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

After a brief introduction to second quantisation notation (section II), which is used extensively, I focus of the “four H’s”: the Hückel (or tight binding; section III), Hubbard (section IV), Heisenberg (section V) and Holstein (section VII) models. These models play central roles in our understanding of condensed matter, particularly for materials where electronic correlations are important, but are less well known to the chemistry community. Some other related models, such as the PariserParr-Pople model, the extended Hubbard model, multiorbital models and the ionic Hubbard model, are also discussed in section VI. As well as their practical applications these models allow us to systematically investigate electronic correlations by ‘turning on’ various correlation terms in the Hamiltonian one at a time. Finally, in section VIII, I discuss the epistemological basis of effective Hamiltonians and compare and contrast this approach with ab initio methods before discussing the problem of the parameterisation of effective Hamiltonians.