT). In the MFT one has the bias that the ground state does have some sort of magnetic order (i.e. ferromagnetic, Néel antiferromagnetic, etc.). The problem is then usually solved by means of a variational ansatz. However, one is usually interested in more than just the ground-state energy, which, after all, is not directly measurable and depends very sensitively on the properties at short distances. Most often we wish to evaluate the long-distance, low-frequency, properties of the correlation and response functions of this theory. Moreover, in some cases, such as in one dimension, the fluctuations overwhelm the MFT predictions. In this chapter we will consider the standard MFT (i.e. Hartree–Fock), which is expected to become accurate at weak coupling. We will consider both ferromagnetic and antiferromagnetic states. We will also rederive these results using path integrals. As a byproduct, we will also have a theory of the fluctuations: the non-linear sigma model.
The Hubbard model is a `highly oversimplified model' for electrons in a solid which interact with each other through extremely short-ranged repulsive (Coulomb) interaction. The Hamiltonian of the Hubbard model consists of two parts: which describes quantum mechanical hopping of electrons, and which describes non-linear repulsive interaction. Either or alone is easy to analyse, and does not favour any specific order. But their sum is believed to exhibit various non-trivial phenomena including metal-insulator transition, antiferromagnetism, ferrimagnetism, ferromagnetism, Tomonaga-Luttinger liquid, and superconductivity. It is believed that we can find various interesting `universality classes' of strongly interacting electron systems by studying the idealized Hubbard model. In the present article we review some mathematically rigorous results relating to the Hubbard model which shed light on the `physics' of this fascinating model. We mainly concentrate on the magnetic properties of the model in its ground states. We discuss the Lieb-Mattis theorem on the absence of ferromagnetism in one dimension, Koma-Tasaki bounds on the decay of correlations at finite temperatures in two dimensions, the Yamanaka-Oshikawa-Affleck theorem on low-lying excitations in one dimension, Lieb's important theorem for the half-filled model on a bipartite lattice, Kubo-Kishi bounds on the charge and superconducting susceptibilities of half-filled models at finite temperatures, and three rigorous examples of saturated ferromagnetism due to Nagaoka, Mielke, and Tasaki. We have tried to make the article accessible to non-experts by giving basic definitions and describing elementary materials in detail.