An introduction to effective low-energy Hamiltonians in condensed matter physics and chemistry

TL;DR

This paper introduces fundamental low-energy Hamiltonians used in condensed matter physics and chemistry, explaining their theoretical basis, applications, and how they bridge physics and chemistry literatures for graduate students.

Contribution

It provides a clear, accessible introduction to key effective Hamiltonians and discusses their parametrization, aiming to unify physics and chemistry approaches for beginners.

Findings

Clarifies the use of the four H's models in condensed matter and chemistry

Discusses the epistemological basis and parametrization of effective Hamiltonians

Compares effective Hamiltonians with ab initio methods

Abstract

These lecture notes introduce some simple effective Hamiltonians (also known as semi-empirical models) that have widespread applications to solid state and molecular systems. They are aimed as an introduction to a beginning graduate student. I also hope that it may help to break down the divide between the physics and chemistry literatures. After a brief introduction to second quantisation notation, which is used extensively, I focus of the "four H's": the Huckel (or tight binding), Hubbard, Heisenberg and Holstein models. Some other related models, such as the Pariser-Parr-Pople model, the extended Hubbard model, multi-orbital models and the ionic Hubbard model, are also discussed. Finally, I discuss the epistemological basis of effective Hamiltonians and compare and contrast this with that of ab initio methods as well as discussing the problem of parametrising effective Hamiltonians.