The Hubbard model was originally introduced as a description of the motion of electrons in transition metals, with the motivation of understanding of their magnetic properties. This original model remains a very active subject of research today: important progress has been made in recent years by examining its properties in the limit of large spatial dimensionality [28,27].
In this chapter, we only examine the much simpler ``boson Hubbard model'', following the analysis of 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-1/2 fermionic electrons in the original model. These bosons could represent Cooper pairs of electrons undergoing Josephson tunneling between superconducting 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 which does not map onto a previously studied classical phase transition in one higher dimension. The continuum theory describing this transition includes complex Berry phase terms, which, in the simplest formulation of the theory, do not become real even after analytic continuation to imaginary time. We shall meet some genuinely new physical phenomena associated with quantum critical points in a relatively simple context, and the insight will be generally applicable to more complicated models in subsequent chapters.