Subir Sachdev A
The Hubbard model was originally introduced as a description of the motion of electrons in transition metals, with the motivation of understanding their magnetic properties. This original model remains a very active subject of research today, and important progress has been made in recent years by examining its properties in the limit of large spatial dimensionality. In this chapter, we shall only examine the much simpler “boson Hubbard model,” following the analysis in an important paper by Fisher, Weichman, Grinstein, and Fisher. As the name implies, the elementary degrees of freedom in this model are spinless bosons, which take the place of the spin-½ fermionic electrons in the original model. These bosons could represent Cooper pairs of electrons undergoing Josephson tunneling between super conducting islands or helium atoms moving on a substrate. Processes in which the Cooper pair boson decays into a pair of electrons are neglected in this simple model, and this caveat must be kept in mind while discussing experimental applications. Many of the results discussed in this chapter were also obtained in early literature on quantum transitions in anisotropic magnets in the presence of an applied magnetic field. These are reviewed by Kaganov and Chubukov, who also gave an extensive discussion of experimental applications. We will, however, not use their formulation here. Apart from its direct physical applications, the importance of the boson Hubbard model lies in providing one of the simplest realizations of a quantum phase transition that does not map onto a previously studied classical phase transition in one higher dimension.
Cambridge University Press eBooks•Jan 13, 2000•0 citations