Elementary Introduction to the Hubbard Model

INTRODUCTION

 The Hubbard model offers one of the most simple ways to get insight into how the interactions between electrons can give rise to insulating, magnetic, and even novel superconducting effects in a solid. It was written down in the early 1960’s and initially applied to understanding the behavior of the transition metal monoxides (FeO, NiO, CoO), compounds which are antiferromagnetic insulators, yet had been predicted to be metallic by methods which treat strong interactions less carefully.

Over the intervening years, the Hubbard model has been applied to the understanding of many systems, from ‘heavy fermion’ systems in the 1980’s, to high temperature superconductors in the 1990’s. Indeed, it is an amazing feature of the model that, despite its simplicity, its exhibits behavior relevant to many of the most subtle and beautiful properties of solid state systems.

 The Hubbard model has been studied by the full range of analytic techniques developed by condensed matter theorists, from simple mean field approaches to field theoretic methods employing Feynman diagrams, expansions in the degeneracy of the number of ‘flavors’ (spin, orbital angular momentum), etc. It has also been extensively attacked with numerical methods like diagonalization and quantum monte carlo.

The objective of these notes is to provide an introduction to the Hubbard model and to a few of the most simple ways in which it is solved. We begin with a discussion of the second quantized operators using which the Hubbard model is written.

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http://quest.ucdavis.edu/tutorial/hubbard7.pdf