Introduction to Effective Low‐Energy Hamiltonians in Condensed Matter Physics and Chemistry

Abstract

In this chapter I discuss some simple effective Hamiltonians that have widespread applications to solid-state and molecular systems. Although meant to be an introduction to a beginning graduate student, I hope that it may also help to break down the divide between the physics and chemistry literatures. After a brief introduction to second quantization notation (Section 10.1), which is used extensively, I focus on the âœfour Hâ™sâ: the Huckel (or tight binding; Section 10.2), Hubbard (Section 10.3), Heisenberg (Section 10.4), and Holstein (Section 10.6) models. These models play central roles in our understanding of condensed matter physics, particularly for materials where electronic correlations are important but are less well known to the chemistry community. Some related models, such as the PariserâParrâPople model, the extended Hubbard model, multiorbital models, and the ionic Hubbard model, are also discussed in Section 10.6. As well as their practical applications, these models allow us to investigate electronic correlations systematically by âœturning onâ various interactions in the Hamiltonian one at a time. Finally, in Section 10.7, I discuss the epistemological basis of effective Hamiltonians and compare and contrast this approach with ab initio methods before discussing the problem of the parameterization of effective Hamiltonians. As this chapter is intended to be introductory, I do not attempt to make frequent comparisons to the latest research problems; rather, I compare the predictions of model Hamiltonians with simple systems chosen for pedagogical reasons. Similarly, references have been chosen for their pedagogical and historical value rather than on the basis of scientific priority. Given the similarity in the problems addressed by theoretical chemistry and theoretical condensed matter physics, surprisingly few advanced texts discuss the interface of two subjects. Unfortunately, this leads to many cultural differences between the fields. Nevertheless, some textbooks do try to bridge the gap, and the reader in search of more than the introductory material presented here is referred to a book by Fulde1 and several other chapters in this book: Chapter 6 describes the state of the art in using density functional theory and ab initio HartreeâFockbased approaches to the a priori evaluation of properties of systems involving strongly correlated electrons, and Chapter 4 describes ab initio approaches based on quantum Monte Carlo.