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

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.